3.2338 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )^3}{d+e x} \, dx\)

Optimal. Leaf size=544 \[ -\frac{(d+e x)^3 \left (A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )\right )}{3 e^8}-\frac{3 c (d+e x)^5 \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{5 e^8}-\frac{3 (d+e x)^2 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{2 e^8}-\frac{(d+e x)^4 \left (B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{4 e^8}-\frac{(B d-A e) \log (d+e x) \left (a e^2-b d e+c d^2\right )^3}{e^8}-\frac{x \left (a e^2-b d e+c d^2\right )^2 \left (3 A e (2 c d-b e)-B \left (7 c d^2-e (4 b d-a e)\right )\right )}{e^7}-\frac{c^2 (d+e x)^6 (-A c e-3 b B e+7 B c d)}{6 e^8}+\frac{B c^3 (d+e x)^7}{7 e^8} \]

[Out]

-(((c*d^2 - b*d*e + a*e^2)^2*(3*A*e*(2*c*d - b*e) - B*(7*c*d^2 - e*(4*b*d - a*e)
))*x)/e^7) - (3*(c*d^2 - b*d*e + a*e^2)*(B*(7*c^2*d^3 - c*d*e*(8*b*d - 3*a*e) +
b*e^2*(2*b*d - a*e)) - A*e*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e)))*(d + e*x)^
2)/(2*e^8) - ((A*e*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))
- B*(35*c^3*d^4 - b^2*e^3*(4*b*d - 3*a*e) - 30*c^2*d^2*e*(2*b*d - a*e) + 3*c*e^2
*(10*b^2*d^2 - 8*a*b*d*e + a^2*e^2)))*(d + e*x)^3)/(3*e^8) - ((B*(35*c^3*d^3 - b
^3*e^3 + 3*b*c*e^2*(5*b*d - 2*a*e) - 15*c^2*d*e*(3*b*d - a*e)) - 3*A*c*e*(5*c^2*
d^2 + b^2*e^2 - c*e*(5*b*d - a*e)))*(d + e*x)^4)/(4*e^8) - (3*c*(A*c*e*(2*c*d -
b*e) - B*(7*c^2*d^2 + b^2*e^2 - c*e*(6*b*d - a*e)))*(d + e*x)^5)/(5*e^8) - (c^2*
(7*B*c*d - 3*b*B*e - A*c*e)*(d + e*x)^6)/(6*e^8) + (B*c^3*(d + e*x)^7)/(7*e^8) -
((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^3*Log[d + e*x])/e^8

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Rubi [A]  time = 3.44741, antiderivative size = 541, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04 \[ -\frac{(d+e x)^3 \left (A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )\right )}{3 e^8}-\frac{3 c (d+e x)^5 \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{5 e^8}-\frac{3 (d+e x)^2 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{2 e^8}-\frac{(d+e x)^4 \left (B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{4 e^8}-\frac{(B d-A e) \log (d+e x) \left (a e^2-b d e+c d^2\right )^3}{e^8}+\frac{x \left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{e^7}-\frac{c^2 (d+e x)^6 (-A c e-3 b B e+7 B c d)}{6 e^8}+\frac{B c^3 (d+e x)^7}{7 e^8} \]

Antiderivative was successfully verified.

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

[Out]

((c*d^2 - b*d*e + a*e^2)^2*(7*B*c*d^2 - B*e*(4*b*d - a*e) - 3*A*e*(2*c*d - b*e))
*x)/e^7 - (3*(c*d^2 - b*d*e + a*e^2)*(B*(7*c^2*d^3 - c*d*e*(8*b*d - 3*a*e) + b*e
^2*(2*b*d - a*e)) - A*e*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e)))*(d + e*x)^2)/
(2*e^8) - ((A*e*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e)) - B
*(35*c^3*d^4 - b^2*e^3*(4*b*d - 3*a*e) - 30*c^2*d^2*e*(2*b*d - a*e) + 3*c*e^2*(1
0*b^2*d^2 - 8*a*b*d*e + a^2*e^2)))*(d + e*x)^3)/(3*e^8) - ((B*(35*c^3*d^3 - b^3*
e^3 + 3*b*c*e^2*(5*b*d - 2*a*e) - 15*c^2*d*e*(3*b*d - a*e)) - 3*A*c*e*(5*c^2*d^2
+ b^2*e^2 - c*e*(5*b*d - a*e)))*(d + e*x)^4)/(4*e^8) - (3*c*(A*c*e*(2*c*d - b*e
) - B*(7*c^2*d^2 + b^2*e^2 - c*e*(6*b*d - a*e)))*(d + e*x)^5)/(5*e^8) - (c^2*(7*
B*c*d - 3*b*B*e - A*c*e)*(d + e*x)^6)/(6*e^8) + (B*c^3*(d + e*x)^7)/(7*e^8) - ((
B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^3*Log[d + e*x])/e^8

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

[Out]

Timed out

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Mathematica [A]  time = 1.44027, size = 700, normalized size = 1.29 \[ \frac{e x \left (7 A e \left (15 c e^2 \left (6 a^2 e^2 (e x-2 d)+4 a b e \left (6 d^2-3 d e x+2 e^2 x^2\right )+b^2 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )+10 b e^3 \left (18 a^2 e^2+9 a b e (e x-2 d)+b^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+3 c^2 e \left (5 a e \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+b \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )+c^3 \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )\right )+B \left (21 c e^2 \left (10 a^2 e^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )+10 a b e \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+b^2 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )+35 e^3 \left (12 a^3 e^3+18 a^2 b e^2 (e x-2 d)+6 a b^2 e \left (6 d^2-3 d e x+2 e^2 x^2\right )+b^3 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )+21 c^2 e \left (a e \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )+b \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )\right )+c^3 \left (420 d^6-210 d^5 e x+140 d^4 e^2 x^2-105 d^3 e^3 x^3+84 d^2 e^4 x^4-70 d e^5 x^5+60 e^6 x^6\right )\right )\right )-420 (B d-A e) \log (d+e x) \left (e (a e-b d)+c d^2\right )^3}{420 e^8} \]

Antiderivative was successfully verified.

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

[Out]

(e*x*(7*A*e*(c^3*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*
e^4*x^4 + 10*e^5*x^5) + 10*b*e^3*(18*a^2*e^2 + 9*a*b*e*(-2*d + e*x) + b^2*(6*d^2
- 3*d*e*x + 2*e^2*x^2)) + 15*c*e^2*(6*a^2*e^2*(-2*d + e*x) + 4*a*b*e*(6*d^2 - 3
*d*e*x + 2*e^2*x^2) + b^2*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3)) + 3*c
^2*e*(5*a*e*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + b*(60*d^4 - 30*d^3
*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4))) + B*(c^3*(420*d^6 - 210*d^5
*e*x + 140*d^4*e^2*x^2 - 105*d^3*e^3*x^3 + 84*d^2*e^4*x^4 - 70*d*e^5*x^5 + 60*e^
6*x^6) + 35*e^3*(12*a^3*e^3 + 18*a^2*b*e^2*(-2*d + e*x) + 6*a*b^2*e*(6*d^2 - 3*d
*e*x + 2*e^2*x^2) + b^3*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3)) + 21*c*
e^2*(10*a^2*e^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + 10*a*b*e*(-12*d^3 + 6*d^2*e*x -
4*d*e^2*x^2 + 3*e^3*x^3) + b^2*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*
x^3 + 12*e^4*x^4)) + 21*c^2*e*(a*e*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*
e^3*x^3 + 12*e^4*x^4) + b*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^
3 - 12*d*e^4*x^4 + 10*e^5*x^5)))) - 420*(B*d - A*e)*(c*d^2 + e*(-(b*d) + a*e))^3
*Log[d + e*x])/(420*e^8)

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Maple [B]  time = 0.013, size = 1319, normalized size = 2.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/e^3*A*b^3*d^2*x+3/e*A*a^2*b*x+1/7*B*c^3*x^7/e-3/e^2*A*x^2*a*b*c*d-6/e^4*ln(e*x
+d)*A*a*b*c*d^3+6/e^5*ln(e*x+d)*B*a*b*c*d^4-6/e^4*B*a*b*c*d^3*x+3/e^3*B*x^2*a*b*
c*d^2+6/e^3*A*a*b*c*d^2*x-2/e^2*B*x^3*a*b*c*d-3/e^4*A*b^2*c*d^3*x+3/e^5*A*b*c^2*
d^4*x-3/e^2*A*a*b^2*d*x-3/e^4*A*a*c^2*d^3*x-3/2/e^2*B*x^2*a*b^2*d-3/2/e^4*B*x^2*
a*c^2*d^3-3/2/e^4*B*x^2*b^2*c*d^3+3/2/e^5*B*x^2*b*c^2*d^4-3/e^2*A*a^2*c*d*x+3/e^
5*B*b^2*c*d^4*x-3/e^6*B*b*c^2*d^5*x-3/4/e^2*B*x^4*a*c^2*d-3/4/e^2*B*x^4*b^2*c*d+
3/4/e^3*B*x^4*b*c^2*d^2-3/2/e^4*A*x^2*b*c^2*d^3-3/2/e^2*B*x^2*a^2*c*d+3/e^3*B*a*
b^2*d^2*x+3/e^5*B*a*c^2*d^4*x+3/e^3*ln(e*x+d)*A*a^2*c*d^2+3/e^3*ln(e*x+d)*A*a*b^
2*d^2+1/e^3*B*x^3*b^2*c*d^2+2/e*A*x^3*a*b*c-1/e^2*A*x^3*a*c^2*d-1/e^2*A*x^3*b^2*
c*d-1/e^4*B*x^3*b*c^2*d^3+3/2/e^3*A*x^2*a*c^2*d^2+3/2/e^3*A*x^2*b^2*c*d^2-3/5/e^
2*B*x^5*b*c^2*d-3/4/e^2*A*x^4*b*c^2*d+3/2/e*B*x^4*a*b*c+1/e^3*A*x^3*b*c^2*d^2+1/
e^3*B*x^3*a*c^2*d^2-3/e^2*B*a^2*b*d*x+3/e^3*B*a^2*c*d^2*x-3/e^4*ln(e*x+d)*B*a*b^
2*d^3+3/e^5*ln(e*x+d)*A*a*c^2*d^4+3/e^5*ln(e*x+d)*A*b^2*c*d^4-3/e^2*ln(e*x+d)*A*
a^2*b*d+3/e^7*ln(e*x+d)*B*b*c^2*d^6-3/e^6*ln(e*x+d)*B*a*c^2*d^5-3/e^6*ln(e*x+d)*
B*b^2*c*d^5-3/e^6*ln(e*x+d)*A*b*c^2*d^5+3/e^3*ln(e*x+d)*B*a^2*b*d^2-3/e^4*ln(e*x
+d)*B*a^2*c*d^3+1/e*B*a^3*x+1/3/e*A*x^3*b^3+1/4/e*B*x^4*b^3+1/6/e*A*x^6*c^3+1/e*
ln(e*x+d)*A*a^3-1/6/e^2*B*x^6*c^3*d+1/e^7*B*c^3*d^6*x+1/e*B*x^3*a^2*c+1/e*B*x^3*
a*b^2+3/2/e*A*x^2*a^2*c+3/2/e*A*x^2*a*b^2-1/2/e^2*A*x^2*b^3*d+1/2/e^5*A*x^2*c^3*
d^4+3/2/e*B*x^2*a^2*b-1/3/e^2*B*x^3*b^3*d-1/2/e^6*B*x^2*c^3*d^5-1/e^6*A*c^3*d^5*
x-1/e^4*B*b^3*d^3*x+1/2/e^3*B*x^2*b^3*d^2-1/e^4*ln(e*x+d)*A*b^3*d^3+1/e^7*ln(e*x
+d)*A*c^3*d^6-1/e^2*ln(e*x+d)*B*a^3*d+1/e^5*ln(e*x+d)*B*b^3*d^4-1/e^8*ln(e*x+d)*
B*c^3*d^7+1/3/e^5*B*x^3*c^3*d^4+1/4/e^3*A*x^4*c^3*d^2-1/4/e^4*B*x^4*c^3*d^3-1/3/
e^4*A*x^3*c^3*d^3+1/5/e^3*B*x^5*c^3*d^2+3/4/e*A*x^4*a*c^2+3/4/e*A*x^4*b^2*c+3/5/
e*A*x^5*b*c^2-1/5/e^2*A*x^5*c^3*d+3/5/e*B*x^5*a*c^2+3/5/e*B*x^5*b^2*c+1/2/e*B*x^
6*b*c^2

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Maxima [A]  time = 0.705666, size = 1137, normalized size = 2.09 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/420*(60*B*c^3*e^6*x^7 - 70*(B*c^3*d*e^5 - (3*B*b*c^2 + A*c^3)*e^6)*x^6 + 84*(B
*c^3*d^2*e^4 - (3*B*b*c^2 + A*c^3)*d*e^5 + 3*(B*b^2*c + (B*a + A*b)*c^2)*e^6)*x^
5 - 105*(B*c^3*d^3*e^3 - (3*B*b*c^2 + A*c^3)*d^2*e^4 + 3*(B*b^2*c + (B*a + A*b)*
c^2)*d*e^5 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*e^6)*x^4 + 140*(B*c^3*d
^4*e^2 - (3*B*b*c^2 + A*c^3)*d^3*e^3 + 3*(B*b^2*c + (B*a + A*b)*c^2)*d^2*e^4 - (
B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d*e^5 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2
+ 2*A*a*b)*c)*e^6)*x^3 - 210*(B*c^3*d^5*e - (3*B*b*c^2 + A*c^3)*d^4*e^2 + 3*(B*
b^2*c + (B*a + A*b)*c^2)*d^3*e^3 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d
^2*e^4 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d*e^5 - 3*(B*a^2*b + A*a*b^
2 + A*a^2*c)*e^6)*x^2 + 420*(B*c^3*d^6 - (3*B*b*c^2 + A*c^3)*d^5*e + 3*(B*b^2*c
+ (B*a + A*b)*c^2)*d^4*e^2 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^3*e^3
+ (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^2*e^4 - 3*(B*a^2*b + A*a*b^2 +
A*a^2*c)*d*e^5 + (B*a^3 + 3*A*a^2*b)*e^6)*x)/e^7 - (B*c^3*d^7 - A*a^3*e^7 - (3*B
*b*c^2 + A*c^3)*d^6*e + 3*(B*b^2*c + (B*a + A*b)*c^2)*d^5*e^2 - (B*b^3 + 3*A*a*c
^2 + 3*(2*B*a*b + A*b^2)*c)*d^4*e^3 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c
)*d^3*e^4 - 3*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^2*e^5 + (B*a^3 + 3*A*a^2*b)*d*e^6)
*log(e*x + d)/e^8

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Fricas [A]  time = 0.263917, size = 1138, normalized size = 2.09 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/420*(60*B*c^3*e^7*x^7 - 70*(B*c^3*d*e^6 - (3*B*b*c^2 + A*c^3)*e^7)*x^6 + 84*(B
*c^3*d^2*e^5 - (3*B*b*c^2 + A*c^3)*d*e^6 + 3*(B*b^2*c + (B*a + A*b)*c^2)*e^7)*x^
5 - 105*(B*c^3*d^3*e^4 - (3*B*b*c^2 + A*c^3)*d^2*e^5 + 3*(B*b^2*c + (B*a + A*b)*
c^2)*d*e^6 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*e^7)*x^4 + 140*(B*c^3*d
^4*e^3 - (3*B*b*c^2 + A*c^3)*d^3*e^4 + 3*(B*b^2*c + (B*a + A*b)*c^2)*d^2*e^5 - (
B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d*e^6 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2
+ 2*A*a*b)*c)*e^7)*x^3 - 210*(B*c^3*d^5*e^2 - (3*B*b*c^2 + A*c^3)*d^4*e^3 + 3*(
B*b^2*c + (B*a + A*b)*c^2)*d^3*e^4 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)
*d^2*e^5 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d*e^6 - 3*(B*a^2*b + A*a*
b^2 + A*a^2*c)*e^7)*x^2 + 420*(B*c^3*d^6*e - (3*B*b*c^2 + A*c^3)*d^5*e^2 + 3*(B*
b^2*c + (B*a + A*b)*c^2)*d^4*e^3 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d
^3*e^4 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^2*e^5 - 3*(B*a^2*b + A*a*
b^2 + A*a^2*c)*d*e^6 + (B*a^3 + 3*A*a^2*b)*e^7)*x - 420*(B*c^3*d^7 - A*a^3*e^7 -
(3*B*b*c^2 + A*c^3)*d^6*e + 3*(B*b^2*c + (B*a + A*b)*c^2)*d^5*e^2 - (B*b^3 + 3*
A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^4*e^3 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a
*b)*c)*d^3*e^4 - 3*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^2*e^5 + (B*a^3 + 3*A*a^2*b)*d
*e^6)*log(e*x + d))/e^8

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Sympy [A]  time = 11.359, size = 949, normalized size = 1.74 \[ \frac{B c^{3} x^{7}}{7 e} + \frac{x^{6} \left (A c^{3} e + 3 B b c^{2} e - B c^{3} d\right )}{6 e^{2}} + \frac{x^{5} \left (3 A b c^{2} e^{2} - A c^{3} d e + 3 B a c^{2} e^{2} + 3 B b^{2} c e^{2} - 3 B b c^{2} d e + B c^{3} d^{2}\right )}{5 e^{3}} + \frac{x^{4} \left (3 A a c^{2} e^{3} + 3 A b^{2} c e^{3} - 3 A b c^{2} d e^{2} + A c^{3} d^{2} e + 6 B a b c e^{3} - 3 B a c^{2} d e^{2} + B b^{3} e^{3} - 3 B b^{2} c d e^{2} + 3 B b c^{2} d^{2} e - B c^{3} d^{3}\right )}{4 e^{4}} + \frac{x^{3} \left (6 A a b c e^{4} - 3 A a c^{2} d e^{3} + A b^{3} e^{4} - 3 A b^{2} c d e^{3} + 3 A b c^{2} d^{2} e^{2} - A c^{3} d^{3} e + 3 B a^{2} c e^{4} + 3 B a b^{2} e^{4} - 6 B a b c d e^{3} + 3 B a c^{2} d^{2} e^{2} - B b^{3} d e^{3} + 3 B b^{2} c d^{2} e^{2} - 3 B b c^{2} d^{3} e + B c^{3} d^{4}\right )}{3 e^{5}} + \frac{x^{2} \left (3 A a^{2} c e^{5} + 3 A a b^{2} e^{5} - 6 A a b c d e^{4} + 3 A a c^{2} d^{2} e^{3} - A b^{3} d e^{4} + 3 A b^{2} c d^{2} e^{3} - 3 A b c^{2} d^{3} e^{2} + A c^{3} d^{4} e + 3 B a^{2} b e^{5} - 3 B a^{2} c d e^{4} - 3 B a b^{2} d e^{4} + 6 B a b c d^{2} e^{3} - 3 B a c^{2} d^{3} e^{2} + B b^{3} d^{2} e^{3} - 3 B b^{2} c d^{3} e^{2} + 3 B b c^{2} d^{4} e - B c^{3} d^{5}\right )}{2 e^{6}} + \frac{x \left (3 A a^{2} b e^{6} - 3 A a^{2} c d e^{5} - 3 A a b^{2} d e^{5} + 6 A a b c d^{2} e^{4} - 3 A a c^{2} d^{3} e^{3} + A b^{3} d^{2} e^{4} - 3 A b^{2} c d^{3} e^{3} + 3 A b c^{2} d^{4} e^{2} - A c^{3} d^{5} e + B a^{3} e^{6} - 3 B a^{2} b d e^{5} + 3 B a^{2} c d^{2} e^{4} + 3 B a b^{2} d^{2} e^{4} - 6 B a b c d^{3} e^{3} + 3 B a c^{2} d^{4} e^{2} - B b^{3} d^{3} e^{3} + 3 B b^{2} c d^{4} e^{2} - 3 B b c^{2} d^{5} e + B c^{3} d^{6}\right )}{e^{7}} - \frac{\left (- A e + B d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{3} \log{\left (d + e x \right )}}{e^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*c**3*x**7/(7*e) + x**6*(A*c**3*e + 3*B*b*c**2*e - B*c**3*d)/(6*e**2) + x**5*(3
*A*b*c**2*e**2 - A*c**3*d*e + 3*B*a*c**2*e**2 + 3*B*b**2*c*e**2 - 3*B*b*c**2*d*e
+ B*c**3*d**2)/(5*e**3) + x**4*(3*A*a*c**2*e**3 + 3*A*b**2*c*e**3 - 3*A*b*c**2*
d*e**2 + A*c**3*d**2*e + 6*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)/(4*e**4) + x**3*(6*A*a*b*c*e**4
- 3*A*a*c**2*d*e**3 + A*b**3*e**4 - 3*A*b**2*c*d*e**3 + 3*A*b*c**2*d**2*e**2 -
A*c**3*d**3*e + 3*B*a**2*c*e**4 + 3*B*a*b**2*e**4 - 6*B*a*b*c*d*e**3 + 3*B*a*c**
2*d**2*e**2 - B*b**3*d*e**3 + 3*B*b**2*c*d**2*e**2 - 3*B*b*c**2*d**3*e + B*c**3*
d**4)/(3*e**5) + x**2*(3*A*a**2*c*e**5 + 3*A*a*b**2*e**5 - 6*A*a*b*c*d*e**4 + 3*
A*a*c**2*d**2*e**3 - A*b**3*d*e**4 + 3*A*b**2*c*d**2*e**3 - 3*A*b*c**2*d**3*e**2
+ A*c**3*d**4*e + 3*B*a**2*b*e**5 - 3*B*a**2*c*d*e**4 - 3*B*a*b**2*d*e**4 + 6*B
*a*b*c*d**2*e**3 - 3*B*a*c**2*d**3*e**2 + B*b**3*d**2*e**3 - 3*B*b**2*c*d**3*e**
2 + 3*B*b*c**2*d**4*e - B*c**3*d**5)/(2*e**6) + x*(3*A*a**2*b*e**6 - 3*A*a**2*c*
d*e**5 - 3*A*a*b**2*d*e**5 + 6*A*a*b*c*d**2*e**4 - 3*A*a*c**2*d**3*e**3 + A*b**3
*d**2*e**4 - 3*A*b**2*c*d**3*e**3 + 3*A*b*c**2*d**4*e**2 - A*c**3*d**5*e + B*a**
3*e**6 - 3*B*a**2*b*d*e**5 + 3*B*a**2*c*d**2*e**4 + 3*B*a*b**2*d**2*e**4 - 6*B*a
*b*c*d**3*e**3 + 3*B*a*c**2*d**4*e**2 - B*b**3*d**3*e**3 + 3*B*b**2*c*d**4*e**2
- 3*B*b*c**2*d**5*e + B*c**3*d**6)/e**7 - (-A*e + B*d)*(a*e**2 - b*d*e + c*d**2)
**3*log(d + e*x)/e**8

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GIAC/XCAS [A]  time = 0.279407, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done