3.538 \(\int \frac {(d+e x)^{3/2}}{x (a+b x+c x^2)} \, dx\)

Optimal. Leaf size=340 \[ -\frac {\sqrt {2} \left (-c d \left (d \sqrt {b^2-4 a c}-4 a e\right )+a e^2 \sqrt {b^2-4 a c}-b \left (a e^2+c d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{a \sqrt {c} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\sqrt {2} \left (-c d \left (d \sqrt {b^2-4 a c}+4 a e\right )+a e^2 \sqrt {b^2-4 a c}+b \left (a e^2+c d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{a \sqrt {c} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {2 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{a} \]

[Out]

-2*d^(3/2)*arctanh((e*x+d)^(1/2)/d^(1/2))/a-arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/
2)))^(1/2))*2^(1/2)*(-b*(a*e^2+c*d^2)+a*e^2*(-4*a*c+b^2)^(1/2)-c*d*(-4*a*e+d*(-4*a*c+b^2)^(1/2)))/a/c^(1/2)/(-
4*a*c+b^2)^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)-arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b+(-4*
a*c+b^2)^(1/2)))^(1/2))*2^(1/2)*(b*(a*e^2+c*d^2)+a*e^2*(-4*a*c+b^2)^(1/2)-c*d*(4*a*e+d*(-4*a*c+b^2)^(1/2)))/a/
c^(1/2)/(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)

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Rubi [A]  time = 1.58, antiderivative size = 340, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {897, 1287, 206, 1166, 208} \[ -\frac {\sqrt {2} \left (-c d \left (d \sqrt {b^2-4 a c}-4 a e\right )+a e^2 \sqrt {b^2-4 a c}-b \left (a e^2+c d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{a \sqrt {c} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\sqrt {2} \left (-c d \left (d \sqrt {b^2-4 a c}+4 a e\right )+a e^2 \sqrt {b^2-4 a c}+b \left (a e^2+c d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{a \sqrt {c} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {2 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(3/2)/(x*(a + b*x + c*x^2)),x]

[Out]

(-2*d^(3/2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/a - (Sqrt[2]*(a*Sqrt[b^2 - 4*a*c]*e^2 - c*d*(Sqrt[b^2 - 4*a*c]*d -
4*a*e) - b*(c*d^2 + a*e^2))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])
/(a*Sqrt[c]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) - (Sqrt[2]*(a*Sqrt[b^2 - 4*a*c]*e^2 - c
*d*(Sqrt[b^2 - 4*a*c]*d + 4*a*e) + b*(c*d^2 + a*e^2))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b
+ Sqrt[b^2 - 4*a*c])*e]])/(a*Sqrt[c]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 897

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - ((2*c*d - b*e)*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 1287

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[Ex
pandIntegrand[((f*x)^m*(d + e*x^2)^q)/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^
2 - 4*a*c, 0] && IntegerQ[q] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {(d+e x)^{3/2}}{x \left (a+b x+c x^2\right )} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {x^4}{\left (-\frac {d}{e}+\frac {x^2}{e}\right ) \left (\frac {c d^2-b d e+a e^2}{e^2}-\frac {(2 c d-b e) x^2}{e^2}+\frac {c x^4}{e^2}\right )} \, dx,x,\sqrt {d+e x}\right )}{e}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (-\frac {d^2 e}{a \left (d-x^2\right )}+\frac {e \left (d \left (c d^2-b d e+a e^2\right )-\left (c d^2-a e^2\right ) x^2\right )}{a \left (c d^2-b d e+a e^2-(2 c d-b e) x^2+c x^4\right )}\right ) \, dx,x,\sqrt {d+e x}\right )}{e}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {d \left (c d^2-b d e+a e^2\right )+\left (-c d^2+a e^2\right ) x^2}{c d^2-b d e+a e^2+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{a}-\frac {\left (2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{d-x^2} \, dx,x,\sqrt {d+e x}\right )}{a}\\ &=-\frac {2 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{a}+\frac {\left (a \sqrt {b^2-4 a c} e^2-c d \left (\sqrt {b^2-4 a c} d-4 a e\right )-b \left (c d^2+a e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{a \sqrt {b^2-4 a c}}+\frac {\left (a \sqrt {b^2-4 a c} e^2-c d \left (\sqrt {b^2-4 a c} d+4 a e\right )+b \left (c d^2+a e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{a \sqrt {b^2-4 a c}}\\ &=-\frac {2 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{a}-\frac {\sqrt {2} \left (a \sqrt {b^2-4 a c} e^2-c d \left (\sqrt {b^2-4 a c} d-4 a e\right )-b \left (c d^2+a e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{a \sqrt {c} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\sqrt {2} \left (a \sqrt {b^2-4 a c} e^2-c d \left (\sqrt {b^2-4 a c} d+4 a e\right )+b \left (c d^2+a e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{a \sqrt {c} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\\ \end {align*}

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Mathematica [A]  time = 1.16, size = 331, normalized size = 0.97 \[ \frac {\frac {\sqrt {2} \left (c d \left (d \sqrt {b^2-4 a c}-4 a e\right )-a e^2 \sqrt {b^2-4 a c}+b \left (a e^2+c d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {e \sqrt {b^2-4 a c}-b e+2 c d}}\right )}{\sqrt {c} \sqrt {b^2-4 a c} \sqrt {e \left (\sqrt {b^2-4 a c}-b\right )+2 c d}}+\frac {\sqrt {2} \left (c d \left (d \sqrt {b^2-4 a c}+4 a e\right )-a e^2 \sqrt {b^2-4 a c}-b \left (a e^2+c d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {c} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-2 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(3/2)/(x*(a + b*x + c*x^2)),x]

[Out]

(-2*d^(3/2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] + (Sqrt[2]*(-(a*Sqrt[b^2 - 4*a*c]*e^2) + c*d*(Sqrt[b^2 - 4*a*c]*d -
4*a*e) + b*(c*d^2 + a*e^2))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])
/(Sqrt[c]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) + (Sqrt[2]*(-(a*Sqrt[b^2 - 4*a*c]*e^2) +
c*d*(Sqrt[b^2 - 4*a*c]*d + 4*a*e) - b*(c*d^2 + a*e^2))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (
b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[c]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/a

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fricas [B]  time = 15.30, size = 5167, normalized size = 15.20 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/x/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

[-1/2*(sqrt(2)*a*sqrt(-(3*a*b*c*d^2*e - 6*a^2*c*d*e^2 + a^2*b*e^3 - (b^2*c - 2*a*c^2)*d^3 + (a^2*b^2*c - 4*a^3
*c^2)*sqrt((b^2*c^2*d^6 - 6*a*b*c^2*d^5*e + 9*a^2*c^2*d^4*e^2 + 2*a^2*b*c*d^3*e^3 - 6*a^3*c*d^2*e^4 + a^4*e^6)
/(a^4*b^2*c^2 - 4*a^5*c^3)))/(a^2*b^2*c - 4*a^3*c^2))*log(sqrt(2)*((b^3*c - 4*a*b*c^2)*d^4 - 3*(a*b^2*c - 4*a^
2*c^2)*d^3*e + (a^2*b^2 - 4*a^3*c)*d*e^3 + ((a^2*b^3*c - 4*a^3*b*c^2)*d - 2*(a^3*b^2*c - 4*a^4*c^2)*e)*sqrt((b
^2*c^2*d^6 - 6*a*b*c^2*d^5*e + 9*a^2*c^2*d^4*e^2 + 2*a^2*b*c*d^3*e^3 - 6*a^3*c*d^2*e^4 + a^4*e^6)/(a^4*b^2*c^2
- 4*a^5*c^3)))*sqrt(-(3*a*b*c*d^2*e - 6*a^2*c*d*e^2 + a^2*b*e^3 - (b^2*c - 2*a*c^2)*d^3 + (a^2*b^2*c - 4*a^3*
c^2)*sqrt((b^2*c^2*d^6 - 6*a*b*c^2*d^5*e + 9*a^2*c^2*d^4*e^2 + 2*a^2*b*c*d^3*e^3 - 6*a^3*c*d^2*e^4 + a^4*e^6)/
(a^4*b^2*c^2 - 4*a^5*c^3)))/(a^2*b^2*c - 4*a^3*c^2)) + 4*(b*c^2*d^5 + 4*a*b*c*d^3*e^2 - 2*a^2*c*d^2*e^3 - a^2*
b*d*e^4 + a^3*e^5 - (b^2*c + 3*a*c^2)*d^4*e)*sqrt(e*x + d)) - sqrt(2)*a*sqrt(-(3*a*b*c*d^2*e - 6*a^2*c*d*e^2 +
a^2*b*e^3 - (b^2*c - 2*a*c^2)*d^3 + (a^2*b^2*c - 4*a^3*c^2)*sqrt((b^2*c^2*d^6 - 6*a*b*c^2*d^5*e + 9*a^2*c^2*d
^4*e^2 + 2*a^2*b*c*d^3*e^3 - 6*a^3*c*d^2*e^4 + a^4*e^6)/(a^4*b^2*c^2 - 4*a^5*c^3)))/(a^2*b^2*c - 4*a^3*c^2))*l
og(-sqrt(2)*((b^3*c - 4*a*b*c^2)*d^4 - 3*(a*b^2*c - 4*a^2*c^2)*d^3*e + (a^2*b^2 - 4*a^3*c)*d*e^3 + ((a^2*b^3*c
- 4*a^3*b*c^2)*d - 2*(a^3*b^2*c - 4*a^4*c^2)*e)*sqrt((b^2*c^2*d^6 - 6*a*b*c^2*d^5*e + 9*a^2*c^2*d^4*e^2 + 2*a
^2*b*c*d^3*e^3 - 6*a^3*c*d^2*e^4 + a^4*e^6)/(a^4*b^2*c^2 - 4*a^5*c^3)))*sqrt(-(3*a*b*c*d^2*e - 6*a^2*c*d*e^2 +
a^2*b*e^3 - (b^2*c - 2*a*c^2)*d^3 + (a^2*b^2*c - 4*a^3*c^2)*sqrt((b^2*c^2*d^6 - 6*a*b*c^2*d^5*e + 9*a^2*c^2*d
^4*e^2 + 2*a^2*b*c*d^3*e^3 - 6*a^3*c*d^2*e^4 + a^4*e^6)/(a^4*b^2*c^2 - 4*a^5*c^3)))/(a^2*b^2*c - 4*a^3*c^2)) +
4*(b*c^2*d^5 + 4*a*b*c*d^3*e^2 - 2*a^2*c*d^2*e^3 - a^2*b*d*e^4 + a^3*e^5 - (b^2*c + 3*a*c^2)*d^4*e)*sqrt(e*x
+ d)) + sqrt(2)*a*sqrt(-(3*a*b*c*d^2*e - 6*a^2*c*d*e^2 + a^2*b*e^3 - (b^2*c - 2*a*c^2)*d^3 - (a^2*b^2*c - 4*a^
3*c^2)*sqrt((b^2*c^2*d^6 - 6*a*b*c^2*d^5*e + 9*a^2*c^2*d^4*e^2 + 2*a^2*b*c*d^3*e^3 - 6*a^3*c*d^2*e^4 + a^4*e^6
)/(a^4*b^2*c^2 - 4*a^5*c^3)))/(a^2*b^2*c - 4*a^3*c^2))*log(sqrt(2)*((b^3*c - 4*a*b*c^2)*d^4 - 3*(a*b^2*c - 4*a
^2*c^2)*d^3*e + (a^2*b^2 - 4*a^3*c)*d*e^3 - ((a^2*b^3*c - 4*a^3*b*c^2)*d - 2*(a^3*b^2*c - 4*a^4*c^2)*e)*sqrt((
b^2*c^2*d^6 - 6*a*b*c^2*d^5*e + 9*a^2*c^2*d^4*e^2 + 2*a^2*b*c*d^3*e^3 - 6*a^3*c*d^2*e^4 + a^4*e^6)/(a^4*b^2*c^
2 - 4*a^5*c^3)))*sqrt(-(3*a*b*c*d^2*e - 6*a^2*c*d*e^2 + a^2*b*e^3 - (b^2*c - 2*a*c^2)*d^3 - (a^2*b^2*c - 4*a^3
*c^2)*sqrt((b^2*c^2*d^6 - 6*a*b*c^2*d^5*e + 9*a^2*c^2*d^4*e^2 + 2*a^2*b*c*d^3*e^3 - 6*a^3*c*d^2*e^4 + a^4*e^6)
/(a^4*b^2*c^2 - 4*a^5*c^3)))/(a^2*b^2*c - 4*a^3*c^2)) + 4*(b*c^2*d^5 + 4*a*b*c*d^3*e^2 - 2*a^2*c*d^2*e^3 - a^2
*b*d*e^4 + a^3*e^5 - (b^2*c + 3*a*c^2)*d^4*e)*sqrt(e*x + d)) - sqrt(2)*a*sqrt(-(3*a*b*c*d^2*e - 6*a^2*c*d*e^2
+ a^2*b*e^3 - (b^2*c - 2*a*c^2)*d^3 - (a^2*b^2*c - 4*a^3*c^2)*sqrt((b^2*c^2*d^6 - 6*a*b*c^2*d^5*e + 9*a^2*c^2*
d^4*e^2 + 2*a^2*b*c*d^3*e^3 - 6*a^3*c*d^2*e^4 + a^4*e^6)/(a^4*b^2*c^2 - 4*a^5*c^3)))/(a^2*b^2*c - 4*a^3*c^2))*
log(-sqrt(2)*((b^3*c - 4*a*b*c^2)*d^4 - 3*(a*b^2*c - 4*a^2*c^2)*d^3*e + (a^2*b^2 - 4*a^3*c)*d*e^3 - ((a^2*b^3*
c - 4*a^3*b*c^2)*d - 2*(a^3*b^2*c - 4*a^4*c^2)*e)*sqrt((b^2*c^2*d^6 - 6*a*b*c^2*d^5*e + 9*a^2*c^2*d^4*e^2 + 2*
a^2*b*c*d^3*e^3 - 6*a^3*c*d^2*e^4 + a^4*e^6)/(a^4*b^2*c^2 - 4*a^5*c^3)))*sqrt(-(3*a*b*c*d^2*e - 6*a^2*c*d*e^2
+ a^2*b*e^3 - (b^2*c - 2*a*c^2)*d^3 - (a^2*b^2*c - 4*a^3*c^2)*sqrt((b^2*c^2*d^6 - 6*a*b*c^2*d^5*e + 9*a^2*c^2*
d^4*e^2 + 2*a^2*b*c*d^3*e^3 - 6*a^3*c*d^2*e^4 + a^4*e^6)/(a^4*b^2*c^2 - 4*a^5*c^3)))/(a^2*b^2*c - 4*a^3*c^2))
+ 4*(b*c^2*d^5 + 4*a*b*c*d^3*e^2 - 2*a^2*c*d^2*e^3 - a^2*b*d*e^4 + a^3*e^5 - (b^2*c + 3*a*c^2)*d^4*e)*sqrt(e*x
+ d)) - 2*d^(3/2)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x))/a, -1/2*(sqrt(2)*a*sqrt(-(3*a*b*c*d^2*e - 6*a
^2*c*d*e^2 + a^2*b*e^3 - (b^2*c - 2*a*c^2)*d^3 + (a^2*b^2*c - 4*a^3*c^2)*sqrt((b^2*c^2*d^6 - 6*a*b*c^2*d^5*e +
9*a^2*c^2*d^4*e^2 + 2*a^2*b*c*d^3*e^3 - 6*a^3*c*d^2*e^4 + a^4*e^6)/(a^4*b^2*c^2 - 4*a^5*c^3)))/(a^2*b^2*c - 4
*a^3*c^2))*log(sqrt(2)*((b^3*c - 4*a*b*c^2)*d^4 - 3*(a*b^2*c - 4*a^2*c^2)*d^3*e + (a^2*b^2 - 4*a^3*c)*d*e^3 +
((a^2*b^3*c - 4*a^3*b*c^2)*d - 2*(a^3*b^2*c - 4*a^4*c^2)*e)*sqrt((b^2*c^2*d^6 - 6*a*b*c^2*d^5*e + 9*a^2*c^2*d^
4*e^2 + 2*a^2*b*c*d^3*e^3 - 6*a^3*c*d^2*e^4 + a^4*e^6)/(a^4*b^2*c^2 - 4*a^5*c^3)))*sqrt(-(3*a*b*c*d^2*e - 6*a^
2*c*d*e^2 + a^2*b*e^3 - (b^2*c - 2*a*c^2)*d^3 + (a^2*b^2*c - 4*a^3*c^2)*sqrt((b^2*c^2*d^6 - 6*a*b*c^2*d^5*e +
9*a^2*c^2*d^4*e^2 + 2*a^2*b*c*d^3*e^3 - 6*a^3*c*d^2*e^4 + a^4*e^6)/(a^4*b^2*c^2 - 4*a^5*c^3)))/(a^2*b^2*c - 4*
a^3*c^2)) + 4*(b*c^2*d^5 + 4*a*b*c*d^3*e^2 - 2*a^2*c*d^2*e^3 - a^2*b*d*e^4 + a^3*e^5 - (b^2*c + 3*a*c^2)*d^4*e
)*sqrt(e*x + d)) - sqrt(2)*a*sqrt(-(3*a*b*c*d^2*e - 6*a^2*c*d*e^2 + a^2*b*e^3 - (b^2*c - 2*a*c^2)*d^3 + (a^2*b
^2*c - 4*a^3*c^2)*sqrt((b^2*c^2*d^6 - 6*a*b*c^2*d^5*e + 9*a^2*c^2*d^4*e^2 + 2*a^2*b*c*d^3*e^3 - 6*a^3*c*d^2*e^
4 + a^4*e^6)/(a^4*b^2*c^2 - 4*a^5*c^3)))/(a^2*b^2*c - 4*a^3*c^2))*log(-sqrt(2)*((b^3*c - 4*a*b*c^2)*d^4 - 3*(a
*b^2*c - 4*a^2*c^2)*d^3*e + (a^2*b^2 - 4*a^3*c)*d*e^3 + ((a^2*b^3*c - 4*a^3*b*c^2)*d - 2*(a^3*b^2*c - 4*a^4*c^
2)*e)*sqrt((b^2*c^2*d^6 - 6*a*b*c^2*d^5*e + 9*a^2*c^2*d^4*e^2 + 2*a^2*b*c*d^3*e^3 - 6*a^3*c*d^2*e^4 + a^4*e^6)
/(a^4*b^2*c^2 - 4*a^5*c^3)))*sqrt(-(3*a*b*c*d^2*e - 6*a^2*c*d*e^2 + a^2*b*e^3 - (b^2*c - 2*a*c^2)*d^3 + (a^2*b
^2*c - 4*a^3*c^2)*sqrt((b^2*c^2*d^6 - 6*a*b*c^2*d^5*e + 9*a^2*c^2*d^4*e^2 + 2*a^2*b*c*d^3*e^3 - 6*a^3*c*d^2*e^
4 + a^4*e^6)/(a^4*b^2*c^2 - 4*a^5*c^3)))/(a^2*b^2*c - 4*a^3*c^2)) + 4*(b*c^2*d^5 + 4*a*b*c*d^3*e^2 - 2*a^2*c*d
^2*e^3 - a^2*b*d*e^4 + a^3*e^5 - (b^2*c + 3*a*c^2)*d^4*e)*sqrt(e*x + d)) + sqrt(2)*a*sqrt(-(3*a*b*c*d^2*e - 6*
a^2*c*d*e^2 + a^2*b*e^3 - (b^2*c - 2*a*c^2)*d^3 - (a^2*b^2*c - 4*a^3*c^2)*sqrt((b^2*c^2*d^6 - 6*a*b*c^2*d^5*e
+ 9*a^2*c^2*d^4*e^2 + 2*a^2*b*c*d^3*e^3 - 6*a^3*c*d^2*e^4 + a^4*e^6)/(a^4*b^2*c^2 - 4*a^5*c^3)))/(a^2*b^2*c -
4*a^3*c^2))*log(sqrt(2)*((b^3*c - 4*a*b*c^2)*d^4 - 3*(a*b^2*c - 4*a^2*c^2)*d^3*e + (a^2*b^2 - 4*a^3*c)*d*e^3 -
((a^2*b^3*c - 4*a^3*b*c^2)*d - 2*(a^3*b^2*c - 4*a^4*c^2)*e)*sqrt((b^2*c^2*d^6 - 6*a*b*c^2*d^5*e + 9*a^2*c^2*d
^4*e^2 + 2*a^2*b*c*d^3*e^3 - 6*a^3*c*d^2*e^4 + a^4*e^6)/(a^4*b^2*c^2 - 4*a^5*c^3)))*sqrt(-(3*a*b*c*d^2*e - 6*a
^2*c*d*e^2 + a^2*b*e^3 - (b^2*c - 2*a*c^2)*d^3 - (a^2*b^2*c - 4*a^3*c^2)*sqrt((b^2*c^2*d^6 - 6*a*b*c^2*d^5*e +
9*a^2*c^2*d^4*e^2 + 2*a^2*b*c*d^3*e^3 - 6*a^3*c*d^2*e^4 + a^4*e^6)/(a^4*b^2*c^2 - 4*a^5*c^3)))/(a^2*b^2*c - 4
*a^3*c^2)) + 4*(b*c^2*d^5 + 4*a*b*c*d^3*e^2 - 2*a^2*c*d^2*e^3 - a^2*b*d*e^4 + a^3*e^5 - (b^2*c + 3*a*c^2)*d^4*
e)*sqrt(e*x + d)) - sqrt(2)*a*sqrt(-(3*a*b*c*d^2*e - 6*a^2*c*d*e^2 + a^2*b*e^3 - (b^2*c - 2*a*c^2)*d^3 - (a^2*
b^2*c - 4*a^3*c^2)*sqrt((b^2*c^2*d^6 - 6*a*b*c^2*d^5*e + 9*a^2*c^2*d^4*e^2 + 2*a^2*b*c*d^3*e^3 - 6*a^3*c*d^2*e
^4 + a^4*e^6)/(a^4*b^2*c^2 - 4*a^5*c^3)))/(a^2*b^2*c - 4*a^3*c^2))*log(-sqrt(2)*((b^3*c - 4*a*b*c^2)*d^4 - 3*(
a*b^2*c - 4*a^2*c^2)*d^3*e + (a^2*b^2 - 4*a^3*c)*d*e^3 - ((a^2*b^3*c - 4*a^3*b*c^2)*d - 2*(a^3*b^2*c - 4*a^4*c
^2)*e)*sqrt((b^2*c^2*d^6 - 6*a*b*c^2*d^5*e + 9*a^2*c^2*d^4*e^2 + 2*a^2*b*c*d^3*e^3 - 6*a^3*c*d^2*e^4 + a^4*e^6
)/(a^4*b^2*c^2 - 4*a^5*c^3)))*sqrt(-(3*a*b*c*d^2*e - 6*a^2*c*d*e^2 + a^2*b*e^3 - (b^2*c - 2*a*c^2)*d^3 - (a^2*
b^2*c - 4*a^3*c^2)*sqrt((b^2*c^2*d^6 - 6*a*b*c^2*d^5*e + 9*a^2*c^2*d^4*e^2 + 2*a^2*b*c*d^3*e^3 - 6*a^3*c*d^2*e
^4 + a^4*e^6)/(a^4*b^2*c^2 - 4*a^5*c^3)))/(a^2*b^2*c - 4*a^3*c^2)) + 4*(b*c^2*d^5 + 4*a*b*c*d^3*e^2 - 2*a^2*c*
d^2*e^3 - a^2*b*d*e^4 + a^3*e^5 - (b^2*c + 3*a*c^2)*d^4*e)*sqrt(e*x + d)) - 4*sqrt(-d)*d*arctan(sqrt(e*x + d)*
sqrt(-d)/d))/a]

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giac [B]  time = 0.41, size = 822, normalized size = 2.42 \[ \frac {2 \, d^{2} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{a \sqrt {-d}} - \frac {{\left ({\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} e - {\left (a b^{2} - 4 \, a^{2} c\right )} e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} a^{2} - 2 \, {\left (\sqrt {b^{2} - 4 \, a c} a c^{2} d^{3} - \sqrt {b^{2} - 4 \, a c} a b c d^{2} e + \sqrt {b^{2} - 4 \, a c} a^{2} c d e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | a \right |} - {\left (2 \, a^{2} b c^{2} d^{3} + 6 \, a^{3} b c d e^{2} - a^{3} b^{2} e^{3} - {\left (a^{2} b^{2} c + 8 \, a^{3} c^{2}\right )} d^{2} e\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, a c d - a b e + \sqrt {-4 \, {\left (a c d^{2} - a b d e + a^{2} e^{2}\right )} a c + {\left (2 \, a c d - a b e\right )}^{2}}}{a c}}}\right )}{4 \, {\left (\sqrt {b^{2} - 4 \, a c} a^{2} c^{2} d^{2} - \sqrt {b^{2} - 4 \, a c} a^{2} b c d e + \sqrt {b^{2} - 4 \, a c} a^{3} c e^{2}\right )} {\left | a \right |} {\left | c \right |}} + \frac {{\left ({\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} e - {\left (a b^{2} - 4 \, a^{2} c\right )} e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} a^{2} + 2 \, {\left (\sqrt {b^{2} - 4 \, a c} a c^{2} d^{3} - \sqrt {b^{2} - 4 \, a c} a b c d^{2} e + \sqrt {b^{2} - 4 \, a c} a^{2} c d e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | a \right |} - {\left (2 \, a^{2} b c^{2} d^{3} + 6 \, a^{3} b c d e^{2} - a^{3} b^{2} e^{3} - {\left (a^{2} b^{2} c + 8 \, a^{3} c^{2}\right )} d^{2} e\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, a c d - a b e - \sqrt {-4 \, {\left (a c d^{2} - a b d e + a^{2} e^{2}\right )} a c + {\left (2 \, a c d - a b e\right )}^{2}}}{a c}}}\right )}{4 \, {\left (\sqrt {b^{2} - 4 \, a c} a^{2} c^{2} d^{2} - \sqrt {b^{2} - 4 \, a c} a^{2} b c d e + \sqrt {b^{2} - 4 \, a c} a^{3} c e^{2}\right )} {\left | a \right |} {\left | c \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/x/(c*x^2+b*x+a),x, algorithm="giac")

[Out]

2*d^2*arctan(sqrt(x*e + d)/sqrt(-d))/(a*sqrt(-d)) - 1/4*(((b^2*c - 4*a*c^2)*d^2*e - (a*b^2 - 4*a^2*c)*e^3)*sqr
t(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*a^2 - 2*(sqrt(b^2 - 4*a*c)*a*c^2*d^3 - sqrt(b^2 - 4*a*c)*a*b*c*d
^2*e + sqrt(b^2 - 4*a*c)*a^2*c*d*e^2)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(a) - (2*a^2*b*c^2*d
^3 + 6*a^3*b*c*d*e^2 - a^3*b^2*e^3 - (a^2*b^2*c + 8*a^3*c^2)*d^2*e)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)
*c)*e))*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*a*c*d - a*b*e + sqrt(-4*(a*c*d^2 - a*b*d*e + a^2*e^2)*a*c +
(2*a*c*d - a*b*e)^2))/(a*c)))/((sqrt(b^2 - 4*a*c)*a^2*c^2*d^2 - sqrt(b^2 - 4*a*c)*a^2*b*c*d*e + sqrt(b^2 - 4*a
*c)*a^3*c*e^2)*abs(a)*abs(c)) + 1/4*(((b^2*c - 4*a*c^2)*d^2*e - (a*b^2 - 4*a^2*c)*e^3)*sqrt(-4*c^2*d + 2*(b*c
+ sqrt(b^2 - 4*a*c)*c)*e)*a^2 + 2*(sqrt(b^2 - 4*a*c)*a*c^2*d^3 - sqrt(b^2 - 4*a*c)*a*b*c*d^2*e + sqrt(b^2 - 4*
a*c)*a^2*c*d*e^2)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*abs(a) - (2*a^2*b*c^2*d^3 + 6*a^3*b*c*d*e^2
- a^3*b^2*e^3 - (a^2*b^2*c + 8*a^3*c^2)*d^2*e)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqr
t(1/2)*sqrt(x*e + d)/sqrt(-(2*a*c*d - a*b*e - sqrt(-4*(a*c*d^2 - a*b*d*e + a^2*e^2)*a*c + (2*a*c*d - a*b*e)^2)
)/(a*c)))/((sqrt(b^2 - 4*a*c)*a^2*c^2*d^2 - sqrt(b^2 - 4*a*c)*a^2*b*c*d*e + sqrt(b^2 - 4*a*c)*a^3*c*e^2)*abs(a
)*abs(c))

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maple [B]  time = 0.04, size = 944, normalized size = 2.78 \[ \frac {\sqrt {2}\, b c \,d^{2} e \arctanh \left (\frac {\sqrt {e x +d}\, \sqrt {2}\, c}{\sqrt {\left (-b e +2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}\right )}{\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\, \sqrt {\left (-b e +2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}\, a}+\frac {\sqrt {2}\, b c \,d^{2} e \arctan \left (\frac {\sqrt {e x +d}\, \sqrt {2}\, c}{\sqrt {\left (b e -2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}\right )}{\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\, \sqrt {\left (b e -2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}\, a}+\frac {\sqrt {2}\, b \,e^{3} \arctanh \left (\frac {\sqrt {e x +d}\, \sqrt {2}\, c}{\sqrt {\left (-b e +2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}\right )}{\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\, \sqrt {\left (-b e +2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}+\frac {\sqrt {2}\, b \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, \sqrt {2}\, c}{\sqrt {\left (b e -2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}\right )}{\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\, \sqrt {\left (b e -2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}-\frac {4 \sqrt {2}\, c d \,e^{2} \arctanh \left (\frac {\sqrt {e x +d}\, \sqrt {2}\, c}{\sqrt {\left (-b e +2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}\right )}{\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\, \sqrt {\left (-b e +2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}-\frac {4 \sqrt {2}\, c d \,e^{2} \arctan \left (\frac {\sqrt {e x +d}\, \sqrt {2}\, c}{\sqrt {\left (b e -2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}\right )}{\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\, \sqrt {\left (b e -2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}+\frac {\sqrt {2}\, c \,d^{2} \arctanh \left (\frac {\sqrt {e x +d}\, \sqrt {2}\, c}{\sqrt {\left (-b e +2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}\right )}{\sqrt {\left (-b e +2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}\, a}-\frac {\sqrt {2}\, c \,d^{2} \arctan \left (\frac {\sqrt {e x +d}\, \sqrt {2}\, c}{\sqrt {\left (b e -2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}\right )}{\sqrt {\left (b e -2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}\, a}-\frac {\sqrt {2}\, e^{2} \arctanh \left (\frac {\sqrt {e x +d}\, \sqrt {2}\, c}{\sqrt {\left (-b e +2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}\right )}{\sqrt {\left (-b e +2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}+\frac {\sqrt {2}\, e^{2} \arctan \left (\frac {\sqrt {e x +d}\, \sqrt {2}\, c}{\sqrt {\left (b e -2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}\right )}{\sqrt {\left (b e -2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}-\frac {2 d^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(3/2)/x/(c*x^2+b*x+a),x)

[Out]

-2*d^(3/2)*arctanh((e*x+d)^(1/2)/d^(1/2))/a+e^3/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^
2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b-4*e^2*c/
(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)
/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*d+e/a*c/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4
*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c
)*b*d^2-e^2*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d
+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)+1/a*c*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((
e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*d^2+e^3/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/
2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^
(1/2))*c)^(1/2)*c)*b-4*e^2*c/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*a
rctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*d+e/a*c/(-(4*a*c-b^2)*e^2)^(1/2)
*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)
*e^2)^(1/2))*c)^(1/2)*c)*b*d^2+e^2*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)
*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)-1/a*c*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))
*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*d^2

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c x^{2} + b x + a\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/x/(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

integrate((e*x + d)^(3/2)/((c*x^2 + b*x + a)*x), x)

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mupad [B]  time = 8.16, size = 20897, normalized size = 61.46 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(3/2)/(x*(a + b*x + c*x^2)),x)

[Out]

atan(((((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3
*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e - 3*a*c*d^2*e*(-(4*a*c - b^2)^
3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*((((b^4
*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 +
4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e - 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 1
2*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*((d + e*x)^(1/2)*((
b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^
2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e - 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2)
+ 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*(512*a^5*c^4*e^1
0 + 32*a^3*b^4*c^2*e^10 - 256*a^4*b^2*c^3*e^10 + 768*a^4*c^5*d^2*e^8 + 64*a^2*b^4*c^3*d^2*e^8 - 448*a^3*b^2*c^
4*d^2*e^8 - 896*a^4*b*c^4*d*e^9 - 64*a^2*b^5*c^2*d*e^9 + 480*a^3*b^3*c^3*d*e^9) - 384*a^3*c^5*d^4*e^8 - 384*a^
4*c^4*d^2*e^10 + 96*a^2*b^2*c^4*d^4*e^8 - 128*a^2*b^3*c^3*d^3*e^9 + 32*a^2*b^4*c^2*d^2*e^10 - 32*a^3*b^2*c^3*d
^2*e^10 + 128*a^4*b*c^3*d*e^11 + 512*a^3*b*c^4*d^3*e^9 - 32*a^3*b^3*c^2*d*e^11) + (d + e*x)^(1/2)*(32*a^3*b^3*
c*e^13 - 128*a^4*b*c^2*e^13 + 704*a^4*c^3*d*e^12 - 576*a^2*c^5*d^5*e^8 + 896*a^3*c^4*d^3*e^10 - 64*b^4*c^3*d^5
*e^8 + 64*b^5*c^2*d^4*e^9 + 192*a^2*b^2*c^3*d^3*e^10 + 448*a^2*b^3*c^2*d^2*e^11 - 64*a^2*b^4*c*d*e^12 + 384*a*
b^2*c^4*d^5*e^8 - 320*a*b^3*c^3*d^4*e^9 - 128*a*b^4*c^2*d^3*e^10 + 384*a^2*b*c^4*d^4*e^9 - 1664*a^3*b*c^3*d^2*
e^11 + 64*a^3*b^2*c^2*d*e^12))*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) -
6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e - 3*a*
c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*
b^2*c^2)))^(1/2) + 96*a*c^5*d^7*e^8 + 32*a^4*c^2*d*e^14 - 672*a^2*c^4*d^5*e^10 - 736*a^3*c^3*d^3*e^12 - 32*b^2
*c^4*d^7*e^8 - 32*b^3*c^3*d^6*e^9 + 64*b^4*c^2*d^5*e^10 - 96*a^2*b^2*c^2*d^3*e^12 + 256*a*b*c^4*d^6*e^9 - 32*a
^3*b^2*c*d*e^14 - 288*a*b^2*c^3*d^5*e^10 - 160*a*b^3*c^2*d^4*e^11 + 1280*a^2*b*c^3*d^4*e^11 + 32*a^2*b^3*c*d^2
*e^13 + 128*a^3*b*c^2*d^2*e^13) + (d + e*x)^(1/2)*(32*a^4*c*e^16 + 96*c^5*d^8*e^8 - 256*a*c^4*d^6*e^10 - 256*b
*c^4*d^7*e^9 + 64*b^4*c*d^4*e^12 + 256*a^2*c^3*d^4*e^12 + 128*a^3*c^2*d^2*e^14 + 384*b^2*c^3*d^6*e^10 - 256*b^
3*c^2*d^5*e^11 - 128*a^3*b*c*d*e^15 - 128*a*b^3*c*d^3*e^13 + 256*a*b^2*c^2*d^4*e^12 - 384*a^2*b*c^2*d^3*e^13 +
192*a^2*b^2*c*d^2*e^14))*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b
^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e - 3*a*c*d^2
*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c
^2)))^(1/2)*1i + (((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d
^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e - 3*a*c*d^2*e*(-(4*
a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1
/2)*((((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*
c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e - 3*a*c*d^2*e*(-(4*a*c - b^2)^3
)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*((d + e*
x)^(1/2)*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a
^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e - 3*a*c*d^2*e*(-(4*a*c - b^2
)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*(512*
a^5*c^4*e^10 + 32*a^3*b^4*c^2*e^10 - 256*a^4*b^2*c^3*e^10 + 768*a^4*c^5*d^2*e^8 + 64*a^2*b^4*c^3*d^2*e^8 - 448
*a^3*b^2*c^4*d^2*e^8 - 896*a^4*b*c^4*d*e^9 - 64*a^2*b^5*c^2*d*e^9 + 480*a^3*b^3*c^3*d*e^9) + 384*a^3*c^5*d^4*e
^8 + 384*a^4*c^4*d^2*e^10 - 96*a^2*b^2*c^4*d^4*e^8 + 128*a^2*b^3*c^3*d^3*e^9 - 32*a^2*b^4*c^2*d^2*e^10 + 32*a^
3*b^2*c^3*d^2*e^10 - 128*a^4*b*c^3*d*e^11 - 512*a^3*b*c^4*d^3*e^9 + 32*a^3*b^3*c^2*d*e^11) + (d + e*x)^(1/2)*(
32*a^3*b^3*c*e^13 - 128*a^4*b*c^2*e^13 + 704*a^4*c^3*d*e^12 - 576*a^2*c^5*d^5*e^8 + 896*a^3*c^4*d^3*e^10 - 64*
b^4*c^3*d^5*e^8 + 64*b^5*c^2*d^4*e^9 + 192*a^2*b^2*c^3*d^3*e^10 + 448*a^2*b^3*c^2*d^2*e^11 - 64*a^2*b^4*c*d*e^
12 + 384*a*b^2*c^4*d^5*e^8 - 320*a*b^3*c^3*d^4*e^9 - 128*a*b^4*c^2*d^3*e^10 + 384*a^2*b*c^4*d^4*e^9 - 1664*a^3
*b*c^3*d^2*e^11 + 64*a^3*b^2*c^2*d*e^12))*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^
3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d
^2*e - 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4
*c - 8*a^3*b^2*c^2)))^(1/2) - 96*a*c^5*d^7*e^8 - 32*a^4*c^2*d*e^14 + 672*a^2*c^4*d^5*e^10 + 736*a^3*c^3*d^3*e^
12 + 32*b^2*c^4*d^7*e^8 + 32*b^3*c^3*d^6*e^9 - 64*b^4*c^2*d^5*e^10 + 96*a^2*b^2*c^2*d^3*e^12 - 256*a*b*c^4*d^6
*e^9 + 32*a^3*b^2*c*d*e^14 + 288*a*b^2*c^3*d^5*e^10 + 160*a*b^3*c^2*d^4*e^11 - 1280*a^2*b*c^3*d^4*e^11 - 32*a^
2*b^3*c*d^2*e^13 - 128*a^3*b*c^2*d^2*e^13) + (d + e*x)^(1/2)*(32*a^4*c*e^16 + 96*c^5*d^8*e^8 - 256*a*c^4*d^6*e
^10 - 256*b*c^4*d^7*e^9 + 64*b^4*c*d^4*e^12 + 256*a^2*c^3*d^4*e^12 + 128*a^3*c^2*d^2*e^14 + 384*b^2*c^3*d^6*e^
10 - 256*b^3*c^2*d^5*e^11 - 128*a^3*b*c*d*e^15 - 128*a*b^3*c*d^3*e^13 + 256*a*b^2*c^2*d^4*e^12 - 384*a^2*b*c^2
*d^3*e^13 + 192*a^2*b^2*c*d^2*e^14))*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^3)^(1
/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e
- 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c -
8*a^3*b^2*c^2)))^(1/2)*1i)/((((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*
a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e - 3*a*c*
d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^
2*c^2)))^(1/2)*((((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^
3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e - 3*a*c*d^2*e*(-(4*a
*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/
2)*((d + e*x)^(1/2)*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2
*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e - 3*a*c*d^2*e*(-(
4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^
(1/2)*(512*a^5*c^4*e^10 + 32*a^3*b^4*c^2*e^10 - 256*a^4*b^2*c^3*e^10 + 768*a^4*c^5*d^2*e^8 + 64*a^2*b^4*c^3*d^
2*e^8 - 448*a^3*b^2*c^4*d^2*e^8 - 896*a^4*b*c^4*d*e^9 - 64*a^2*b^5*c^2*d*e^9 + 480*a^3*b^3*c^3*d*e^9) + 384*a^
3*c^5*d^4*e^8 + 384*a^4*c^4*d^2*e^10 - 96*a^2*b^2*c^4*d^4*e^8 + 128*a^2*b^3*c^3*d^3*e^9 - 32*a^2*b^4*c^2*d^2*e
^10 + 32*a^3*b^2*c^3*d^2*e^10 - 128*a^4*b*c^3*d*e^11 - 512*a^3*b*c^4*d^3*e^9 + 32*a^3*b^3*c^2*d*e^11) + (d + e
*x)^(1/2)*(32*a^3*b^3*c*e^13 - 128*a^4*b*c^2*e^13 + 704*a^4*c^3*d*e^12 - 576*a^2*c^5*d^5*e^8 + 896*a^3*c^4*d^3
*e^10 - 64*b^4*c^3*d^5*e^8 + 64*b^5*c^2*d^4*e^9 + 192*a^2*b^2*c^3*d^3*e^10 + 448*a^2*b^3*c^2*d^2*e^11 - 64*a^2
*b^4*c*d*e^12 + 384*a*b^2*c^4*d^5*e^8 - 320*a*b^3*c^3*d^4*e^9 - 128*a*b^4*c^2*d^3*e^10 + 384*a^2*b*c^4*d^4*e^9
- 1664*a^3*b*c^3*d^2*e^11 + 64*a^3*b^2*c^2*d*e^12))*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*
a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) -
3*a*b^3*c*d^2*e - 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^
3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2) - 96*a*c^5*d^7*e^8 - 32*a^4*c^2*d*e^14 + 672*a^2*c^4*d^5*e^10 + 736*a^3
*c^3*d^3*e^12 + 32*b^2*c^4*d^7*e^8 + 32*b^3*c^3*d^6*e^9 - 64*b^4*c^2*d^5*e^10 + 96*a^2*b^2*c^2*d^3*e^12 - 256*
a*b*c^4*d^6*e^9 + 32*a^3*b^2*c*d*e^14 + 288*a*b^2*c^3*d^5*e^10 + 160*a*b^3*c^2*d^4*e^11 - 1280*a^2*b*c^3*d^4*e
^11 - 32*a^2*b^3*c*d^2*e^13 - 128*a^3*b*c^2*d^2*e^13) + (d + e*x)^(1/2)*(32*a^4*c*e^16 + 96*c^5*d^8*e^8 - 256*
a*c^4*d^6*e^10 - 256*b*c^4*d^7*e^9 + 64*b^4*c*d^4*e^12 + 256*a^2*c^3*d^4*e^12 + 128*a^3*c^2*d^2*e^14 + 384*b^2
*c^3*d^6*e^10 - 256*b^3*c^2*d^5*e^11 - 128*a^3*b*c*d*e^15 - 128*a*b^3*c*d^3*e^13 + 256*a*b^2*c^2*d^4*e^12 - 38
4*a^2*b*c^2*d^3*e^13 + 192*a^2*b^2*c*d^2*e^14))*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c -
b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b
^3*c*d^2*e - 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a
^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2) - (((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^3)^(1
/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e
- 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c -
8*a^3*b^2*c^2)))^(1/2)*((((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^
2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e - 3*a*c*d^2*
e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^
2)))^(1/2)*((d + e*x)^(1/2)*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a
*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e - 3*a*c*d
^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2
*c^2)))^(1/2)*(512*a^5*c^4*e^10 + 32*a^3*b^4*c^2*e^10 - 256*a^4*b^2*c^3*e^10 + 768*a^4*c^5*d^2*e^8 + 64*a^2*b^
4*c^3*d^2*e^8 - 448*a^3*b^2*c^4*d^2*e^8 - 896*a^4*b*c^4*d*e^9 - 64*a^2*b^5*c^2*d*e^9 + 480*a^3*b^3*c^3*d*e^9)
- 384*a^3*c^5*d^4*e^8 - 384*a^4*c^4*d^2*e^10 + 96*a^2*b^2*c^4*d^4*e^8 - 128*a^2*b^3*c^3*d^3*e^9 + 32*a^2*b^4*c
^2*d^2*e^10 - 32*a^3*b^2*c^3*d^2*e^10 + 128*a^4*b*c^3*d*e^11 + 512*a^3*b*c^4*d^3*e^9 - 32*a^3*b^3*c^2*d*e^11)
+ (d + e*x)^(1/2)*(32*a^3*b^3*c*e^13 - 128*a^4*b*c^2*e^13 + 704*a^4*c^3*d*e^12 - 576*a^2*c^5*d^5*e^8 + 896*a^3
*c^4*d^3*e^10 - 64*b^4*c^3*d^5*e^8 + 64*b^5*c^2*d^4*e^9 + 192*a^2*b^2*c^3*d^3*e^10 + 448*a^2*b^3*c^2*d^2*e^11
- 64*a^2*b^4*c*d*e^12 + 384*a*b^2*c^4*d^5*e^8 - 320*a*b^3*c^3*d^4*e^9 - 128*a*b^4*c^2*d^3*e^10 + 384*a^2*b*c^4
*d^4*e^9 - 1664*a^3*b*c^3*d^2*e^11 + 64*a^3*b^2*c^2*d*e^12))*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e
^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^
(1/2) - 3*a*b^3*c*d^2*e - 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(1
6*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2) + 96*a*c^5*d^7*e^8 + 32*a^4*c^2*d*e^14 - 672*a^2*c^4*d^5*e^10 -
736*a^3*c^3*d^3*e^12 - 32*b^2*c^4*d^7*e^8 - 32*b^3*c^3*d^6*e^9 + 64*b^4*c^2*d^5*e^10 - 96*a^2*b^2*c^2*d^3*e^1
2 + 256*a*b*c^4*d^6*e^9 - 32*a^3*b^2*c*d*e^14 - 288*a*b^2*c^3*d^5*e^10 - 160*a*b^3*c^2*d^4*e^11 + 1280*a^2*b*c
^3*d^4*e^11 + 32*a^2*b^3*c*d^2*e^13 + 128*a^3*b*c^2*d^2*e^13) + (d + e*x)^(1/2)*(32*a^4*c*e^16 + 96*c^5*d^8*e^
8 - 256*a*c^4*d^6*e^10 - 256*b*c^4*d^7*e^9 + 64*b^4*c*d^4*e^12 + 256*a^2*c^3*d^4*e^12 + 128*a^3*c^2*d^2*e^14 +
384*b^2*c^3*d^6*e^10 - 256*b^3*c^2*d^5*e^11 - 128*a^3*b*c*d*e^15 - 128*a*b^3*c*d^3*e^13 + 256*a*b^2*c^2*d^4*e
^12 - 384*a^2*b*c^2*d^3*e^13 + 192*a^2*b^2*c*d^2*e^14))*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-
(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2)
- 3*a*b^3*c*d^2*e - 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4
*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2) + 192*c^4*d^8*e^10 + 448*a*c^3*d^6*e^12 + 64*a^3*c*d^2*e^16 - 512*b*
c^3*d^7*e^11 - 128*b^3*c*d^5*e^13 + 320*a^2*c^2*d^4*e^14 + 448*b^2*c^2*d^6*e^12 - 768*a*b*c^2*d^5*e^13 + 320*a
*b^2*c*d^4*e^14 - 256*a^2*b*c*d^3*e^15))*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^3
)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^
2*e - 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*
c - 8*a^3*b^2*c^2)))^(1/2)*2i + atan(((((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 - a^2*e^3*(-(4*a*c - b^2)^3)^
(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*
e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c
- 8*a^3*b^2*c^2)))^(1/2)*((((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 - a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*
b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e + 3*a*c*d^
2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*
c^2)))^(1/2)*((d + e*x)^(1/2)*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 - a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6
*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e + 3*a*c
*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b
^2*c^2)))^(1/2)*(512*a^5*c^4*e^10 + 32*a^3*b^4*c^2*e^10 - 256*a^4*b^2*c^3*e^10 + 768*a^4*c^5*d^2*e^8 + 64*a^2*
b^4*c^3*d^2*e^8 - 448*a^3*b^2*c^4*d^2*e^8 - 896*a^4*b*c^4*d*e^9 - 64*a^2*b^5*c^2*d*e^9 + 480*a^3*b^3*c^3*d*e^9
) - 384*a^3*c^5*d^4*e^8 - 384*a^4*c^4*d^2*e^10 + 96*a^2*b^2*c^4*d^4*e^8 - 128*a^2*b^3*c^3*d^3*e^9 + 32*a^2*b^4
*c^2*d^2*e^10 - 32*a^3*b^2*c^3*d^2*e^10 + 128*a^4*b*c^3*d*e^11 + 512*a^3*b*c^4*d^3*e^9 - 32*a^3*b^3*c^2*d*e^11
) + (d + e*x)^(1/2)*(32*a^3*b^3*c*e^13 - 128*a^4*b*c^2*e^13 + 704*a^4*c^3*d*e^12 - 576*a^2*c^5*d^5*e^8 + 896*a
^3*c^4*d^3*e^10 - 64*b^4*c^3*d^5*e^8 + 64*b^5*c^2*d^4*e^9 + 192*a^2*b^2*c^3*d^3*e^10 + 448*a^2*b^3*c^2*d^2*e^1
1 - 64*a^2*b^4*c*d*e^12 + 384*a*b^2*c^4*d^5*e^8 - 320*a*b^3*c^3*d^4*e^9 - 128*a*b^4*c^2*d^3*e^10 + 384*a^2*b*c
^4*d^4*e^9 - 1664*a^3*b*c^3*d^2*e^11 + 64*a^3*b^2*c^2*d*e^12))*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 - a^2
*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a*c - b^2)^3
)^(1/2) - 3*a*b^3*c*d^2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*
(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2) + 96*a*c^5*d^7*e^8 + 32*a^4*c^2*d*e^14 - 672*a^2*c^4*d^5*e^10
- 736*a^3*c^3*d^3*e^12 - 32*b^2*c^4*d^7*e^8 - 32*b^3*c^3*d^6*e^9 + 64*b^4*c^2*d^5*e^10 - 96*a^2*b^2*c^2*d^3*e
^12 + 256*a*b*c^4*d^6*e^9 - 32*a^3*b^2*c*d*e^14 - 288*a*b^2*c^3*d^5*e^10 - 160*a*b^3*c^2*d^4*e^11 + 1280*a^2*b
*c^3*d^4*e^11 + 32*a^2*b^3*c*d^2*e^13 + 128*a^3*b*c^2*d^2*e^13) + (d + e*x)^(1/2)*(32*a^4*c*e^16 + 96*c^5*d^8*
e^8 - 256*a*c^4*d^6*e^10 - 256*b*c^4*d^7*e^9 + 64*b^4*c*d^4*e^12 + 256*a^2*c^3*d^4*e^12 + 128*a^3*c^2*d^2*e^14
+ 384*b^2*c^3*d^6*e^10 - 256*b^3*c^2*d^5*e^11 - 128*a^3*b*c*d*e^15 - 128*a*b^3*c*d^3*e^13 + 256*a*b^2*c^2*d^4
*e^12 - 384*a^2*b*c^2*d^3*e^13 + 192*a^2*b^2*c*d^2*e^14))*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 - a^2*e^3*
(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a*c - b^2)^3)^(1/
2) - 3*a*b^3*c*d^2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a
^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*1i + (((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 - a^2*e^3*(-(4*a*c
- b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a
*b^3*c*d^2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 +
a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*((((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 - a^2*e^3*(-(4*a*c - b^2)^3)^(
1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e
+ 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c -
8*a^3*b^2*c^2)))^(1/2)*((d + e*x)^(1/2)*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 - a^2*e^3*(-(4*a*c - b^2)^3
)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^
2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*
c - 8*a^3*b^2*c^2)))^(1/2)*(512*a^5*c^4*e^10 + 32*a^3*b^4*c^2*e^10 - 256*a^4*b^2*c^3*e^10 + 768*a^4*c^5*d^2*e^
8 + 64*a^2*b^4*c^3*d^2*e^8 - 448*a^3*b^2*c^4*d^2*e^8 - 896*a^4*b*c^4*d*e^9 - 64*a^2*b^5*c^2*d*e^9 + 480*a^3*b^
3*c^3*d*e^9) + 384*a^3*c^5*d^4*e^8 + 384*a^4*c^4*d^2*e^10 - 96*a^2*b^2*c^4*d^4*e^8 + 128*a^2*b^3*c^3*d^3*e^9 -
32*a^2*b^4*c^2*d^2*e^10 + 32*a^3*b^2*c^3*d^2*e^10 - 128*a^4*b*c^3*d*e^11 - 512*a^3*b*c^4*d^3*e^9 + 32*a^3*b^3
*c^2*d*e^11) + (d + e*x)^(1/2)*(32*a^3*b^3*c*e^13 - 128*a^4*b*c^2*e^13 + 704*a^4*c^3*d*e^12 - 576*a^2*c^5*d^5*
e^8 + 896*a^3*c^4*d^3*e^10 - 64*b^4*c^3*d^5*e^8 + 64*b^5*c^2*d^4*e^9 + 192*a^2*b^2*c^3*d^3*e^10 + 448*a^2*b^3*
c^2*d^2*e^11 - 64*a^2*b^4*c*d*e^12 + 384*a*b^2*c^4*d^5*e^8 - 320*a*b^3*c^3*d^4*e^9 - 128*a*b^4*c^2*d^3*e^10 +
384*a^2*b*c^4*d^4*e^9 - 1664*a^3*b*c^3*d^2*e^11 + 64*a^3*b^2*c^2*d*e^12))*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^
3*d^3 - a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a
*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c
*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2) - 96*a*c^5*d^7*e^8 - 32*a^4*c^2*d*e^14 + 672*a^2*c
^4*d^5*e^10 + 736*a^3*c^3*d^3*e^12 + 32*b^2*c^4*d^7*e^8 + 32*b^3*c^3*d^6*e^9 - 64*b^4*c^2*d^5*e^10 + 96*a^2*b^
2*c^2*d^3*e^12 - 256*a*b*c^4*d^6*e^9 + 32*a^3*b^2*c*d*e^14 + 288*a*b^2*c^3*d^5*e^10 + 160*a*b^3*c^2*d^4*e^11 -
1280*a^2*b*c^3*d^4*e^11 - 32*a^2*b^3*c*d^2*e^13 - 128*a^3*b*c^2*d^2*e^13) + (d + e*x)^(1/2)*(32*a^4*c*e^16 +
96*c^5*d^8*e^8 - 256*a*c^4*d^6*e^10 - 256*b*c^4*d^7*e^9 + 64*b^4*c*d^4*e^12 + 256*a^2*c^3*d^4*e^12 + 128*a^3*c
^2*d^2*e^14 + 384*b^2*c^3*d^6*e^10 - 256*b^3*c^2*d^5*e^11 - 128*a^3*b*c*d*e^15 - 128*a*b^3*c*d^3*e^13 + 256*a*
b^2*c^2*d^4*e^12 - 384*a^2*b*c^2*d^3*e^13 + 192*a^2*b^2*c*d^2*e^14))*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3
- a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a*c -
b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^
2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*1i)/((((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 - a^2*e
^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a*c - b^2)^3)^
(1/2) - 3*a*b^3*c*d^2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(1
6*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*((((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 - a^2*e^3*(-(4*a*c
- b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*
b^3*c*d^2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 +
a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*((d + e*x)^(1/2)*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 - a^2*e^3*(-(4*a
*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3
*a*b^3*c*d^2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3
+ a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*(512*a^5*c^4*e^10 + 32*a^3*b^4*c^2*e^10 - 256*a^4*b^2*c^3*e^10 + 768*a^4
*c^5*d^2*e^8 + 64*a^2*b^4*c^3*d^2*e^8 - 448*a^3*b^2*c^4*d^2*e^8 - 896*a^4*b*c^4*d*e^9 - 64*a^2*b^5*c^2*d*e^9 +
480*a^3*b^3*c^3*d*e^9) + 384*a^3*c^5*d^4*e^8 + 384*a^4*c^4*d^2*e^10 - 96*a^2*b^2*c^4*d^4*e^8 + 128*a^2*b^3*c^
3*d^3*e^9 - 32*a^2*b^4*c^2*d^2*e^10 + 32*a^3*b^2*c^3*d^2*e^10 - 128*a^4*b*c^3*d*e^11 - 512*a^3*b*c^4*d^3*e^9 +
32*a^3*b^3*c^2*d*e^11) + (d + e*x)^(1/2)*(32*a^3*b^3*c*e^13 - 128*a^4*b*c^2*e^13 + 704*a^4*c^3*d*e^12 - 576*a
^2*c^5*d^5*e^8 + 896*a^3*c^4*d^3*e^10 - 64*b^4*c^3*d^5*e^8 + 64*b^5*c^2*d^4*e^9 + 192*a^2*b^2*c^3*d^3*e^10 + 4
48*a^2*b^3*c^2*d^2*e^11 - 64*a^2*b^4*c*d*e^12 + 384*a*b^2*c^4*d^5*e^8 - 320*a*b^3*c^3*d^4*e^9 - 128*a*b^4*c^2*
d^3*e^10 + 384*a^2*b*c^4*d^4*e^9 - 1664*a^3*b*c^3*d^2*e^11 + 64*a^3*b^2*c^2*d*e^12))*((b^4*c*d^3 - a^2*b^3*e^3
+ 8*a^2*c^3*d^3 - a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c
*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e +
6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2) - 96*a*c^5*d^7*e^8 - 32*a^4*c^2*d*e^14
+ 672*a^2*c^4*d^5*e^10 + 736*a^3*c^3*d^3*e^12 + 32*b^2*c^4*d^7*e^8 + 32*b^3*c^3*d^6*e^9 - 64*b^4*c^2*d^5*e^10
+ 96*a^2*b^2*c^2*d^3*e^12 - 256*a*b*c^4*d^6*e^9 + 32*a^3*b^2*c*d*e^14 + 288*a*b^2*c^3*d^5*e^10 + 160*a*b^3*c^2
*d^4*e^11 - 1280*a^2*b*c^3*d^4*e^11 - 32*a^2*b^3*c*d^2*e^13 - 128*a^3*b*c^2*d^2*e^13) + (d + e*x)^(1/2)*(32*a^
4*c*e^16 + 96*c^5*d^8*e^8 - 256*a*c^4*d^6*e^10 - 256*b*c^4*d^7*e^9 + 64*b^4*c*d^4*e^12 + 256*a^2*c^3*d^4*e^12
+ 128*a^3*c^2*d^2*e^14 + 384*b^2*c^3*d^6*e^10 - 256*b^3*c^2*d^5*e^11 - 128*a^3*b*c*d*e^15 - 128*a*b^3*c*d^3*e^
13 + 256*a*b^2*c^2*d^4*e^12 - 384*a^2*b*c^2*d^3*e^13 + 192*a^2*b^2*c*d^2*e^14))*((b^4*c*d^3 - a^2*b^3*e^3 + 8*
a^2*c^3*d^3 - a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*
(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2
*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2) - (((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3
- a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a*c -
b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^
2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*((((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 - a^2*e^3*(
-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a*c - b^2)^3)^(1/2
) - 3*a*b^3*c*d^2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^
4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*((d + e*x)^(1/2)*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 - a^2*e^
3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a*c - b^2)^3)^(
1/2) - 3*a*b^3*c*d^2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16
*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*(512*a^5*c^4*e^10 + 32*a^3*b^4*c^2*e^10 - 256*a^4*b^2*c^3*e^10 +
768*a^4*c^5*d^2*e^8 + 64*a^2*b^4*c^3*d^2*e^8 - 448*a^3*b^2*c^4*d^2*e^8 - 896*a^4*b*c^4*d*e^9 - 64*a^2*b^5*c^2
*d*e^9 + 480*a^3*b^3*c^3*d*e^9) - 384*a^3*c^5*d^4*e^8 - 384*a^4*c^4*d^2*e^10 + 96*a^2*b^2*c^4*d^4*e^8 - 128*a^
2*b^3*c^3*d^3*e^9 + 32*a^2*b^4*c^2*d^2*e^10 - 32*a^3*b^2*c^3*d^2*e^10 + 128*a^4*b*c^3*d*e^11 + 512*a^3*b*c^4*d
^3*e^9 - 32*a^3*b^3*c^2*d*e^11) + (d + e*x)^(1/2)*(32*a^3*b^3*c*e^13 - 128*a^4*b*c^2*e^13 + 704*a^4*c^3*d*e^12
- 576*a^2*c^5*d^5*e^8 + 896*a^3*c^4*d^3*e^10 - 64*b^4*c^3*d^5*e^8 + 64*b^5*c^2*d^4*e^9 + 192*a^2*b^2*c^3*d^3*
e^10 + 448*a^2*b^3*c^2*d^2*e^11 - 64*a^2*b^4*c*d*e^12 + 384*a*b^2*c^4*d^5*e^8 - 320*a*b^3*c^3*d^4*e^9 - 128*a*
b^4*c^2*d^3*e^10 + 384*a^2*b*c^4*d^4*e^9 - 1664*a^3*b*c^3*d^2*e^11 + 64*a^3*b^2*c^2*d*e^12))*((b^4*c*d^3 - a^2
*b^3*e^3 + 8*a^2*c^3*d^3 - a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e
^3 - b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*
d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2) + 96*a*c^5*d^7*e^8 + 32*a^4*c^2
*d*e^14 - 672*a^2*c^4*d^5*e^10 - 736*a^3*c^3*d^3*e^12 - 32*b^2*c^4*d^7*e^8 - 32*b^3*c^3*d^6*e^9 + 64*b^4*c^2*d
^5*e^10 - 96*a^2*b^2*c^2*d^3*e^12 + 256*a*b*c^4*d^6*e^9 - 32*a^3*b^2*c*d*e^14 - 288*a*b^2*c^3*d^5*e^10 - 160*a
*b^3*c^2*d^4*e^11 + 1280*a^2*b*c^3*d^4*e^11 + 32*a^2*b^3*c*d^2*e^13 + 128*a^3*b*c^2*d^2*e^13) + (d + e*x)^(1/2
)*(32*a^4*c*e^16 + 96*c^5*d^8*e^8 - 256*a*c^4*d^6*e^10 - 256*b*c^4*d^7*e^9 + 64*b^4*c*d^4*e^12 + 256*a^2*c^3*d
^4*e^12 + 128*a^3*c^2*d^2*e^14 + 384*b^2*c^3*d^6*e^10 - 256*b^3*c^2*d^5*e^11 - 128*a^3*b*c*d*e^15 - 128*a*b^3*
c*d^3*e^13 + 256*a*b^2*c^2*d^4*e^12 - 384*a^2*b*c^2*d^3*e^13 + 192*a^2*b^2*c*d^2*e^14))*((b^4*c*d^3 - a^2*b^3*
e^3 + 8*a^2*c^3*d^3 - a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 -
b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e
+ 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2) + 192*c^4*d^8*e^10 + 448*a*c^3*d^6*e
^12 + 64*a^3*c*d^2*e^16 - 512*b*c^3*d^7*e^11 - 128*b^3*c*d^5*e^13 + 320*a^2*c^2*d^4*e^14 + 448*b^2*c^2*d^6*e^1
2 - 768*a*b*c^2*d^5*e^13 + 320*a*b^2*c*d^4*e^14 - 256*a^2*b*c*d^3*e^15))*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3
*d^3 - a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a*
c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*
d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*2i - (2*atanh((64*a^3*c*e^16*(d^3)^(1/2)*(d + e*x)^
(1/2))/(2304*c^4*d^8*e^10 + 1920*a*c^3*d^6*e^12 + 64*a^3*c*d^2*e^16 - 3328*b*c^3*d^7*e^11 - 192*b^3*c*d^5*e^13
+ 256*a^2*c^2*d^4*e^14 + (576*c^5*d^10*e^8)/a + 640*b^2*c^2*d^6*e^12 + (640*b^2*c^3*d^8*e^10)/a + (384*b^3*c^
2*d^7*e^11)/a - (128*b^2*c^4*d^10*e^8)/a^2 + (320*b^3*c^3*d^9*e^9)/a^2 - (192*b^4*c^2*d^8*e^10)/a^2 - 1024*a*b
*c^2*d^5*e^13 + 384*a*b^2*c*d^4*e^14 - 256*a^2*b*c*d^3*e^15 - (1536*b*c^4*d^9*e^9)/a) + (576*c^5*d^8*e^8*(d^3)
^(1/2)*(d + e*x)^(1/2))/(576*c^5*d^10*e^8 + 2304*a*c^4*d^8*e^10 + 64*a^4*c*d^2*e^16 - 1536*b*c^4*d^9*e^9 + 192
0*a^2*c^3*d^6*e^12 + 256*a^3*c^2*d^4*e^14 + 640*b^2*c^3*d^8*e^10 + 384*b^3*c^2*d^7*e^11 - (128*b^2*c^4*d^10*e^
8)/a + (320*b^3*c^3*d^9*e^9)/a - (192*b^4*c^2*d^8*e^10)/a - 3328*a*b*c^3*d^7*e^11 - 192*a*b^3*c*d^5*e^13 - 256
*a^3*b*c*d^3*e^15 + 640*a*b^2*c^2*d^6*e^12 - 1024*a^2*b*c^2*d^5*e^13 + 384*a^2*b^2*c*d^4*e^14) + (2304*c^4*d^6
*e^10*(d^3)^(1/2)*(d + e*x)^(1/2))/(2304*c^4*d^8*e^10 + 1920*a*c^3*d^6*e^12 + 64*a^3*c*d^2*e^16 - 3328*b*c^3*d
^7*e^11 - 192*b^3*c*d^5*e^13 + 256*a^2*c^2*d^4*e^14 + (576*c^5*d^10*e^8)/a + 640*b^2*c^2*d^6*e^12 + (640*b^2*c
^3*d^8*e^10)/a + (384*b^3*c^2*d^7*e^11)/a - (128*b^2*c^4*d^10*e^8)/a^2 + (320*b^3*c^3*d^9*e^9)/a^2 - (192*b^4*
c^2*d^8*e^10)/a^2 - 1024*a*b*c^2*d^5*e^13 + 384*a*b^2*c*d^4*e^14 - 256*a^2*b*c*d^3*e^15 - (1536*b*c^4*d^9*e^9)
/a) - (128*b^2*c^4*d^8*e^8*(d^3)^(1/2)*(d + e*x)^(1/2))/(576*a*c^5*d^10*e^8 + 64*a^5*c*d^2*e^16 + 2304*a^2*c^4
*d^8*e^10 + 1920*a^3*c^3*d^6*e^12 + 256*a^4*c^2*d^4*e^14 - 128*b^2*c^4*d^10*e^8 + 320*b^3*c^3*d^9*e^9 - 192*b^
4*c^2*d^8*e^10 + 640*a^2*b^2*c^2*d^6*e^12 - 1536*a*b*c^4*d^9*e^9 - 256*a^4*b*c*d^3*e^15 + 640*a*b^2*c^3*d^8*e^
10 + 384*a*b^3*c^2*d^7*e^11 - 3328*a^2*b*c^3*d^7*e^11 - 192*a^2*b^3*c*d^5*e^13 - 1024*a^3*b*c^2*d^5*e^13 + 384
*a^3*b^2*c*d^4*e^14) + (320*b^3*c^3*d^7*e^9*(d^3)^(1/2)*(d + e*x)^(1/2))/(576*a*c^5*d^10*e^8 + 64*a^5*c*d^2*e^
16 + 2304*a^2*c^4*d^8*e^10 + 1920*a^3*c^3*d^6*e^12 + 256*a^4*c^2*d^4*e^14 - 128*b^2*c^4*d^10*e^8 + 320*b^3*c^3
*d^9*e^9 - 192*b^4*c^2*d^8*e^10 + 640*a^2*b^2*c^2*d^6*e^12 - 1536*a*b*c^4*d^9*e^9 - 256*a^4*b*c*d^3*e^15 + 640
*a*b^2*c^3*d^8*e^10 + 384*a*b^3*c^2*d^7*e^11 - 3328*a^2*b*c^3*d^7*e^11 - 192*a^2*b^3*c*d^5*e^13 - 1024*a^3*b*c
^2*d^5*e^13 + 384*a^3*b^2*c*d^4*e^14) - (192*b^4*c^2*d^6*e^10*(d^3)^(1/2)*(d + e*x)^(1/2))/(576*a*c^5*d^10*e^8
+ 64*a^5*c*d^2*e^16 + 2304*a^2*c^4*d^8*e^10 + 1920*a^3*c^3*d^6*e^12 + 256*a^4*c^2*d^4*e^14 - 128*b^2*c^4*d^10
*e^8 + 320*b^3*c^3*d^9*e^9 - 192*b^4*c^2*d^8*e^10 + 640*a^2*b^2*c^2*d^6*e^12 - 1536*a*b*c^4*d^9*e^9 - 256*a^4*
b*c*d^3*e^15 + 640*a*b^2*c^3*d^8*e^10 + 384*a*b^3*c^2*d^7*e^11 - 3328*a^2*b*c^3*d^7*e^11 - 192*a^2*b^3*c*d^5*e
^13 - 1024*a^3*b*c^2*d^5*e^13 + 384*a^3*b^2*c*d^4*e^14) + (1920*a*c^3*d^4*e^12*(d^3)^(1/2)*(d + e*x)^(1/2))/(2
304*c^4*d^8*e^10 + 1920*a*c^3*d^6*e^12 + 64*a^3*c*d^2*e^16 - 3328*b*c^3*d^7*e^11 - 192*b^3*c*d^5*e^13 + 256*a^
2*c^2*d^4*e^14 + (576*c^5*d^10*e^8)/a + 640*b^2*c^2*d^6*e^12 + (640*b^2*c^3*d^8*e^10)/a + (384*b^3*c^2*d^7*e^1
1)/a - (128*b^2*c^4*d^10*e^8)/a^2 + (320*b^3*c^3*d^9*e^9)/a^2 - (192*b^4*c^2*d^8*e^10)/a^2 - 1024*a*b*c^2*d^5*
e^13 + 384*a*b^2*c*d^4*e^14 - 256*a^2*b*c*d^3*e^15 - (1536*b*c^4*d^9*e^9)/a) - (3328*b*c^3*d^5*e^11*(d^3)^(1/2
)*(d + e*x)^(1/2))/(2304*c^4*d^8*e^10 + 1920*a*c^3*d^6*e^12 + 64*a^3*c*d^2*e^16 - 3328*b*c^3*d^7*e^11 - 192*b^
3*c*d^5*e^13 + 256*a^2*c^2*d^4*e^14 + (576*c^5*d^10*e^8)/a + 640*b^2*c^2*d^6*e^12 + (640*b^2*c^3*d^8*e^10)/a +
(384*b^3*c^2*d^7*e^11)/a - (128*b^2*c^4*d^10*e^8)/a^2 + (320*b^3*c^3*d^9*e^9)/a^2 - (192*b^4*c^2*d^8*e^10)/a^
2 - 1024*a*b*c^2*d^5*e^13 + 384*a*b^2*c*d^4*e^14 - 256*a^2*b*c*d^3*e^15 - (1536*b*c^4*d^9*e^9)/a) - (192*b^3*c
*d^3*e^13*(d^3)^(1/2)*(d + e*x)^(1/2))/(2304*c^4*d^8*e^10 + 1920*a*c^3*d^6*e^12 + 64*a^3*c*d^2*e^16 - 3328*b*c
^3*d^7*e^11 - 192*b^3*c*d^5*e^13 + 256*a^2*c^2*d^4*e^14 + (576*c^5*d^10*e^8)/a + 640*b^2*c^2*d^6*e^12 + (640*b
^2*c^3*d^8*e^10)/a + (384*b^3*c^2*d^7*e^11)/a - (128*b^2*c^4*d^10*e^8)/a^2 + (320*b^3*c^3*d^9*e^9)/a^2 - (192*
b^4*c^2*d^8*e^10)/a^2 - 1024*a*b*c^2*d^5*e^13 + 384*a*b^2*c*d^4*e^14 - 256*a^2*b*c*d^3*e^15 - (1536*b*c^4*d^9*
e^9)/a) + (640*b^2*c^3*d^6*e^10*(d^3)^(1/2)*(d + e*x)^(1/2))/(576*c^5*d^10*e^8 + 2304*a*c^4*d^8*e^10 + 64*a^4*
c*d^2*e^16 - 1536*b*c^4*d^9*e^9 + 1920*a^2*c^3*d^6*e^12 + 256*a^3*c^2*d^4*e^14 + 640*b^2*c^3*d^8*e^10 + 384*b^
3*c^2*d^7*e^11 - (128*b^2*c^4*d^10*e^8)/a + (320*b^3*c^3*d^9*e^9)/a - (192*b^4*c^2*d^8*e^10)/a - 3328*a*b*c^3*
d^7*e^11 - 192*a*b^3*c*d^5*e^13 - 256*a^3*b*c*d^3*e^15 + 640*a*b^2*c^2*d^6*e^12 - 1024*a^2*b*c^2*d^5*e^13 + 38
4*a^2*b^2*c*d^4*e^14) + (384*b^3*c^2*d^5*e^11*(d^3)^(1/2)*(d + e*x)^(1/2))/(576*c^5*d^10*e^8 + 2304*a*c^4*d^8*
e^10 + 64*a^4*c*d^2*e^16 - 1536*b*c^4*d^9*e^9 + 1920*a^2*c^3*d^6*e^12 + 256*a^3*c^2*d^4*e^14 + 640*b^2*c^3*d^8
*e^10 + 384*b^3*c^2*d^7*e^11 - (128*b^2*c^4*d^10*e^8)/a + (320*b^3*c^3*d^9*e^9)/a - (192*b^4*c^2*d^8*e^10)/a -
3328*a*b*c^3*d^7*e^11 - 192*a*b^3*c*d^5*e^13 - 256*a^3*b*c*d^3*e^15 + 640*a*b^2*c^2*d^6*e^12 - 1024*a^2*b*c^2
*d^5*e^13 + 384*a^2*b^2*c*d^4*e^14) + (256*a^2*c^2*d^2*e^14*(d^3)^(1/2)*(d + e*x)^(1/2))/(2304*c^4*d^8*e^10 +
1920*a*c^3*d^6*e^12 + 64*a^3*c*d^2*e^16 - 3328*b*c^3*d^7*e^11 - 192*b^3*c*d^5*e^13 + 256*a^2*c^2*d^4*e^14 + (5
76*c^5*d^10*e^8)/a + 640*b^2*c^2*d^6*e^12 + (640*b^2*c^3*d^8*e^10)/a + (384*b^3*c^2*d^7*e^11)/a - (128*b^2*c^4
*d^10*e^8)/a^2 + (320*b^3*c^3*d^9*e^9)/a^2 - (192*b^4*c^2*d^8*e^10)/a^2 - 1024*a*b*c^2*d^5*e^13 + 384*a*b^2*c*
d^4*e^14 - 256*a^2*b*c*d^3*e^15 - (1536*b*c^4*d^9*e^9)/a) + (640*b^2*c^2*d^4*e^12*(d^3)^(1/2)*(d + e*x)^(1/2))
/(2304*c^4*d^8*e^10 + 1920*a*c^3*d^6*e^12 + 64*a^3*c*d^2*e^16 - 3328*b*c^3*d^7*e^11 - 192*b^3*c*d^5*e^13 + 256
*a^2*c^2*d^4*e^14 + (576*c^5*d^10*e^8)/a + 640*b^2*c^2*d^6*e^12 + (640*b^2*c^3*d^8*e^10)/a + (384*b^3*c^2*d^7*
e^11)/a - (128*b^2*c^4*d^10*e^8)/a^2 + (320*b^3*c^3*d^9*e^9)/a^2 - (192*b^4*c^2*d^8*e^10)/a^2 - 1024*a*b*c^2*d
^5*e^13 + 384*a*b^2*c*d^4*e^14 - 256*a^2*b*c*d^3*e^15 - (1536*b*c^4*d^9*e^9)/a) - (1536*b*c^4*d^7*e^9*(d^3)^(1
/2)*(d + e*x)^(1/2))/(576*c^5*d^10*e^8 + 2304*a*c^4*d^8*e^10 + 64*a^4*c*d^2*e^16 - 1536*b*c^4*d^9*e^9 + 1920*a
^2*c^3*d^6*e^12 + 256*a^3*c^2*d^4*e^14 + 640*b^2*c^3*d^8*e^10 + 384*b^3*c^2*d^7*e^11 - (128*b^2*c^4*d^10*e^8)/
a + (320*b^3*c^3*d^9*e^9)/a - (192*b^4*c^2*d^8*e^10)/a - 3328*a*b*c^3*d^7*e^11 - 192*a*b^3*c*d^5*e^13 - 256*a^
3*b*c*d^3*e^15 + 640*a*b^2*c^2*d^6*e^12 - 1024*a^2*b*c^2*d^5*e^13 + 384*a^2*b^2*c*d^4*e^14) - (256*a^2*b*c*d*e
^15*(d^3)^(1/2)*(d + e*x)^(1/2))/(2304*c^4*d^8*e^10 + 1920*a*c^3*d^6*e^12 + 64*a^3*c*d^2*e^16 - 3328*b*c^3*d^7
*e^11 - 192*b^3*c*d^5*e^13 + 256*a^2*c^2*d^4*e^14 + (576*c^5*d^10*e^8)/a + 640*b^2*c^2*d^6*e^12 + (640*b^2*c^3
*d^8*e^10)/a + (384*b^3*c^2*d^7*e^11)/a - (128*b^2*c^4*d^10*e^8)/a^2 + (320*b^3*c^3*d^9*e^9)/a^2 - (192*b^4*c^
2*d^8*e^10)/a^2 - 1024*a*b*c^2*d^5*e^13 + 384*a*b^2*c*d^4*e^14 - 256*a^2*b*c*d^3*e^15 - (1536*b*c^4*d^9*e^9)/a
) - (1024*a*b*c^2*d^3*e^13*(d^3)^(1/2)*(d + e*x)^(1/2))/(2304*c^4*d^8*e^10 + 1920*a*c^3*d^6*e^12 + 64*a^3*c*d^
2*e^16 - 3328*b*c^3*d^7*e^11 - 192*b^3*c*d^5*e^13 + 256*a^2*c^2*d^4*e^14 + (576*c^5*d^10*e^8)/a + 640*b^2*c^2*
d^6*e^12 + (640*b^2*c^3*d^8*e^10)/a + (384*b^3*c^2*d^7*e^11)/a - (128*b^2*c^4*d^10*e^8)/a^2 + (320*b^3*c^3*d^9
*e^9)/a^2 - (192*b^4*c^2*d^8*e^10)/a^2 - 1024*a*b*c^2*d^5*e^13 + 384*a*b^2*c*d^4*e^14 - 256*a^2*b*c*d^3*e^15 -
(1536*b*c^4*d^9*e^9)/a) + (384*a*b^2*c*d^2*e^14*(d^3)^(1/2)*(d + e*x)^(1/2))/(2304*c^4*d^8*e^10 + 1920*a*c^3*
d^6*e^12 + 64*a^3*c*d^2*e^16 - 3328*b*c^3*d^7*e^11 - 192*b^3*c*d^5*e^13 + 256*a^2*c^2*d^4*e^14 + (576*c^5*d^10
*e^8)/a + 640*b^2*c^2*d^6*e^12 + (640*b^2*c^3*d^8*e^10)/a + (384*b^3*c^2*d^7*e^11)/a - (128*b^2*c^4*d^10*e^8)/
a^2 + (320*b^3*c^3*d^9*e^9)/a^2 - (192*b^4*c^2*d^8*e^10)/a^2 - 1024*a*b*c^2*d^5*e^13 + 384*a*b^2*c*d^4*e^14 -
256*a^2*b*c*d^3*e^15 - (1536*b*c^4*d^9*e^9)/a))*(d^3)^(1/2))/a

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(3/2)/x/(c*x**2+b*x+a),x)

[Out]

Timed out

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