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

Optimal. Leaf size=159 \[ -\frac{3 b f^3 \left (b+2 c (d+e x)^2\right )}{4 e \left (b^2-4 a c\right )^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac{f^3 \left (2 a+b (d+e x)^2\right )}{4 e \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac{3 b c f^3 \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{e \left (b^2-4 a c\right )^{5/2}} \]

[Out]

(f^3*(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*f^3*(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*f^3*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 = 0.402498, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{3 b f^3 \left (b+2 c (d+e x)^2\right )}{4 e \left (b^2-4 a c\right )^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac{f^3 \left (2 a+b (d+e x)^2\right )}{4 e \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac{3 b c f^3 \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{e \left (b^2-4 a c\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(f^3*(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*f^3*(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*f^3*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 = 39.0261, size = 144, normalized size = 0.91 \[ \frac{3 b c f^{3} \operatorname{atanh}{\left (\frac{b + 2 c \left (d + e x\right )^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{e \left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} - \frac{3 b f^{3} \left (b + 2 c \left (d + e x\right )^{2}\right )}{4 e \left (- 4 a c + b^{2}\right )^{2} \left (a + b \left (d + e x\right )^{2} + c \left (d + e x\right )^{4}\right )} + \frac{f^{3} \left (2 a + b \left (d + e x\right )^{2}\right )}{4 e \left (- 4 a c + b^{2}\right ) \left (a + b \left (d + e x\right )^{2} + c \left (d + e x\right )^{4}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*b*c*f**3*atanh((b + 2*c*(d + e*x)**2)/sqrt(-4*a*c + b**2))/(e*(-4*a*c + b**2)*
*(5/2)) - 3*b*f**3*(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)) + f**3*(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 = 0.355179, size = 149, normalized size = 0.94 \[ \frac{f^3 \left (\frac{\left (b^2-4 a c\right ) \left (2 a+b (d+e x)^2\right )}{\left (a+(d+e x)^2 \left (b+c (d+e x)^2\right )\right )^2}-\frac{12 b c \tan ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}-\frac{3 b \left (b+2 c (d+e x)^2\right )}{a+b (d+e x)^2+c (d+e x)^4}\right )}{4 e \left (b^2-4 a c\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(f^3*((-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
*ArcTan[(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 = 0.016, size = 2181, normalized size = 13.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/2*f^3/(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*c^2*e^5*b/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6-9*f^3/(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*e^4*
b*c^2*d/(16*a^2*c^2-8*a*b^2*c+b^4)*x^5-45/2*f^3/(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*b*c^2*e^3/(16*a^2*c^2-
8*a*b^2*c+b^4)*x^4*d^2-9/4*f^3/(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*b^2*c*e^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^
4-30*f^3/(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*c^2*d^3*e^2*b/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3-9*f^3/(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*
c*d*e^2*b^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3-45/2*f^3/(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*b*e/(16*a^2*c^2-8*
a*b^2*c+b^4)*x^2*c^2*d^4-27/2*f^3/(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*b^2*e/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2
*c*d^2-5/2*f^3/(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*b*e/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*a*c-1/2*f^3/(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*b^3*e/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2-9*f^3/(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*d^5*b/(16*a^2*c^2-8*a*b^2
*c+b^4)*x*c^2-9*f^3/(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*d^3*b^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x*c-5*f^3/(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*d*b/(16*a^2*c^2-8*a*b^2*c+b^4)*x*a*c-f^3/(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*d*b^3/(16*a^2*c^2-8*a*b^
2*c+b^4)*x-3/2*f^3/(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/e/(16*a^2*c^2-8*a*b^2*c+b^4)*b*c^2*d^6-9/4*f^3/(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/e/(16*a^2*c^2-8*a*b^2*c+b^4)*b^2*c*d^4-5/2*f^3/(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/e/(16*a^2*c^2-8*a
*b^2*c+b^4)*a*b*c*d^2-1/2*f^3/(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/e/(16*a^2*c^2-8*a*b^2*c+b^4)*b^3*d^2-2*f
^3/(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/e/(16*a^2*c^2-8*a*b^2*c+b^4)*a^2*c-1/4*f^3/(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/e/(16*a^2*c^2-
8*a*b^2*c+b^4)*a*b^2+3/2*f^3*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*f*x + d*f)^3/((e*x + d)^4*c + (e*x + d)^2*b + a)^3,x, algorithm="maxima")

[Out]

-3*b*c*f^3*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*f^3*x^6 + 36*b*c^2*d*e^5*f^3*x^5 + 9*(10*b*c^2*d^2 + b^2*c)
*e^4*f^3*x^4 + 12*(10*b*c^2*d^3 + 3*b^2*c*d)*e^3*f^3*x^3 + 2*(45*b*c^2*d^4 + 27*
b^2*c*d^2 + b^3 + 5*a*b*c)*e^2*f^3*x^2 + 4*(9*b*c^2*d^5 + 9*b^2*c*d^3 + (b^3 + 5
*a*b*c)*d)*e*f^3*x + (6*b*c^2*d^6 + 9*b^2*c*d^4 + a*b^2 + 8*a^2*c + 2*(b^3 + 5*a
*b*c)*d^2)*f^3)/((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 = 0.43656, 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*f*x + d*f)^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*f^3*x^8 + 8*b*c^3*d*e^7*f^3*x^7 + 2*(14*b*c^3*d^2 + b^2*c^2)*
e^6*f^3*x^6 + 4*(14*b*c^3*d^3 + 3*b^2*c^2*d)*e^5*f^3*x^5 + (70*b*c^3*d^4 + 30*b^
2*c^2*d^2 + b^3*c + 2*a*b*c^2)*e^4*f^3*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*f^3*x^3 + 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*f^3*x^2 + 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*f^3*x + (b*c^3*d^8 + 2*b^2*c^2*d^6 + 2*a*b^2*
c*d^2 + (b^3*c + 2*a*b*c^2)*d^4 + a^2*b*c)*f^3)*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*f^3*x^6 + 36*b*c^2*d*e^5*f^3*x^5 + 9*(10*b*c^2*d^2 + b^2*c)*e^4*f^
3*x^4 + 12*(10*b*c^2*d^3 + 3*b^2*c*d)*e^3*f^3*x^3 + 2*(45*b*c^2*d^4 + 27*b^2*c*d
^2 + b^3 + 5*a*b*c)*e^2*f^3*x^2 + 4*(9*b*c^2*d^5 + 9*b^2*c*d^3 + (b^3 + 5*a*b*c)
*d)*e*f^3*x + (6*b*c^2*d^6 + 9*b^2*c*d^4 + a*b^2 + 8*a^2*c + 2*(b^3 + 5*a*b*c)*d
^2)*f^3)*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)), -1/4*(12*(b*c^3*
e^8*f^3*x^8 + 8*b*c^3*d*e^7*f^3*x^7 + 2*(14*b*c^3*d^2 + b^2*c^2)*e^6*f^3*x^6 + 4
*(14*b*c^3*d^3 + 3*b^2*c^2*d)*e^5*f^3*x^5 + (70*b*c^3*d^4 + 30*b^2*c^2*d^2 + b^3
*c + 2*a*b*c^2)*e^4*f^3*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*f^3*x^3 + 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*f^3*x^2 + 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*f^3*x + (b*c^3*d^8 + 2*b^2*c^2*d^6 + 2*a*b^2*c*d^2 + (b^3*c
+ 2*a*b*c^2)*d^4 + a^2*b*c)*f^3)*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*f^3*x^6 + 36*b*c^2*d*e^5*f^3*x
^5 + 9*(10*b*c^2*d^2 + b^2*c)*e^4*f^3*x^4 + 12*(10*b*c^2*d^3 + 3*b^2*c*d)*e^3*f^
3*x^3 + 2*(45*b*c^2*d^4 + 27*b^2*c*d^2 + b^3 + 5*a*b*c)*e^2*f^3*x^2 + 4*(9*b*c^2
*d^5 + 9*b^2*c*d^3 + (b^3 + 5*a*b*c)*d)*e*f^3*x + (6*b*c^2*d^6 + 9*b^2*c*d^4 + a
*b^2 + 8*a^2*c + 2*(b^3 + 5*a*b*c)*d^2)*f^3)*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 + 1
5*(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*f*x+d*f)**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. \[ \int \frac{{\left (e f x + d f\right )}^{3}}{{\left ({\left (e x + d\right )}^{4} c +{\left (e x + d\right )}^{2} b + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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