3.631 \(\int \frac{(d+e x)^3}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx\)

Optimal. Leaf size=150 \[ \frac{2 a+b (d+e x)^2}{4 e \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}-\frac{3 b \left (b+2 c (d+e x)^2\right )}{4 e \left (b^2-4 a c\right )^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac{3 b c \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{e \left (b^2-4 a c\right )^{5/2}} \]

[Out]

(2*a + b*(d + e*x)^2)/(4*(b^2 - 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2)
- (3*b*(b + 2*c*(d + e*x)^2))/(4*(b^2 - 4*a*c)^2*e*(a + b*(d + e*x)^2 + c*(d + e
*x)^4)) + (3*b*c*ArcTanh[(b + 2*c*(d + e*x)^2)/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c
)^(5/2)*e)

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Rubi [A]  time = 0.372762, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2 a+b (d+e x)^2}{4 e \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}-\frac{3 b \left (b+2 c (d+e x)^2\right )}{4 e \left (b^2-4 a c\right )^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac{3 b c \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{e \left (b^2-4 a c\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^3/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3,x]

[Out]

(2*a + b*(d + e*x)^2)/(4*(b^2 - 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2)
- (3*b*(b + 2*c*(d + e*x)^2))/(4*(b^2 - 4*a*c)^2*e*(a + b*(d + e*x)^2 + c*(d + e
*x)^4)) + (3*b*c*ArcTanh[(b + 2*c*(d + e*x)^2)/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c
)^(5/2)*e)

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Rubi in Sympy [A]  time = 36.178, size = 134, normalized size = 0.89 \[ \frac{3 b c \operatorname{atanh}{\left (\frac{b + 2 c \left (d + e x\right )^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{e \left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} - \frac{3 b \left (b + 2 c \left (d + e x\right )^{2}\right )}{4 e \left (- 4 a c + b^{2}\right )^{2} \left (a + b \left (d + e x\right )^{2} + c \left (d + e x\right )^{4}\right )} + \frac{2 a + b \left (d + e x\right )^{2}}{4 e \left (- 4 a c + b^{2}\right ) \left (a + b \left (d + e x\right )^{2} + c \left (d + e x\right )^{4}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3/(a+b*(e*x+d)**2+c*(e*x+d)**4)**3,x)

[Out]

3*b*c*atanh((b + 2*c*(d + e*x)**2)/sqrt(-4*a*c + b**2))/(e*(-4*a*c + b**2)**(5/2
)) - 3*b*(b + 2*c*(d + e*x)**2)/(4*e*(-4*a*c + b**2)**2*(a + b*(d + e*x)**2 + c*
(d + e*x)**4)) + (2*a + b*(d + e*x)**2)/(4*e*(-4*a*c + b**2)*(a + b*(d + e*x)**2
+ c*(d + e*x)**4)**2)

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Mathematica [A]  time = 0.348234, size = 146, normalized size = 0.97 \[ \frac{\frac{\left (b^2-4 a c\right ) \left (2 a+b (d+e x)^2\right )}{\left (a+(d+e x)^2 \left (b+c (d+e x)^2\right )\right )^2}-\frac{12 b c \tan ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}-\frac{3 b \left (b+2 c (d+e x)^2\right )}{a+b (d+e x)^2+c (d+e x)^4}}{4 e \left (b^2-4 a c\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^3/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3,x]

[Out]

((-3*b*(b + 2*c*(d + e*x)^2))/(a + b*(d + e*x)^2 + c*(d + e*x)^4) + ((b^2 - 4*a*
c)*(2*a + b*(d + e*x)^2))/(a + (d + e*x)^2*(b + c*(d + e*x)^2))^2 - (12*b*c*ArcT
an[(b + 2*c*(d + e*x)^2)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(4*(b^2 - 4*a*
c)^2*e)

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Maple [C]  time = 0.064, size = 544, normalized size = 3.6 \[{\frac{1}{ \left ( c{e}^{4}{x}^{4}+4\,cd{e}^{3}{x}^{3}+6\,c{d}^{2}{e}^{2}{x}^{2}+4\,c{d}^{3}ex+b{e}^{2}{x}^{2}+c{d}^{4}+2\,bdex+b{d}^{2}+a \right ) ^{2}} \left ( -{\frac{3\,{c}^{2}{e}^{5}b{x}^{6}}{32\,{a}^{2}{c}^{2}-16\,a{b}^{2}c+2\,{b}^{4}}}-9\,{\frac{{c}^{2}d{e}^{4}b{x}^{5}}{16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4}}}-{\frac{9\,b{e}^{3}c \left ( 10\,c{d}^{2}+b \right ){x}^{4}}{64\,{a}^{2}{c}^{2}-32\,a{b}^{2}c+4\,{b}^{4}}}-3\,{\frac{bd{e}^{2}c \left ( 10\,c{d}^{2}+3\,b \right ){x}^{3}}{16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4}}}-{\frac{be \left ( 45\,{c}^{2}{d}^{4}+27\,bc{d}^{2}+5\,ac+{b}^{2} \right ){x}^{2}}{32\,{a}^{2}{c}^{2}-16\,a{b}^{2}c+2\,{b}^{4}}}-{\frac{bd \left ( 9\,{c}^{2}{d}^{4}+9\,bc{d}^{2}+5\,ac+{b}^{2} \right ) x}{16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4}}}-{\frac{6\,b{c}^{2}{d}^{6}+9\,{b}^{2}c{d}^{4}+10\,abc{d}^{2}+2\,{b}^{3}{d}^{2}+8\,{a}^{2}c+a{b}^{2}}{4\,e \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) }} \right ) }+{\frac{3\,bc}{2\,e}\sum _{{\it \_R}={\it RootOf} \left ( c{e}^{4}{{\it \_Z}}^{4}+4\,cd{e}^{3}{{\it \_Z}}^{3}+ \left ( 6\,c{d}^{2}{e}^{2}+b{e}^{2} \right ){{\it \_Z}}^{2}+ \left ( 4\,c{d}^{3}e+2\,bde \right ){\it \_Z}+c{d}^{4}+b{d}^{2}+a \right ) }{\frac{ \left ( -e{\it \_R}-d \right ) \ln \left ( x-{\it \_R} \right ) }{ \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) \left ( 2\,c{e}^{3}{{\it \_R}}^{3}+6\,cd{e}^{2}{{\it \_R}}^{2}+6\,{d}^{2}ec{\it \_R}+2\,c{d}^{3}+be{\it \_R}+bd \right ) }}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3/(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x)

[Out]

(-3/2*c^2*e^5*b/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6-9*e^4*b*c^2*d/(16*a^2*c^2-8*a*b^2
*c+b^4)*x^5-9/4*b*c*e^3*(10*c*d^2+b)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4-3*c*d*e^2*b*
(10*c*d^2+3*b)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3-1/2*b*e*(45*c^2*d^4+27*b*c*d^2+5*a
*c+b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2-d*b*(9*c^2*d^4+9*b*c*d^2+5*a*c+b^2)/(16*a
^2*c^2-8*a*b^2*c+b^4)*x-1/4/e*(6*b*c^2*d^6+9*b^2*c*d^4+10*a*b*c*d^2+2*b^3*d^2+8*
a^2*c+a*b^2)/(16*a^2*c^2-8*a*b^2*c+b^4))/(c*e^4*x^4+4*c*d*e^3*x^3+6*c*d^2*e^2*x^
2+4*c*d^3*e*x+b*e^2*x^2+c*d^4+2*b*d*e*x+b*d^2+a)^2+3/2*b*c/e*sum((-_R*e-d)/(16*a
^2*c^2-8*a*b^2*c+b^4)/(2*_R^3*c*e^3+6*_R^2*c*d*e^2+6*_R*c*d^2*e+2*c*d^3+_R*b*e+b
*d)*ln(x-_R),_R=RootOf(c*e^4*_Z^4+4*c*d*e^3*_Z^3+(6*c*d^2*e^2+b*e^2)*_Z^2+(4*c*d
^3*e+2*b*d*e)*_Z+c*d^4+b*d^2+a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/((e*x + d)^4*c + (e*x + d)^2*b + a)^3,x, algorithm="maxima")

[Out]

-3*b*c*integrate((e*x + d)/(c*e^4*x^4 + 4*c*d*e^3*x^3 + c*d^4 + (6*c*d^2 + b)*e^
2*x^2 + b*d^2 + 2*(2*c*d^3 + b*d)*e*x + a), x)/(b^4 - 8*a*b^2*c + 16*a^2*c^2) -
1/4*(6*b*c^2*e^6*x^6 + 36*b*c^2*d*e^5*x^5 + 6*b*c^2*d^6 + 9*(10*b*c^2*d^2 + b^2*
c)*e^4*x^4 + 9*b^2*c*d^4 + 12*(10*b*c^2*d^3 + 3*b^2*c*d)*e^3*x^3 + 2*(45*b*c^2*d
^4 + 27*b^2*c*d^2 + b^3 + 5*a*b*c)*e^2*x^2 + a*b^2 + 8*a^2*c + 2*(b^3 + 5*a*b*c)
*d^2 + 4*(9*b*c^2*d^5 + 9*b^2*c*d^3 + (b^3 + 5*a*b*c)*d)*e*x)/((b^4*c^2 - 8*a*b^
2*c^3 + 16*a^2*c^4)*e^9*x^8 + 8*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d*e^8*x^7 +
2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3 + 14*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)
*d^2)*e^7*x^6 + 4*(14*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^3 + 3*(b^5*c - 8*a*
b^3*c^2 + 16*a^2*b*c^3)*d)*e^6*x^5 + (b^6 - 6*a*b^4*c + 32*a^3*c^3 + 70*(b^4*c^2
- 8*a*b^2*c^3 + 16*a^2*c^4)*d^4 + 30*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^2)*
e^5*x^4 + 4*(14*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^5 + 10*(b^5*c - 8*a*b^3*c
^2 + 16*a^2*b*c^3)*d^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*d)*e^4*x^3 + 2*(14*(b^4*
c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^6 + a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2 + 15*(b
^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^4 + 3*(b^6 - 6*a*b^4*c + 32*a^3*c^3)*d^2)*e
^3*x^2 + 4*(2*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^7 + 3*(b^5*c - 8*a*b^3*c^2
+ 16*a^2*b*c^3)*d^5 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*d^3 + (a*b^5 - 8*a^2*b^3*c
+ 16*a^3*b*c^2)*d)*e^2*x + ((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^8 + 2*(b^5*c
- 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^6 + a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^6 -
6*a*b^4*c + 32*a^3*c^3)*d^4 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d^2)*e)

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Fricas [A]  time = 0.44027, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/((e*x + d)^4*c + (e*x + d)^2*b + a)^3,x, algorithm="fricas")

[Out]

[1/4*(6*(b*c^3*e^8*x^8 + 8*b*c^3*d*e^7*x^7 + 2*(14*b*c^3*d^2 + b^2*c^2)*e^6*x^6
+ b*c^3*d^8 + 4*(14*b*c^3*d^3 + 3*b^2*c^2*d)*e^5*x^5 + 2*b^2*c^2*d^6 + (70*b*c^3
*d^4 + 30*b^2*c^2*d^2 + b^3*c + 2*a*b*c^2)*e^4*x^4 + 4*(14*b*c^3*d^5 + 10*b^2*c^
2*d^3 + (b^3*c + 2*a*b*c^2)*d)*e^3*x^3 + 2*a*b^2*c*d^2 + (b^3*c + 2*a*b*c^2)*d^4
+ 2*(14*b*c^3*d^6 + 15*b^2*c^2*d^4 + a*b^2*c + 3*(b^3*c + 2*a*b*c^2)*d^2)*e^2*x
^2 + a^2*b*c + 4*(2*b*c^3*d^7 + 3*b^2*c^2*d^5 + a*b^2*c*d + (b^3*c + 2*a*b*c^2)*
d^3)*e*x)*log((2*(b^2*c - 4*a*c^2)*e^2*x^2 + 4*(b^2*c - 4*a*c^2)*d*e*x + b^3 - 4
*a*b*c + 2*(b^2*c - 4*a*c^2)*d^2 + (2*c^2*e^4*x^4 + 8*c^2*d*e^3*x^3 + 2*c^2*d^4
+ 2*(6*c^2*d^2 + b*c)*e^2*x^2 + 2*b*c*d^2 + 4*(2*c^2*d^3 + b*c*d)*e*x + b^2 - 2*
a*c)*sqrt(b^2 - 4*a*c))/(c*e^4*x^4 + 4*c*d*e^3*x^3 + c*d^4 + (6*c*d^2 + b)*e^2*x
^2 + b*d^2 + 2*(2*c*d^3 + b*d)*e*x + a)) - (6*b*c^2*e^6*x^6 + 36*b*c^2*d*e^5*x^5
+ 6*b*c^2*d^6 + 9*(10*b*c^2*d^2 + b^2*c)*e^4*x^4 + 9*b^2*c*d^4 + 12*(10*b*c^2*d
^3 + 3*b^2*c*d)*e^3*x^3 + 2*(45*b*c^2*d^4 + 27*b^2*c*d^2 + b^3 + 5*a*b*c)*e^2*x^
2 + a*b^2 + 8*a^2*c + 2*(b^3 + 5*a*b*c)*d^2 + 4*(9*b*c^2*d^5 + 9*b^2*c*d^3 + (b^
3 + 5*a*b*c)*d)*e*x)*sqrt(b^2 - 4*a*c))/(((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*e
^9*x^8 + 8*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d*e^8*x^7 + 2*(b^5*c - 8*a*b^3*c
^2 + 16*a^2*b*c^3 + 14*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^2)*e^7*x^6 + 4*(14
*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^3 + 3*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^
3)*d)*e^6*x^5 + (b^6 - 6*a*b^4*c + 32*a^3*c^3 + 70*(b^4*c^2 - 8*a*b^2*c^3 + 16*a
^2*c^4)*d^4 + 30*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^2)*e^5*x^4 + 4*(14*(b^4*
c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^5 + 10*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^
3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*d)*e^4*x^3 + 2*(14*(b^4*c^2 - 8*a*b^2*c^3 + 1
6*a^2*c^4)*d^6 + a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2 + 15*(b^5*c - 8*a*b^3*c^2 +
16*a^2*b*c^3)*d^4 + 3*(b^6 - 6*a*b^4*c + 32*a^3*c^3)*d^2)*e^3*x^2 + 4*(2*(b^4*c^
2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^7 + 3*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^5 +
(b^6 - 6*a*b^4*c + 32*a^3*c^3)*d^3 + (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d)*e^
2*x + ((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^8 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^
2*b*c^3)*d^6 + a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^6 - 6*a*b^4*c + 32*a^3*c^
3)*d^4 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d^2)*e)*sqrt(b^2 - 4*a*c)), -1/4
*(12*(b*c^3*e^8*x^8 + 8*b*c^3*d*e^7*x^7 + 2*(14*b*c^3*d^2 + b^2*c^2)*e^6*x^6 + b
*c^3*d^8 + 4*(14*b*c^3*d^3 + 3*b^2*c^2*d)*e^5*x^5 + 2*b^2*c^2*d^6 + (70*b*c^3*d^
4 + 30*b^2*c^2*d^2 + b^3*c + 2*a*b*c^2)*e^4*x^4 + 4*(14*b*c^3*d^5 + 10*b^2*c^2*d
^3 + (b^3*c + 2*a*b*c^2)*d)*e^3*x^3 + 2*a*b^2*c*d^2 + (b^3*c + 2*a*b*c^2)*d^4 +
2*(14*b*c^3*d^6 + 15*b^2*c^2*d^4 + a*b^2*c + 3*(b^3*c + 2*a*b*c^2)*d^2)*e^2*x^2
+ a^2*b*c + 4*(2*b*c^3*d^7 + 3*b^2*c^2*d^5 + a*b^2*c*d + (b^3*c + 2*a*b*c^2)*d^3
)*e*x)*arctan(-(2*c*e^2*x^2 + 4*c*d*e*x + 2*c*d^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 -
4*a*c)) + (6*b*c^2*e^6*x^6 + 36*b*c^2*d*e^5*x^5 + 6*b*c^2*d^6 + 9*(10*b*c^2*d^2
+ b^2*c)*e^4*x^4 + 9*b^2*c*d^4 + 12*(10*b*c^2*d^3 + 3*b^2*c*d)*e^3*x^3 + 2*(45*
b*c^2*d^4 + 27*b^2*c*d^2 + b^3 + 5*a*b*c)*e^2*x^2 + a*b^2 + 8*a^2*c + 2*(b^3 + 5
*a*b*c)*d^2 + 4*(9*b*c^2*d^5 + 9*b^2*c*d^3 + (b^3 + 5*a*b*c)*d)*e*x)*sqrt(-b^2 +
4*a*c))/(((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*e^9*x^8 + 8*(b^4*c^2 - 8*a*b^2*c
^3 + 16*a^2*c^4)*d*e^8*x^7 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3 + 14*(b^4*c^2
- 8*a*b^2*c^3 + 16*a^2*c^4)*d^2)*e^7*x^6 + 4*(14*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^
2*c^4)*d^3 + 3*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d)*e^6*x^5 + (b^6 - 6*a*b^4*
c + 32*a^3*c^3 + 70*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^4 + 30*(b^5*c - 8*a*b
^3*c^2 + 16*a^2*b*c^3)*d^2)*e^5*x^4 + 4*(14*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)
*d^5 + 10*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^3 + (b^6 - 6*a*b^4*c + 32*a^3*c
^3)*d)*e^4*x^3 + 2*(14*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^6 + a*b^5 - 8*a^2*
b^3*c + 16*a^3*b*c^2 + 15*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^4 + 3*(b^6 - 6*
a*b^4*c + 32*a^3*c^3)*d^2)*e^3*x^2 + 4*(2*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d
^7 + 3*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^5 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)
*d^3 + (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d)*e^2*x + ((b^4*c^2 - 8*a*b^2*c^3 +
16*a^2*c^4)*d^8 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^6 + a^2*b^4 - 8*a^3*
b^2*c + 16*a^4*c^2 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*d^4 + 2*(a*b^5 - 8*a^2*b^3*c
+ 16*a^3*b*c^2)*d^2)*e)*sqrt(-b^2 + 4*a*c))]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3/(a+b*(e*x+d)**2+c*(e*x+d)**4)**3,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{3}}{{\left ({\left (e x + d\right )}^{4} c +{\left (e x + d\right )}^{2} b + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/((e*x + d)^4*c + (e*x + d)^2*b + a)^3,x, algorithm="giac")

[Out]

integrate((e*x + d)^3/((e*x + d)^4*c + (e*x + d)^2*b + a)^3, x)