3.2191 \(\int \frac{(d+e x)^4}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=169 \[ \frac{3 (d+e x) \left (a e^2-b d e+c d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{12 \left (a e^2-b d e+c d^2\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac{(d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

[Out]

-((d + e*x)^3*(b*d - 2*a*e + (2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2
)^2) + (3*(c*d^2 - b*d*e + a*e^2)*(d + e*x)*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b
^2 - 4*a*c)^2*(a + b*x + c*x^2)) - (12*(c*d^2 - b*d*e + a*e^2)^2*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 = 0.287504, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{3 (d+e x) \left (a e^2-b d e+c d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{12 \left (a e^2-b d e+c d^2\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac{(d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-((d + e*x)^3*(b*d - 2*a*e + (2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2
)^2) + (3*(c*d^2 - b*d*e + a*e^2)*(d + e*x)*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b
^2 - 4*a*c)^2*(a + b*x + c*x^2)) - (12*(c*d^2 - b*d*e + a*e^2)^2*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 = 38.0727, size = 158, normalized size = 0.93 \[ \frac{\left (d + e x\right )^{3} \left (2 a e - b d + x \left (b e - 2 c d\right )\right )}{2 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{2}} - \frac{3 \left (d + e x\right ) \left (2 a e - b d + x \left (b e - 2 c d\right )\right ) \left (a e^{2} - b d e + c d^{2}\right )}{\left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )} - \frac{12 \left (a e^{2} - b d e + c d^{2}\right )^{2} \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d + e*x)**3*(2*a*e - b*d + x*(b*e - 2*c*d))/(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**2 - b*d*e + c*d**2)/(
(-4*a*c + b**2)**2*(a + b*x + c*x**2)) - 12*(a*e**2 - b*d*e + c*d**2)**2*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 = 1.18094, size = 413, normalized size = 2.44 \[ \frac{1}{2} \left (\frac{b c \left (-3 a^2 e^4+6 a c d e^2 (d+2 e x)+c^2 d^3 (d-4 e x)\right )+2 c^2 \left (a^2 e^3 (4 d+e x)-2 a c d^2 e (2 d+3 e x)+c^2 d^4 x\right )+b^3 e^3 (a e-4 c d x)+2 b^2 c e^2 \left (3 c d^2 x-2 a e (d+e x)\right )+b^4 e^4 x}{c^3 \left (4 a c-b^2\right ) (a+x (b+c x))^2}+\frac{2 b c^2 \left (11 a^2 e^4+6 a c d e^2 (d-2 e x)+3 c^2 d^3 (d-4 e x)\right )+4 c^3 \left (-a^2 e^3 (16 d+5 e x)+6 a c d^2 e^2 x+3 c^2 d^4 x\right )+2 b^3 c e^2 \left (3 c d^2-4 a e^2\right )+4 b^2 c^2 e \left (a e^2 (5 d+4 e x)-3 c d^2 (d-e x)\right )+b^5 e^4-2 b^4 c e^3 (2 d+e x)}{c^3 \left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac{24 \left (e (a e-b d)+c d^2\right )^2 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((b^4*e^4*x + b^3*e^3*(a*e - 4*c*d*x) + 2*b^2*c*e^2*(3*c*d^2*x - 2*a*e*(d + e*x)
) + b*c*(-3*a^2*e^4 + c^2*d^3*(d - 4*e*x) + 6*a*c*d*e^2*(d + 2*e*x)) + 2*c^2*(c^
2*d^4*x + a^2*e^3*(4*d + e*x) - 2*a*c*d^2*e*(2*d + 3*e*x)))/(c^3*(-b^2 + 4*a*c)*
(a + x*(b + c*x))^2) + (b^5*e^4 + 2*b^3*c*e^2*(3*c*d^2 - 4*a*e^2) - 2*b^4*c*e^3*
(2*d + e*x) + 2*b*c^2*(11*a^2*e^4 + 3*c^2*d^3*(d - 4*e*x) + 6*a*c*d*e^2*(d - 2*e
*x)) + 4*b^2*c^2*e*(-3*c*d^2*(d - e*x) + a*e^2*(5*d + 4*e*x)) + 4*c^3*(3*c^2*d^4
*x + 6*a*c*d^2*e^2*x - a^2*e^3*(16*d + 5*e*x)))/(c^3*(b^2 - 4*a*c)^2*(a + x*(b +
c*x))) + (24*(c*d^2 + e*(-(b*d) + a*e))^2*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 = 932, normalized size = 5.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-(10*a^2*c^2*e^4-8*a*b^2*c*e^4+12*a*b*c^2*d*e^3-12*a*c^3*d^2*e^2+b^4*e^4-6*b^2*
c^2*d^2*e^2+12*b*c^3*d^3*e-6*c^4*d^4)/c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3+1/2*(2*a^
2*b*c^2*e^4-64*a^2*c^3*d*e^3+8*a*b^3*c*e^4-4*a*b^2*c^2*d*e^3+36*a*b*c^3*d^2*e^2-
b^5*e^4-4*b^4*c*d*e^3+18*b^3*c^2*d^2*e^2-36*b^2*c^3*d^3*e+18*b*c^4*d^4)/c^2/(16*
a^2*c^2-8*a*b^2*c+b^4)*x^2-(6*a^3*c^2*e^4-10*a^2*b^2*c*e^4+20*a^2*b*c^2*d*e^3+12
*a^2*c^3*d^2*e^2+a*b^4*e^4+4*a*b^3*c*d*e^3-30*a*b^2*c^2*d^2*e^2+20*a*b*c^3*d^3*e
-10*a*c^4*d^4+4*b^3*c^2*d^3*e-2*b^2*c^3*d^4)/(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^4-32*a^3*c^2*d*e^3-a^2*b^3*e^4-4*a^2*b^2*c*d*e^3+36*a^2*b*c^
2*d^2*e^2-32*a^2*c^3*d^3*e-4*a*b^2*c^2*d^3*e+10*a*b*c^3*d^4-b^3*c^2*d^4)/(16*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^4-24/(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^3+24/(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^2+1
2/(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^2-24/(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*e+12/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arc
tan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c^2*d^4

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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)^4/(c*x^2 + b*x + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.263561, 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)^4/(c*x^2 + b*x + a)^3,x, algorithm="fricas")

[Out]

[1/2*(12*(a^2*c^4*d^4 - 2*a^2*b*c^3*d^3*e - 2*a^3*b*c^2*d*e^3 + a^4*c^2*e^4 + (a
^2*b^2*c^2 + 2*a^3*c^3)*d^2*e^2 + (c^6*d^4 - 2*b*c^5*d^3*e - 2*a*b*c^4*d*e^3 + a
^2*c^4*e^4 + (b^2*c^4 + 2*a*c^5)*d^2*e^2)*x^4 + 2*(b*c^5*d^4 - 2*b^2*c^4*d^3*e -
2*a*b^2*c^3*d*e^3 + a^2*b*c^3*e^4 + (b^3*c^3 + 2*a*b*c^4)*d^2*e^2)*x^3 + ((b^2*
c^4 + 2*a*c^5)*d^4 - 2*(b^3*c^3 + 2*a*b*c^4)*d^3*e + (b^4*c^2 + 4*a*b^2*c^3 + 4*
a^2*c^4)*d^2*e^2 - 2*(a*b^3*c^2 + 2*a^2*b*c^3)*d*e^3 + (a^2*b^2*c^2 + 2*a^3*c^3)
*e^4)*x^2 + 2*(a*b*c^4*d^4 - 2*a*b^2*c^3*d^3*e - 2*a^2*b^2*c^2*d*e^3 + a^3*b*c^2
*e^4 + (a*b^3*c^2 + 2*a^2*b*c^3)*d^2*e^2)*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)) + (36*a^2*b*c^2*d^2*e^2 - (b^3*c^2 - 10*a*b*c^3)*d^4 - 4*(a*b^2*c^2 + 8*a^2
*c^3)*d^3*e - 4*(a^2*b^2*c + 8*a^3*c^2)*d*e^3 - (a^2*b^3 - 10*a^3*b*c)*e^4 + 2*(
6*c^5*d^4 - 12*b*c^4*d^3*e - 12*a*b*c^3*d*e^3 + 6*(b^2*c^3 + 2*a*c^4)*d^2*e^2 -
(b^4*c - 8*a*b^2*c^2 + 10*a^2*c^3)*e^4)*x^3 + (18*b*c^4*d^4 - 36*b^2*c^3*d^3*e +
18*(b^3*c^2 + 2*a*b*c^3)*d^2*e^2 - 4*(b^4*c + a*b^2*c^2 + 16*a^2*c^3)*d*e^3 - (
b^5 - 8*a*b^3*c - 2*a^2*b*c^2)*e^4)*x^2 + 2*(2*(b^2*c^3 + 5*a*c^4)*d^4 - 4*(b^3*
c^2 + 5*a*b*c^3)*d^3*e + 6*(5*a*b^2*c^2 - 2*a^2*c^3)*d^2*e^2 - 4*(a*b^3*c + 5*a^
2*b*c^2)*d*e^3 - (a*b^4 - 10*a^2*b^2*c + 6*a^3*c^2)*e^4)*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 + 3
2*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*(24*(a^2*c^4*d^4 - 2*a^2*b*c^3*d^3*e - 2*a^3*b*c^2*d*e^3 + a^4*c^2*e^4
+ (a^2*b^2*c^2 + 2*a^3*c^3)*d^2*e^2 + (c^6*d^4 - 2*b*c^5*d^3*e - 2*a*b*c^4*d*e^3
+ a^2*c^4*e^4 + (b^2*c^4 + 2*a*c^5)*d^2*e^2)*x^4 + 2*(b*c^5*d^4 - 2*b^2*c^4*d^3
*e - 2*a*b^2*c^3*d*e^3 + a^2*b*c^3*e^4 + (b^3*c^3 + 2*a*b*c^4)*d^2*e^2)*x^3 + ((
b^2*c^4 + 2*a*c^5)*d^4 - 2*(b^3*c^3 + 2*a*b*c^4)*d^3*e + (b^4*c^2 + 4*a*b^2*c^3
+ 4*a^2*c^4)*d^2*e^2 - 2*(a*b^3*c^2 + 2*a^2*b*c^3)*d*e^3 + (a^2*b^2*c^2 + 2*a^3*
c^3)*e^4)*x^2 + 2*(a*b*c^4*d^4 - 2*a*b^2*c^3*d^3*e - 2*a^2*b^2*c^2*d*e^3 + a^3*b
*c^2*e^4 + (a*b^3*c^2 + 2*a^2*b*c^3)*d^2*e^2)*x)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c
*x + b)/(b^2 - 4*a*c)) + (36*a^2*b*c^2*d^2*e^2 - (b^3*c^2 - 10*a*b*c^3)*d^4 - 4*
(a*b^2*c^2 + 8*a^2*c^3)*d^3*e - 4*(a^2*b^2*c + 8*a^3*c^2)*d*e^3 - (a^2*b^3 - 10*
a^3*b*c)*e^4 + 2*(6*c^5*d^4 - 12*b*c^4*d^3*e - 12*a*b*c^3*d*e^3 + 6*(b^2*c^3 + 2
*a*c^4)*d^2*e^2 - (b^4*c - 8*a*b^2*c^2 + 10*a^2*c^3)*e^4)*x^3 + (18*b*c^4*d^4 -
36*b^2*c^3*d^3*e + 18*(b^3*c^2 + 2*a*b*c^3)*d^2*e^2 - 4*(b^4*c + a*b^2*c^2 + 16*
a^2*c^3)*d*e^3 - (b^5 - 8*a*b^3*c - 2*a^2*b*c^2)*e^4)*x^2 + 2*(2*(b^2*c^3 + 5*a*
c^4)*d^4 - 4*(b^3*c^2 + 5*a*b*c^3)*d^3*e + 6*(5*a*b^2*c^2 - 2*a^2*c^3)*d^2*e^2 -
4*(a*b^3*c + 5*a^2*b*c^2)*d*e^3 - (a*b^4 - 10*a^2*b^2*c + 6*a^3*c^2)*e^4)*x)*sq
rt(-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 [A]  time = 113.783, size = 1355, normalized size = 8.02 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6*sqrt(-1/(4*a*c - b**2)**5)*(a*e**2 - b*d*e + c*d**2)**2*log(x + (-384*a**3*c*
*3*sqrt(-1/(4*a*c - b**2)**5)*(a*e**2 - b*d*e + c*d**2)**2 + 288*a**2*b**2*c**2*
sqrt(-1/(4*a*c - b**2)**5)*(a*e**2 - b*d*e + c*d**2)**2 + 6*a**2*b*e**4 - 72*a*b
**4*c*sqrt(-1/(4*a*c - b**2)**5)*(a*e**2 - b*d*e + c*d**2)**2 - 12*a*b**2*d*e**3
+ 12*a*b*c*d**2*e**2 + 6*b**6*sqrt(-1/(4*a*c - b**2)**5)*(a*e**2 - b*d*e + c*d*
*2)**2 + 6*b**3*d**2*e**2 - 12*b**2*c*d**3*e + 6*b*c**2*d**4)/(12*a**2*c*e**4 -
24*a*b*c*d*e**3 + 24*a*c**2*d**2*e**2 + 12*b**2*c*d**2*e**2 - 24*b*c**2*d**3*e +
12*c**3*d**4)) + 6*sqrt(-1/(4*a*c - b**2)**5)*(a*e**2 - b*d*e + c*d**2)**2*log(
x + (384*a**3*c**3*sqrt(-1/(4*a*c - b**2)**5)*(a*e**2 - b*d*e + c*d**2)**2 - 288
*a**2*b**2*c**2*sqrt(-1/(4*a*c - b**2)**5)*(a*e**2 - b*d*e + c*d**2)**2 + 6*a**2
*b*e**4 + 72*a*b**4*c*sqrt(-1/(4*a*c - b**2)**5)*(a*e**2 - b*d*e + c*d**2)**2 -
12*a*b**2*d*e**3 + 12*a*b*c*d**2*e**2 - 6*b**6*sqrt(-1/(4*a*c - b**2)**5)*(a*e**
2 - b*d*e + c*d**2)**2 + 6*b**3*d**2*e**2 - 12*b**2*c*d**3*e + 6*b*c**2*d**4)/(1
2*a**2*c*e**4 - 24*a*b*c*d*e**3 + 24*a*c**2*d**2*e**2 + 12*b**2*c*d**2*e**2 - 24
*b*c**2*d**3*e + 12*c**3*d**4)) - (-10*a**3*b*c*e**4 + 32*a**3*c**2*d*e**3 + a**
2*b**3*e**4 + 4*a**2*b**2*c*d*e**3 - 36*a**2*b*c**2*d**2*e**2 + 32*a**2*c**3*d**
3*e + 4*a*b**2*c**2*d**3*e - 10*a*b*c**3*d**4 + b**3*c**2*d**4 + x**3*(20*a**2*c
**3*e**4 - 16*a*b**2*c**2*e**4 + 24*a*b*c**3*d*e**3 - 24*a*c**4*d**2*e**2 + 2*b*
*4*c*e**4 - 12*b**2*c**3*d**2*e**2 + 24*b*c**4*d**3*e - 12*c**5*d**4) + x**2*(-2
*a**2*b*c**2*e**4 + 64*a**2*c**3*d*e**3 - 8*a*b**3*c*e**4 + 4*a*b**2*c**2*d*e**3
- 36*a*b*c**3*d**2*e**2 + b**5*e**4 + 4*b**4*c*d*e**3 - 18*b**3*c**2*d**2*e**2
+ 36*b**2*c**3*d**3*e - 18*b*c**4*d**4) + x*(12*a**3*c**2*e**4 - 20*a**2*b**2*c*
e**4 + 40*a**2*b*c**2*d*e**3 + 24*a**2*c**3*d**2*e**2 + 2*a*b**4*e**4 + 8*a*b**3
*c*d*e**3 - 60*a*b**2*c**2*d**2*e**2 + 40*a*b*c**3*d**3*e - 20*a*c**4*d**4 + 8*b
**3*c**2*d**3*e - 4*b**2*c**3*d**4))/(32*a**4*c**4 - 16*a**3*b**2*c**3 + 2*a**2*
b**4*c**2 + x**4*(32*a**2*c**6 - 16*a*b**2*c**5 + 2*b**4*c**4) + x**3*(64*a**2*b
*c**5 - 32*a*b**3*c**4 + 4*b**5*c**3) + x**2*(64*a**3*c**5 - 12*a*b**4*c**3 + 2*
b**6*c**2) + x*(64*a**3*b*c**4 - 32*a**2*b**3*c**3 + 4*a*b**5*c**2))

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GIAC/XCAS [A]  time = 0.211176, size = 845, normalized size = 5. \[ \frac{12 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{12 \, c^{5} d^{4} x^{3} - 24 \, b c^{4} d^{3} x^{3} e + 18 \, b c^{4} d^{4} x^{2} + 12 \, b^{2} c^{3} d^{2} x^{3} e^{2} + 24 \, a c^{4} d^{2} x^{3} e^{2} - 36 \, b^{2} c^{3} d^{3} x^{2} e + 4 \, b^{2} c^{3} d^{4} x + 20 \, a c^{4} d^{4} x - 24 \, a b c^{3} d x^{3} e^{3} + 18 \, b^{3} c^{2} d^{2} x^{2} e^{2} + 36 \, a b c^{3} d^{2} x^{2} e^{2} - 8 \, b^{3} c^{2} d^{3} x e - 40 \, a b c^{3} d^{3} x e - b^{3} c^{2} d^{4} + 10 \, a b c^{3} d^{4} - 2 \, b^{4} c x^{3} e^{4} + 16 \, a b^{2} c^{2} x^{3} e^{4} - 20 \, a^{2} c^{3} x^{3} e^{4} - 4 \, b^{4} c d x^{2} e^{3} - 4 \, a b^{2} c^{2} d x^{2} e^{3} - 64 \, a^{2} c^{3} d x^{2} e^{3} + 60 \, a b^{2} c^{2} d^{2} x e^{2} - 24 \, a^{2} c^{3} d^{2} x e^{2} - 4 \, a b^{2} c^{2} d^{3} e - 32 \, a^{2} c^{3} d^{3} e - b^{5} x^{2} e^{4} + 8 \, a b^{3} c x^{2} e^{4} + 2 \, a^{2} b c^{2} x^{2} e^{4} - 8 \, a b^{3} c d x e^{3} - 40 \, a^{2} b c^{2} d x e^{3} + 36 \, a^{2} b c^{2} d^{2} e^{2} - 2 \, a b^{4} x e^{4} + 20 \, a^{2} b^{2} c x e^{4} - 12 \, a^{3} c^{2} x e^{4} - 4 \, a^{2} b^{2} c d e^{3} - 32 \, a^{3} c^{2} d e^{3} - a^{2} b^{3} e^{4} + 10 \, a^{3} b c e^{4}}{2 \,{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )}{\left (c x^{2} + b x + a\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

12*(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)
*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt(-b^
2 + 4*a*c)) + 1/2*(12*c^5*d^4*x^3 - 24*b*c^4*d^3*x^3*e + 18*b*c^4*d^4*x^2 + 12*b
^2*c^3*d^2*x^3*e^2 + 24*a*c^4*d^2*x^3*e^2 - 36*b^2*c^3*d^3*x^2*e + 4*b^2*c^3*d^4
*x + 20*a*c^4*d^4*x - 24*a*b*c^3*d*x^3*e^3 + 18*b^3*c^2*d^2*x^2*e^2 + 36*a*b*c^3
*d^2*x^2*e^2 - 8*b^3*c^2*d^3*x*e - 40*a*b*c^3*d^3*x*e - b^3*c^2*d^4 + 10*a*b*c^3
*d^4 - 2*b^4*c*x^3*e^4 + 16*a*b^2*c^2*x^3*e^4 - 20*a^2*c^3*x^3*e^4 - 4*b^4*c*d*x
^2*e^3 - 4*a*b^2*c^2*d*x^2*e^3 - 64*a^2*c^3*d*x^2*e^3 + 60*a*b^2*c^2*d^2*x*e^2 -
24*a^2*c^3*d^2*x*e^2 - 4*a*b^2*c^2*d^3*e - 32*a^2*c^3*d^3*e - b^5*x^2*e^4 + 8*a
*b^3*c*x^2*e^4 + 2*a^2*b*c^2*x^2*e^4 - 8*a*b^3*c*d*x*e^3 - 40*a^2*b*c^2*d*x*e^3
+ 36*a^2*b*c^2*d^2*e^2 - 2*a*b^4*x*e^4 + 20*a^2*b^2*c*x*e^4 - 12*a^3*c^2*x*e^4 -
4*a^2*b^2*c*d*e^3 - 32*a^3*c^2*d*e^3 - a^2*b^3*e^4 + 10*a^3*b*c*e^4)/((b^4*c^2
- 8*a*b^2*c^3 + 16*a^2*c^4)*(c*x^2 + b*x + a)^2)