3.2363 \(\int \frac{(A+B x) (d+e x)^3}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=357 \[ \frac{e x \left (B \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )+A c e (3 c d-b e)\right )}{c^3}+\frac{\log \left (a+b x+c x^2\right ) \left (A c e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )+B \left (-3 c^2 d e (a e+b d)+b c e^2 (2 a e+3 b d)-b^3 e^3+c^3 d^3\right )\right )}{2 c^4}-\frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (b^2 c e \left (-4 a B e^2+3 A c d e+3 B c d^2\right )-b c^2 \left (-3 a A e^3-9 a B d e^2+3 A c d^2 e+B c d^3\right )+2 c^2 \left (A c d \left (c d^2-3 a e^2\right )-a B e \left (3 c d^2-a e^2\right )\right )-b^3 c e^2 (A e+3 B d)+b^4 B e^3\right )}{c^4 \sqrt{b^2-4 a c}}+\frac{e^2 x^2 (A c e-b B e+3 B c d)}{2 c^2}+\frac{B e^3 x^3}{3 c} \]

[Out]

(e*(A*c*e*(3*c*d - b*e) + B*(3*c^2*d^2 + b^2*e^2 - c*e*(3*b*d + a*e)))*x)/c^3 +
(e^2*(3*B*c*d - b*B*e + A*c*e)*x^2)/(2*c^2) + (B*e^3*x^3)/(3*c) - ((b^4*B*e^3 -
b^3*c*e^2*(3*B*d + A*e) + b^2*c*e*(3*B*c*d^2 + 3*A*c*d*e - 4*a*B*e^2) - b*c^2*(B
*c*d^3 + 3*A*c*d^2*e - 9*a*B*d*e^2 - 3*a*A*e^3) + 2*c^2*(A*c*d*(c*d^2 - 3*a*e^2)
- a*B*e*(3*c*d^2 - a*e^2)))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^4*Sqrt[b
^2 - 4*a*c]) + ((A*c*e*(3*c^2*d^2 + b^2*e^2 - c*e*(3*b*d + a*e)) + B*(c^3*d^3 -
b^3*e^3 - 3*c^2*d*e*(b*d + a*e) + b*c*e^2*(3*b*d + 2*a*e)))*Log[a + b*x + c*x^2]
)/(2*c^4)

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Rubi [A]  time = 1.34849, antiderivative size = 357, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{e x \left (B \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )+A c e (3 c d-b e)\right )}{c^3}+\frac{\log \left (a+b x+c x^2\right ) \left (A c e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )+B \left (-3 c^2 d e (a e+b d)+b c e^2 (2 a e+3 b d)-b^3 e^3+c^3 d^3\right )\right )}{2 c^4}-\frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (b^2 c e \left (-4 a B e^2+3 A c d e+3 B c d^2\right )-b c^2 \left (-3 a A e^3-9 a B d e^2+3 A c d^2 e+B c d^3\right )+2 c^2 \left (A c d \left (c d^2-3 a e^2\right )-a B e \left (3 c d^2-a e^2\right )\right )-b^3 c e^2 (A e+3 B d)+b^4 B e^3\right )}{c^4 \sqrt{b^2-4 a c}}+\frac{e^2 x^2 (A c e-b B e+3 B c d)}{2 c^2}+\frac{B e^3 x^3}{3 c} \]

Antiderivative was successfully verified.

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

[Out]

(e*(A*c*e*(3*c*d - b*e) + B*(3*c^2*d^2 + b^2*e^2 - c*e*(3*b*d + a*e)))*x)/c^3 +
(e^2*(3*B*c*d - b*B*e + A*c*e)*x^2)/(2*c^2) + (B*e^3*x^3)/(3*c) - ((b^4*B*e^3 -
b^3*c*e^2*(3*B*d + A*e) + b^2*c*e*(3*B*c*d^2 + 3*A*c*d*e - 4*a*B*e^2) - b*c^2*(B
*c*d^3 + 3*A*c*d^2*e - 9*a*B*d*e^2 - 3*a*A*e^3) + 2*c^2*(A*c*d*(c*d^2 - 3*a*e^2)
- a*B*e*(3*c*d^2 - a*e^2)))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^4*Sqrt[b
^2 - 4*a*c]) + ((A*c*e*(3*c^2*d^2 + b^2*e^2 - c*e*(3*b*d + a*e)) + B*(c^3*d^3 -
b^3*e^3 - 3*c^2*d*e*(b*d + a*e) + b*c*e^2*(3*b*d + 2*a*e)))*Log[a + b*x + c*x^2]
)/(2*c^4)

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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((B*x+A)*(e*x+d)**3/(c*x**2+b*x+a),x)

[Out]

Timed out

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Mathematica [A]  time = 0.634982, size = 352, normalized size = 0.99 \[ \frac{6 c e x \left (B \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )+A c e (3 c d-b e)\right )+3 \log (a+x (b+c x)) \left (A c e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )+B \left (-3 c^2 d e (a e+b d)+b c e^2 (2 a e+3 b d)-b^3 e^3+c^3 d^3\right )\right )+\frac{6 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right ) \left (b^2 c e \left (-4 a B e^2+3 A c d e+3 B c d^2\right )+b c^2 \left (3 a A e^3+9 a B d e^2-3 A c d^2 e-B c d^3\right )+2 c^2 \left (A c d \left (c d^2-3 a e^2\right )+a B e \left (a e^2-3 c d^2\right )\right )-b^3 c e^2 (A e+3 B d)+b^4 B e^3\right )}{\sqrt{4 a c-b^2}}+3 c^2 e^2 x^2 (A c e-b B e+3 B c d)+2 B c^3 e^3 x^3}{6 c^4} \]

Antiderivative was successfully verified.

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

[Out]

(6*c*e*(A*c*e*(3*c*d - b*e) + B*(3*c^2*d^2 + b^2*e^2 - c*e*(3*b*d + a*e)))*x + 3
*c^2*e^2*(3*B*c*d - b*B*e + A*c*e)*x^2 + 2*B*c^3*e^3*x^3 + (6*(b^4*B*e^3 - b^3*c
*e^2*(3*B*d + A*e) + b^2*c*e*(3*B*c*d^2 + 3*A*c*d*e - 4*a*B*e^2) + b*c^2*(-(B*c*
d^3) - 3*A*c*d^2*e + 9*a*B*d*e^2 + 3*a*A*e^3) + 2*c^2*(A*c*d*(c*d^2 - 3*a*e^2) +
a*B*e*(-3*c*d^2 + a*e^2)))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 +
4*a*c] + 3*(A*c*e*(3*c^2*d^2 + b^2*e^2 - c*e*(3*b*d + a*e)) + B*(c^3*d^3 - b^3*e
^3 - 3*c^2*d*e*(b*d + a*e) + b*c*e^2*(3*b*d + 2*a*e)))*Log[a + x*(b + c*x)])/(6*
c^4)

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Maple [B]  time = 0.011, size = 946, normalized size = 2.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*B*e^3*x^3/c+3/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^2*
B*d^2*e+3/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*A*a*b*e^3-6/
c/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*A*a*d*e^2-4/c^3/(4*a*c-b
^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*B*a*b^2*e^3-6/c/(4*a*c-b^2)^(1/2)*
arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*B*a*d^2*e+3/c^2/(4*a*c-b^2)^(1/2)*arctan((2*
c*x+b)/(4*a*c-b^2)^(1/2))*b^2*A*d*e^2-3/c/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*
a*c-b^2)^(1/2))*b*A*d^2*e-3/c^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(
1/2))*b^3*B*d*e^2+1/c^4/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^
4*B*e^3-1/c/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*B*d^3-3/2/c^
2*ln(c*x^2+b*x+a)*B*d^2*b*e+2/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)
^(1/2))*B*a^2*e^3-1/c^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*A*
b^3*e^3+3/2/c^3*ln(c*x^2+b*x+a)*B*d*b^2*e^2+1/c^3*ln(c*x^2+b*x+a)*a*b*B*e^3-3/2/
c^2*ln(c*x^2+b*x+a)*B*d*a*e^2-3/2/c^2*ln(c*x^2+b*x+a)*A*d*b*e^2-3*e^2/c^2*B*b*d*
x+1/2/c*ln(c*x^2+b*x+a)*B*d^3+2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(
1/2))*A*d^3+1/2*e^3/c*A*x^2+9/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)
^(1/2))*B*a*b*d*e^2+e^3/c^3*B*b^2*x-1/2*e^3/c^2*B*x^2*b+3/2*e^2/c*B*x^2*d+3*e/c*
B*d^2*x-1/2/c^4*ln(c*x^2+b*x+a)*b^3*B*e^3-1/2/c^2*ln(c*x^2+b*x+a)*a*A*e^3+1/2/c^
3*ln(c*x^2+b*x+a)*A*b^2*e^3+3/2/c*ln(c*x^2+b*x+a)*A*d^2*e-e^3/c^2*A*b*x+3*e^2/c*
A*d*x-e^3/c^2*B*a*x

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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((B*x + A)*(e*x + d)^3/(c*x^2 + b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.392134, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/6*(3*((B*b*c^3 - 2*A*c^4)*d^3 - 3*(B*b^2*c^2 - (2*B*a + A*b)*c^3)*d^2*e + 3*
(B*b^3*c + 2*A*a*c^3 - (3*B*a*b + A*b^2)*c^2)*d*e^2 - (B*b^4 + (2*B*a^2 + 3*A*a*
b)*c^2 - (4*B*a*b^2 + A*b^3)*c)*e^3)*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*B*c^3*e^3*x^3 + 3*(3*B*c^3*d*e^2 - (B*b*c^2 - A*c^3)*e^3)*x^2 + 6*(3*B*c^3*d^2
*e - 3*(B*b*c^2 - A*c^3)*d*e^2 + (B*b^2*c - (B*a + A*b)*c^2)*e^3)*x + 3*(B*c^3*d
^3 - 3*(B*b*c^2 - A*c^3)*d^2*e + 3*(B*b^2*c - (B*a + A*b)*c^2)*d*e^2 - (B*b^3 +
A*a*c^2 - (2*B*a*b + A*b^2)*c)*e^3)*log(c*x^2 + b*x + a))*sqrt(b^2 - 4*a*c))/(sq
rt(b^2 - 4*a*c)*c^4), -1/6*(6*((B*b*c^3 - 2*A*c^4)*d^3 - 3*(B*b^2*c^2 - (2*B*a +
A*b)*c^3)*d^2*e + 3*(B*b^3*c + 2*A*a*c^3 - (3*B*a*b + A*b^2)*c^2)*d*e^2 - (B*b^
4 + (2*B*a^2 + 3*A*a*b)*c^2 - (4*B*a*b^2 + A*b^3)*c)*e^3)*arctan(-sqrt(-b^2 + 4*
a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - (2*B*c^3*e^3*x^3 + 3*(3*B*c^3*d*e^2 - (B*b*c^2
- A*c^3)*e^3)*x^2 + 6*(3*B*c^3*d^2*e - 3*(B*b*c^2 - A*c^3)*d*e^2 + (B*b^2*c - (
B*a + A*b)*c^2)*e^3)*x + 3*(B*c^3*d^3 - 3*(B*b*c^2 - A*c^3)*d^2*e + 3*(B*b^2*c -
(B*a + A*b)*c^2)*d*e^2 - (B*b^3 + A*a*c^2 - (2*B*a*b + A*b^2)*c)*e^3)*log(c*x^2
+ b*x + a))*sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^4)]

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Sympy [A]  time = 71.2072, size = 2754, normalized size = 7.71 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*e**3*x**3/(3*c) + (-sqrt(-4*a*c + b**2)*(3*A*a*b*c**2*e**3 - 6*A*a*c**3*d*e**2
- A*b**3*c*e**3 + 3*A*b**2*c**2*d*e**2 - 3*A*b*c**3*d**2*e + 2*A*c**4*d**3 + 2*
B*a**2*c**2*e**3 - 4*B*a*b**2*c*e**3 + 9*B*a*b*c**2*d*e**2 - 6*B*a*c**3*d**2*e +
B*b**4*e**3 - 3*B*b**3*c*d*e**2 + 3*B*b**2*c**2*d**2*e - B*b*c**3*d**3)/(2*c**4
*(4*a*c - b**2)) + (-A*a*c**2*e**3 + A*b**2*c*e**3 - 3*A*b*c**2*d*e**2 + 3*A*c**
3*d**2*e + 2*B*a*b*c*e**3 - 3*B*a*c**2*d*e**2 - B*b**3*e**3 + 3*B*b**2*c*d*e**2
- 3*B*b*c**2*d**2*e + B*c**3*d**3)/(2*c**4))*log(x + (2*A*a**2*c**2*e**3 - A*a*b
**2*c*e**3 + 3*A*a*b*c**2*d*e**2 - 6*A*a*c**3*d**2*e + A*b*c**3*d**3 - 3*B*a**2*
b*c*e**3 + 6*B*a**2*c**2*d*e**2 + B*a*b**3*e**3 - 3*B*a*b**2*c*d*e**2 + 3*B*a*b*
c**2*d**2*e - 2*B*a*c**3*d**3 + 4*a*c**4*(-sqrt(-4*a*c + b**2)*(3*A*a*b*c**2*e**
3 - 6*A*a*c**3*d*e**2 - A*b**3*c*e**3 + 3*A*b**2*c**2*d*e**2 - 3*A*b*c**3*d**2*e
+ 2*A*c**4*d**3 + 2*B*a**2*c**2*e**3 - 4*B*a*b**2*c*e**3 + 9*B*a*b*c**2*d*e**2
- 6*B*a*c**3*d**2*e + B*b**4*e**3 - 3*B*b**3*c*d*e**2 + 3*B*b**2*c**2*d**2*e - B
*b*c**3*d**3)/(2*c**4*(4*a*c - b**2)) + (-A*a*c**2*e**3 + A*b**2*c*e**3 - 3*A*b*
c**2*d*e**2 + 3*A*c**3*d**2*e + 2*B*a*b*c*e**3 - 3*B*a*c**2*d*e**2 - B*b**3*e**3
+ 3*B*b**2*c*d*e**2 - 3*B*b*c**2*d**2*e + B*c**3*d**3)/(2*c**4)) - b**2*c**3*(-
sqrt(-4*a*c + b**2)*(3*A*a*b*c**2*e**3 - 6*A*a*c**3*d*e**2 - A*b**3*c*e**3 + 3*A
*b**2*c**2*d*e**2 - 3*A*b*c**3*d**2*e + 2*A*c**4*d**3 + 2*B*a**2*c**2*e**3 - 4*B
*a*b**2*c*e**3 + 9*B*a*b*c**2*d*e**2 - 6*B*a*c**3*d**2*e + B*b**4*e**3 - 3*B*b**
3*c*d*e**2 + 3*B*b**2*c**2*d**2*e - B*b*c**3*d**3)/(2*c**4*(4*a*c - b**2)) + (-A
*a*c**2*e**3 + A*b**2*c*e**3 - 3*A*b*c**2*d*e**2 + 3*A*c**3*d**2*e + 2*B*a*b*c*e
**3 - 3*B*a*c**2*d*e**2 - B*b**3*e**3 + 3*B*b**2*c*d*e**2 - 3*B*b*c**2*d**2*e +
B*c**3*d**3)/(2*c**4)))/(3*A*a*b*c**2*e**3 - 6*A*a*c**3*d*e**2 - A*b**3*c*e**3 +
3*A*b**2*c**2*d*e**2 - 3*A*b*c**3*d**2*e + 2*A*c**4*d**3 + 2*B*a**2*c**2*e**3 -
4*B*a*b**2*c*e**3 + 9*B*a*b*c**2*d*e**2 - 6*B*a*c**3*d**2*e + B*b**4*e**3 - 3*B
*b**3*c*d*e**2 + 3*B*b**2*c**2*d**2*e - B*b*c**3*d**3)) + (sqrt(-4*a*c + b**2)*(
3*A*a*b*c**2*e**3 - 6*A*a*c**3*d*e**2 - A*b**3*c*e**3 + 3*A*b**2*c**2*d*e**2 - 3
*A*b*c**3*d**2*e + 2*A*c**4*d**3 + 2*B*a**2*c**2*e**3 - 4*B*a*b**2*c*e**3 + 9*B*
a*b*c**2*d*e**2 - 6*B*a*c**3*d**2*e + B*b**4*e**3 - 3*B*b**3*c*d*e**2 + 3*B*b**2
*c**2*d**2*e - B*b*c**3*d**3)/(2*c**4*(4*a*c - b**2)) + (-A*a*c**2*e**3 + A*b**2
*c*e**3 - 3*A*b*c**2*d*e**2 + 3*A*c**3*d**2*e + 2*B*a*b*c*e**3 - 3*B*a*c**2*d*e*
*2 - B*b**3*e**3 + 3*B*b**2*c*d*e**2 - 3*B*b*c**2*d**2*e + B*c**3*d**3)/(2*c**4)
)*log(x + (2*A*a**2*c**2*e**3 - A*a*b**2*c*e**3 + 3*A*a*b*c**2*d*e**2 - 6*A*a*c*
*3*d**2*e + A*b*c**3*d**3 - 3*B*a**2*b*c*e**3 + 6*B*a**2*c**2*d*e**2 + B*a*b**3*
e**3 - 3*B*a*b**2*c*d*e**2 + 3*B*a*b*c**2*d**2*e - 2*B*a*c**3*d**3 + 4*a*c**4*(s
qrt(-4*a*c + b**2)*(3*A*a*b*c**2*e**3 - 6*A*a*c**3*d*e**2 - A*b**3*c*e**3 + 3*A*
b**2*c**2*d*e**2 - 3*A*b*c**3*d**2*e + 2*A*c**4*d**3 + 2*B*a**2*c**2*e**3 - 4*B*
a*b**2*c*e**3 + 9*B*a*b*c**2*d*e**2 - 6*B*a*c**3*d**2*e + B*b**4*e**3 - 3*B*b**3
*c*d*e**2 + 3*B*b**2*c**2*d**2*e - B*b*c**3*d**3)/(2*c**4*(4*a*c - b**2)) + (-A*
a*c**2*e**3 + A*b**2*c*e**3 - 3*A*b*c**2*d*e**2 + 3*A*c**3*d**2*e + 2*B*a*b*c*e*
*3 - 3*B*a*c**2*d*e**2 - B*b**3*e**3 + 3*B*b**2*c*d*e**2 - 3*B*b*c**2*d**2*e + B
*c**3*d**3)/(2*c**4)) - b**2*c**3*(sqrt(-4*a*c + b**2)*(3*A*a*b*c**2*e**3 - 6*A*
a*c**3*d*e**2 - A*b**3*c*e**3 + 3*A*b**2*c**2*d*e**2 - 3*A*b*c**3*d**2*e + 2*A*c
**4*d**3 + 2*B*a**2*c**2*e**3 - 4*B*a*b**2*c*e**3 + 9*B*a*b*c**2*d*e**2 - 6*B*a*
c**3*d**2*e + B*b**4*e**3 - 3*B*b**3*c*d*e**2 + 3*B*b**2*c**2*d**2*e - B*b*c**3*
d**3)/(2*c**4*(4*a*c - b**2)) + (-A*a*c**2*e**3 + A*b**2*c*e**3 - 3*A*b*c**2*d*e
**2 + 3*A*c**3*d**2*e + 2*B*a*b*c*e**3 - 3*B*a*c**2*d*e**2 - B*b**3*e**3 + 3*B*b
**2*c*d*e**2 - 3*B*b*c**2*d**2*e + B*c**3*d**3)/(2*c**4)))/(3*A*a*b*c**2*e**3 -
6*A*a*c**3*d*e**2 - A*b**3*c*e**3 + 3*A*b**2*c**2*d*e**2 - 3*A*b*c**3*d**2*e + 2
*A*c**4*d**3 + 2*B*a**2*c**2*e**3 - 4*B*a*b**2*c*e**3 + 9*B*a*b*c**2*d*e**2 - 6*
B*a*c**3*d**2*e + B*b**4*e**3 - 3*B*b**3*c*d*e**2 + 3*B*b**2*c**2*d**2*e - B*b*c
**3*d**3)) - x**2*(-A*c*e**3 + B*b*e**3 - 3*B*c*d*e**2)/(2*c**2) - x*(A*b*c*e**3
- 3*A*c**2*d*e**2 + B*a*c*e**3 - B*b**2*e**3 + 3*B*b*c*d*e**2 - 3*B*c**2*d**2*e
)/c**3

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GIAC/XCAS [A]  time = 0.279061, size = 543, normalized size = 1.52 \[ \frac{2 \, B c^{2} x^{3} e^{3} + 9 \, B c^{2} d x^{2} e^{2} + 18 \, B c^{2} d^{2} x e - 3 \, B b c x^{2} e^{3} + 3 \, A c^{2} x^{2} e^{3} - 18 \, B b c d x e^{2} + 18 \, A c^{2} d x e^{2} + 6 \, B b^{2} x e^{3} - 6 \, B a c x e^{3} - 6 \, A b c x e^{3}}{6 \, c^{3}} + \frac{{\left (B c^{3} d^{3} - 3 \, B b c^{2} d^{2} e + 3 \, A c^{3} d^{2} e + 3 \, B b^{2} c d e^{2} - 3 \, B a c^{2} d e^{2} - 3 \, A b c^{2} d e^{2} - B b^{3} e^{3} + 2 \, B a b c e^{3} + A b^{2} c e^{3} - A a c^{2} e^{3}\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, c^{4}} - \frac{{\left (B b c^{3} d^{3} - 2 \, A c^{4} d^{3} - 3 \, B b^{2} c^{2} d^{2} e + 6 \, B a c^{3} d^{2} e + 3 \, A b c^{3} d^{2} e + 3 \, B b^{3} c d e^{2} - 9 \, B a b c^{2} d e^{2} - 3 \, A b^{2} c^{2} d e^{2} + 6 \, A a c^{3} d e^{2} - B b^{4} e^{3} + 4 \, B a b^{2} c e^{3} + A b^{3} c e^{3} - 2 \, B a^{2} c^{2} e^{3} - 3 \, A a b c^{2} e^{3}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(2*B*c^2*x^3*e^3 + 9*B*c^2*d*x^2*e^2 + 18*B*c^2*d^2*x*e - 3*B*b*c*x^2*e^3 +
3*A*c^2*x^2*e^3 - 18*B*b*c*d*x*e^2 + 18*A*c^2*d*x*e^2 + 6*B*b^2*x*e^3 - 6*B*a*c*
x*e^3 - 6*A*b*c*x*e^3)/c^3 + 1/2*(B*c^3*d^3 - 3*B*b*c^2*d^2*e + 3*A*c^3*d^2*e +
3*B*b^2*c*d*e^2 - 3*B*a*c^2*d*e^2 - 3*A*b*c^2*d*e^2 - B*b^3*e^3 + 2*B*a*b*c*e^3
+ A*b^2*c*e^3 - A*a*c^2*e^3)*ln(c*x^2 + b*x + a)/c^4 - (B*b*c^3*d^3 - 2*A*c^4*d^
3 - 3*B*b^2*c^2*d^2*e + 6*B*a*c^3*d^2*e + 3*A*b*c^3*d^2*e + 3*B*b^3*c*d*e^2 - 9*
B*a*b*c^2*d*e^2 - 3*A*b^2*c^2*d*e^2 + 6*A*a*c^3*d*e^2 - B*b^4*e^3 + 4*B*a*b^2*c*
e^3 + A*b^3*c*e^3 - 2*B*a^2*c^2*e^3 - 3*A*a*b*c^2*e^3)*arctan((2*c*x + b)/sqrt(-
b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^4)