3.1625 \(\int \frac {(b+2 c x) (d+e x)^{3/2}}{(a+b x+c x^2)^3} \, dx\)

Optimal. Leaf size=322 \[ -\frac {3 e (b+2 c x) \sqrt {d+e x}}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {3 \sqrt {c} e \left (4 c d-e \left (2 b-\sqrt {b^2-4 a c}\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{2 \sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {3 \sqrt {c} e \left (4 c d-e \left (\sqrt {b^2-4 a c}+2 b\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{2 \sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {(d+e x)^{3/2}}{2 \left (a+b x+c x^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.79, antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {768, 736, 826, 1166, 208} \[ -\frac {3 e (b+2 c x) \sqrt {d+e x}}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {3 \sqrt {c} e \left (4 c d-e \left (2 b-\sqrt {b^2-4 a c}\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{2 \sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {3 \sqrt {c} e \left (4 c d-e \left (\sqrt {b^2-4 a c}+2 b\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{2 \sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {(d+e x)^{3/2}}{2 \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 736

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^m*(b + 2*
c*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m
- 1)*(b*e*m + 2*c*d*(2*p + 3) + 2*c*e*(m + 2*p + 3)*x)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d
, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m
, 0] && (LtQ[m, 1] || (ILtQ[m + 2*p + 3, 0] && NeQ[m, 2])) && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 768

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Sim
p[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(2*c*(p + 1)), x] - Dist[(e*g*m)/(2*c*(p + 1)), Int[(d + e*x)^(m -
1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[2*c*f - b*g, 0] && LtQ[p, -1]
&& GtQ[m, 0]

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

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]

Rubi steps

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

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Mathematica [B]  time = 6.46, size = 1069, normalized size = 3.32 \[ -\frac {\left (-2 a c (2 c d-b e)+b \left (-e b^2+c d b+2 a c e\right )+c (b (2 c d-b e)-2 c (b d-2 a e)) x\right ) (d+e x)^{5/2}}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^2}-\frac {-\frac {\left (\frac {1}{2} a c \left (b^2-4 a c\right ) (2 c d-b e) e^2-\frac {1}{2} \left (b^2-4 a c\right ) (3 c d-b e) \left (-e b^2+c d b+2 a c e\right ) e+c \left (\frac {1}{2} c \left (b^2-4 a c\right ) e^2 (b d-2 a e)-\frac {1}{2} \left (b^2-4 a c\right ) e (2 c d-b e) (3 c d-b e)\right ) x\right ) (d+e x)^{5/2}}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )}-\frac {\frac {1}{2} \left (b^2-4 a c\right ) \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{3/2} e^2+\frac {2 \left (\frac {9}{4} c \left (b^2-4 a c\right ) (2 c d-b e) \left (c d^2-e (b d-a e)\right ) \sqrt {d+e x} e^2+\frac {4 \left (\frac {\sqrt {2 c d-b e-\sqrt {b^2-4 a c} e} \left (-\frac {9}{8} \left (b^2-4 a c\right ) e^2 \left (c d^2-e (b d-a e)\right )^2 c^3-\frac {\frac {9}{8} \left (b^2-4 a c\right ) e^2 (b e-2 c d) \left (c d^2-e (b d-a e)\right )^2 c^3+2 \left (\frac {9}{8} c^3 \left (b^2-4 a c\right ) d e^2 \left (c d^2-e (b d-a e)\right )^2-\frac {9}{16} c^2 \left (b^2-4 a c\right ) e^2 (4 c d-b e) \left (c d^2-e (b d-a e)\right )^2\right ) c}{\sqrt {b^2-4 a c} e}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {2} \sqrt {c} \left (-2 c d+b e+\sqrt {b^2-4 a c} e\right )}+\frac {\sqrt {2 c d-b e+\sqrt {b^2-4 a c} e} \left (\frac {\frac {9}{8} \left (b^2-4 a c\right ) e^2 (b e-2 c d) \left (c d^2-e (b d-a e)\right )^2 c^3+2 \left (\frac {9}{8} c^3 \left (b^2-4 a c\right ) d e^2 \left (c d^2-e (b d-a e)\right )^2-\frac {9}{16} c^2 \left (b^2-4 a c\right ) e^2 (4 c d-b e) \left (c d^2-e (b d-a e)\right )^2\right ) c}{\sqrt {b^2-4 a c} e}-\frac {9}{8} c^3 \left (b^2-4 a c\right ) e^2 \left (c d^2-e (b d-a e)\right )^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-b e+\sqrt {b^2-4 a c} e}}\right )}{\sqrt {2} \sqrt {c} \left (-2 c d+b e-\sqrt {b^2-4 a c} e\right )}\right )}{c}\right )}{3 c}}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((d + e*x)^(5/2)*(-2*a*c*(2*c*d - b*e) + b*(b*c*d - b^2*e + 2*a*c*e) + c*(-2*c*(b*d - 2*a*e) + b*(2*c*d -
b*e))*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^2) - (-(((d + e*x)^(5/2)*((a*c*(b^2 - 4*a*
c)*e^2*(2*c*d - b*e))/2 - ((b^2 - 4*a*c)*e*(3*c*d - b*e)*(b*c*d - b^2*e + 2*a*c*e))/2 + c*((c*(b^2 - 4*a*c)*e^
2*(b*d - 2*a*e))/2 - ((b^2 - 4*a*c)*e*(2*c*d - b*e)*(3*c*d - b*e))/2)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^
2)*(a + b*x + c*x^2))) - (((b^2 - 4*a*c)*e^2*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))*(d + e*x)^(3/2))/2 +
(2*((9*c*(b^2 - 4*a*c)*e^2*(2*c*d - b*e)*(c*d^2 - e*(b*d - a*e))*Sqrt[d + e*x])/4 + (4*((Sqrt[2*c*d - b*e - Sq
rt[b^2 - 4*a*c]*e]*((-9*c^3*(b^2 - 4*a*c)*e^2*(c*d^2 - e*(b*d - a*e))^2)/8 - ((9*c^3*(b^2 - 4*a*c)*e^2*(-2*c*d
+ b*e)*(c*d^2 - e*(b*d - a*e))^2)/8 + 2*c*((9*c^3*(b^2 - 4*a*c)*d*e^2*(c*d^2 - e*(b*d - a*e))^2)/8 - (9*c^2*(
b^2 - 4*a*c)*e^2*(4*c*d - b*e)*(c*d^2 - e*(b*d - a*e))^2)/16))/(Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]
*Sqrt[d + e*x])/Sqrt[2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[2]*Sqrt[c]*(-2*c*d + b*e + Sqrt[b^2 - 4*a*c]*e
)) + (Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]*((-9*c^3*(b^2 - 4*a*c)*e^2*(c*d^2 - e*(b*d - a*e))^2)/8 + ((9*c^
3*(b^2 - 4*a*c)*e^2*(-2*c*d + b*e)*(c*d^2 - e*(b*d - a*e))^2)/8 + 2*c*((9*c^3*(b^2 - 4*a*c)*d*e^2*(c*d^2 - e*(
b*d - a*e))^2)/8 - (9*c^2*(b^2 - 4*a*c)*e^2*(4*c*d - b*e)*(c*d^2 - e*(b*d - a*e))^2)/16))/(Sqrt[b^2 - 4*a*c]*e
))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[2]*Sqrt[c]*(-2*c*d
+ b*e - Sqrt[b^2 - 4*a*c]*e))))/c))/(3*c))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)))/(2*(b^2 - 4*a*c)*(c*d^2 -
b*d*e + a*e^2))

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fricas [B]  time = 1.02, size = 6728, normalized size = 20.89 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(3*sqrt(1/2)*((b^2*c^2 - 4*a*c^3)*x^4 + a^2*b^2 - 4*a^3*c + 2*(b^3*c - 4*a*b*c^2)*x^3 + (b^4 - 2*a*b^2*c -
8*a^2*c^2)*x^2 + 2*(a*b^3 - 4*a^2*b*c)*x)*sqrt((32*c^3*d^3*e^2 - 48*b*c^2*d^2*e^3 + 6*(3*b^2*c + 4*a*c^2)*d*e
^4 - (b^3 + 12*a*b*c)*e^5 + sqrt(e^10/((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 - 2*(b^7*c -
12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3*e + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 1
28*a^4*c^4)*d^2*e^2 - 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*e^3 + (a^2*b^6 - 12*a^3*b^4*c
+ 48*a^4*b^2*c^2 - 64*a^5*c^3)*e^4))*((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*
b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2))/((b^6*
c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e +
(a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2))*log(27/2*sqrt(1/2)*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^
2*c^3)*d*e^6 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*e^7 - sqrt(e^10/((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64
*a^3*c^5)*d^4 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3*e + (b^8 - 10*a*b^6*c + 24*a^2*b^
4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2*e^2 - 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*e^3
+ (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*e^4))*(8*(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 -
64*a^3*c^6)*d^4 - 16*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^3*e + 3*(3*b^8*c - 32*a*b^6*c^
2 + 96*a^2*b^4*c^3 - 256*a^4*c^5)*d^2*e^2 - (b^9 - 96*a^2*b^5*c^2 + 512*a^3*b^3*c^3 - 768*a^4*b*c^4)*d*e^3 + (
a*b^8 - 8*a^2*b^6*c + 128*a^4*b^2*c^3 - 256*a^5*c^4)*e^4))*sqrt((32*c^3*d^3*e^2 - 48*b*c^2*d^2*e^3 + 6*(3*b^2*
c + 4*a*c^2)*d*e^4 - (b^3 + 12*a*b*c)*e^5 + sqrt(e^10/((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*
d^4 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3*e + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32
*a^3*b^2*c^3 - 128*a^4*c^4)*d^2*e^2 - 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*e^3 + (a^2*b^
6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*e^4))*((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^
2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c
^3)*e^2))/((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*
a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2)) + 27*(16*c^3*d^2*e^6 - 16*b*c^2*d*
e^7 + (3*b^2*c + 4*a*c^2)*e^8)*sqrt(e*x + d)) - 3*sqrt(1/2)*((b^2*c^2 - 4*a*c^3)*x^4 + a^2*b^2 - 4*a^3*c + 2*(
b^3*c - 4*a*b*c^2)*x^3 + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*x^2 + 2*(a*b^3 - 4*a^2*b*c)*x)*sqrt((32*c^3*d^3*e^2 - 4
8*b*c^2*d^2*e^3 + 6*(3*b^2*c + 4*a*c^2)*d*e^4 - (b^3 + 12*a*b*c)*e^5 + sqrt(e^10/((b^6*c^2 - 12*a*b^4*c^3 + 48
*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3*e + (b^8 - 10*a*
b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2*e^2 - 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64
*a^4*b*c^3)*d*e^3 + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*e^4))*((b^6*c - 12*a*b^4*c^2 + 48*a
^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c
+ 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2))/((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b
^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2))*log(-27
/2*sqrt(1/2)*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*e^6 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*e^7 - sqrt(e^10/((
b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*
b*c^4)*d^3*e + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2*e^2 - 2*(a*b^7 - 12*a^2*
b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*e^3 + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*e^4))*(8
*(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*d^4 - 16*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64
*a^3*b*c^5)*d^3*e + 3*(3*b^8*c - 32*a*b^6*c^2 + 96*a^2*b^4*c^3 - 256*a^4*c^5)*d^2*e^2 - (b^9 - 96*a^2*b^5*c^2
+ 512*a^3*b^3*c^3 - 768*a^4*b*c^4)*d*e^3 + (a*b^8 - 8*a^2*b^6*c + 128*a^4*b^2*c^3 - 256*a^5*c^4)*e^4))*sqrt((3
2*c^3*d^3*e^2 - 48*b*c^2*d^2*e^3 + 6*(3*b^2*c + 4*a*c^2)*d*e^4 - (b^3 + 12*a*b*c)*e^5 + sqrt(e^10/((b^6*c^2 -
12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3
*e + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2*e^2 - 2*(a*b^7 - 12*a^2*b^5*c + 48
*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*e^3 + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*e^4))*((b^6*c - 12
*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^
6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2))/((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^
2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c
^3)*e^2)) + 27*(16*c^3*d^2*e^6 - 16*b*c^2*d*e^7 + (3*b^2*c + 4*a*c^2)*e^8)*sqrt(e*x + d)) + 3*sqrt(1/2)*((b^2*
c^2 - 4*a*c^3)*x^4 + a^2*b^2 - 4*a^3*c + 2*(b^3*c - 4*a*b*c^2)*x^3 + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*x^2 + 2*(a*
b^3 - 4*a^2*b*c)*x)*sqrt((32*c^3*d^3*e^2 - 48*b*c^2*d^2*e^3 + 6*(3*b^2*c + 4*a*c^2)*d*e^4 - (b^3 + 12*a*b*c)*e
^5 - sqrt(e^10/((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*
b^3*c^3 - 64*a^3*b*c^4)*d^3*e + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2*e^2 - 2
*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*e^3 + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*
a^5*c^3)*e^4))*((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2
- 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2))/((b^6*c - 12*a*b^4*c^2 + 48*a
^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c
+ 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2))*log(27/2*sqrt(1/2)*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*e^6 - (b^5 - 8
*a*b^3*c + 16*a^2*b*c^2)*e^7 + sqrt(e^10/((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 - 2*(b^7*
c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3*e + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3
- 128*a^4*c^4)*d^2*e^2 - 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*e^3 + (a^2*b^6 - 12*a^3*b^
4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*e^4))*(8*(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*d^4 - 16*(b
^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^3*e + 3*(3*b^8*c - 32*a*b^6*c^2 + 96*a^2*b^4*c^3 - 25
6*a^4*c^5)*d^2*e^2 - (b^9 - 96*a^2*b^5*c^2 + 512*a^3*b^3*c^3 - 768*a^4*b*c^4)*d*e^3 + (a*b^8 - 8*a^2*b^6*c + 1
28*a^4*b^2*c^3 - 256*a^5*c^4)*e^4))*sqrt((32*c^3*d^3*e^2 - 48*b*c^2*d^2*e^3 + 6*(3*b^2*c + 4*a*c^2)*d*e^4 - (b
^3 + 12*a*b*c)*e^5 - sqrt(e^10/((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 - 2*(b^7*c - 12*a*b
^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3*e + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*
c^4)*d^2*e^2 - 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*e^3 + (a^2*b^6 - 12*a^3*b^4*c + 48*a
^4*b^2*c^2 - 64*a^5*c^3)*e^4))*((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c +
48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2))/((b^6*c - 12*
a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6
- 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2)) + 27*(16*c^3*d^2*e^6 - 16*b*c^2*d*e^7 + (3*b^2*c + 4*a*c^
2)*e^8)*sqrt(e*x + d)) - 3*sqrt(1/2)*((b^2*c^2 - 4*a*c^3)*x^4 + a^2*b^2 - 4*a^3*c + 2*(b^3*c - 4*a*b*c^2)*x^3
+ (b^4 - 2*a*b^2*c - 8*a^2*c^2)*x^2 + 2*(a*b^3 - 4*a^2*b*c)*x)*sqrt((32*c^3*d^3*e^2 - 48*b*c^2*d^2*e^3 + 6*(3*
b^2*c + 4*a*c^2)*d*e^4 - (b^3 + 12*a*b*c)*e^5 - sqrt(e^10/((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c
^5)*d^4 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3*e + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2
+ 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2*e^2 - 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*e^3 + (a^
2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*e^4))*((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4
)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a
^4*c^3)*e^2))/((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 -
64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2))*log(-27/2*sqrt(1/2)*(2*(b^4*c
- 8*a*b^2*c^2 + 16*a^2*c^3)*d*e^6 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*e^7 + sqrt(e^10/((b^6*c^2 - 12*a*b^4*c^3
+ 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3*e + (b^8 - 1
0*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2*e^2 - 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2
- 64*a^4*b*c^3)*d*e^3 + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*e^4))*(8*(b^6*c^3 - 12*a*b^4*c^
4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*d^4 - 16*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^3*e + 3*(
3*b^8*c - 32*a*b^6*c^2 + 96*a^2*b^4*c^3 - 256*a^4*c^5)*d^2*e^2 - (b^9 - 96*a^2*b^5*c^2 + 512*a^3*b^3*c^3 - 768
*a^4*b*c^4)*d*e^3 + (a*b^8 - 8*a^2*b^6*c + 128*a^4*b^2*c^3 - 256*a^5*c^4)*e^4))*sqrt((32*c^3*d^3*e^2 - 48*b*c^
2*d^2*e^3 + 6*(3*b^2*c + 4*a*c^2)*d*e^4 - (b^3 + 12*a*b*c)*e^5 - sqrt(e^10/((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b
^2*c^4 - 64*a^3*c^5)*d^4 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3*e + (b^8 - 10*a*b^6*c
+ 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2*e^2 - 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b
*c^3)*d*e^3 + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*e^4))*((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2
*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a
^3*b^2*c^2 - 64*a^4*c^3)*e^2))/((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c +
48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2)) + 27*(16*c^3*
d^2*e^6 - 16*b*c^2*d*e^7 + (3*b^2*c + 4*a*c^2)*e^8)*sqrt(e*x + d)) - 2*(6*c^2*e*x^3 + 9*b*c*e*x^2 + 3*a*b*e +
(5*b^2 - 2*a*c)*e*x + 2*(b^2 - 4*a*c)*d)*sqrt(e*x + d))/((b^2*c^2 - 4*a*c^3)*x^4 + a^2*b^2 - 4*a^3*c + 2*(b^3*
c - 4*a*b*c^2)*x^3 + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*x^2 + 2*(a*b^3 - 4*a^2*b*c)*x)

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giac [B]  time = 3.70, size = 1189, normalized size = 3.69 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/16*(sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*(b^2*e - 4*a*c*e)^2*e^2 + (2*sqrt(b^2 - 4*a*c)*c*d*e^2
- sqrt(b^2 - 4*a*c)*b*e^3)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(b^2*e - 4*a*c*e) - 2*(4*(b^2*
c^2 - 4*a*c^3)*d^2*e^2 - 4*(b^3*c - 4*a*b*c^2)*d*e^3 + (b^4 - 4*a*b^2*c)*e^4)*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*b^2*c*d - 8*a*c^2*d - b^3*e + 4*a*b*c*e + sqrt((2*
b^2*c*d - 8*a*c^2*d - b^3*e + 4*a*b*c*e)^2 - 4*(b^2*c*d^2 - 4*a*c^2*d^2 - b^3*d*e + 4*a*b*c*d*e + a*b^2*e^2 -
4*a^2*c*e^2)*(b^2*c - 4*a*c^2)))/(b^2*c - 4*a*c^2)))/(((b^2*c - 4*a*c^2)*sqrt(b^2 - 4*a*c)*d^2 - (b^3 - 4*a*b*
c)*sqrt(b^2 - 4*a*c)*d*e + (a*b^2 - 4*a^2*c)*sqrt(b^2 - 4*a*c)*e^2)*abs(b^2*e - 4*a*c*e)*abs(c)) + 3/16*(sqrt(
-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*(b^2*e - 4*a*c*e)^2*e^2 - (2*sqrt(b^2 - 4*a*c)*c*d*e^2 - sqrt(b^2
- 4*a*c)*b*e^3)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*abs(b^2*e - 4*a*c*e) - 2*(4*(b^2*c^2 - 4*a*c^
3)*d^2*e^2 - 4*(b^3*c - 4*a*b*c^2)*d*e^3 + (b^4 - 4*a*b^2*c)*e^4)*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*b^2*c*d - 8*a*c^2*d - b^3*e + 4*a*b*c*e - sqrt((2*b^2*c*d - 8*
a*c^2*d - b^3*e + 4*a*b*c*e)^2 - 4*(b^2*c*d^2 - 4*a*c^2*d^2 - b^3*d*e + 4*a*b*c*d*e + a*b^2*e^2 - 4*a^2*c*e^2)
*(b^2*c - 4*a*c^2)))/(b^2*c - 4*a*c^2)))/(((b^2*c - 4*a*c^2)*sqrt(b^2 - 4*a*c)*d^2 - (b^3 - 4*a*b*c)*sqrt(b^2
- 4*a*c)*d*e + (a*b^2 - 4*a^2*c)*sqrt(b^2 - 4*a*c)*e^2)*abs(b^2*e - 4*a*c*e)*abs(c)) - 1/4*(6*(x*e + d)^(7/2)*
c^2*e^2 - 18*(x*e + d)^(5/2)*c^2*d*e^2 + 18*(x*e + d)^(3/2)*c^2*d^2*e^2 - 6*sqrt(x*e + d)*c^2*d^3*e^2 + 9*(x*e
+ d)^(5/2)*b*c*e^3 - 18*(x*e + d)^(3/2)*b*c*d*e^3 + 9*sqrt(x*e + d)*b*c*d^2*e^3 + 5*(x*e + d)^(3/2)*b^2*e^4 -
2*(x*e + d)^(3/2)*a*c*e^4 - 3*sqrt(x*e + d)*b^2*d*e^4 - 6*sqrt(x*e + d)*a*c*d*e^4 + 3*sqrt(x*e + d)*a*b*e^5)/
(((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e + a*e^2)^2*(b^2 - 4*a*c))

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maple [B]  time = 0.18, size = 1231, normalized size = 3.82 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)^2*c^2/(4*a*c-b^2)*(e*x+d)^(7/2)+9/4*e^3/(c*e^2*x^2+b*e^2*x+a*e^2)^2*c/(4*a*c
-b^2)*(e*x+d)^(5/2)*b-9/2*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)^2*c^2/(4*a*c-b^2)*(e*x+d)^(5/2)*d-1/2*e^4/(c*e^2*x^2+b
*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(3/2)*a*c+5/4*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(3/2)*b^
2-9/2*e^3/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(3/2)*b*c*d+9/2*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a
*c-b^2)*(e*x+d)^(3/2)*c^2*d^2+3/4*e^5/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(1/2)*a*b-3/2*e^4/(c*e^2
*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(1/2)*a*c*d-3/4*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^
(1/2)*b^2*d+9/4*e^3/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(1/2)*b*c*d^2-3/2*e^2/(c*e^2*x^2+b*e^2*x+a
*e^2)^2/(4*a*c-b^2)*(e*x+d)^(1/2)*c^2*d^3+3/2*e^3/(4*a*c-b^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)*b-3*e^2/(4*a*c-b^2)*c^2/(-(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-3/4*e^2/(4*a*c-b^2)*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)+3/2*e^3/(4*a*c-b^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)*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-3*e^2/(4*a*c-b
^2)*c^2/(-(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)*d+3/4*e^2/(4*a*c-b^2)*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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 7.15, size = 16393, normalized size = 50.91 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((d + e*x)^(3/2)*(5*b^2*e^4 + 18*c^2*d^2*e^2 - 2*a*c*e^4 - 18*b*c*d*e^3))/(4*(4*a*c - b^2)) - (3*(d + e*x)^(1
/2)*(b^2*d*e^4 + 2*c^2*d^3*e^2 - a*b*e^5 + 2*a*c*d*e^4 - 3*b*c*d^2*e^3))/(4*(4*a*c - b^2)) + (3*c^2*e^2*(d + e
*x)^(7/2))/(2*(4*a*c - b^2)) + (9*c*e^2*(b*e - 2*c*d)*(d + e*x)^(5/2))/(4*(4*a*c - b^2)))/(c^2*(d + e*x)^4 - (
d + e*x)*(4*c^2*d^3 + 2*b^2*d*e^2 - 2*a*b*e^3 + 4*a*c*d*e^2 - 6*b*c*d^2*e) - (4*c^2*d - 2*b*c*e)*(d + e*x)^3 +
(d + e*x)^2*(b^2*e^2 + 6*c^2*d^2 + 2*a*c*e^2 - 6*b*c*d*e) + a^2*e^4 + c^2*d^4 + b^2*d^2*e^2 - 2*a*b*d*e^3 - 2
*b*c*d^3*e + 2*a*c*d^2*e^2) - atan(((((3*(32*b^7*c^2*e^5 - 384*a*b^5*c^3*e^5 - 2048*a^3*b*c^5*e^5 + 4096*a^3*c
^6*d*e^4 - 64*b^6*c^3*d*e^4 + 1536*a^2*b^3*c^4*e^5 + 768*a*b^4*c^4*d*e^4 - 3072*a^2*b^2*c^5*d*e^4))/(32*(b^6 -
64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) - ((d + e*x)^(1/2)*((9*(e^5*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^5 + 7
68*a^4*b*c^4*e^5 - 1536*a^4*c^5*d*e^4 + 96*a^2*b^5*c^2*e^5 - 512*a^3*b^3*c^3*e^5 - 2048*a^3*c^6*d^3*e^2 + 32*b
^6*c^3*d^3*e^2 - 48*b^7*c^2*d^2*e^3 + 18*b^8*c*d*e^4 + 1536*a^2*b^2*c^5*d^3*e^2 - 2304*a^2*b^3*c^4*d^2*e^3 - 1
92*a*b^6*c^2*d*e^4 - 384*a*b^4*c^4*d^3*e^2 + 576*a*b^5*c^3*d^2*e^3 + 576*a^2*b^4*c^3*d*e^4 + 3072*a^3*b*c^5*d^
2*e^3))/(128*(a*b^12*e^2 + b^12*c*d^2 + 4096*a^6*c^7*d^2 + 4096*a^7*c^6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2 - 2
4*a^2*b^10*c*e^2 + 240*a^2*b^8*c^3*d^2 - 1280*a^3*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 +
240*a^3*b^8*c^2*e^2 - 1280*a^4*b^6*c^3*e^2 + 3840*a^5*b^4*c^4*e^2 - 6144*a^6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e
- 240*a^2*b^9*c^2*d*e + 1280*a^3*b^7*c^3*d*e - 3840*a^4*b^5*c^4*d*e + 6144*a^5*b^3*c^5*d*e + 24*a*b^11*c*d*e))
)^(1/2)*(32*b^7*c^2*e^3 - 384*a*b^5*c^3*e^3 - 2048*a^3*b*c^5*e^3 + 4096*a^3*c^6*d*e^2 - 64*b^6*c^3*d*e^2 + 153
6*a^2*b^3*c^4*e^3 + 768*a*b^4*c^4*d*e^2 - 3072*a^2*b^2*c^5*d*e^2))/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*((9*(e^
5*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^5 + 768*a^4*b*c^4*e^5 - 1536*a^4*c^5*d*e^4 + 96*a^2*b^5*c^2*e^5 - 512*a^3*b
^3*c^3*e^5 - 2048*a^3*c^6*d^3*e^2 + 32*b^6*c^3*d^3*e^2 - 48*b^7*c^2*d^2*e^3 + 18*b^8*c*d*e^4 + 1536*a^2*b^2*c^
5*d^3*e^2 - 2304*a^2*b^3*c^4*d^2*e^3 - 192*a*b^6*c^2*d*e^4 - 384*a*b^4*c^4*d^3*e^2 + 576*a*b^5*c^3*d^2*e^3 + 5
76*a^2*b^4*c^3*d*e^4 + 3072*a^3*b*c^5*d^2*e^3))/(128*(a*b^12*e^2 + b^12*c*d^2 + 4096*a^6*c^7*d^2 + 4096*a^7*c^
6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2 - 24*a^2*b^10*c*e^2 + 240*a^2*b^8*c^3*d^2 - 1280*a^3*b^6*c^4*d^2 + 3840*a
^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 + 240*a^3*b^8*c^2*e^2 - 1280*a^4*b^6*c^3*e^2 + 3840*a^5*b^4*c^4*e^2 - 61
44*a^6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e - 240*a^2*b^9*c^2*d*e + 1280*a^3*b^7*c^3*d*e - 3840*a^4*b^5*c^4*d*e +
6144*a^5*b^3*c^5*d*e + 24*a*b^11*c*d*e)))^(1/2) + ((d + e*x)^(1/2)*(36*a*c^4*e^6 - 45*b^2*c^3*e^6 - 144*c^5*d^
2*e^4 + 144*b*c^4*d*e^5))/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*((9*(e^5*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^5 + 76
8*a^4*b*c^4*e^5 - 1536*a^4*c^5*d*e^4 + 96*a^2*b^5*c^2*e^5 - 512*a^3*b^3*c^3*e^5 - 2048*a^3*c^6*d^3*e^2 + 32*b^
6*c^3*d^3*e^2 - 48*b^7*c^2*d^2*e^3 + 18*b^8*c*d*e^4 + 1536*a^2*b^2*c^5*d^3*e^2 - 2304*a^2*b^3*c^4*d^2*e^3 - 19
2*a*b^6*c^2*d*e^4 - 384*a*b^4*c^4*d^3*e^2 + 576*a*b^5*c^3*d^2*e^3 + 576*a^2*b^4*c^3*d*e^4 + 3072*a^3*b*c^5*d^2
*e^3))/(128*(a*b^12*e^2 + b^12*c*d^2 + 4096*a^6*c^7*d^2 + 4096*a^7*c^6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2 - 24
*a^2*b^10*c*e^2 + 240*a^2*b^8*c^3*d^2 - 1280*a^3*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 + 2
40*a^3*b^8*c^2*e^2 - 1280*a^4*b^6*c^3*e^2 + 3840*a^5*b^4*c^4*e^2 - 6144*a^6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e -
240*a^2*b^9*c^2*d*e + 1280*a^3*b^7*c^3*d*e - 3840*a^4*b^5*c^4*d*e + 6144*a^5*b^3*c^5*d*e + 24*a*b^11*c*d*e)))
^(1/2)*1i - (((3*(32*b^7*c^2*e^5 - 384*a*b^5*c^3*e^5 - 2048*a^3*b*c^5*e^5 + 4096*a^3*c^6*d*e^4 - 64*b^6*c^3*d*
e^4 + 1536*a^2*b^3*c^4*e^5 + 768*a*b^4*c^4*d*e^4 - 3072*a^2*b^2*c^5*d*e^4))/(32*(b^6 - 64*a^3*c^3 + 48*a^2*b^2
*c^2 - 12*a*b^4*c)) + ((d + e*x)^(1/2)*((9*(e^5*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^5 + 768*a^4*b*c^4*e^5 - 1536*
a^4*c^5*d*e^4 + 96*a^2*b^5*c^2*e^5 - 512*a^3*b^3*c^3*e^5 - 2048*a^3*c^6*d^3*e^2 + 32*b^6*c^3*d^3*e^2 - 48*b^7*
c^2*d^2*e^3 + 18*b^8*c*d*e^4 + 1536*a^2*b^2*c^5*d^3*e^2 - 2304*a^2*b^3*c^4*d^2*e^3 - 192*a*b^6*c^2*d*e^4 - 384
*a*b^4*c^4*d^3*e^2 + 576*a*b^5*c^3*d^2*e^3 + 576*a^2*b^4*c^3*d*e^4 + 3072*a^3*b*c^5*d^2*e^3))/(128*(a*b^12*e^2
+ b^12*c*d^2 + 4096*a^6*c^7*d^2 + 4096*a^7*c^6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2 - 24*a^2*b^10*c*e^2 + 240*a
^2*b^8*c^3*d^2 - 1280*a^3*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 + 240*a^3*b^8*c^2*e^2 - 12
80*a^4*b^6*c^3*e^2 + 3840*a^5*b^4*c^4*e^2 - 6144*a^6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e - 240*a^2*b^9*c^2*d*e +
1280*a^3*b^7*c^3*d*e - 3840*a^4*b^5*c^4*d*e + 6144*a^5*b^3*c^5*d*e + 24*a*b^11*c*d*e)))^(1/2)*(32*b^7*c^2*e^3
- 384*a*b^5*c^3*e^3 - 2048*a^3*b*c^5*e^3 + 4096*a^3*c^6*d*e^2 - 64*b^6*c^3*d*e^2 + 1536*a^2*b^3*c^4*e^3 + 768*
a*b^4*c^4*d*e^2 - 3072*a^2*b^2*c^5*d*e^2))/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*((9*(e^5*(-(4*a*c - b^2)^9)^(1/
2) - b^9*e^5 + 768*a^4*b*c^4*e^5 - 1536*a^4*c^5*d*e^4 + 96*a^2*b^5*c^2*e^5 - 512*a^3*b^3*c^3*e^5 - 2048*a^3*c^
6*d^3*e^2 + 32*b^6*c^3*d^3*e^2 - 48*b^7*c^2*d^2*e^3 + 18*b^8*c*d*e^4 + 1536*a^2*b^2*c^5*d^3*e^2 - 2304*a^2*b^3
*c^4*d^2*e^3 - 192*a*b^6*c^2*d*e^4 - 384*a*b^4*c^4*d^3*e^2 + 576*a*b^5*c^3*d^2*e^3 + 576*a^2*b^4*c^3*d*e^4 + 3
072*a^3*b*c^5*d^2*e^3))/(128*(a*b^12*e^2 + b^12*c*d^2 + 4096*a^6*c^7*d^2 + 4096*a^7*c^6*e^2 - b^13*d*e - 24*a*
b^10*c^2*d^2 - 24*a^2*b^10*c*e^2 + 240*a^2*b^8*c^3*d^2 - 1280*a^3*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d^2 - 6144*a^
5*b^2*c^6*d^2 + 240*a^3*b^8*c^2*e^2 - 1280*a^4*b^6*c^3*e^2 + 3840*a^5*b^4*c^4*e^2 - 6144*a^6*b^2*c^5*e^2 - 409
6*a^6*b*c^6*d*e - 240*a^2*b^9*c^2*d*e + 1280*a^3*b^7*c^3*d*e - 3840*a^4*b^5*c^4*d*e + 6144*a^5*b^3*c^5*d*e + 2
4*a*b^11*c*d*e)))^(1/2) - ((d + e*x)^(1/2)*(36*a*c^4*e^6 - 45*b^2*c^3*e^6 - 144*c^5*d^2*e^4 + 144*b*c^4*d*e^5)
)/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*((9*(e^5*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^5 + 768*a^4*b*c^4*e^5 - 1536*a
^4*c^5*d*e^4 + 96*a^2*b^5*c^2*e^5 - 512*a^3*b^3*c^3*e^5 - 2048*a^3*c^6*d^3*e^2 + 32*b^6*c^3*d^3*e^2 - 48*b^7*c
^2*d^2*e^3 + 18*b^8*c*d*e^4 + 1536*a^2*b^2*c^5*d^3*e^2 - 2304*a^2*b^3*c^4*d^2*e^3 - 192*a*b^6*c^2*d*e^4 - 384*
a*b^4*c^4*d^3*e^2 + 576*a*b^5*c^3*d^2*e^3 + 576*a^2*b^4*c^3*d*e^4 + 3072*a^3*b*c^5*d^2*e^3))/(128*(a*b^12*e^2
+ b^12*c*d^2 + 4096*a^6*c^7*d^2 + 4096*a^7*c^6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2 - 24*a^2*b^10*c*e^2 + 240*a^
2*b^8*c^3*d^2 - 1280*a^3*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 + 240*a^3*b^8*c^2*e^2 - 128
0*a^4*b^6*c^3*e^2 + 3840*a^5*b^4*c^4*e^2 - 6144*a^6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e - 240*a^2*b^9*c^2*d*e + 1
280*a^3*b^7*c^3*d*e - 3840*a^4*b^5*c^4*d*e + 6144*a^5*b^3*c^5*d*e + 24*a*b^11*c*d*e)))^(1/2)*1i)/((((3*(32*b^7
*c^2*e^5 - 384*a*b^5*c^3*e^5 - 2048*a^3*b*c^5*e^5 + 4096*a^3*c^6*d*e^4 - 64*b^6*c^3*d*e^4 + 1536*a^2*b^3*c^4*e
^5 + 768*a*b^4*c^4*d*e^4 - 3072*a^2*b^2*c^5*d*e^4))/(32*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) - ((
d + e*x)^(1/2)*((9*(e^5*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^5 + 768*a^4*b*c^4*e^5 - 1536*a^4*c^5*d*e^4 + 96*a^2*b
^5*c^2*e^5 - 512*a^3*b^3*c^3*e^5 - 2048*a^3*c^6*d^3*e^2 + 32*b^6*c^3*d^3*e^2 - 48*b^7*c^2*d^2*e^3 + 18*b^8*c*d
*e^4 + 1536*a^2*b^2*c^5*d^3*e^2 - 2304*a^2*b^3*c^4*d^2*e^3 - 192*a*b^6*c^2*d*e^4 - 384*a*b^4*c^4*d^3*e^2 + 576
*a*b^5*c^3*d^2*e^3 + 576*a^2*b^4*c^3*d*e^4 + 3072*a^3*b*c^5*d^2*e^3))/(128*(a*b^12*e^2 + b^12*c*d^2 + 4096*a^6
*c^7*d^2 + 4096*a^7*c^6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2 - 24*a^2*b^10*c*e^2 + 240*a^2*b^8*c^3*d^2 - 1280*a^
3*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 + 240*a^3*b^8*c^2*e^2 - 1280*a^4*b^6*c^3*e^2 + 384
0*a^5*b^4*c^4*e^2 - 6144*a^6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e - 240*a^2*b^9*c^2*d*e + 1280*a^3*b^7*c^3*d*e - 3
840*a^4*b^5*c^4*d*e + 6144*a^5*b^3*c^5*d*e + 24*a*b^11*c*d*e)))^(1/2)*(32*b^7*c^2*e^3 - 384*a*b^5*c^3*e^3 - 20
48*a^3*b*c^5*e^3 + 4096*a^3*c^6*d*e^2 - 64*b^6*c^3*d*e^2 + 1536*a^2*b^3*c^4*e^3 + 768*a*b^4*c^4*d*e^2 - 3072*a
^2*b^2*c^5*d*e^2))/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*((9*(e^5*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^5 + 768*a^4*b
*c^4*e^5 - 1536*a^4*c^5*d*e^4 + 96*a^2*b^5*c^2*e^5 - 512*a^3*b^3*c^3*e^5 - 2048*a^3*c^6*d^3*e^2 + 32*b^6*c^3*d
^3*e^2 - 48*b^7*c^2*d^2*e^3 + 18*b^8*c*d*e^4 + 1536*a^2*b^2*c^5*d^3*e^2 - 2304*a^2*b^3*c^4*d^2*e^3 - 192*a*b^6
*c^2*d*e^4 - 384*a*b^4*c^4*d^3*e^2 + 576*a*b^5*c^3*d^2*e^3 + 576*a^2*b^4*c^3*d*e^4 + 3072*a^3*b*c^5*d^2*e^3))/
(128*(a*b^12*e^2 + b^12*c*d^2 + 4096*a^6*c^7*d^2 + 4096*a^7*c^6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2 - 24*a^2*b^
10*c*e^2 + 240*a^2*b^8*c^3*d^2 - 1280*a^3*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 + 240*a^3*
b^8*c^2*e^2 - 1280*a^4*b^6*c^3*e^2 + 3840*a^5*b^4*c^4*e^2 - 6144*a^6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e - 240*a^
2*b^9*c^2*d*e + 1280*a^3*b^7*c^3*d*e - 3840*a^4*b^5*c^4*d*e + 6144*a^5*b^3*c^5*d*e + 24*a*b^11*c*d*e)))^(1/2)
+ ((d + e*x)^(1/2)*(36*a*c^4*e^6 - 45*b^2*c^3*e^6 - 144*c^5*d^2*e^4 + 144*b*c^4*d*e^5))/(4*(b^4 + 16*a^2*c^2 -
8*a*b^2*c)))*((9*(e^5*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^5 + 768*a^4*b*c^4*e^5 - 1536*a^4*c^5*d*e^4 + 96*a^2*b^
5*c^2*e^5 - 512*a^3*b^3*c^3*e^5 - 2048*a^3*c^6*d^3*e^2 + 32*b^6*c^3*d^3*e^2 - 48*b^7*c^2*d^2*e^3 + 18*b^8*c*d*
e^4 + 1536*a^2*b^2*c^5*d^3*e^2 - 2304*a^2*b^3*c^4*d^2*e^3 - 192*a*b^6*c^2*d*e^4 - 384*a*b^4*c^4*d^3*e^2 + 576*
a*b^5*c^3*d^2*e^3 + 576*a^2*b^4*c^3*d*e^4 + 3072*a^3*b*c^5*d^2*e^3))/(128*(a*b^12*e^2 + b^12*c*d^2 + 4096*a^6*
c^7*d^2 + 4096*a^7*c^6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2 - 24*a^2*b^10*c*e^2 + 240*a^2*b^8*c^3*d^2 - 1280*a^3
*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 + 240*a^3*b^8*c^2*e^2 - 1280*a^4*b^6*c^3*e^2 + 3840
*a^5*b^4*c^4*e^2 - 6144*a^6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e - 240*a^2*b^9*c^2*d*e + 1280*a^3*b^7*c^3*d*e - 38
40*a^4*b^5*c^4*d*e + 6144*a^5*b^3*c^5*d*e + 24*a*b^11*c*d*e)))^(1/2) + (((3*(32*b^7*c^2*e^5 - 384*a*b^5*c^3*e^
5 - 2048*a^3*b*c^5*e^5 + 4096*a^3*c^6*d*e^4 - 64*b^6*c^3*d*e^4 + 1536*a^2*b^3*c^4*e^5 + 768*a*b^4*c^4*d*e^4 -
3072*a^2*b^2*c^5*d*e^4))/(32*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + ((d + e*x)^(1/2)*((9*(e^5*(-(
4*a*c - b^2)^9)^(1/2) - b^9*e^5 + 768*a^4*b*c^4*e^5 - 1536*a^4*c^5*d*e^4 + 96*a^2*b^5*c^2*e^5 - 512*a^3*b^3*c^
3*e^5 - 2048*a^3*c^6*d^3*e^2 + 32*b^6*c^3*d^3*e^2 - 48*b^7*c^2*d^2*e^3 + 18*b^8*c*d*e^4 + 1536*a^2*b^2*c^5*d^3
*e^2 - 2304*a^2*b^3*c^4*d^2*e^3 - 192*a*b^6*c^2*d*e^4 - 384*a*b^4*c^4*d^3*e^2 + 576*a*b^5*c^3*d^2*e^3 + 576*a^
2*b^4*c^3*d*e^4 + 3072*a^3*b*c^5*d^2*e^3))/(128*(a*b^12*e^2 + b^12*c*d^2 + 4096*a^6*c^7*d^2 + 4096*a^7*c^6*e^2
- b^13*d*e - 24*a*b^10*c^2*d^2 - 24*a^2*b^10*c*e^2 + 240*a^2*b^8*c^3*d^2 - 1280*a^3*b^6*c^4*d^2 + 3840*a^4*b^
4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 + 240*a^3*b^8*c^2*e^2 - 1280*a^4*b^6*c^3*e^2 + 3840*a^5*b^4*c^4*e^2 - 6144*a^
6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e - 240*a^2*b^9*c^2*d*e + 1280*a^3*b^7*c^3*d*e - 3840*a^4*b^5*c^4*d*e + 6144*
a^5*b^3*c^5*d*e + 24*a*b^11*c*d*e)))^(1/2)*(32*b^7*c^2*e^3 - 384*a*b^5*c^3*e^3 - 2048*a^3*b*c^5*e^3 + 4096*a^3
*c^6*d*e^2 - 64*b^6*c^3*d*e^2 + 1536*a^2*b^3*c^4*e^3 + 768*a*b^4*c^4*d*e^2 - 3072*a^2*b^2*c^5*d*e^2))/(4*(b^4
+ 16*a^2*c^2 - 8*a*b^2*c)))*((9*(e^5*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^5 + 768*a^4*b*c^4*e^5 - 1536*a^4*c^5*d*e
^4 + 96*a^2*b^5*c^2*e^5 - 512*a^3*b^3*c^3*e^5 - 2048*a^3*c^6*d^3*e^2 + 32*b^6*c^3*d^3*e^2 - 48*b^7*c^2*d^2*e^3
+ 18*b^8*c*d*e^4 + 1536*a^2*b^2*c^5*d^3*e^2 - 2304*a^2*b^3*c^4*d^2*e^3 - 192*a*b^6*c^2*d*e^4 - 384*a*b^4*c^4*
d^3*e^2 + 576*a*b^5*c^3*d^2*e^3 + 576*a^2*b^4*c^3*d*e^4 + 3072*a^3*b*c^5*d^2*e^3))/(128*(a*b^12*e^2 + b^12*c*d
^2 + 4096*a^6*c^7*d^2 + 4096*a^7*c^6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2 - 24*a^2*b^10*c*e^2 + 240*a^2*b^8*c^3*
d^2 - 1280*a^3*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 + 240*a^3*b^8*c^2*e^2 - 1280*a^4*b^6*
c^3*e^2 + 3840*a^5*b^4*c^4*e^2 - 6144*a^6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e - 240*a^2*b^9*c^2*d*e + 1280*a^3*b^
7*c^3*d*e - 3840*a^4*b^5*c^4*d*e + 6144*a^5*b^3*c^5*d*e + 24*a*b^11*c*d*e)))^(1/2) - ((d + e*x)^(1/2)*(36*a*c^
4*e^6 - 45*b^2*c^3*e^6 - 144*c^5*d^2*e^4 + 144*b*c^4*d*e^5))/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*((9*(e^5*(-(4
*a*c - b^2)^9)^(1/2) - b^9*e^5 + 768*a^4*b*c^4*e^5 - 1536*a^4*c^5*d*e^4 + 96*a^2*b^5*c^2*e^5 - 512*a^3*b^3*c^3
*e^5 - 2048*a^3*c^6*d^3*e^2 + 32*b^6*c^3*d^3*e^2 - 48*b^7*c^2*d^2*e^3 + 18*b^8*c*d*e^4 + 1536*a^2*b^2*c^5*d^3*
e^2 - 2304*a^2*b^3*c^4*d^2*e^3 - 192*a*b^6*c^2*d*e^4 - 384*a*b^4*c^4*d^3*e^2 + 576*a*b^5*c^3*d^2*e^3 + 576*a^2
*b^4*c^3*d*e^4 + 3072*a^3*b*c^5*d^2*e^3))/(128*(a*b^12*e^2 + b^12*c*d^2 + 4096*a^6*c^7*d^2 + 4096*a^7*c^6*e^2
- b^13*d*e - 24*a*b^10*c^2*d^2 - 24*a^2*b^10*c*e^2 + 240*a^2*b^8*c^3*d^2 - 1280*a^3*b^6*c^4*d^2 + 3840*a^4*b^4
*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 + 240*a^3*b^8*c^2*e^2 - 1280*a^4*b^6*c^3*e^2 + 3840*a^5*b^4*c^4*e^2 - 6144*a^6
*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e - 240*a^2*b^9*c^2*d*e + 1280*a^3*b^7*c^3*d*e - 3840*a^4*b^5*c^4*d*e + 6144*a
^5*b^3*c^5*d*e + 24*a*b^11*c*d*e)))^(1/2) + (3*(36*a*c^4*e^8 + 27*b^2*c^3*e^8 + 144*c^5*d^2*e^6 - 144*b*c^4*d*
e^7))/(16*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c))))*((9*(e^5*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^5 + 76
8*a^4*b*c^4*e^5 - 1536*a^4*c^5*d*e^4 + 96*a^2*b^5*c^2*e^5 - 512*a^3*b^3*c^3*e^5 - 2048*a^3*c^6*d^3*e^2 + 32*b^
6*c^3*d^3*e^2 - 48*b^7*c^2*d^2*e^3 + 18*b^8*c*d*e^4 + 1536*a^2*b^2*c^5*d^3*e^2 - 2304*a^2*b^3*c^4*d^2*e^3 - 19
2*a*b^6*c^2*d*e^4 - 384*a*b^4*c^4*d^3*e^2 + 576*a*b^5*c^3*d^2*e^3 + 576*a^2*b^4*c^3*d*e^4 + 3072*a^3*b*c^5*d^2
*e^3))/(128*(a*b^12*e^2 + b^12*c*d^2 + 4096*a^6*c^7*d^2 + 4096*a^7*c^6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2 - 24
*a^2*b^10*c*e^2 + 240*a^2*b^8*c^3*d^2 - 1280*a^3*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 + 2
40*a^3*b^8*c^2*e^2 - 1280*a^4*b^6*c^3*e^2 + 3840*a^5*b^4*c^4*e^2 - 6144*a^6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e -
240*a^2*b^9*c^2*d*e + 1280*a^3*b^7*c^3*d*e - 3840*a^4*b^5*c^4*d*e + 6144*a^5*b^3*c^5*d*e + 24*a*b^11*c*d*e)))
^(1/2)*2i - atan(((((3*(32*b^7*c^2*e^5 - 384*a*b^5*c^3*e^5 - 2048*a^3*b*c^5*e^5 + 4096*a^3*c^6*d*e^4 - 64*b^6*
c^3*d*e^4 + 1536*a^2*b^3*c^4*e^5 + 768*a*b^4*c^4*d*e^4 - 3072*a^2*b^2*c^5*d*e^4))/(32*(b^6 - 64*a^3*c^3 + 48*a
^2*b^2*c^2 - 12*a*b^4*c)) - ((d + e*x)^(1/2)*(-(9*(b^9*e^5 + e^5*(-(4*a*c - b^2)^9)^(1/2) - 768*a^4*b*c^4*e^5
+ 1536*a^4*c^5*d*e^4 - 96*a^2*b^5*c^2*e^5 + 512*a^3*b^3*c^3*e^5 + 2048*a^3*c^6*d^3*e^2 - 32*b^6*c^3*d^3*e^2 +
48*b^7*c^2*d^2*e^3 - 18*b^8*c*d*e^4 - 1536*a^2*b^2*c^5*d^3*e^2 + 2304*a^2*b^3*c^4*d^2*e^3 + 192*a*b^6*c^2*d*e^
4 + 384*a*b^4*c^4*d^3*e^2 - 576*a*b^5*c^3*d^2*e^3 - 576*a^2*b^4*c^3*d*e^4 - 3072*a^3*b*c^5*d^2*e^3))/(128*(a*b
^12*e^2 + b^12*c*d^2 + 4096*a^6*c^7*d^2 + 4096*a^7*c^6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2 - 24*a^2*b^10*c*e^2
+ 240*a^2*b^8*c^3*d^2 - 1280*a^3*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 + 240*a^3*b^8*c^2*e
^2 - 1280*a^4*b^6*c^3*e^2 + 3840*a^5*b^4*c^4*e^2 - 6144*a^6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e - 240*a^2*b^9*c^2
*d*e + 1280*a^3*b^7*c^3*d*e - 3840*a^4*b^5*c^4*d*e + 6144*a^5*b^3*c^5*d*e + 24*a*b^11*c*d*e)))^(1/2)*(32*b^7*c
^2*e^3 - 384*a*b^5*c^3*e^3 - 2048*a^3*b*c^5*e^3 + 4096*a^3*c^6*d*e^2 - 64*b^6*c^3*d*e^2 + 1536*a^2*b^3*c^4*e^3
+ 768*a*b^4*c^4*d*e^2 - 3072*a^2*b^2*c^5*d*e^2))/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*(-(9*(b^9*e^5 + e^5*(-(4
*a*c - b^2)^9)^(1/2) - 768*a^4*b*c^4*e^5 + 1536*a^4*c^5*d*e^4 - 96*a^2*b^5*c^2*e^5 + 512*a^3*b^3*c^3*e^5 + 204
8*a^3*c^6*d^3*e^2 - 32*b^6*c^3*d^3*e^2 + 48*b^7*c^2*d^2*e^3 - 18*b^8*c*d*e^4 - 1536*a^2*b^2*c^5*d^3*e^2 + 2304
*a^2*b^3*c^4*d^2*e^3 + 192*a*b^6*c^2*d*e^4 + 384*a*b^4*c^4*d^3*e^2 - 576*a*b^5*c^3*d^2*e^3 - 576*a^2*b^4*c^3*d
*e^4 - 3072*a^3*b*c^5*d^2*e^3))/(128*(a*b^12*e^2 + b^12*c*d^2 + 4096*a^6*c^7*d^2 + 4096*a^7*c^6*e^2 - b^13*d*e
- 24*a*b^10*c^2*d^2 - 24*a^2*b^10*c*e^2 + 240*a^2*b^8*c^3*d^2 - 1280*a^3*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d^2 -
6144*a^5*b^2*c^6*d^2 + 240*a^3*b^8*c^2*e^2 - 1280*a^4*b^6*c^3*e^2 + 3840*a^5*b^4*c^4*e^2 - 6144*a^6*b^2*c^5*e
^2 - 4096*a^6*b*c^6*d*e - 240*a^2*b^9*c^2*d*e + 1280*a^3*b^7*c^3*d*e - 3840*a^4*b^5*c^4*d*e + 6144*a^5*b^3*c^5
*d*e + 24*a*b^11*c*d*e)))^(1/2) + ((d + e*x)^(1/2)*(36*a*c^4*e^6 - 45*b^2*c^3*e^6 - 144*c^5*d^2*e^4 + 144*b*c^
4*d*e^5))/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*(-(9*(b^9*e^5 + e^5*(-(4*a*c - b^2)^9)^(1/2) - 768*a^4*b*c^4*e^5
+ 1536*a^4*c^5*d*e^4 - 96*a^2*b^5*c^2*e^5 + 512*a^3*b^3*c^3*e^5 + 2048*a^3*c^6*d^3*e^2 - 32*b^6*c^3*d^3*e^2 +
48*b^7*c^2*d^2*e^3 - 18*b^8*c*d*e^4 - 1536*a^2*b^2*c^5*d^3*e^2 + 2304*a^2*b^3*c^4*d^2*e^3 + 192*a*b^6*c^2*d*e
^4 + 384*a*b^4*c^4*d^3*e^2 - 576*a*b^5*c^3*d^2*e^3 - 576*a^2*b^4*c^3*d*e^4 - 3072*a^3*b*c^5*d^2*e^3))/(128*(a*
b^12*e^2 + b^12*c*d^2 + 4096*a^6*c^7*d^2 + 4096*a^7*c^6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2 - 24*a^2*b^10*c*e^2
+ 240*a^2*b^8*c^3*d^2 - 1280*a^3*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 + 240*a^3*b^8*c^2*
e^2 - 1280*a^4*b^6*c^3*e^2 + 3840*a^5*b^4*c^4*e^2 - 6144*a^6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e - 240*a^2*b^9*c^
2*d*e + 1280*a^3*b^7*c^3*d*e - 3840*a^4*b^5*c^4*d*e + 6144*a^5*b^3*c^5*d*e + 24*a*b^11*c*d*e)))^(1/2)*1i - (((
3*(32*b^7*c^2*e^5 - 384*a*b^5*c^3*e^5 - 2048*a^3*b*c^5*e^5 + 4096*a^3*c^6*d*e^4 - 64*b^6*c^3*d*e^4 + 1536*a^2*
b^3*c^4*e^5 + 768*a*b^4*c^4*d*e^4 - 3072*a^2*b^2*c^5*d*e^4))/(32*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4
*c)) + ((d + e*x)^(1/2)*(-(9*(b^9*e^5 + e^5*(-(4*a*c - b^2)^9)^(1/2) - 768*a^4*b*c^4*e^5 + 1536*a^4*c^5*d*e^4
- 96*a^2*b^5*c^2*e^5 + 512*a^3*b^3*c^3*e^5 + 2048*a^3*c^6*d^3*e^2 - 32*b^6*c^3*d^3*e^2 + 48*b^7*c^2*d^2*e^3 -
18*b^8*c*d*e^4 - 1536*a^2*b^2*c^5*d^3*e^2 + 2304*a^2*b^3*c^4*d^2*e^3 + 192*a*b^6*c^2*d*e^4 + 384*a*b^4*c^4*d^3
*e^2 - 576*a*b^5*c^3*d^2*e^3 - 576*a^2*b^4*c^3*d*e^4 - 3072*a^3*b*c^5*d^2*e^3))/(128*(a*b^12*e^2 + b^12*c*d^2
+ 4096*a^6*c^7*d^2 + 4096*a^7*c^6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2 - 24*a^2*b^10*c*e^2 + 240*a^2*b^8*c^3*d^2
- 1280*a^3*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 + 240*a^3*b^8*c^2*e^2 - 1280*a^4*b^6*c^3
*e^2 + 3840*a^5*b^4*c^4*e^2 - 6144*a^6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e - 240*a^2*b^9*c^2*d*e + 1280*a^3*b^7*c
^3*d*e - 3840*a^4*b^5*c^4*d*e + 6144*a^5*b^3*c^5*d*e + 24*a*b^11*c*d*e)))^(1/2)*(32*b^7*c^2*e^3 - 384*a*b^5*c^
3*e^3 - 2048*a^3*b*c^5*e^3 + 4096*a^3*c^6*d*e^2 - 64*b^6*c^3*d*e^2 + 1536*a^2*b^3*c^4*e^3 + 768*a*b^4*c^4*d*e^
2 - 3072*a^2*b^2*c^5*d*e^2))/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*(-(9*(b^9*e^5 + e^5*(-(4*a*c - b^2)^9)^(1/2)
- 768*a^4*b*c^4*e^5 + 1536*a^4*c^5*d*e^4 - 96*a^2*b^5*c^2*e^5 + 512*a^3*b^3*c^3*e^5 + 2048*a^3*c^6*d^3*e^2 - 3
2*b^6*c^3*d^3*e^2 + 48*b^7*c^2*d^2*e^3 - 18*b^8*c*d*e^4 - 1536*a^2*b^2*c^5*d^3*e^2 + 2304*a^2*b^3*c^4*d^2*e^3
+ 192*a*b^6*c^2*d*e^4 + 384*a*b^4*c^4*d^3*e^2 - 576*a*b^5*c^3*d^2*e^3 - 576*a^2*b^4*c^3*d*e^4 - 3072*a^3*b*c^5
*d^2*e^3))/(128*(a*b^12*e^2 + b^12*c*d^2 + 4096*a^6*c^7*d^2 + 4096*a^7*c^6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2
- 24*a^2*b^10*c*e^2 + 240*a^2*b^8*c^3*d^2 - 1280*a^3*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2
+ 240*a^3*b^8*c^2*e^2 - 1280*a^4*b^6*c^3*e^2 + 3840*a^5*b^4*c^4*e^2 - 6144*a^6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d
*e - 240*a^2*b^9*c^2*d*e + 1280*a^3*b^7*c^3*d*e - 3840*a^4*b^5*c^4*d*e + 6144*a^5*b^3*c^5*d*e + 24*a*b^11*c*d*
e)))^(1/2) - ((d + e*x)^(1/2)*(36*a*c^4*e^6 - 45*b^2*c^3*e^6 - 144*c^5*d^2*e^4 + 144*b*c^4*d*e^5))/(4*(b^4 + 1
6*a^2*c^2 - 8*a*b^2*c)))*(-(9*(b^9*e^5 + e^5*(-(4*a*c - b^2)^9)^(1/2) - 768*a^4*b*c^4*e^5 + 1536*a^4*c^5*d*e^4
- 96*a^2*b^5*c^2*e^5 + 512*a^3*b^3*c^3*e^5 + 2048*a^3*c^6*d^3*e^2 - 32*b^6*c^3*d^3*e^2 + 48*b^7*c^2*d^2*e^3 -
18*b^8*c*d*e^4 - 1536*a^2*b^2*c^5*d^3*e^2 + 2304*a^2*b^3*c^4*d^2*e^3 + 192*a*b^6*c^2*d*e^4 + 384*a*b^4*c^4*d^
3*e^2 - 576*a*b^5*c^3*d^2*e^3 - 576*a^2*b^4*c^3*d*e^4 - 3072*a^3*b*c^5*d^2*e^3))/(128*(a*b^12*e^2 + b^12*c*d^2
+ 4096*a^6*c^7*d^2 + 4096*a^7*c^6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2 - 24*a^2*b^10*c*e^2 + 240*a^2*b^8*c^3*d^
2 - 1280*a^3*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 + 240*a^3*b^8*c^2*e^2 - 1280*a^4*b^6*c^
3*e^2 + 3840*a^5*b^4*c^4*e^2 - 6144*a^6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e - 240*a^2*b^9*c^2*d*e + 1280*a^3*b^7*
c^3*d*e - 3840*a^4*b^5*c^4*d*e + 6144*a^5*b^3*c^5*d*e + 24*a*b^11*c*d*e)))^(1/2)*1i)/((((3*(32*b^7*c^2*e^5 - 3
84*a*b^5*c^3*e^5 - 2048*a^3*b*c^5*e^5 + 4096*a^3*c^6*d*e^4 - 64*b^6*c^3*d*e^4 + 1536*a^2*b^3*c^4*e^5 + 768*a*b
^4*c^4*d*e^4 - 3072*a^2*b^2*c^5*d*e^4))/(32*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) - ((d + e*x)^(1/
2)*(-(9*(b^9*e^5 + e^5*(-(4*a*c - b^2)^9)^(1/2) - 768*a^4*b*c^4*e^5 + 1536*a^4*c^5*d*e^4 - 96*a^2*b^5*c^2*e^5
+ 512*a^3*b^3*c^3*e^5 + 2048*a^3*c^6*d^3*e^2 - 32*b^6*c^3*d^3*e^2 + 48*b^7*c^2*d^2*e^3 - 18*b^8*c*d*e^4 - 1536
*a^2*b^2*c^5*d^3*e^2 + 2304*a^2*b^3*c^4*d^2*e^3 + 192*a*b^6*c^2*d*e^4 + 384*a*b^4*c^4*d^3*e^2 - 576*a*b^5*c^3*
d^2*e^3 - 576*a^2*b^4*c^3*d*e^4 - 3072*a^3*b*c^5*d^2*e^3))/(128*(a*b^12*e^2 + b^12*c*d^2 + 4096*a^6*c^7*d^2 +
4096*a^7*c^6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2 - 24*a^2*b^10*c*e^2 + 240*a^2*b^8*c^3*d^2 - 1280*a^3*b^6*c^4*d
^2 + 3840*a^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 + 240*a^3*b^8*c^2*e^2 - 1280*a^4*b^6*c^3*e^2 + 3840*a^5*b^4*c
^4*e^2 - 6144*a^6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e - 240*a^2*b^9*c^2*d*e + 1280*a^3*b^7*c^3*d*e - 3840*a^4*b^5
*c^4*d*e + 6144*a^5*b^3*c^5*d*e + 24*a*b^11*c*d*e)))^(1/2)*(32*b^7*c^2*e^3 - 384*a*b^5*c^3*e^3 - 2048*a^3*b*c^
5*e^3 + 4096*a^3*c^6*d*e^2 - 64*b^6*c^3*d*e^2 + 1536*a^2*b^3*c^4*e^3 + 768*a*b^4*c^4*d*e^2 - 3072*a^2*b^2*c^5*
d*e^2))/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*(-(9*(b^9*e^5 + e^5*(-(4*a*c - b^2)^9)^(1/2) - 768*a^4*b*c^4*e^5 +
1536*a^4*c^5*d*e^4 - 96*a^2*b^5*c^2*e^5 + 512*a^3*b^3*c^3*e^5 + 2048*a^3*c^6*d^3*e^2 - 32*b^6*c^3*d^3*e^2 + 4
8*b^7*c^2*d^2*e^3 - 18*b^8*c*d*e^4 - 1536*a^2*b^2*c^5*d^3*e^2 + 2304*a^2*b^3*c^4*d^2*e^3 + 192*a*b^6*c^2*d*e^4
+ 384*a*b^4*c^4*d^3*e^2 - 576*a*b^5*c^3*d^2*e^3 - 576*a^2*b^4*c^3*d*e^4 - 3072*a^3*b*c^5*d^2*e^3))/(128*(a*b^
12*e^2 + b^12*c*d^2 + 4096*a^6*c^7*d^2 + 4096*a^7*c^6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2 - 24*a^2*b^10*c*e^2 +
240*a^2*b^8*c^3*d^2 - 1280*a^3*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 + 240*a^3*b^8*c^2*e^
2 - 1280*a^4*b^6*c^3*e^2 + 3840*a^5*b^4*c^4*e^2 - 6144*a^6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e - 240*a^2*b^9*c^2*
d*e + 1280*a^3*b^7*c^3*d*e - 3840*a^4*b^5*c^4*d*e + 6144*a^5*b^3*c^5*d*e + 24*a*b^11*c*d*e)))^(1/2) + ((d + e*
x)^(1/2)*(36*a*c^4*e^6 - 45*b^2*c^3*e^6 - 144*c^5*d^2*e^4 + 144*b*c^4*d*e^5))/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c
)))*(-(9*(b^9*e^5 + e^5*(-(4*a*c - b^2)^9)^(1/2) - 768*a^4*b*c^4*e^5 + 1536*a^4*c^5*d*e^4 - 96*a^2*b^5*c^2*e^5
+ 512*a^3*b^3*c^3*e^5 + 2048*a^3*c^6*d^3*e^2 - 32*b^6*c^3*d^3*e^2 + 48*b^7*c^2*d^2*e^3 - 18*b^8*c*d*e^4 - 153
6*a^2*b^2*c^5*d^3*e^2 + 2304*a^2*b^3*c^4*d^2*e^3 + 192*a*b^6*c^2*d*e^4 + 384*a*b^4*c^4*d^3*e^2 - 576*a*b^5*c^3
*d^2*e^3 - 576*a^2*b^4*c^3*d*e^4 - 3072*a^3*b*c^5*d^2*e^3))/(128*(a*b^12*e^2 + b^12*c*d^2 + 4096*a^6*c^7*d^2 +
4096*a^7*c^6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2 - 24*a^2*b^10*c*e^2 + 240*a^2*b^8*c^3*d^2 - 1280*a^3*b^6*c^4*
d^2 + 3840*a^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 + 240*a^3*b^8*c^2*e^2 - 1280*a^4*b^6*c^3*e^2 + 3840*a^5*b^4*
c^4*e^2 - 6144*a^6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e - 240*a^2*b^9*c^2*d*e + 1280*a^3*b^7*c^3*d*e - 3840*a^4*b^
5*c^4*d*e + 6144*a^5*b^3*c^5*d*e + 24*a*b^11*c*d*e)))^(1/2) + (((3*(32*b^7*c^2*e^5 - 384*a*b^5*c^3*e^5 - 2048*
a^3*b*c^5*e^5 + 4096*a^3*c^6*d*e^4 - 64*b^6*c^3*d*e^4 + 1536*a^2*b^3*c^4*e^5 + 768*a*b^4*c^4*d*e^4 - 3072*a^2*
b^2*c^5*d*e^4))/(32*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + ((d + e*x)^(1/2)*(-(9*(b^9*e^5 + e^5*(
-(4*a*c - b^2)^9)^(1/2) - 768*a^4*b*c^4*e^5 + 1536*a^4*c^5*d*e^4 - 96*a^2*b^5*c^2*e^5 + 512*a^3*b^3*c^3*e^5 +
2048*a^3*c^6*d^3*e^2 - 32*b^6*c^3*d^3*e^2 + 48*b^7*c^2*d^2*e^3 - 18*b^8*c*d*e^4 - 1536*a^2*b^2*c^5*d^3*e^2 + 2
304*a^2*b^3*c^4*d^2*e^3 + 192*a*b^6*c^2*d*e^4 + 384*a*b^4*c^4*d^3*e^2 - 576*a*b^5*c^3*d^2*e^3 - 576*a^2*b^4*c^
3*d*e^4 - 3072*a^3*b*c^5*d^2*e^3))/(128*(a*b^12*e^2 + b^12*c*d^2 + 4096*a^6*c^7*d^2 + 4096*a^7*c^6*e^2 - b^13*
d*e - 24*a*b^10*c^2*d^2 - 24*a^2*b^10*c*e^2 + 240*a^2*b^8*c^3*d^2 - 1280*a^3*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d^
2 - 6144*a^5*b^2*c^6*d^2 + 240*a^3*b^8*c^2*e^2 - 1280*a^4*b^6*c^3*e^2 + 3840*a^5*b^4*c^4*e^2 - 6144*a^6*b^2*c^
5*e^2 - 4096*a^6*b*c^6*d*e - 240*a^2*b^9*c^2*d*e + 1280*a^3*b^7*c^3*d*e - 3840*a^4*b^5*c^4*d*e + 6144*a^5*b^3*
c^5*d*e + 24*a*b^11*c*d*e)))^(1/2)*(32*b^7*c^2*e^3 - 384*a*b^5*c^3*e^3 - 2048*a^3*b*c^5*e^3 + 4096*a^3*c^6*d*e
^2 - 64*b^6*c^3*d*e^2 + 1536*a^2*b^3*c^4*e^3 + 768*a*b^4*c^4*d*e^2 - 3072*a^2*b^2*c^5*d*e^2))/(4*(b^4 + 16*a^2
*c^2 - 8*a*b^2*c)))*(-(9*(b^9*e^5 + e^5*(-(4*a*c - b^2)^9)^(1/2) - 768*a^4*b*c^4*e^5 + 1536*a^4*c^5*d*e^4 - 96
*a^2*b^5*c^2*e^5 + 512*a^3*b^3*c^3*e^5 + 2048*a^3*c^6*d^3*e^2 - 32*b^6*c^3*d^3*e^2 + 48*b^7*c^2*d^2*e^3 - 18*b
^8*c*d*e^4 - 1536*a^2*b^2*c^5*d^3*e^2 + 2304*a^2*b^3*c^4*d^2*e^3 + 192*a*b^6*c^2*d*e^4 + 384*a*b^4*c^4*d^3*e^2
- 576*a*b^5*c^3*d^2*e^3 - 576*a^2*b^4*c^3*d*e^4 - 3072*a^3*b*c^5*d^2*e^3))/(128*(a*b^12*e^2 + b^12*c*d^2 + 40
96*a^6*c^7*d^2 + 4096*a^7*c^6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2 - 24*a^2*b^10*c*e^2 + 240*a^2*b^8*c^3*d^2 - 1
280*a^3*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 + 240*a^3*b^8*c^2*e^2 - 1280*a^4*b^6*c^3*e^2
+ 3840*a^5*b^4*c^4*e^2 - 6144*a^6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e - 240*a^2*b^9*c^2*d*e + 1280*a^3*b^7*c^3*d
*e - 3840*a^4*b^5*c^4*d*e + 6144*a^5*b^3*c^5*d*e + 24*a*b^11*c*d*e)))^(1/2) - ((d + e*x)^(1/2)*(36*a*c^4*e^6 -
45*b^2*c^3*e^6 - 144*c^5*d^2*e^4 + 144*b*c^4*d*e^5))/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*(-(9*(b^9*e^5 + e^5*
(-(4*a*c - b^2)^9)^(1/2) - 768*a^4*b*c^4*e^5 + 1536*a^4*c^5*d*e^4 - 96*a^2*b^5*c^2*e^5 + 512*a^3*b^3*c^3*e^5 +
2048*a^3*c^6*d^3*e^2 - 32*b^6*c^3*d^3*e^2 + 48*b^7*c^2*d^2*e^3 - 18*b^8*c*d*e^4 - 1536*a^2*b^2*c^5*d^3*e^2 +
2304*a^2*b^3*c^4*d^2*e^3 + 192*a*b^6*c^2*d*e^4 + 384*a*b^4*c^4*d^3*e^2 - 576*a*b^5*c^3*d^2*e^3 - 576*a^2*b^4*c
^3*d*e^4 - 3072*a^3*b*c^5*d^2*e^3))/(128*(a*b^12*e^2 + b^12*c*d^2 + 4096*a^6*c^7*d^2 + 4096*a^7*c^6*e^2 - b^13
*d*e - 24*a*b^10*c^2*d^2 - 24*a^2*b^10*c*e^2 + 240*a^2*b^8*c^3*d^2 - 1280*a^3*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d
^2 - 6144*a^5*b^2*c^6*d^2 + 240*a^3*b^8*c^2*e^2 - 1280*a^4*b^6*c^3*e^2 + 3840*a^5*b^4*c^4*e^2 - 6144*a^6*b^2*c
^5*e^2 - 4096*a^6*b*c^6*d*e - 240*a^2*b^9*c^2*d*e + 1280*a^3*b^7*c^3*d*e - 3840*a^4*b^5*c^4*d*e + 6144*a^5*b^3
*c^5*d*e + 24*a*b^11*c*d*e)))^(1/2) + (3*(36*a*c^4*e^8 + 27*b^2*c^3*e^8 + 144*c^5*d^2*e^6 - 144*b*c^4*d*e^7))/
(16*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c))))*(-(9*(b^9*e^5 + e^5*(-(4*a*c - b^2)^9)^(1/2) - 768*a^4
*b*c^4*e^5 + 1536*a^4*c^5*d*e^4 - 96*a^2*b^5*c^2*e^5 + 512*a^3*b^3*c^3*e^5 + 2048*a^3*c^6*d^3*e^2 - 32*b^6*c^3
*d^3*e^2 + 48*b^7*c^2*d^2*e^3 - 18*b^8*c*d*e^4 - 1536*a^2*b^2*c^5*d^3*e^2 + 2304*a^2*b^3*c^4*d^2*e^3 + 192*a*b
^6*c^2*d*e^4 + 384*a*b^4*c^4*d^3*e^2 - 576*a*b^5*c^3*d^2*e^3 - 576*a^2*b^4*c^3*d*e^4 - 3072*a^3*b*c^5*d^2*e^3)
)/(128*(a*b^12*e^2 + b^12*c*d^2 + 4096*a^6*c^7*d^2 + 4096*a^7*c^6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2 - 24*a^2*
b^10*c*e^2 + 240*a^2*b^8*c^3*d^2 - 1280*a^3*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 + 240*a^
3*b^8*c^2*e^2 - 1280*a^4*b^6*c^3*e^2 + 3840*a^5*b^4*c^4*e^2 - 6144*a^6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e - 240*
a^2*b^9*c^2*d*e + 1280*a^3*b^7*c^3*d*e - 3840*a^4*b^5*c^4*d*e + 6144*a^5*b^3*c^5*d*e + 24*a*b^11*c*d*e)))^(1/2
)*2i

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

[Out]

Timed out

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