3.2371 \(\int \frac{(d+e x)^3 (f+g x)}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=195 \[ -\frac{6 \left (a e^2-b d e+c d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) (2 a e g-b d g-b e f+2 c d f)}{\left (b^2-4 a c\right )^{5/2}}-\frac{(d+e x)^3 (-2 a g+x (2 c f-b g)+b f)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{3 (d+e x) (-2 a e+x (2 c d-b e)+b d) (2 a e g-b d g-b e f+2 c d f)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )} \]

[Out]

-((d + e*x)^3*(b*f - 2*a*g + (2*c*f - b*g)*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2
)^2) + (3*(2*c*d*f - b*e*f - b*d*g + 2*a*e*g)*(d + e*x)*(b*d - 2*a*e + (2*c*d -
b*e)*x))/(2*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) - (6*(c*d^2 - b*d*e + a*e^2)*(2*c
*d*f - b*e*f - b*d*g + 2*a*e*g)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4
*a*c)^(5/2)

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Rubi [A]  time = 0.395821, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ -\frac{6 \left (a e^2-b d e+c d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) (2 a e g-b d g-b e f+2 c d f)}{\left (b^2-4 a c\right )^{5/2}}-\frac{(d+e x)^3 (-2 a g+x (2 c f-b g)+b f)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{3 (d+e x) (-2 a e+x (2 c d-b e)+b d) (2 a e g-b d g-b e f+2 c d f)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-((d + e*x)^3*(b*f - 2*a*g + (2*c*f - b*g)*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2
)^2) + (3*(2*c*d*f - b*e*f - b*d*g + 2*a*e*g)*(d + e*x)*(b*d - 2*a*e + (2*c*d -
b*e)*x))/(2*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) - (6*(c*d^2 - b*d*e + a*e^2)*(2*c
*d*f - b*e*f - b*d*g + 2*a*e*g)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4
*a*c)^(5/2)

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Rubi in Sympy [A]  time = 48.1822, size = 184, normalized size = 0.94 \[ \frac{\left (d + e x\right )^{3} \left (2 a g - b f + x \left (b g - 2 c f\right )\right )}{2 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{2}} + \frac{3 \left (d + e x\right ) \left (2 a e - b d + x \left (b e - 2 c d\right )\right ) \left (- a e g + \frac{b \left (d g + e f\right )}{2} - c d f\right )}{\left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )} + \frac{6 \left (a e^{2} - b d e + c d^{2}\right ) \left (- 2 a e g + b \left (d g + e f\right ) - 2 c d f\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d + e*x)**3*(2*a*g - b*f + x*(b*g - 2*c*f))/(2*(-4*a*c + b**2)*(a + b*x + c*x**
2)**2) + 3*(d + e*x)*(2*a*e - b*d + x*(b*e - 2*c*d))*(-a*e*g + b*(d*g + e*f)/2 -
c*d*f)/((-4*a*c + b**2)**2*(a + b*x + c*x**2)) + 6*(a*e**2 - b*d*e + c*d**2)*(-
2*a*e*g + b*(d*g + e*f) - 2*c*d*f)*atanh((b + 2*c*x)/sqrt(-4*a*c + b**2))/(-4*a*
c + b**2)**(5/2)

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Mathematica [B]  time = 2.79065, size = 550, normalized size = 2.82 \[ \frac{1}{2} \left (\frac{b c \left (-3 a^2 e^3 g+3 a c e \left (d^2 g+d e (f+3 g x)+e^2 f x\right )+c^2 d^2 (d (f-g x)-3 e f x)\right )+2 c^2 \left (a^2 e^2 (3 d g+e (f+g x))-a c d \left (d^2 g+3 d e (f+g x)+3 e^2 f x\right )+c^2 d^3 f x\right )+b^3 e^2 (a e g-c x (3 d g+e f))-b^2 c e (a e (3 d g+e f+4 e g x)-3 c d x (d g+e f))+b^4 e^3 g x}{c^3 \left (4 a c-b^2\right ) (a+x (b+c x))^2}+\frac{2 b c^2 \left (11 a^2 e^3 g+3 a c e \left (d^2 g+d e (f-3 g x)-e^2 f x\right )+3 c^2 d^2 (d (f-g x)-3 e f x)\right )-4 c^3 \left (a^2 e^2 (12 d g+4 e f+5 e g x)-3 a c d e x (d g+e f)-3 c^2 d^3 f x\right )+b^3 c e \left (3 c d (d g+e f)-8 a e^2 g\right )+b^2 c^2 \left (a e^2 (15 d g+5 e f+16 e g x)-3 c d \left (d^2 g+3 d e f-2 d e g x-2 e^2 f x\right )\right )+b^5 e^3 g-b^4 c e^2 (3 d g+e (f+2 g x))}{c^3 \left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac{12 \left (e (a e-b d)+c d^2\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right ) (2 a e g-b (d g+e f)+2 c d f)}{\left (4 a c-b^2\right )^{5/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((b^5*e^3*g + b^3*c*e*(-8*a*e^2*g + 3*c*d*(e*f + d*g)) - b^4*c*e^2*(3*d*g + e*(f
+ 2*g*x)) - 4*c^3*(-3*c^2*d^3*f*x - 3*a*c*d*e*(e*f + d*g)*x + a^2*e^2*(4*e*f +
12*d*g + 5*e*g*x)) + b^2*c^2*(a*e^2*(5*e*f + 15*d*g + 16*e*g*x) - 3*c*d*(3*d*e*f
+ d^2*g - 2*e^2*f*x - 2*d*e*g*x)) + 2*b*c^2*(11*a^2*e^3*g + 3*a*c*e*(d^2*g - e^
2*f*x + d*e*(f - 3*g*x)) + 3*c^2*d^2*(-3*e*f*x + d*(f - g*x))))/(c^3*(b^2 - 4*a*
c)^2*(a + x*(b + c*x))) + (b^4*e^3*g*x + b^3*e^2*(a*e*g - c*(e*f + 3*d*g)*x) - b
^2*c*e*(-3*c*d*(e*f + d*g)*x + a*e*(e*f + 3*d*g + 4*e*g*x)) + 2*c^2*(c^2*d^3*f*x
+ a^2*e^2*(3*d*g + e*(f + g*x)) - a*c*d*(d^2*g + 3*e^2*f*x + 3*d*e*(f + g*x)))
+ b*c*(-3*a^2*e^3*g + c^2*d^2*(-3*e*f*x + d*(f - g*x)) + 3*a*c*e*(d^2*g + e^2*f*
x + d*e*(f + 3*g*x))))/(c^3*(-b^2 + 4*a*c)*(a + x*(b + c*x))^2) + (12*(c*d^2 + e
*(-(b*d) + a*e))*(2*c*d*f + 2*a*e*g - b*(e*f + d*g))*ArcTan[(b + 2*c*x)/Sqrt[-b^
2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2))/2

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Maple [B]  time = 0.019, size = 1458, normalized size = 7.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-(10*a^2*c^2*e^3*g-8*a*b^2*c*e^3*g+9*a*b*c^2*d*e^2*g+3*a*b*c^2*e^3*f-6*a*c^3*d^
2*e*g-6*a*c^3*d*e^2*f+b^4*e^3*g-3*b^2*c^2*d^2*e*g-3*b^2*c^2*d*e^2*f+3*b*c^3*d^3*
g+9*b*c^3*d^2*e*f-6*c^4*d^3*f)/c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3+1/2*(2*a^2*b*c^2
*e^3*g-48*a^2*c^3*d*e^2*g-16*a^2*c^3*e^3*f+8*a*b^3*c*e^3*g-3*a*b^2*c^2*d*e^2*g-a
*b^2*c^2*e^3*f+18*a*b*c^3*d^2*e*g+18*a*b*c^3*d*e^2*f-b^5*e^3*g-3*b^4*c*d*e^2*g-b
^4*c*e^3*f+9*b^3*c^2*d^2*e*g+9*b^3*c^2*d*e^2*f-9*b^2*c^3*d^3*g-27*b^2*c^3*d^2*e*
f+18*b*c^4*d^3*f)/(16*a^2*c^2-8*a*b^2*c+b^4)/c^2*x^2-(6*a^3*c^2*e^3*g-10*a^2*b^2
*c*e^3*g+15*a^2*b*c^2*d*e^2*g+5*a^2*b*c^2*e^3*f+6*a^2*c^3*d^2*e*g+6*a^2*c^3*d*e^
2*f+a*b^4*e^3*g+3*a*b^3*c*d*e^2*g+a*b^3*c*e^3*f-15*a*b^2*c^2*d^2*e*g-15*a*b^2*c^
2*d*e^2*f+5*a*b*c^3*d^3*g+15*a*b*c^3*d^2*e*f-10*a*c^4*d^3*f+b^3*c^2*d^3*g+3*b^3*
c^2*d^2*e*f-2*b^2*c^3*d^3*f)/(16*a^2*c^2-8*a*b^2*c+b^4)/c^2*x+1/2/c^2*(10*a^3*b*
c*e^3*g-24*a^3*c^2*d*e^2*g-8*a^3*c^2*e^3*f-a^2*b^3*e^3*g-3*a^2*b^2*c*d*e^2*g-a^2
*b^2*c*e^3*f+18*a^2*b*c^2*d^2*e*g+18*a^2*b*c^2*d*e^2*f-8*a^2*c^3*d^3*g-24*a^2*c^
3*d^2*e*f-a*b^2*c^2*d^3*g-3*a*b^2*c^2*d^2*e*f+10*a*b*c^3*d^3*f-b^3*c^2*d^3*f)/(1
6*a^2*c^2-8*a*b^2*c+b^4))/(c*x^2+b*x+a)^2+12/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b
^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*e^3*g-18/(16*a^2*c^2-8*a*b^2*c
+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b*d*e^2*g-6/(16*a^
2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b*e
^3*f+12/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2
)^(1/2))*a*c*d^2*e*g+12/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c
*x+b)/(4*a*c-b^2)^(1/2))*a*c*d*e^2*f+6/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1
/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^2*d^2*e*g+6/(16*a^2*c^2-8*a*b^2*c+b^4)
/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^2*d*e^2*f-6/(16*a^2*c^2
-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*c*d^3*g-
18/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/
2))*b*c*d^2*e*f+12/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)
/(4*a*c-b^2)^(1/2))*c^2*d^3*f

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.360985, 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*(g*x + f)/(c*x^2 + b*x + a)^3,x, algorithm="fricas")

[Out]

[1/2*(6*(((2*c^6*d^3 - 3*b*c^5*d^2*e - a*b*c^4*e^3 + (b^2*c^4 + 2*a*c^5)*d*e^2)*
f - (b*c^5*d^3 + 3*a*b*c^4*d*e^2 - 2*a^2*c^4*e^3 - (b^2*c^4 + 2*a*c^5)*d^2*e)*g)
*x^4 + 2*((2*b*c^5*d^3 - 3*b^2*c^4*d^2*e - a*b^2*c^3*e^3 + (b^3*c^3 + 2*a*b*c^4)
*d*e^2)*f - (b^2*c^4*d^3 + 3*a*b^2*c^3*d*e^2 - 2*a^2*b*c^3*e^3 - (b^3*c^3 + 2*a*
b*c^4)*d^2*e)*g)*x^3 + ((2*(b^2*c^4 + 2*a*c^5)*d^3 - 3*(b^3*c^3 + 2*a*b*c^4)*d^2
*e + (b^4*c^2 + 4*a*b^2*c^3 + 4*a^2*c^4)*d*e^2 - (a*b^3*c^2 + 2*a^2*b*c^3)*e^3)*
f - ((b^3*c^3 + 2*a*b*c^4)*d^3 - (b^4*c^2 + 4*a*b^2*c^3 + 4*a^2*c^4)*d^2*e + 3*(
a*b^3*c^2 + 2*a^2*b*c^3)*d*e^2 - 2*(a^2*b^2*c^2 + 2*a^3*c^3)*e^3)*g)*x^2 + (2*a^
2*c^4*d^3 - 3*a^2*b*c^3*d^2*e - a^3*b*c^2*e^3 + (a^2*b^2*c^2 + 2*a^3*c^3)*d*e^2)
*f - (a^2*b*c^3*d^3 + 3*a^3*b*c^2*d*e^2 - 2*a^4*c^2*e^3 - (a^2*b^2*c^2 + 2*a^3*c
^3)*d^2*e)*g + 2*((2*a*b*c^4*d^3 - 3*a*b^2*c^3*d^2*e - a^2*b^2*c^2*e^3 + (a*b^3*
c^2 + 2*a^2*b*c^3)*d*e^2)*f - (a*b^2*c^3*d^3 + 3*a^2*b^2*c^2*d*e^2 - 2*a^3*b*c^2
*e^3 - (a*b^3*c^2 + 2*a^2*b*c^3)*d^2*e)*g)*x)*log(-(b^3 - 4*a*b*c + 2*(b^2*c - 4
*a*c^2)*x - (2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c*x^2 + b*x
+ a)) + (2*(3*(2*c^5*d^3 - 3*b*c^4*d^2*e - a*b*c^3*e^3 + (b^2*c^3 + 2*a*c^4)*d*e
^2)*f - (3*b*c^4*d^3 + 9*a*b*c^3*d*e^2 - 3*(b^2*c^3 + 2*a*c^4)*d^2*e + (b^4*c -
8*a*b^2*c^2 + 10*a^2*c^3)*e^3)*g)*x^3 + ((18*b*c^4*d^3 - 27*b^2*c^3*d^2*e + 9*(b
^3*c^2 + 2*a*b*c^3)*d*e^2 - (b^4*c + a*b^2*c^2 + 16*a^2*c^3)*e^3)*f - (9*b^2*c^3
*d^3 - 9*(b^3*c^2 + 2*a*b*c^3)*d^2*e + 3*(b^4*c + a*b^2*c^2 + 16*a^2*c^3)*d*e^2
+ (b^5 - 8*a*b^3*c - 2*a^2*b*c^2)*e^3)*g)*x^2 + (18*a^2*b*c^2*d*e^2 - (b^3*c^2 -
10*a*b*c^3)*d^3 - 3*(a*b^2*c^2 + 8*a^2*c^3)*d^2*e - (a^2*b^2*c + 8*a^3*c^2)*e^3
)*f + (18*a^2*b*c^2*d^2*e - (a*b^2*c^2 + 8*a^2*c^3)*d^3 - 3*(a^2*b^2*c + 8*a^3*c
^2)*d*e^2 - (a^2*b^3 - 10*a^3*b*c)*e^3)*g + 2*((2*(b^2*c^3 + 5*a*c^4)*d^3 - 3*(b
^3*c^2 + 5*a*b*c^3)*d^2*e + 3*(5*a*b^2*c^2 - 2*a^2*c^3)*d*e^2 - (a*b^3*c + 5*a^2
*b*c^2)*e^3)*f - ((b^3*c^2 + 5*a*b*c^3)*d^3 - 3*(5*a*b^2*c^2 - 2*a^2*c^3)*d^2*e
+ 3*(a*b^3*c + 5*a^2*b*c^2)*d*e^2 + (a*b^4 - 10*a^2*b^2*c + 6*a^3*c^2)*e^3)*g)*x
)*sqrt(b^2 - 4*a*c))/((a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4 + (b^4*c^4 - 8*a
*b^2*c^5 + 16*a^2*c^6)*x^4 + 2*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*x^3 + (b^6
*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*x^2 + 2*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c
^4)*x)*sqrt(b^2 - 4*a*c)), 1/2*(12*(((2*c^6*d^3 - 3*b*c^5*d^2*e - a*b*c^4*e^3 +
(b^2*c^4 + 2*a*c^5)*d*e^2)*f - (b*c^5*d^3 + 3*a*b*c^4*d*e^2 - 2*a^2*c^4*e^3 - (b
^2*c^4 + 2*a*c^5)*d^2*e)*g)*x^4 + 2*((2*b*c^5*d^3 - 3*b^2*c^4*d^2*e - a*b^2*c^3*
e^3 + (b^3*c^3 + 2*a*b*c^4)*d*e^2)*f - (b^2*c^4*d^3 + 3*a*b^2*c^3*d*e^2 - 2*a^2*
b*c^3*e^3 - (b^3*c^3 + 2*a*b*c^4)*d^2*e)*g)*x^3 + ((2*(b^2*c^4 + 2*a*c^5)*d^3 -
3*(b^3*c^3 + 2*a*b*c^4)*d^2*e + (b^4*c^2 + 4*a*b^2*c^3 + 4*a^2*c^4)*d*e^2 - (a*b
^3*c^2 + 2*a^2*b*c^3)*e^3)*f - ((b^3*c^3 + 2*a*b*c^4)*d^3 - (b^4*c^2 + 4*a*b^2*c
^3 + 4*a^2*c^4)*d^2*e + 3*(a*b^3*c^2 + 2*a^2*b*c^3)*d*e^2 - 2*(a^2*b^2*c^2 + 2*a
^3*c^3)*e^3)*g)*x^2 + (2*a^2*c^4*d^3 - 3*a^2*b*c^3*d^2*e - a^3*b*c^2*e^3 + (a^2*
b^2*c^2 + 2*a^3*c^3)*d*e^2)*f - (a^2*b*c^3*d^3 + 3*a^3*b*c^2*d*e^2 - 2*a^4*c^2*e
^3 - (a^2*b^2*c^2 + 2*a^3*c^3)*d^2*e)*g + 2*((2*a*b*c^4*d^3 - 3*a*b^2*c^3*d^2*e
- a^2*b^2*c^2*e^3 + (a*b^3*c^2 + 2*a^2*b*c^3)*d*e^2)*f - (a*b^2*c^3*d^3 + 3*a^2*
b^2*c^2*d*e^2 - 2*a^3*b*c^2*e^3 - (a*b^3*c^2 + 2*a^2*b*c^3)*d^2*e)*g)*x)*arctan(
-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (2*(3*(2*c^5*d^3 - 3*b*c^4*d^2*
e - a*b*c^3*e^3 + (b^2*c^3 + 2*a*c^4)*d*e^2)*f - (3*b*c^4*d^3 + 9*a*b*c^3*d*e^2
- 3*(b^2*c^3 + 2*a*c^4)*d^2*e + (b^4*c - 8*a*b^2*c^2 + 10*a^2*c^3)*e^3)*g)*x^3 +
((18*b*c^4*d^3 - 27*b^2*c^3*d^2*e + 9*(b^3*c^2 + 2*a*b*c^3)*d*e^2 - (b^4*c + a*
b^2*c^2 + 16*a^2*c^3)*e^3)*f - (9*b^2*c^3*d^3 - 9*(b^3*c^2 + 2*a*b*c^3)*d^2*e +
3*(b^4*c + a*b^2*c^2 + 16*a^2*c^3)*d*e^2 + (b^5 - 8*a*b^3*c - 2*a^2*b*c^2)*e^3)*
g)*x^2 + (18*a^2*b*c^2*d*e^2 - (b^3*c^2 - 10*a*b*c^3)*d^3 - 3*(a*b^2*c^2 + 8*a^2
*c^3)*d^2*e - (a^2*b^2*c + 8*a^3*c^2)*e^3)*f + (18*a^2*b*c^2*d^2*e - (a*b^2*c^2
+ 8*a^2*c^3)*d^3 - 3*(a^2*b^2*c + 8*a^3*c^2)*d*e^2 - (a^2*b^3 - 10*a^3*b*c)*e^3)
*g + 2*((2*(b^2*c^3 + 5*a*c^4)*d^3 - 3*(b^3*c^2 + 5*a*b*c^3)*d^2*e + 3*(5*a*b^2*
c^2 - 2*a^2*c^3)*d*e^2 - (a*b^3*c + 5*a^2*b*c^2)*e^3)*f - ((b^3*c^2 + 5*a*b*c^3)
*d^3 - 3*(5*a*b^2*c^2 - 2*a^2*c^3)*d^2*e + 3*(a*b^3*c + 5*a^2*b*c^2)*d*e^2 + (a*
b^4 - 10*a^2*b^2*c + 6*a^3*c^2)*e^3)*g)*x)*sqrt(-b^2 + 4*a*c))/((a^2*b^4*c^2 - 8
*a^3*b^2*c^3 + 16*a^4*c^4 + (b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*x^4 + 2*(b^5*c^
3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*x^3 + (b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*x^2 +
2*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*x)*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*(g*x+f)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.368328, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done