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3.2180 \(\int \frac{(d+e x)^4}{\left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=260 \[ \frac{2 e^2 x \left (-c e (3 a e+2 b d)+b^2 e^2+3 c^2 d^2\right )}{c^2 \left (b^2-4 a c\right )}+\frac{e^3 x^2 (2 c d-b e)}{c \left (b^2-4 a c\right )}-\frac{(d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{2 \left (2 b^2 c e^3 (3 a e+b d)-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (a e+2 b d)-b^4 e^4+2 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}+\frac{e^3 (2 c d-b e) \log \left (a+b x+c x^2\right )}{c^3} \]

[Out]

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

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Rubi [A]  time = 1.23319, antiderivative size = 260, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{2 e^2 x \left (-c e (3 a e+2 b d)+b^2 e^2+3 c^2 d^2\right )}{c^2 \left (b^2-4 a c\right )}+\frac{e^3 x^2 (2 c d-b e)}{c \left (b^2-4 a c\right )}-\frac{(d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{2 \left (2 b^2 c e^3 (3 a e+b d)-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (a e+2 b d)-b^4 e^4+2 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}+\frac{e^3 (2 c d-b e) \log \left (a+b x+c x^2\right )}{c^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.913255, size = 298, normalized size = 1.15 \[ \frac{\frac{-b c \left (-3 a^2 e^4+6 a c d e^2 (d+2 e x)+c^2 d^3 (d-4 e x)\right )-2 c^2 \left (a^2 e^3 (4 d+e x)-2 a c d^2 e (2 d+3 e x)+c^2 d^4 x\right )+b^3 e^3 (4 c d x-a e)+2 b^2 c e^2 \left (2 a e (d+e x)-3 c d^2 x\right )-b^4 e^4 x}{\left (b^2-4 a c\right ) (a+x (b+c x))}-\frac{2 \left (-2 b^2 c e^3 (3 a e+b d)+4 c^3 d^2 e (b d-3 a e)+6 a c^2 e^3 (a e+2 b d)+b^4 e^4-2 c^4 d^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+e^3 (2 c d-b e) \log (a+x (b+c x))+c e^4 x}{c^3} \]

Antiderivative was successfully verified.

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

[Out]

(c*e^4*x + (-(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)))/((b
^2 - 4*a*c)*(a + x*(b + c*x))) - (2*(-2*c^4*d^4 + b^4*e^4 + 4*c^3*d^2*e*(b*d - 3
*a*e) + 6*a*c^2*e^3*(2*b*d + a*e) - 2*b^2*c*e^3*(b*d + 3*a*e))*ArcTan[(b + 2*c*x
)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + e^3*(2*c*d - b*e)*Log[a + x*(b + c
*x)])/c^3

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Maple [B]  time = 0.017, size = 1601, normalized size = 6.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

12/c/(c*x^2+b*x+a)/(4*a*c-b^2)*x*a*b*d*e^3+12/c^2/(64*a^3*c^3-48*a^2*b^2*c^2+12*
a*b^4*c-b^6)^(1/2)*arctan((2*c*(4*a*c-b^2)*x+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b
^2*c^2+12*a*b^4*c-b^6)^(1/2))*a*b^2*e^4+4/c^2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^
4*c-b^6)^(1/2)*arctan((2*c*(4*a*c-b^2)*x+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c
^2+12*a*b^4*c-b^6)^(1/2))*b^3*d*e^3-2/c^2/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^2+b*x+
a))*b^2*d*e^3-24/c/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*c*
(4*a*c-b^2)*x+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*a
*b*d*e^3+6/c/(c*x^2+b*x+a)/(4*a*c-b^2)*x*b^2*d^2*e^2+6/c/(c*x^2+b*x+a)/(4*a*c-b^
2)*a*b*d^2*e^2-4/c^2/(c*x^2+b*x+a)/(4*a*c-b^2)*x*a*b^2*e^4-4/c^2/(c*x^2+b*x+a)/(
4*a*c-b^2)*x*b^3*d*e^3-4/c^2/(c*x^2+b*x+a)/(4*a*c-b^2)*a*b^2*d*e^3-12/c/(64*a^3*
c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*c*(4*a*c-b^2)*x+(4*a*c-b^2)*b
)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*a^2*e^4-8/(64*a^3*c^3-48*a^2
*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*c*(4*a*c-b^2)*x+(4*a*c-b^2)*b)/(64*a^3*
c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*d^3*e*b+24/(64*a^3*c^3-48*a^2*b^2*c^2+
12*a*b^4*c-b^6)^(1/2)*arctan((2*c*(4*a*c-b^2)*x+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^
2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*a*d^2*e^2+2/c/(c*x^2+b*x+a)/(4*a*c-b^2)*x*e^4*a
^2+1/c^3/(c*x^2+b*x+a)/(4*a*c-b^2)*x*b^4*e^4+8/c/(c*x^2+b*x+a)/(4*a*c-b^2)*a^2*d
*e^3+1/c^3/(c*x^2+b*x+a)/(4*a*c-b^2)*a*b^3*e^4+2*c/(c*x^2+b*x+a)/(4*a*c-b^2)*x*d
^4-12/(c*x^2+b*x+a)/(4*a*c-b^2)*x*a*e^2*d^2+1/c^3/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*
x^2+b*x+a))*b^3*e^4+1/(c*x^2+b*x+a)/(4*a*c-b^2)*d^4*b+4*c/(64*a^3*c^3-48*a^2*b^2
*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*c*(4*a*c-b^2)*x+(4*a*c-b^2)*b)/(64*a^3*c^3-
48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*d^4-4/(c*x^2+b*x+a)/(4*a*c-b^2)*x*b*d^3*e+
8/c/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^2+b*x+a))*a*d*e^3-3/c^2/(c*x^2+b*x+a)/(4*a*c
-b^2)*a^2*b*e^4-4/c^2/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^2+b*x+a))*a*b*e^4+e^4*x/c^
2-8/(c*x^2+b*x+a)/(4*a*c-b^2)*a*d^3*e-2/c^3/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*
c-b^6)^(1/2)*arctan((2*c*(4*a*c-b^2)*x+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2
+12*a*b^4*c-b^6)^(1/2))*b^4*e^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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.273316, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[((2*a*c^4*d^4 - 4*a*b*c^3*d^3*e + 12*a^2*c^3*d^2*e^2 + 2*(a*b^3*c - 6*a^2*b*c^2
)*d*e^3 - (a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2)*e^4 + (2*c^5*d^4 - 4*b*c^4*d^3*e + 1
2*a*c^4*d^2*e^2 + 2*(b^3*c^2 - 6*a*b*c^3)*d*e^3 - (b^4*c - 6*a*b^2*c^2 + 6*a^2*c
^3)*e^4)*x^2 + (2*b*c^4*d^4 - 4*b^2*c^3*d^3*e + 12*a*b*c^3*d^2*e^2 + 2*(b^4*c -
6*a*b^2*c^2)*d*e^3 - (b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*e^4)*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)) - (b*c^3*d^4 - 8*a*c^3*d^3*e + 6*a*b*c^2*d^2*e^2 - (b^2*c^2
- 4*a*c^3)*e^4*x^3 - (b^3*c - 4*a*b*c^2)*e^4*x^2 - 4*(a*b^2*c - 2*a^2*c^2)*d*e^3
 + (a*b^3 - 3*a^2*b*c)*e^4 + (2*c^4*d^4 - 4*b*c^3*d^3*e + 6*(b^2*c^2 - 2*a*c^3)*
d^2*e^2 - 4*(b^3*c - 3*a*b*c^2)*d*e^3 + (b^4 - 5*a*b^2*c + 6*a^2*c^2)*e^4)*x - (
2*(a*b^2*c - 4*a^2*c^2)*d*e^3 - (a*b^3 - 4*a^2*b*c)*e^4 + (2*(b^2*c^2 - 4*a*c^3)
*d*e^3 - (b^3*c - 4*a*b*c^2)*e^4)*x^2 + (2*(b^3*c - 4*a*b*c^2)*d*e^3 - (b^4 - 4*
a*b^2*c)*e^4)*x)*log(c*x^2 + b*x + a))*sqrt(b^2 - 4*a*c))/((a*b^2*c^3 - 4*a^2*c^
4 + (b^2*c^4 - 4*a*c^5)*x^2 + (b^3*c^3 - 4*a*b*c^4)*x)*sqrt(b^2 - 4*a*c)), -(2*(
2*a*c^4*d^4 - 4*a*b*c^3*d^3*e + 12*a^2*c^3*d^2*e^2 + 2*(a*b^3*c - 6*a^2*b*c^2)*d
*e^3 - (a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2)*e^4 + (2*c^5*d^4 - 4*b*c^4*d^3*e + 12*a
*c^4*d^2*e^2 + 2*(b^3*c^2 - 6*a*b*c^3)*d*e^3 - (b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)
*e^4)*x^2 + (2*b*c^4*d^4 - 4*b^2*c^3*d^3*e + 12*a*b*c^3*d^2*e^2 + 2*(b^4*c - 6*a
*b^2*c^2)*d*e^3 - (b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*e^4)*x)*arctan(-sqrt(-b^2 + 4*
a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (b*c^3*d^4 - 8*a*c^3*d^3*e + 6*a*b*c^2*d^2*e^2
 - (b^2*c^2 - 4*a*c^3)*e^4*x^3 - (b^3*c - 4*a*b*c^2)*e^4*x^2 - 4*(a*b^2*c - 2*a^
2*c^2)*d*e^3 + (a*b^3 - 3*a^2*b*c)*e^4 + (2*c^4*d^4 - 4*b*c^3*d^3*e + 6*(b^2*c^2
 - 2*a*c^3)*d^2*e^2 - 4*(b^3*c - 3*a*b*c^2)*d*e^3 + (b^4 - 5*a*b^2*c + 6*a^2*c^2
)*e^4)*x - (2*(a*b^2*c - 4*a^2*c^2)*d*e^3 - (a*b^3 - 4*a^2*b*c)*e^4 + (2*(b^2*c^
2 - 4*a*c^3)*d*e^3 - (b^3*c - 4*a*b*c^2)*e^4)*x^2 + (2*(b^3*c - 4*a*b*c^2)*d*e^3
 - (b^4 - 4*a*b^2*c)*e^4)*x)*log(c*x^2 + b*x + a))*sqrt(-b^2 + 4*a*c))/((a*b^2*c
^3 - 4*a^2*c^4 + (b^2*c^4 - 4*a*c^5)*x^2 + (b^3*c^3 - 4*a*b*c^4)*x)*sqrt(-b^2 +
4*a*c))]

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Sympy [A]  time = 44.5131, size = 1924, normalized size = 7.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-e**3*(b*e - 2*c*d)/c**3 - sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2*e**4 - 6*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 - 2*b**3*c*d*e**
3 + 4*b*c**3*d**3*e - 2*c**4*d**4)/(c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*
a*b**4*c - b**6)))*log(x + (-10*a**2*b*c*e**4 - 16*a**2*c**4*(-e**3*(b*e - 2*c*d
)/c**3 - sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2*e**4 - 6*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 - 2*b**3*c*d*e**3 + 4*b*c**3*d**3*e
 - 2*c**4*d**4)/(c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))
+ 32*a**2*c**2*d*e**3 + 2*a*b**3*e**4 + 8*a*b**2*c**3*(-e**3*(b*e - 2*c*d)/c**3
- sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2*e**4 - 6*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 - 2*b**3*c*d*e**3 + 4*b*c**3*d**3*e - 2*c*
*4*d**4)/(c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) - 4*a*b
**2*c*d*e**3 - 12*a*b*c**2*d**2*e**2 - b**4*c**2*(-e**3*(b*e - 2*c*d)/c**3 - sqr
t(-(4*a*c - b**2)**3)*(6*a**2*c**2*e**4 - 6*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 - 2*b**3*c*d*e**3 + 4*b*c**3*d**3*e - 2*c**4*d*
*4)/(c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) + 4*b**2*c**
2*d**3*e - 2*b*c**3*d**4)/(12*a**2*c**2*e**4 - 12*a*b**2*c*e**4 + 24*a*b*c**2*d*
e**3 - 24*a*c**3*d**2*e**2 + 2*b**4*e**4 - 4*b**3*c*d*e**3 + 8*b*c**3*d**3*e - 4
*c**4*d**4)) + (-e**3*(b*e - 2*c*d)/c**3 + sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2
*e**4 - 6*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 -
 2*b**3*c*d*e**3 + 4*b*c**3*d**3*e - 2*c**4*d**4)/(c**3*(64*a**3*c**3 - 48*a**2*
b**2*c**2 + 12*a*b**4*c - b**6)))*log(x + (-10*a**2*b*c*e**4 - 16*a**2*c**4*(-e*
*3*(b*e - 2*c*d)/c**3 + sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2*e**4 - 6*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 - 2*b**3*c*d*e**3 +
4*b*c**3*d**3*e - 2*c**4*d**4)/(c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b*
*4*c - b**6))) + 32*a**2*c**2*d*e**3 + 2*a*b**3*e**4 + 8*a*b**2*c**3*(-e**3*(b*e
 - 2*c*d)/c**3 + sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2*e**4 - 6*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 - 2*b**3*c*d*e**3 + 4*b*c**
3*d**3*e - 2*c**4*d**4)/(c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c -
b**6))) - 4*a*b**2*c*d*e**3 - 12*a*b*c**2*d**2*e**2 - b**4*c**2*(-e**3*(b*e - 2*
c*d)/c**3 + sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2*e**4 - 6*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 - 2*b**3*c*d*e**3 + 4*b*c**3*d**
3*e - 2*c**4*d**4)/(c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)
)) + 4*b**2*c**2*d**3*e - 2*b*c**3*d**4)/(12*a**2*c**2*e**4 - 12*a*b**2*c*e**4 +
 24*a*b*c**2*d*e**3 - 24*a*c**3*d**2*e**2 + 2*b**4*e**4 - 4*b**3*c*d*e**3 + 8*b*
c**3*d**3*e - 4*c**4*d**4)) + (-3*a**2*b*c*e**4 + 8*a**2*c**2*d*e**3 + a*b**3*e*
*4 - 4*a*b**2*c*d*e**3 + 6*a*b*c**2*d**2*e**2 - 8*a*c**3*d**3*e + b*c**3*d**4 +
x*(2*a**2*c**2*e**4 - 4*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 - 4*b**3*c*d*e**3 + 6*b**2*c**2*d**2*e**2 - 4*b*c**3*d**3*e + 2*c**
4*d**4))/(4*a**2*c**4 - a*b**2*c**3 + x**2*(4*a*c**5 - b**2*c**4) + x*(4*a*b*c**
4 - b**3*c**3)) + e**4*x/c**2

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GIAC/XCAS [A]  time = 0.20856, size = 479, normalized size = 1.84 \[ -\frac{2 \,{\left (2 \, c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 12 \, a c^{3} d^{2} e^{2} + 2 \, b^{3} c d e^{3} - 12 \, a b c^{2} d e^{3} - b^{4} e^{4} + 6 \, a b^{2} c e^{4} - 6 \, a^{2} c^{2} e^{4}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{x e^{4}}{c^{2}} + \frac{{\left (2 \, c d e^{3} - b e^{4}\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{c^{3}} - \frac{\frac{{\left (2 \, c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \, b^{2} c^{2} d^{2} e^{2} - 12 \, a c^{3} d^{2} e^{2} - 4 \, b^{3} c d e^{3} + 12 \, a b c^{2} d e^{3} + b^{4} e^{4} - 4 \, a b^{2} c e^{4} + 2 \, a^{2} c^{2} e^{4}\right )} x}{c} + \frac{b c^{3} d^{4} - 8 \, a c^{3} d^{3} e + 6 \, a b c^{2} d^{2} e^{2} - 4 \, a b^{2} c d e^{3} + 8 \, a^{2} c^{2} d e^{3} + a b^{3} e^{4} - 3 \, a^{2} b c e^{4}}{c}}{{\left (c x^{2} + b x + a\right )}{\left (b^{2} - 4 \, a c\right )} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(2*c^4*d^4 - 4*b*c^3*d^3*e + 12*a*c^3*d^2*e^2 + 2*b^3*c*d*e^3 - 12*a*b*c^2*d*
e^3 - b^4*e^4 + 6*a*b^2*c*e^4 - 6*a^2*c^2*e^4)*arctan((2*c*x + b)/sqrt(-b^2 + 4*
a*c))/((b^2*c^3 - 4*a*c^4)*sqrt(-b^2 + 4*a*c)) + x*e^4/c^2 + (2*c*d*e^3 - b*e^4)
*ln(c*x^2 + b*x + a)/c^3 - ((2*c^4*d^4 - 4*b*c^3*d^3*e + 6*b^2*c^2*d^2*e^2 - 12*
a*c^3*d^2*e^2 - 4*b^3*c*d*e^3 + 12*a*b*c^2*d*e^3 + b^4*e^4 - 4*a*b^2*c*e^4 + 2*a
^2*c^2*e^4)*x/c + (b*c^3*d^4 - 8*a*c^3*d^3*e + 6*a*b*c^2*d^2*e^2 - 4*a*b^2*c*d*e
^3 + 8*a^2*c^2*d*e^3 + a*b^3*e^4 - 3*a^2*b*c*e^4)/c)/((c*x^2 + b*x + a)*(b^2 - 4
*a*c)*c^2)

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