3.525 \(\int \frac {x^4 \sqrt {d+e x}}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=490 \[ \frac {\sqrt {2} \left (-\frac {-5 a^2 b c^2 e+2 a^2 c^3 d+5 a b^3 c e-4 a b^2 c^2 d+b^5 (-e)+b^4 c d}{\sqrt {b^2-4 a c}}-a^2 c^2 e+3 a b^2 c e-2 a b c^2 d+b^4 (-e)+b^3 c d\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{c^{9/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {\sqrt {2} \left (\frac {-5 a^2 b c^2 e+2 a^2 c^3 d+5 a b^3 c e-4 a b^2 c^2 d+b^5 (-e)+b^4 c d}{\sqrt {b^2-4 a c}}-a^2 c^2 e+3 a b^2 c e-2 a b c^2 d+b^4 (-e)+b^3 c d\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{c^{9/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {2 b \left (b^2-2 a c\right ) \sqrt {d+e x}}{c^4}+\frac {2 (d+e x)^{3/2} \left (c e (b d-a e)+b^2 e^2+c^2 d^2\right )}{3 c^3 e^3}-\frac {2 (d+e x)^{5/2} (b e+2 c d)}{5 c^2 e^3}+\frac {2 (d+e x)^{7/2}}{7 c e^3} \]

[Out]

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

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Rubi [A]  time = 14.85, antiderivative size = 490, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {897, 1287, 1166, 208} \[ \frac {\sqrt {2} \left (-\frac {-5 a^2 b c^2 e+2 a^2 c^3 d-4 a b^2 c^2 d+5 a b^3 c e+b^4 c d+b^5 (-e)}{\sqrt {b^2-4 a c}}-a^2 c^2 e+3 a b^2 c e-2 a b c^2 d+b^3 c d+b^4 (-e)\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{c^{9/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {\sqrt {2} \left (\frac {-5 a^2 b c^2 e+2 a^2 c^3 d-4 a b^2 c^2 d+5 a b^3 c e+b^4 c d+b^5 (-e)}{\sqrt {b^2-4 a c}}-a^2 c^2 e+3 a b^2 c e-2 a b c^2 d+b^3 c d+b^4 (-e)\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{c^{9/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {2 (d+e x)^{3/2} \left (c e (b d-a e)+b^2 e^2+c^2 d^2\right )}{3 c^3 e^3}-\frac {2 b \left (b^2-2 a c\right ) \sqrt {d+e x}}{c^4}-\frac {2 (d+e x)^{5/2} (b e+2 c d)}{5 c^2 e^3}+\frac {2 (d+e x)^{7/2}}{7 c e^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*Sqrt[d + e*x])/(a + b*x + c*x^2),x]

[Out]

(-2*b*(b^2 - 2*a*c)*Sqrt[d + e*x])/c^4 + (2*(c^2*d^2 + b^2*e^2 + c*e*(b*d - a*e))*(d + e*x)^(3/2))/(3*c^3*e^3)
- (2*(2*c*d + b*e)*(d + e*x)^(5/2))/(5*c^2*e^3) + (2*(d + e*x)^(7/2))/(7*c*e^3) + (Sqrt[2]*(b^3*c*d - 2*a*b*c
^2*d - b^4*e + 3*a*b^2*c*e - a^2*c^2*e - (b^4*c*d - 4*a*b^2*c^2*d + 2*a^2*c^3*d - b^5*e + 5*a*b^3*c*e - 5*a^2*
b*c^2*e)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/
(c^(9/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) + (Sqrt[2]*(b^3*c*d - 2*a*b*c^2*d - b^4*e + 3*a*b^2*c*e - a^
2*c^2*e + (b^4*c*d - 4*a*b^2*c^2*d + 2*a^2*c^3*d - b^5*e + 5*a*b^3*c*e - 5*a^2*b*c^2*e)/Sqrt[b^2 - 4*a*c])*Arc
Tanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(c^(9/2)*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 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 {x^4 \sqrt {d+e x}}{a+b x+c x^2} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {x^2 \left (-\frac {d}{e}+\frac {x^2}{e}\right )^4}{\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}} \, dx,x,\sqrt {d+e x}\right )}{e}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (-\frac {\left (b^3-2 a b c\right ) e}{c^4}+\frac {\left (c^2 d^2+b^2 e^2+c e (b d-a e)\right ) x^2}{c^3 e^2}-\frac {(2 c d+b e) x^4}{c^2 e^2}+\frac {x^6}{c e^2}+\frac {b \left (b^2-2 a c\right ) \left (c d^2-b d e+a e^2\right )-\left (b^3 c d-2 a b c^2 d-b^4 e+3 a b^2 c e-a^2 c^2 e\right ) x^2}{c^4 e \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 )}\right ) \, dx,x,\sqrt {d+e x}\right )}{e}\\ &=-\frac {2 b \left (b^2-2 a c\right ) \sqrt {d+e x}}{c^4}+\frac {2 \left (c^2 d^2+b^2 e^2+c e (b d-a e)\right ) (d+e x)^{3/2}}{3 c^3 e^3}-\frac {2 (2 c d+b e) (d+e x)^{5/2}}{5 c^2 e^3}+\frac {2 (d+e x)^{7/2}}{7 c e^3}+\frac {2 \operatorname {Subst}\left (\int \frac {b \left (b^2-2 a c\right ) \left (c d^2-b d e+a e^2\right )+\left (-b^3 c d+2 a b c^2 d+b^4 e-3 a b^2 c e+a^2 c^2 e\right ) x^2}{\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}} \, dx,x,\sqrt {d+e x}\right )}{c^4 e^2}\\ &=-\frac {2 b \left (b^2-2 a c\right ) \sqrt {d+e x}}{c^4}+\frac {2 \left (c^2 d^2+b^2 e^2+c e (b d-a e)\right ) (d+e x)^{3/2}}{3 c^3 e^3}-\frac {2 (2 c d+b e) (d+e x)^{5/2}}{5 c^2 e^3}+\frac {2 (d+e x)^{7/2}}{7 c e^3}-\frac {\left (b^3 c d-2 a b c^2 d-b^4 e+3 a b^2 c e-a^2 c^2 e-\frac {b^4 c d-4 a b^2 c^2 d+2 a^2 c^3 d-b^5 e+5 a b^3 c e-5 a^2 b c^2 e}{\sqrt {b^2-4 a c}}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {\sqrt {b^2-4 a c}}{2 e}-\frac {2 c d-b e}{2 e^2}+\frac {c x^2}{e^2}} \, dx,x,\sqrt {d+e x}\right )}{c^4 e^2}-\frac {\left (b^3 c d-2 a b c^2 d-b^4 e+3 a b^2 c e-a^2 c^2 e+\frac {b^4 c d-4 a b^2 c^2 d+2 a^2 c^3 d-b^5 e+5 a b^3 c e-5 a^2 b c^2 e}{\sqrt {b^2-4 a c}}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {b^2-4 a c}}{2 e}-\frac {2 c d-b e}{2 e^2}+\frac {c x^2}{e^2}} \, dx,x,\sqrt {d+e x}\right )}{c^4 e^2}\\ &=-\frac {2 b \left (b^2-2 a c\right ) \sqrt {d+e x}}{c^4}+\frac {2 \left (c^2 d^2+b^2 e^2+c e (b d-a e)\right ) (d+e x)^{3/2}}{3 c^3 e^3}-\frac {2 (2 c d+b e) (d+e x)^{5/2}}{5 c^2 e^3}+\frac {2 (d+e x)^{7/2}}{7 c e^3}+\frac {\sqrt {2} \left (b^3 c d-2 a b c^2 d-b^4 e+3 a b^2 c e-a^2 c^2 e-\frac {b^4 c d-4 a b^2 c^2 d+2 a^2 c^3 d-b^5 e+5 a b^3 c e-5 a^2 b c^2 e}{\sqrt {b^2-4 a c}}\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 )}{c^{9/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \left (b^3 c d-2 a b c^2 d-b^4 e+3 a b^2 c e-a^2 c^2 e+\frac {b^4 c d-4 a b^2 c^2 d+2 a^2 c^3 d-b^5 e+5 a b^3 c e-5 a^2 b c^2 e}{\sqrt {b^2-4 a c}}\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 )}{c^{9/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\\ \end {align*}

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Mathematica [A]  time = 0.75, size = 568, normalized size = 1.16 \[ -\frac {\sqrt {2} \left (a^2 c^2 \left (e \sqrt {b^2-4 a c}+2 c d\right )+a b c^2 \left (2 d \sqrt {b^2-4 a c}-5 a e\right )-a b^2 c \left (3 e \sqrt {b^2-4 a c}+4 c d\right )+b^4 \left (e \sqrt {b^2-4 a c}+c d\right )+b^3 c \left (5 a e-d \sqrt {b^2-4 a c}\right )+b^5 (-e)\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {e \sqrt {b^2-4 a c}-b e+2 c d}}\right )}{c^{9/2} \sqrt {b^2-4 a c} \sqrt {e \left (\sqrt {b^2-4 a c}-b\right )+2 c d}}-\frac {\sqrt {2} \left (a^2 c^2 \left (e \sqrt {b^2-4 a c}-2 c d\right )+a b c^2 \left (2 d \sqrt {b^2-4 a c}+5 a e\right )+a b^2 c \left (4 c d-3 e \sqrt {b^2-4 a c}\right )+b^4 \left (e \sqrt {b^2-4 a c}-c d\right )-b^3 c \left (d \sqrt {b^2-4 a c}+5 a e\right )+b^5 e\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{c^{9/2} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {2 \sqrt {d+e x} \left (-7 c^2 e (d+e x) (5 a e-2 b d+3 b e x)+35 b c e^2 (6 a e+b (d+e x))-105 b^3 e^3+c^3 \left (8 d^3-4 d^2 e x+3 d e^2 x^2+15 e^3 x^3\right )\right )}{105 c^4 e^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*Sqrt[d + e*x])/(a + b*x + c*x^2),x]

[Out]

(2*Sqrt[d + e*x]*(-105*b^3*e^3 - 7*c^2*e*(d + e*x)*(-2*b*d + 5*a*e + 3*b*e*x) + c^3*(8*d^3 - 4*d^2*e*x + 3*d*e
^2*x^2 + 15*e^3*x^3) + 35*b*c*e^2*(6*a*e + b*(d + e*x))))/(105*c^4*e^3) - (Sqrt[2]*(-(b^5*e) + a*b*c^2*(2*Sqrt
[b^2 - 4*a*c]*d - 5*a*e) + b^3*c*(-(Sqrt[b^2 - 4*a*c]*d) + 5*a*e) + b^4*(c*d + Sqrt[b^2 - 4*a*c]*e) + a^2*c^2*
(2*c*d + Sqrt[b^2 - 4*a*c]*e) - a*b^2*c*(4*c*d + 3*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]])/(c^(9/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*
e]) - (Sqrt[2]*(b^5*e - b^3*c*(Sqrt[b^2 - 4*a*c]*d + 5*a*e) + a*b*c^2*(2*Sqrt[b^2 - 4*a*c]*d + 5*a*e) + a*b^2*
c*(4*c*d - 3*Sqrt[b^2 - 4*a*c]*e) + a^2*c^2*(-2*c*d + Sqrt[b^2 - 4*a*c]*e) + b^4*(-(c*d) + 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]])/(c^(9/2)*Sqrt[b^2 - 4*a*c
]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

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fricas [B]  time = 2.52, size = 5507, normalized size = 11.24 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)^(1/2)/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

1/210*(105*sqrt(2)*c^4*e^3*sqrt(((b^8*c - 8*a*b^6*c^2 + 20*a^2*b^4*c^3 - 16*a^3*b^2*c^4 + 2*a^4*c^5)*d - (b^9
- 9*a*b^7*c + 27*a^2*b^5*c^2 - 30*a^3*b^3*c^3 + 9*a^4*b*c^4)*e + (b^2*c^9 - 4*a*c^10)*sqrt(((b^14*c^2 - 12*a*b
^12*c^3 + 56*a^2*b^10*c^4 - 128*a^3*b^8*c^5 + 148*a^4*b^6*c^6 - 80*a^5*b^4*c^7 + 16*a^6*b^2*c^8)*d^2 - 2*(b^15
*c - 13*a*b^13*c^2 + 67*a^2*b^11*c^3 - 174*a^3*b^9*c^4 + 239*a^4*b^7*c^5 - 166*a^5*b^5*c^6 + 50*a^6*b^3*c^7 -
4*a^7*b*c^8)*d*e + (b^16 - 14*a*b^14*c + 79*a^2*b^12*c^2 - 230*a^3*b^10*c^3 + 367*a^4*b^8*c^4 - 314*a^5*b^6*c^
5 + 130*a^6*b^4*c^6 - 20*a^7*b^2*c^7 + a^8*c^8)*e^2)/(b^2*c^18 - 4*a*c^19)))/(b^2*c^9 - 4*a*c^10))*log(sqrt(2)
*((b^12*c - 12*a*b^10*c^2 + 54*a^2*b^8*c^3 - 112*a^3*b^6*c^4 + 104*a^4*b^4*c^5 - 32*a^5*b^2*c^6)*d - (b^13 - 1
3*a*b^11*c + 65*a^2*b^9*c^2 - 156*a^3*b^7*c^3 + 181*a^4*b^5*c^4 - 86*a^5*b^3*c^5 + 8*a^6*b*c^6)*e - (b^6*c^9 -
8*a*b^4*c^10 + 18*a^2*b^2*c^11 - 8*a^3*c^12)*sqrt(((b^14*c^2 - 12*a*b^12*c^3 + 56*a^2*b^10*c^4 - 128*a^3*b^8*
c^5 + 148*a^4*b^6*c^6 - 80*a^5*b^4*c^7 + 16*a^6*b^2*c^8)*d^2 - 2*(b^15*c - 13*a*b^13*c^2 + 67*a^2*b^11*c^3 - 1
74*a^3*b^9*c^4 + 239*a^4*b^7*c^5 - 166*a^5*b^5*c^6 + 50*a^6*b^3*c^7 - 4*a^7*b*c^8)*d*e + (b^16 - 14*a*b^14*c +
79*a^2*b^12*c^2 - 230*a^3*b^10*c^3 + 367*a^4*b^8*c^4 - 314*a^5*b^6*c^5 + 130*a^6*b^4*c^6 - 20*a^7*b^2*c^7 + a
^8*c^8)*e^2)/(b^2*c^18 - 4*a*c^19)))*sqrt(((b^8*c - 8*a*b^6*c^2 + 20*a^2*b^4*c^3 - 16*a^3*b^2*c^4 + 2*a^4*c^5)
*d - (b^9 - 9*a*b^7*c + 27*a^2*b^5*c^2 - 30*a^3*b^3*c^3 + 9*a^4*b*c^4)*e + (b^2*c^9 - 4*a*c^10)*sqrt(((b^14*c^
2 - 12*a*b^12*c^3 + 56*a^2*b^10*c^4 - 128*a^3*b^8*c^5 + 148*a^4*b^6*c^6 - 80*a^5*b^4*c^7 + 16*a^6*b^2*c^8)*d^2
- 2*(b^15*c - 13*a*b^13*c^2 + 67*a^2*b^11*c^3 - 174*a^3*b^9*c^4 + 239*a^4*b^7*c^5 - 166*a^5*b^5*c^6 + 50*a^6*
b^3*c^7 - 4*a^7*b*c^8)*d*e + (b^16 - 14*a*b^14*c + 79*a^2*b^12*c^2 - 230*a^3*b^10*c^3 + 367*a^4*b^8*c^4 - 314*
a^5*b^6*c^5 + 130*a^6*b^4*c^6 - 20*a^7*b^2*c^7 + a^8*c^8)*e^2)/(b^2*c^18 - 4*a*c^19)))/(b^2*c^9 - 4*a*c^10)) -
4*((a^4*b^7*c - 6*a^5*b^5*c^2 + 10*a^6*b^3*c^3 - 4*a^7*b*c^4)*d - (a^4*b^8 - 7*a^5*b^6*c + 15*a^6*b^4*c^2 - 1
0*a^7*b^2*c^3 + a^8*c^4)*e)*sqrt(e*x + d)) - 105*sqrt(2)*c^4*e^3*sqrt(((b^8*c - 8*a*b^6*c^2 + 20*a^2*b^4*c^3 -
16*a^3*b^2*c^4 + 2*a^4*c^5)*d - (b^9 - 9*a*b^7*c + 27*a^2*b^5*c^2 - 30*a^3*b^3*c^3 + 9*a^4*b*c^4)*e + (b^2*c^
9 - 4*a*c^10)*sqrt(((b^14*c^2 - 12*a*b^12*c^3 + 56*a^2*b^10*c^4 - 128*a^3*b^8*c^5 + 148*a^4*b^6*c^6 - 80*a^5*b
^4*c^7 + 16*a^6*b^2*c^8)*d^2 - 2*(b^15*c - 13*a*b^13*c^2 + 67*a^2*b^11*c^3 - 174*a^3*b^9*c^4 + 239*a^4*b^7*c^5
- 166*a^5*b^5*c^6 + 50*a^6*b^3*c^7 - 4*a^7*b*c^8)*d*e + (b^16 - 14*a*b^14*c + 79*a^2*b^12*c^2 - 230*a^3*b^10*
c^3 + 367*a^4*b^8*c^4 - 314*a^5*b^6*c^5 + 130*a^6*b^4*c^6 - 20*a^7*b^2*c^7 + a^8*c^8)*e^2)/(b^2*c^18 - 4*a*c^1
9)))/(b^2*c^9 - 4*a*c^10))*log(-sqrt(2)*((b^12*c - 12*a*b^10*c^2 + 54*a^2*b^8*c^3 - 112*a^3*b^6*c^4 + 104*a^4*
b^4*c^5 - 32*a^5*b^2*c^6)*d - (b^13 - 13*a*b^11*c + 65*a^2*b^9*c^2 - 156*a^3*b^7*c^3 + 181*a^4*b^5*c^4 - 86*a^
5*b^3*c^5 + 8*a^6*b*c^6)*e - (b^6*c^9 - 8*a*b^4*c^10 + 18*a^2*b^2*c^11 - 8*a^3*c^12)*sqrt(((b^14*c^2 - 12*a*b^
12*c^3 + 56*a^2*b^10*c^4 - 128*a^3*b^8*c^5 + 148*a^4*b^6*c^6 - 80*a^5*b^4*c^7 + 16*a^6*b^2*c^8)*d^2 - 2*(b^15*
c - 13*a*b^13*c^2 + 67*a^2*b^11*c^3 - 174*a^3*b^9*c^4 + 239*a^4*b^7*c^5 - 166*a^5*b^5*c^6 + 50*a^6*b^3*c^7 - 4
*a^7*b*c^8)*d*e + (b^16 - 14*a*b^14*c + 79*a^2*b^12*c^2 - 230*a^3*b^10*c^3 + 367*a^4*b^8*c^4 - 314*a^5*b^6*c^5
+ 130*a^6*b^4*c^6 - 20*a^7*b^2*c^7 + a^8*c^8)*e^2)/(b^2*c^18 - 4*a*c^19)))*sqrt(((b^8*c - 8*a*b^6*c^2 + 20*a^
2*b^4*c^3 - 16*a^3*b^2*c^4 + 2*a^4*c^5)*d - (b^9 - 9*a*b^7*c + 27*a^2*b^5*c^2 - 30*a^3*b^3*c^3 + 9*a^4*b*c^4)*
e + (b^2*c^9 - 4*a*c^10)*sqrt(((b^14*c^2 - 12*a*b^12*c^3 + 56*a^2*b^10*c^4 - 128*a^3*b^8*c^5 + 148*a^4*b^6*c^6
- 80*a^5*b^4*c^7 + 16*a^6*b^2*c^8)*d^2 - 2*(b^15*c - 13*a*b^13*c^2 + 67*a^2*b^11*c^3 - 174*a^3*b^9*c^4 + 239*
a^4*b^7*c^5 - 166*a^5*b^5*c^6 + 50*a^6*b^3*c^7 - 4*a^7*b*c^8)*d*e + (b^16 - 14*a*b^14*c + 79*a^2*b^12*c^2 - 23
0*a^3*b^10*c^3 + 367*a^4*b^8*c^4 - 314*a^5*b^6*c^5 + 130*a^6*b^4*c^6 - 20*a^7*b^2*c^7 + a^8*c^8)*e^2)/(b^2*c^1
8 - 4*a*c^19)))/(b^2*c^9 - 4*a*c^10)) - 4*((a^4*b^7*c - 6*a^5*b^5*c^2 + 10*a^6*b^3*c^3 - 4*a^7*b*c^4)*d - (a^4
*b^8 - 7*a^5*b^6*c + 15*a^6*b^4*c^2 - 10*a^7*b^2*c^3 + a^8*c^4)*e)*sqrt(e*x + d)) + 105*sqrt(2)*c^4*e^3*sqrt((
(b^8*c - 8*a*b^6*c^2 + 20*a^2*b^4*c^3 - 16*a^3*b^2*c^4 + 2*a^4*c^5)*d - (b^9 - 9*a*b^7*c + 27*a^2*b^5*c^2 - 30
*a^3*b^3*c^3 + 9*a^4*b*c^4)*e - (b^2*c^9 - 4*a*c^10)*sqrt(((b^14*c^2 - 12*a*b^12*c^3 + 56*a^2*b^10*c^4 - 128*a
^3*b^8*c^5 + 148*a^4*b^6*c^6 - 80*a^5*b^4*c^7 + 16*a^6*b^2*c^8)*d^2 - 2*(b^15*c - 13*a*b^13*c^2 + 67*a^2*b^11*
c^3 - 174*a^3*b^9*c^4 + 239*a^4*b^7*c^5 - 166*a^5*b^5*c^6 + 50*a^6*b^3*c^7 - 4*a^7*b*c^8)*d*e + (b^16 - 14*a*b
^14*c + 79*a^2*b^12*c^2 - 230*a^3*b^10*c^3 + 367*a^4*b^8*c^4 - 314*a^5*b^6*c^5 + 130*a^6*b^4*c^6 - 20*a^7*b^2*
c^7 + a^8*c^8)*e^2)/(b^2*c^18 - 4*a*c^19)))/(b^2*c^9 - 4*a*c^10))*log(sqrt(2)*((b^12*c - 12*a*b^10*c^2 + 54*a^
2*b^8*c^3 - 112*a^3*b^6*c^4 + 104*a^4*b^4*c^5 - 32*a^5*b^2*c^6)*d - (b^13 - 13*a*b^11*c + 65*a^2*b^9*c^2 - 156
*a^3*b^7*c^3 + 181*a^4*b^5*c^4 - 86*a^5*b^3*c^5 + 8*a^6*b*c^6)*e + (b^6*c^9 - 8*a*b^4*c^10 + 18*a^2*b^2*c^11 -
8*a^3*c^12)*sqrt(((b^14*c^2 - 12*a*b^12*c^3 + 56*a^2*b^10*c^4 - 128*a^3*b^8*c^5 + 148*a^4*b^6*c^6 - 80*a^5*b^
4*c^7 + 16*a^6*b^2*c^8)*d^2 - 2*(b^15*c - 13*a*b^13*c^2 + 67*a^2*b^11*c^3 - 174*a^3*b^9*c^4 + 239*a^4*b^7*c^5
- 166*a^5*b^5*c^6 + 50*a^6*b^3*c^7 - 4*a^7*b*c^8)*d*e + (b^16 - 14*a*b^14*c + 79*a^2*b^12*c^2 - 230*a^3*b^10*c
^3 + 367*a^4*b^8*c^4 - 314*a^5*b^6*c^5 + 130*a^6*b^4*c^6 - 20*a^7*b^2*c^7 + a^8*c^8)*e^2)/(b^2*c^18 - 4*a*c^19
)))*sqrt(((b^8*c - 8*a*b^6*c^2 + 20*a^2*b^4*c^3 - 16*a^3*b^2*c^4 + 2*a^4*c^5)*d - (b^9 - 9*a*b^7*c + 27*a^2*b^
5*c^2 - 30*a^3*b^3*c^3 + 9*a^4*b*c^4)*e - (b^2*c^9 - 4*a*c^10)*sqrt(((b^14*c^2 - 12*a*b^12*c^3 + 56*a^2*b^10*c
^4 - 128*a^3*b^8*c^5 + 148*a^4*b^6*c^6 - 80*a^5*b^4*c^7 + 16*a^6*b^2*c^8)*d^2 - 2*(b^15*c - 13*a*b^13*c^2 + 67
*a^2*b^11*c^3 - 174*a^3*b^9*c^4 + 239*a^4*b^7*c^5 - 166*a^5*b^5*c^6 + 50*a^6*b^3*c^7 - 4*a^7*b*c^8)*d*e + (b^1
6 - 14*a*b^14*c + 79*a^2*b^12*c^2 - 230*a^3*b^10*c^3 + 367*a^4*b^8*c^4 - 314*a^5*b^6*c^5 + 130*a^6*b^4*c^6 - 2
0*a^7*b^2*c^7 + a^8*c^8)*e^2)/(b^2*c^18 - 4*a*c^19)))/(b^2*c^9 - 4*a*c^10)) - 4*((a^4*b^7*c - 6*a^5*b^5*c^2 +
10*a^6*b^3*c^3 - 4*a^7*b*c^4)*d - (a^4*b^8 - 7*a^5*b^6*c + 15*a^6*b^4*c^2 - 10*a^7*b^2*c^3 + a^8*c^4)*e)*sqrt(
e*x + d)) - 105*sqrt(2)*c^4*e^3*sqrt(((b^8*c - 8*a*b^6*c^2 + 20*a^2*b^4*c^3 - 16*a^3*b^2*c^4 + 2*a^4*c^5)*d -
(b^9 - 9*a*b^7*c + 27*a^2*b^5*c^2 - 30*a^3*b^3*c^3 + 9*a^4*b*c^4)*e - (b^2*c^9 - 4*a*c^10)*sqrt(((b^14*c^2 - 1
2*a*b^12*c^3 + 56*a^2*b^10*c^4 - 128*a^3*b^8*c^5 + 148*a^4*b^6*c^6 - 80*a^5*b^4*c^7 + 16*a^6*b^2*c^8)*d^2 - 2*
(b^15*c - 13*a*b^13*c^2 + 67*a^2*b^11*c^3 - 174*a^3*b^9*c^4 + 239*a^4*b^7*c^5 - 166*a^5*b^5*c^6 + 50*a^6*b^3*c
^7 - 4*a^7*b*c^8)*d*e + (b^16 - 14*a*b^14*c + 79*a^2*b^12*c^2 - 230*a^3*b^10*c^3 + 367*a^4*b^8*c^4 - 314*a^5*b
^6*c^5 + 130*a^6*b^4*c^6 - 20*a^7*b^2*c^7 + a^8*c^8)*e^2)/(b^2*c^18 - 4*a*c^19)))/(b^2*c^9 - 4*a*c^10))*log(-s
qrt(2)*((b^12*c - 12*a*b^10*c^2 + 54*a^2*b^8*c^3 - 112*a^3*b^6*c^4 + 104*a^4*b^4*c^5 - 32*a^5*b^2*c^6)*d - (b^
13 - 13*a*b^11*c + 65*a^2*b^9*c^2 - 156*a^3*b^7*c^3 + 181*a^4*b^5*c^4 - 86*a^5*b^3*c^5 + 8*a^6*b*c^6)*e + (b^6
*c^9 - 8*a*b^4*c^10 + 18*a^2*b^2*c^11 - 8*a^3*c^12)*sqrt(((b^14*c^2 - 12*a*b^12*c^3 + 56*a^2*b^10*c^4 - 128*a^
3*b^8*c^5 + 148*a^4*b^6*c^6 - 80*a^5*b^4*c^7 + 16*a^6*b^2*c^8)*d^2 - 2*(b^15*c - 13*a*b^13*c^2 + 67*a^2*b^11*c
^3 - 174*a^3*b^9*c^4 + 239*a^4*b^7*c^5 - 166*a^5*b^5*c^6 + 50*a^6*b^3*c^7 - 4*a^7*b*c^8)*d*e + (b^16 - 14*a*b^
14*c + 79*a^2*b^12*c^2 - 230*a^3*b^10*c^3 + 367*a^4*b^8*c^4 - 314*a^5*b^6*c^5 + 130*a^6*b^4*c^6 - 20*a^7*b^2*c
^7 + a^8*c^8)*e^2)/(b^2*c^18 - 4*a*c^19)))*sqrt(((b^8*c - 8*a*b^6*c^2 + 20*a^2*b^4*c^3 - 16*a^3*b^2*c^4 + 2*a^
4*c^5)*d - (b^9 - 9*a*b^7*c + 27*a^2*b^5*c^2 - 30*a^3*b^3*c^3 + 9*a^4*b*c^4)*e - (b^2*c^9 - 4*a*c^10)*sqrt(((b
^14*c^2 - 12*a*b^12*c^3 + 56*a^2*b^10*c^4 - 128*a^3*b^8*c^5 + 148*a^4*b^6*c^6 - 80*a^5*b^4*c^7 + 16*a^6*b^2*c^
8)*d^2 - 2*(b^15*c - 13*a*b^13*c^2 + 67*a^2*b^11*c^3 - 174*a^3*b^9*c^4 + 239*a^4*b^7*c^5 - 166*a^5*b^5*c^6 + 5
0*a^6*b^3*c^7 - 4*a^7*b*c^8)*d*e + (b^16 - 14*a*b^14*c + 79*a^2*b^12*c^2 - 230*a^3*b^10*c^3 + 367*a^4*b^8*c^4
- 314*a^5*b^6*c^5 + 130*a^6*b^4*c^6 - 20*a^7*b^2*c^7 + a^8*c^8)*e^2)/(b^2*c^18 - 4*a*c^19)))/(b^2*c^9 - 4*a*c^
10)) - 4*((a^4*b^7*c - 6*a^5*b^5*c^2 + 10*a^6*b^3*c^3 - 4*a^7*b*c^4)*d - (a^4*b^8 - 7*a^5*b^6*c + 15*a^6*b^4*c
^2 - 10*a^7*b^2*c^3 + a^8*c^4)*e)*sqrt(e*x + d)) + 4*(15*c^3*e^3*x^3 + 8*c^3*d^3 + 14*b*c^2*d^2*e + 35*(b^2*c
- a*c^2)*d*e^2 - 105*(b^3 - 2*a*b*c)*e^3 + 3*(c^3*d*e^2 - 7*b*c^2*e^3)*x^2 - (4*c^3*d^2*e + 7*b*c^2*d*e^2 - 35
*(b^2*c - a*c^2)*e^3)*x)*sqrt(e*x + d))/(c^4*e^3)

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giac [B]  time = 0.83, size = 1171, normalized size = 2.39 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)^(1/2)/(c*x^2+b*x+a),x, algorithm="giac")

[Out]

-1/4*(sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*((b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*d*e - (b^6 - 7*a*b
^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*e^2)*c^2 - 2*((b^3*c^3 - 2*a*b*c^4)*sqrt(b^2 - 4*a*c)*d^2 - (b^4*c^2 - 2*a*
b^2*c^3)*sqrt(b^2 - 4*a*c)*d*e + (a*b^3*c^2 - 2*a^2*b*c^3)*sqrt(b^2 - 4*a*c)*e^2)*sqrt(-4*c^2*d + 2*(b*c + sqr
t(b^2 - 4*a*c)*c)*e)*abs(c) + sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*(2*(b^4*c^4 - 4*a*b^2*c^5 + 2*a
^2*c^6)*d^2 - (3*b^5*c^3 - 14*a*b^3*c^4 + 12*a^2*b*c^5)*d*e + (b^6*c^2 - 5*a*b^4*c^3 + 5*a^2*b^2*c^4)*e^2))*ar
ctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*c^8*d*e^24 - b*c^7*e^25 + sqrt(-4*(c^8*d^2*e^24 - b*c^7*d*e^25 + a*c^7
*e^26)*c^8*e^24 + (2*c^8*d*e^24 - b*c^7*e^25)^2))*e^(-24)/c^8))/((sqrt(b^2 - 4*a*c)*c^7*d^2 - sqrt(b^2 - 4*a*c
)*b*c^6*d*e + sqrt(b^2 - 4*a*c)*a*c^6*e^2)*c^2) + 1/4*(sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*((b^5*
c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*d*e - (b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*e^2)*c^2 + 2*((b^3*c^3 - 2
*a*b*c^4)*sqrt(b^2 - 4*a*c)*d^2 - (b^4*c^2 - 2*a*b^2*c^3)*sqrt(b^2 - 4*a*c)*d*e + (a*b^3*c^2 - 2*a^2*b*c^3)*sq
rt(b^2 - 4*a*c)*e^2)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(c) + sqrt(-4*c^2*d + 2*(b*c - sqrt(b
^2 - 4*a*c)*c)*e)*(2*(b^4*c^4 - 4*a*b^2*c^5 + 2*a^2*c^6)*d^2 - (3*b^5*c^3 - 14*a*b^3*c^4 + 12*a^2*b*c^5)*d*e +
(b^6*c^2 - 5*a*b^4*c^3 + 5*a^2*b^2*c^4)*e^2))*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*c^8*d*e^24 - b*c^7*e^
25 - sqrt(-4*(c^8*d^2*e^24 - b*c^7*d*e^25 + a*c^7*e^26)*c^8*e^24 + (2*c^8*d*e^24 - b*c^7*e^25)^2))*e^(-24)/c^8
))/((sqrt(b^2 - 4*a*c)*c^7*d^2 - sqrt(b^2 - 4*a*c)*b*c^6*d*e + sqrt(b^2 - 4*a*c)*a*c^6*e^2)*c^2) + 2/105*(15*(
x*e + d)^(7/2)*c^6*e^18 - 42*(x*e + d)^(5/2)*c^6*d*e^18 + 35*(x*e + d)^(3/2)*c^6*d^2*e^18 - 21*(x*e + d)^(5/2)
*b*c^5*e^19 + 35*(x*e + d)^(3/2)*b*c^5*d*e^19 + 35*(x*e + d)^(3/2)*b^2*c^4*e^20 - 35*(x*e + d)^(3/2)*a*c^5*e^2
0 - 105*sqrt(x*e + d)*b^3*c^3*e^21 + 210*sqrt(x*e + d)*a*b*c^4*e^21)*e^(-21)/c^7

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maple [B]  time = 0.11, size = 2218, normalized size = 4.53 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(e*x+d)^(1/2)/(c*x^2+b*x+a),x)

[Out]

2/7*(e*x+d)^(7/2)/c/e^3-2/5/e^2/c^2*(e*x+d)^(5/2)*b-2/3/e/c^2*(e*x+d)^(3/2)*a+2/3/e/c^3*(e*x+d)^(3/2)*b^2-4/5/
e^3/c*(e*x+d)^(5/2)*d+2/3/e^3/c*(e*x+d)^(3/2)*d^2+4/c^3*a*b*(e*x+d)^(1/2)+4*e/c^2/(-e^2*(4*a*c-b^2))^(1/2)*2^(
1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^
2))^(1/2))*c)^(1/2))*a*b^2*d+4*e/c^2/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c
)^(1/2)*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*a*b^2*d+5*e^2/c^2/(-e
^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((
b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*a^2*b-2*e/c/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*
(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*
a^2*d-e/c^3/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)
^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^4*d-2/c^4*b^3*(e*x+d)^(1/2)+2/c^2*2^(1/2)/((
b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/
2))*c)^(1/2))*a*b*d-2/c^2*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)*2^(1
/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*a*b*d+e^2/c^4/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+2*c
*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c
)^(1/2))*b^5+3*e/c^3*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/(
(-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*a*b^2+e^2/c^4/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((b*e-2*c*d+(-e
^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))
*b^5-3*e/c^3*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d
+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*a*b^2-5*e^2/c^3/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*
c-b^2))^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*a*b^3
+e/c^4*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2
*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^4+1/c^3*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh(c*(e*
x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^3*d-1/c^3*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a
*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^3*d-
e/c^2*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e
^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*a^2-e/c^4*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh(c*(
e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^4+e/c^2*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a
*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*a^2+5*
e^2/c^2/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/
2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*a^2*b-2*e/c/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((b*e
-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))
*c)^(1/2))*a^2*d-5*e^2/c^3/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arc
tan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*a*b^3-e/c^3/(-e^2*(4*a*c-b^2))^(1/
2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*
a*c-b^2))^(1/2))*c)^(1/2))*b^4*d+2/3/e^2/c^2*(e*x+d)^(3/2)*b*d

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e x + d} x^{4}}{c x^{2} + b x + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)^(1/2)/(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

integrate(sqrt(e*x + d)*x^4/(c*x^2 + b*x + a), x)

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mupad [B]  time = 4.86, size = 13879, normalized size = 28.32 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(d + e*x)^(1/2))/(a + b*x + c*x^2),x)

[Out]

(d + e*x)^(3/2)*((4*d^2)/(c*e^3) - (2*(a*e^5 + c*d^2*e^3 - b*d*e^4))/(3*c^2*e^6) + (((8*d)/(c*e^3) + (2*(b*e^4
- 2*c*d*e^3))/(c^2*e^6))*(b*e^4 - 2*c*d*e^3))/(3*c*e^3)) - atan(((((8*(a*b^5*c^5*e^4 + 8*a^3*b*c^7*e^4 - b^6*
c^5*d*e^3 - 6*a^2*b^3*c^6*e^4 + b^5*c^6*d^2*e^2 + 6*a*b^4*c^6*d*e^3 - 6*a*b^3*c^7*d^2*e^2 + 8*a^2*b*c^8*d^2*e^
2 - 8*a^2*b^2*c^7*d*e^3))/c^7 - (8*(d + e*x)^(1/2)*(-(b^11*e + 8*a^5*c^6*d + b^8*e*(-(4*a*c - b^2)^3)^(1/2) -
b^10*c*d - 52*a^2*b^6*c^3*d + 96*a^3*b^4*c^4*d - 66*a^4*b^2*c^5*d + 63*a^2*b^7*c^2*e - 138*a^3*b^5*c^3*e + 129
*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) - 13*a*b^9*c*e + 12*a*b^8*c^2*d - 36*a^5*b*c^5*e - b^7*c*d
*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*
a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c -
b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2)*
(b^3*c^9*e^3 - 2*b^2*c^10*d*e^2 - 4*a*b*c^10*e^3 + 8*a*c^11*d*e^2))/c^7)*(-(b^11*e + 8*a^5*c^6*d + b^8*e*(-(4*
a*c - b^2)^3)^(1/2) - b^10*c*d - 52*a^2*b^6*c^3*d + 96*a^3*b^4*c^4*d - 66*a^4*b^2*c^5*d + 63*a^2*b^7*c^2*e - 1
38*a^3*b^5*c^3*e + 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) - 13*a*b^9*c*e + 12*a*b^8*c^2*d - 36
*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*
c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^
2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 -
8*a*b^2*c^10)))^(1/2) - (8*(d + e*x)^(1/2)*(b^10*e^4 - 2*a^5*c^5*e^4 + 35*a^2*b^6*c^2*e^4 - 50*a^3*b^4*c^3*e^4
+ 25*a^4*b^2*c^4*e^4 + 2*a^4*c^6*d^2*e^2 + b^8*c^2*d^2*e^2 - 10*a*b^8*c*e^4 - 2*b^9*c*d*e^3 + 20*a^2*b^4*c^4*
d^2*e^2 - 16*a^3*b^2*c^5*d^2*e^2 + 18*a*b^7*c^2*d*e^3 - 18*a^4*b*c^5*d*e^3 - 8*a*b^6*c^3*d^2*e^2 - 54*a^2*b^5*
c^3*d*e^3 + 60*a^3*b^3*c^4*d*e^3))/c^7)*(-(b^11*e + 8*a^5*c^6*d + b^8*e*(-(4*a*c - b^2)^3)^(1/2) - b^10*c*d -
52*a^2*b^6*c^3*d + 96*a^3*b^4*c^4*d - 66*a^4*b^2*c^5*d + 63*a^2*b^7*c^2*e - 138*a^3*b^5*c^3*e + 129*a^4*b^3*c^
4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) - 13*a*b^9*c*e + 12*a*b^8*c^2*d - 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c -
b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d
*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1
/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2)*1i - (((8*(
a*b^5*c^5*e^4 + 8*a^3*b*c^7*e^4 - b^6*c^5*d*e^3 - 6*a^2*b^3*c^6*e^4 + b^5*c^6*d^2*e^2 + 6*a*b^4*c^6*d*e^3 - 6*
a*b^3*c^7*d^2*e^2 + 8*a^2*b*c^8*d^2*e^2 - 8*a^2*b^2*c^7*d*e^3))/c^7 + (8*(d + e*x)^(1/2)*(-(b^11*e + 8*a^5*c^6
*d + b^8*e*(-(4*a*c - b^2)^3)^(1/2) - b^10*c*d - 52*a^2*b^6*c^3*d + 96*a^3*b^4*c^4*d - 66*a^4*b^2*c^5*d + 63*a
^2*b^7*c^2*e - 138*a^3*b^5*c^3*e + 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) - 13*a*b^9*c*e + 12*
a*b^8*c^2*d - 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b
^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^
3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c
^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2)*(b^3*c^9*e^3 - 2*b^2*c^10*d*e^2 - 4*a*b*c^10*e^3 + 8*a*c^11*d*e^2))/c^7)
*(-(b^11*e + 8*a^5*c^6*d + b^8*e*(-(4*a*c - b^2)^3)^(1/2) - b^10*c*d - 52*a^2*b^6*c^3*d + 96*a^3*b^4*c^4*d - 6
6*a^4*b^2*c^5*d + 63*a^2*b^7*c^2*e - 138*a^3*b^5*c^3*e + 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2
) - 13*a*b^9*c*e + 12*a*b^8*c^2*d - 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c -
b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*
c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^
3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2) + (8*(d + e*x)^(1/2)*(b^10*e^4 - 2*a^5*c^5*e^4 + 3
5*a^2*b^6*c^2*e^4 - 50*a^3*b^4*c^3*e^4 + 25*a^4*b^2*c^4*e^4 + 2*a^4*c^6*d^2*e^2 + b^8*c^2*d^2*e^2 - 10*a*b^8*c
*e^4 - 2*b^9*c*d*e^3 + 20*a^2*b^4*c^4*d^2*e^2 - 16*a^3*b^2*c^5*d^2*e^2 + 18*a*b^7*c^2*d*e^3 - 18*a^4*b*c^5*d*e
^3 - 8*a*b^6*c^3*d^2*e^2 - 54*a^2*b^5*c^3*d*e^3 + 60*a^3*b^3*c^4*d*e^3))/c^7)*(-(b^11*e + 8*a^5*c^6*d + b^8*e*
(-(4*a*c - b^2)^3)^(1/2) - b^10*c*d - 52*a^2*b^6*c^3*d + 96*a^3*b^4*c^4*d - 66*a^4*b^2*c^5*d + 63*a^2*b^7*c^2*
e - 138*a^3*b^5*c^3*e + 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) - 13*a*b^9*c*e + 12*a*b^8*c^2*d
- 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-
(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) +
15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c
^9 - 8*a*b^2*c^10)))^(1/2)*1i)/((((8*(a*b^5*c^5*e^4 + 8*a^3*b*c^7*e^4 - b^6*c^5*d*e^3 - 6*a^2*b^3*c^6*e^4 + b^
5*c^6*d^2*e^2 + 6*a*b^4*c^6*d*e^3 - 6*a*b^3*c^7*d^2*e^2 + 8*a^2*b*c^8*d^2*e^2 - 8*a^2*b^2*c^7*d*e^3))/c^7 - (8
*(d + e*x)^(1/2)*(-(b^11*e + 8*a^5*c^6*d + b^8*e*(-(4*a*c - b^2)^3)^(1/2) - b^10*c*d - 52*a^2*b^6*c^3*d + 96*a
^3*b^4*c^4*d - 66*a^4*b^2*c^5*d + 63*a^2*b^7*c^2*e - 138*a^3*b^5*c^3*e + 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*
c - b^2)^3)^(1/2) - 13*a*b^9*c*e + 12*a*b^8*c^2*d - 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^
6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/
2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*
(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2)*(b^3*c^9*e^3 - 2*b^2*c^10*d*e^2 -
4*a*b*c^10*e^3 + 8*a*c^11*d*e^2))/c^7)*(-(b^11*e + 8*a^5*c^6*d + b^8*e*(-(4*a*c - b^2)^3)^(1/2) - b^10*c*d - 5
2*a^2*b^6*c^3*d + 96*a^3*b^4*c^4*d - 66*a^4*b^2*c^5*d + 63*a^2*b^7*c^2*e - 138*a^3*b^5*c^3*e + 129*a^4*b^3*c^4
*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) - 13*a*b^9*c*e + 12*a*b^8*c^2*d - 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c -
b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*
(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/
2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2) - (8*(d + e*
x)^(1/2)*(b^10*e^4 - 2*a^5*c^5*e^4 + 35*a^2*b^6*c^2*e^4 - 50*a^3*b^4*c^3*e^4 + 25*a^4*b^2*c^4*e^4 + 2*a^4*c^6*
d^2*e^2 + b^8*c^2*d^2*e^2 - 10*a*b^8*c*e^4 - 2*b^9*c*d*e^3 + 20*a^2*b^4*c^4*d^2*e^2 - 16*a^3*b^2*c^5*d^2*e^2 +
18*a*b^7*c^2*d*e^3 - 18*a^4*b*c^5*d*e^3 - 8*a*b^6*c^3*d^2*e^2 - 54*a^2*b^5*c^3*d*e^3 + 60*a^3*b^3*c^4*d*e^3))
/c^7)*(-(b^11*e + 8*a^5*c^6*d + b^8*e*(-(4*a*c - b^2)^3)^(1/2) - b^10*c*d - 52*a^2*b^6*c^3*d + 96*a^3*b^4*c^4*
d - 66*a^4*b^2*c^5*d + 63*a^2*b^7*c^2*e - 138*a^3*b^5*c^3*e + 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)
^(1/2) - 13*a*b^9*c*e + 12*a*b^8*c^2*d - 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*
a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2
*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c -
b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2) - (16*(a^5*b^4*e^5 + a^7*c^2*e^5 - 3*a^6*b^2*
c*e^5 - a^4*b^5*d*e^4 + a^6*c^3*d^2*e^3 - a^4*b^3*c^2*d^3*e^2 - 5*a^5*b^2*c^2*d^2*e^3 + 2*a^5*b^3*c*d*e^4 + a^
6*b*c^2*d*e^4 + 2*a^4*b^4*c*d^2*e^3 + 2*a^5*b*c^3*d^3*e^2))/c^7 + (((8*(a*b^5*c^5*e^4 + 8*a^3*b*c^7*e^4 - b^6*
c^5*d*e^3 - 6*a^2*b^3*c^6*e^4 + b^5*c^6*d^2*e^2 + 6*a*b^4*c^6*d*e^3 - 6*a*b^3*c^7*d^2*e^2 + 8*a^2*b*c^8*d^2*e^
2 - 8*a^2*b^2*c^7*d*e^3))/c^7 + (8*(d + e*x)^(1/2)*(-(b^11*e + 8*a^5*c^6*d + b^8*e*(-(4*a*c - b^2)^3)^(1/2) -
b^10*c*d - 52*a^2*b^6*c^3*d + 96*a^3*b^4*c^4*d - 66*a^4*b^2*c^5*d + 63*a^2*b^7*c^2*e - 138*a^3*b^5*c^3*e + 129
*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) - 13*a*b^9*c*e + 12*a*b^8*c^2*d - 36*a^5*b*c^5*e - b^7*c*d
*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*
a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c -
b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2)*
(b^3*c^9*e^3 - 2*b^2*c^10*d*e^2 - 4*a*b*c^10*e^3 + 8*a*c^11*d*e^2))/c^7)*(-(b^11*e + 8*a^5*c^6*d + b^8*e*(-(4*
a*c - b^2)^3)^(1/2) - b^10*c*d - 52*a^2*b^6*c^3*d + 96*a^3*b^4*c^4*d - 66*a^4*b^2*c^5*d + 63*a^2*b^7*c^2*e - 1
38*a^3*b^5*c^3*e + 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) - 13*a*b^9*c*e + 12*a*b^8*c^2*d - 36
*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*
c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^
2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 -
8*a*b^2*c^10)))^(1/2) + (8*(d + e*x)^(1/2)*(b^10*e^4 - 2*a^5*c^5*e^4 + 35*a^2*b^6*c^2*e^4 - 50*a^3*b^4*c^3*e^4
+ 25*a^4*b^2*c^4*e^4 + 2*a^4*c^6*d^2*e^2 + b^8*c^2*d^2*e^2 - 10*a*b^8*c*e^4 - 2*b^9*c*d*e^3 + 20*a^2*b^4*c^4*
d^2*e^2 - 16*a^3*b^2*c^5*d^2*e^2 + 18*a*b^7*c^2*d*e^3 - 18*a^4*b*c^5*d*e^3 - 8*a*b^6*c^3*d^2*e^2 - 54*a^2*b^5*
c^3*d*e^3 + 60*a^3*b^3*c^4*d*e^3))/c^7)*(-(b^11*e + 8*a^5*c^6*d + b^8*e*(-(4*a*c - b^2)^3)^(1/2) - b^10*c*d -
52*a^2*b^6*c^3*d + 96*a^3*b^4*c^4*d - 66*a^4*b^2*c^5*d + 63*a^2*b^7*c^2*e - 138*a^3*b^5*c^3*e + 129*a^4*b^3*c^
4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) - 13*a*b^9*c*e + 12*a*b^8*c^2*d - 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c -
b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d
*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1
/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2)))*(-(b^11*e
+ 8*a^5*c^6*d + b^8*e*(-(4*a*c - b^2)^3)^(1/2) - b^10*c*d - 52*a^2*b^6*c^3*d + 96*a^3*b^4*c^4*d - 66*a^4*b^2*
c^5*d + 63*a^2*b^7*c^2*e - 138*a^3*b^5*c^3*e + 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) - 13*a*b
^9*c*e + 12*a*b^8*c^2*d - 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(
1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4
*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/
(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2)*2i - atan(((((8*(a*b^5*c^5*e^4 + 8*a^3*b*c^7*e^4 - b^6*c^5*d
*e^3 - 6*a^2*b^3*c^6*e^4 + b^5*c^6*d^2*e^2 + 6*a*b^4*c^6*d*e^3 - 6*a*b^3*c^7*d^2*e^2 + 8*a^2*b*c^8*d^2*e^2 - 8
*a^2*b^2*c^7*d*e^3))/c^7 - (8*(d + e*x)^(1/2)*((b^8*e*(-(4*a*c - b^2)^3)^(1/2) - 8*a^5*c^6*d - b^11*e + b^10*c
*d + 52*a^2*b^6*c^3*d - 96*a^3*b^4*c^4*d + 66*a^4*b^2*c^5*d - 63*a^2*b^7*c^2*e + 138*a^3*b^5*c^3*e - 129*a^4*b
^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) + 13*a*b^9*c*e - 12*a*b^8*c^2*d + 36*a^5*b*c^5*e - b^7*c*d*(-(4*
a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*
c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^
3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2)*(b^3*c
^9*e^3 - 2*b^2*c^10*d*e^2 - 4*a*b*c^10*e^3 + 8*a*c^11*d*e^2))/c^7)*((b^8*e*(-(4*a*c - b^2)^3)^(1/2) - 8*a^5*c^
6*d - b^11*e + b^10*c*d + 52*a^2*b^6*c^3*d - 96*a^3*b^4*c^4*d + 66*a^4*b^2*c^5*d - 63*a^2*b^7*c^2*e + 138*a^3*
b^5*c^3*e - 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) + 13*a*b^9*c*e - 12*a*b^8*c^2*d + 36*a^5*b*
c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2
)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c
^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2
*c^10)))^(1/2) - (8*(d + e*x)^(1/2)*(b^10*e^4 - 2*a^5*c^5*e^4 + 35*a^2*b^6*c^2*e^4 - 50*a^3*b^4*c^3*e^4 + 25*a
^4*b^2*c^4*e^4 + 2*a^4*c^6*d^2*e^2 + b^8*c^2*d^2*e^2 - 10*a*b^8*c*e^4 - 2*b^9*c*d*e^3 + 20*a^2*b^4*c^4*d^2*e^2
- 16*a^3*b^2*c^5*d^2*e^2 + 18*a*b^7*c^2*d*e^3 - 18*a^4*b*c^5*d*e^3 - 8*a*b^6*c^3*d^2*e^2 - 54*a^2*b^5*c^3*d*e
^3 + 60*a^3*b^3*c^4*d*e^3))/c^7)*((b^8*e*(-(4*a*c - b^2)^3)^(1/2) - 8*a^5*c^6*d - b^11*e + b^10*c*d + 52*a^2*b
^6*c^3*d - 96*a^3*b^4*c^4*d + 66*a^4*b^2*c^5*d - 63*a^2*b^7*c^2*e + 138*a^3*b^5*c^3*e - 129*a^4*b^3*c^4*e + a^
4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) + 13*a*b^9*c*e - 12*a*b^8*c^2*d + 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)
^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*
c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10
*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2)*1i - (((8*(a*b^5*c^
5*e^4 + 8*a^3*b*c^7*e^4 - b^6*c^5*d*e^3 - 6*a^2*b^3*c^6*e^4 + b^5*c^6*d^2*e^2 + 6*a*b^4*c^6*d*e^3 - 6*a*b^3*c^
7*d^2*e^2 + 8*a^2*b*c^8*d^2*e^2 - 8*a^2*b^2*c^7*d*e^3))/c^7 + (8*(d + e*x)^(1/2)*((b^8*e*(-(4*a*c - b^2)^3)^(1
/2) - 8*a^5*c^6*d - b^11*e + b^10*c*d + 52*a^2*b^6*c^3*d - 96*a^3*b^4*c^4*d + 66*a^4*b^2*c^5*d - 63*a^2*b^7*c^
2*e + 138*a^3*b^5*c^3*e - 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) + 13*a*b^9*c*e - 12*a*b^8*c^2
*d + 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*
(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2)
+ 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4
*c^9 - 8*a*b^2*c^10)))^(1/2)*(b^3*c^9*e^3 - 2*b^2*c^10*d*e^2 - 4*a*b*c^10*e^3 + 8*a*c^11*d*e^2))/c^7)*((b^8*e*
(-(4*a*c - b^2)^3)^(1/2) - 8*a^5*c^6*d - b^11*e + b^10*c*d + 52*a^2*b^6*c^3*d - 96*a^3*b^4*c^4*d + 66*a^4*b^2*
c^5*d - 63*a^2*b^7*c^2*e + 138*a^3*b^5*c^3*e - 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) + 13*a*b
^9*c*e - 12*a*b^8*c^2*d + 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(
1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4
*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/
(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2) + (8*(d + e*x)^(1/2)*(b^10*e^4 - 2*a^5*c^5*e^4 + 35*a^2*b^6*
c^2*e^4 - 50*a^3*b^4*c^3*e^4 + 25*a^4*b^2*c^4*e^4 + 2*a^4*c^6*d^2*e^2 + b^8*c^2*d^2*e^2 - 10*a*b^8*c*e^4 - 2*b
^9*c*d*e^3 + 20*a^2*b^4*c^4*d^2*e^2 - 16*a^3*b^2*c^5*d^2*e^2 + 18*a*b^7*c^2*d*e^3 - 18*a^4*b*c^5*d*e^3 - 8*a*b
^6*c^3*d^2*e^2 - 54*a^2*b^5*c^3*d*e^3 + 60*a^3*b^3*c^4*d*e^3))/c^7)*((b^8*e*(-(4*a*c - b^2)^3)^(1/2) - 8*a^5*c
^6*d - b^11*e + b^10*c*d + 52*a^2*b^6*c^3*d - 96*a^3*b^4*c^4*d + 66*a^4*b^2*c^5*d - 63*a^2*b^7*c^2*e + 138*a^3
*b^5*c^3*e - 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) + 13*a*b^9*c*e - 12*a*b^8*c^2*d + 36*a^5*b
*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^
2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*
c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^
2*c^10)))^(1/2)*1i)/((((8*(a*b^5*c^5*e^4 + 8*a^3*b*c^7*e^4 - b^6*c^5*d*e^3 - 6*a^2*b^3*c^6*e^4 + b^5*c^6*d^2*e
^2 + 6*a*b^4*c^6*d*e^3 - 6*a*b^3*c^7*d^2*e^2 + 8*a^2*b*c^8*d^2*e^2 - 8*a^2*b^2*c^7*d*e^3))/c^7 - (8*(d + e*x)^
(1/2)*((b^8*e*(-(4*a*c - b^2)^3)^(1/2) - 8*a^5*c^6*d - b^11*e + b^10*c*d + 52*a^2*b^6*c^3*d - 96*a^3*b^4*c^4*d
+ 66*a^4*b^2*c^5*d - 63*a^2*b^7*c^2*e + 138*a^3*b^5*c^3*e - 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^
(1/2) + 13*a*b^9*c*e - 12*a*b^8*c^2*d + 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a
*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*
b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b
^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2)*(b^3*c^9*e^3 - 2*b^2*c^10*d*e^2 - 4*a*b*c^10*e
^3 + 8*a*c^11*d*e^2))/c^7)*((b^8*e*(-(4*a*c - b^2)^3)^(1/2) - 8*a^5*c^6*d - b^11*e + b^10*c*d + 52*a^2*b^6*c^3
*d - 96*a^3*b^4*c^4*d + 66*a^4*b^2*c^5*d - 63*a^2*b^7*c^2*e + 138*a^3*b^5*c^3*e - 129*a^4*b^3*c^4*e + a^4*c^4*
e*(-(4*a*c - b^2)^3)^(1/2) + 13*a*b^9*c*e - 12*a*b^8*c^2*d + 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2)
- 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^
2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b
^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2) - (8*(d + e*x)^(1/2)*(b^1
0*e^4 - 2*a^5*c^5*e^4 + 35*a^2*b^6*c^2*e^4 - 50*a^3*b^4*c^3*e^4 + 25*a^4*b^2*c^4*e^4 + 2*a^4*c^6*d^2*e^2 + b^8
*c^2*d^2*e^2 - 10*a*b^8*c*e^4 - 2*b^9*c*d*e^3 + 20*a^2*b^4*c^4*d^2*e^2 - 16*a^3*b^2*c^5*d^2*e^2 + 18*a*b^7*c^2
*d*e^3 - 18*a^4*b*c^5*d*e^3 - 8*a*b^6*c^3*d^2*e^2 - 54*a^2*b^5*c^3*d*e^3 + 60*a^3*b^3*c^4*d*e^3))/c^7)*((b^8*e
*(-(4*a*c - b^2)^3)^(1/2) - 8*a^5*c^6*d - b^11*e + b^10*c*d + 52*a^2*b^6*c^3*d - 96*a^3*b^4*c^4*d + 66*a^4*b^2
*c^5*d - 63*a^2*b^7*c^2*e + 138*a^3*b^5*c^3*e - 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) + 13*a*
b^9*c*e - 12*a*b^8*c^2*d + 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^
(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(
4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))
/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2) - (16*(a^5*b^4*e^5 + a^7*c^2*e^5 - 3*a^6*b^2*c*e^5 - a^4*b^
5*d*e^4 + a^6*c^3*d^2*e^3 - a^4*b^3*c^2*d^3*e^2 - 5*a^5*b^2*c^2*d^2*e^3 + 2*a^5*b^3*c*d*e^4 + a^6*b*c^2*d*e^4
+ 2*a^4*b^4*c*d^2*e^3 + 2*a^5*b*c^3*d^3*e^2))/c^7 + (((8*(a*b^5*c^5*e^4 + 8*a^3*b*c^7*e^4 - b^6*c^5*d*e^3 - 6*
a^2*b^3*c^6*e^4 + b^5*c^6*d^2*e^2 + 6*a*b^4*c^6*d*e^3 - 6*a*b^3*c^7*d^2*e^2 + 8*a^2*b*c^8*d^2*e^2 - 8*a^2*b^2*
c^7*d*e^3))/c^7 + (8*(d + e*x)^(1/2)*((b^8*e*(-(4*a*c - b^2)^3)^(1/2) - 8*a^5*c^6*d - b^11*e + b^10*c*d + 52*a
^2*b^6*c^3*d - 96*a^3*b^4*c^4*d + 66*a^4*b^2*c^5*d - 63*a^2*b^7*c^2*e + 138*a^3*b^5*c^3*e - 129*a^4*b^3*c^4*e
+ a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) + 13*a*b^9*c*e - 12*a*b^8*c^2*d + 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2
)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(
4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2)
- 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2)*(b^3*c^9*e^3 -
2*b^2*c^10*d*e^2 - 4*a*b*c^10*e^3 + 8*a*c^11*d*e^2))/c^7)*((b^8*e*(-(4*a*c - b^2)^3)^(1/2) - 8*a^5*c^6*d - b^1
1*e + b^10*c*d + 52*a^2*b^6*c^3*d - 96*a^3*b^4*c^4*d + 66*a^4*b^2*c^5*d - 63*a^2*b^7*c^2*e + 138*a^3*b^5*c^3*e
- 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) + 13*a*b^9*c*e - 12*a*b^8*c^2*d + 36*a^5*b*c^5*e - b
^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2
) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4
*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^
(1/2) + (8*(d + e*x)^(1/2)*(b^10*e^4 - 2*a^5*c^5*e^4 + 35*a^2*b^6*c^2*e^4 - 50*a^3*b^4*c^3*e^4 + 25*a^4*b^2*c^
4*e^4 + 2*a^4*c^6*d^2*e^2 + b^8*c^2*d^2*e^2 - 10*a*b^8*c*e^4 - 2*b^9*c*d*e^3 + 20*a^2*b^4*c^4*d^2*e^2 - 16*a^3
*b^2*c^5*d^2*e^2 + 18*a*b^7*c^2*d*e^3 - 18*a^4*b*c^5*d*e^3 - 8*a*b^6*c^3*d^2*e^2 - 54*a^2*b^5*c^3*d*e^3 + 60*a
^3*b^3*c^4*d*e^3))/c^7)*((b^8*e*(-(4*a*c - b^2)^3)^(1/2) - 8*a^5*c^6*d - b^11*e + b^10*c*d + 52*a^2*b^6*c^3*d
- 96*a^3*b^4*c^4*d + 66*a^4*b^2*c^5*d - 63*a^2*b^7*c^2*e + 138*a^3*b^5*c^3*e - 129*a^4*b^3*c^4*e + a^4*c^4*e*(
-(4*a*c - b^2)^3)^(1/2) + 13*a*b^9*c*e - 12*a*b^8*c^2*d + 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) -
7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^
3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*
c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2)))*((b^8*e*(-(4*a*c - b^2)^3)
^(1/2) - 8*a^5*c^6*d - b^11*e + b^10*c*d + 52*a^2*b^6*c^3*d - 96*a^3*b^4*c^4*d + 66*a^4*b^2*c^5*d - 63*a^2*b^7
*c^2*e + 138*a^3*b^5*c^3*e - 129*a^4*b^3*c^4*e + a^4*c^4*e*(-(4*a*c - b^2)^3)^(1/2) + 13*a*b^9*c*e - 12*a*b^8*
c^2*d + 36*a^5*b*c^5*e - b^7*c*d*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c^2
*d*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^4*d*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^3*d*(-(4*a*c - b^2)^3)^(1/
2) + 15*a^2*b^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 +
b^4*c^9 - 8*a*b^2*c^10)))^(1/2)*2i - ((8*d)/(5*c*e^3) + (2*(b*e^4 - 2*c*d*e^3))/(5*c^2*e^6))*(d + e*x)^(5/2) -
(d + e*x)^(1/2)*((8*d^3)/(c*e^3) - (((8*d)/(c*e^3) + (2*(b*e^4 - 2*c*d*e^3))/(c^2*e^6))*(a*e^5 + c*d^2*e^3 -
b*d*e^4))/(c*e^3) + ((b*e^4 - 2*c*d*e^3)*((12*d^2)/(c*e^3) - (2*(a*e^5 + c*d^2*e^3 - b*d*e^4))/(c^2*e^6) + (((
8*d)/(c*e^3) + (2*(b*e^4 - 2*c*d*e^3))/(c^2*e^6))*(b*e^4 - 2*c*d*e^3))/(c*e^3)))/(c*e^3)) + (2*(d + e*x)^(7/2)
)/(7*c*e^3)

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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(x**4*(e*x+d)**(1/2)/(c*x**2+b*x+a),x)

[Out]

Timed out

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