3.2343 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx\)

Optimal. Leaf size=534 \[ \frac{A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )}{2 e^8 (d+e x)^2}-\frac{3 c \log (d+e x) \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{e^8}+\frac{\left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{e^8 (d+e x)^3}+\frac{B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^8 (d+e x)}+\frac{\left (a e^2-b d e+c d^2\right )^2 \left (3 A e (2 c d-b e)-B \left (7 c d^2-e (4 b d-a e)\right )\right )}{4 e^8 (d+e x)^4}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{5 e^8 (d+e x)^5}-\frac{c^2 x (-A c e-3 b B e+6 B c d)}{e^7}+\frac{B c^3 x^2}{2 e^6} \]

[Out]

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

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Rubi [A]  time = 3.75393, antiderivative size = 532, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04 \[ \frac{A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )}{2 e^8 (d+e x)^2}-\frac{3 c \log (d+e x) \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{e^8}+\frac{\left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{e^8 (d+e x)^3}+\frac{B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^8 (d+e x)}-\frac{\left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{4 e^8 (d+e x)^4}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{5 e^8 (d+e x)^5}-\frac{c^2 x (-A c e-3 b B e+6 B c d)}{e^7}+\frac{B c^3 x^2}{2 e^6} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^6,x]

[Out]

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

[Out]

Timed out

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Mathematica [A]  time = 1.97448, size = 885, normalized size = 1.66 \[ \frac{60 c \left (A c e (b e-2 c d)+B \left (7 c^2 d^2+b^2 e^2+c e (a e-6 b d)\right )\right ) \log (d+e x) (d+e x)^5+A e \left (-2 \left (87 d^6+375 e x d^5+600 e^2 x^2 d^4+400 e^3 x^3 d^3+50 e^4 x^4 d^2-50 e^5 x^5 d-10 e^6 x^6\right ) c^3+e \left (b d \left (137 d^4+625 e x d^3+1100 e^2 x^2 d^2+900 e^3 x^3 d+300 e^4 x^4\right )-12 a e \left (d^4+5 e x d^3+10 e^2 x^2 d^2+10 e^3 x^3 d+5 e^4 x^4\right )\right ) c^2-2 e^2 \left (6 \left (d^4+5 e x d^3+10 e^2 x^2 d^2+10 e^3 x^3 d+5 e^4 x^4\right ) b^2+3 a e \left (d^3+5 e x d^2+10 e^2 x^2 d+10 e^3 x^3\right ) b+a^2 e^2 \left (d^2+5 e x d+10 e^2 x^2\right )\right ) c-e^3 \left (\left (d^3+5 e x d^2+10 e^2 x^2 d+10 e^3 x^3\right ) b^3+2 a e \left (d^2+5 e x d+10 e^2 x^2\right ) b^2+3 a^2 e^2 (d+5 e x) b+4 a^3 e^3\right )\right )+B \left (\left (459 d^7+1875 e x d^6+2700 e^2 x^2 d^5+1300 e^3 x^3 d^4-400 e^4 x^4 d^3-500 e^5 x^5 d^2-70 e^6 x^6 d+10 e^7 x^7\right ) c^3+e \left (a d e \left (137 d^4+625 e x d^3+1100 e^2 x^2 d^2+900 e^3 x^3 d+300 e^4 x^4\right )-6 b \left (87 d^6+375 e x d^5+600 e^2 x^2 d^4+400 e^3 x^3 d^3+50 e^4 x^4 d^2-50 e^5 x^5 d-10 e^6 x^6\right )\right ) c^2+e^2 \left (d \left (137 d^4+625 e x d^3+1100 e^2 x^2 d^2+900 e^3 x^3 d+300 e^4 x^4\right ) b^2-24 a e \left (d^4+5 e x d^3+10 e^2 x^2 d^2+10 e^3 x^3 d+5 e^4 x^4\right ) b-3 a^2 e^2 \left (d^3+5 e x d^2+10 e^2 x^2 d+10 e^3 x^3\right )\right ) c-e^3 \left (4 \left (d^4+5 e x d^3+10 e^2 x^2 d^2+10 e^3 x^3 d+5 e^4 x^4\right ) b^3+3 a e \left (d^3+5 e x d^2+10 e^2 x^2 d+10 e^3 x^3\right ) b^2+2 a^2 e^2 \left (d^2+5 e x d+10 e^2 x^2\right ) b+a^3 e^3 (d+5 e x)\right )\right )}{20 e^8 (d+e x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^6,x]

[Out]

(A*e*(-2*c^3*(87*d^6 + 375*d^5*e*x + 600*d^4*e^2*x^2 + 400*d^3*e^3*x^3 + 50*d^2*
e^4*x^4 - 50*d*e^5*x^5 - 10*e^6*x^6) - e^3*(4*a^3*e^3 + 3*a^2*b*e^2*(d + 5*e*x)
+ 2*a*b^2*e*(d^2 + 5*d*e*x + 10*e^2*x^2) + b^3*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 +
10*e^3*x^3)) - 2*c*e^2*(a^2*e^2*(d^2 + 5*d*e*x + 10*e^2*x^2) + 3*a*b*e*(d^3 + 5
*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + 6*b^2*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2
+ 10*d*e^3*x^3 + 5*e^4*x^4)) + c^2*e*(-12*a*e*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2
+ 10*d*e^3*x^3 + 5*e^4*x^4) + b*d*(137*d^4 + 625*d^3*e*x + 1100*d^2*e^2*x^2 + 90
0*d*e^3*x^3 + 300*e^4*x^4))) + B*(c^3*(459*d^7 + 1875*d^6*e*x + 2700*d^5*e^2*x^2
+ 1300*d^4*e^3*x^3 - 400*d^3*e^4*x^4 - 500*d^2*e^5*x^5 - 70*d*e^6*x^6 + 10*e^7*
x^7) - e^3*(a^3*e^3*(d + 5*e*x) + 2*a^2*b*e^2*(d^2 + 5*d*e*x + 10*e^2*x^2) + 3*a
*b^2*e*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + 4*b^3*(d^4 + 5*d^3*e*x +
10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4)) + c*e^2*(-3*a^2*e^2*(d^3 + 5*d^2*e*x
+ 10*d*e^2*x^2 + 10*e^3*x^3) - 24*a*b*e*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*
d*e^3*x^3 + 5*e^4*x^4) + b^2*d*(137*d^4 + 625*d^3*e*x + 1100*d^2*e^2*x^2 + 900*d
*e^3*x^3 + 300*e^4*x^4)) + c^2*e*(a*d*e*(137*d^4 + 625*d^3*e*x + 1100*d^2*e^2*x^
2 + 900*d*e^3*x^3 + 300*e^4*x^4) - 6*b*(87*d^6 + 375*d^5*e*x + 600*d^4*e^2*x^2 +
400*d^3*e^3*x^3 + 50*d^2*e^4*x^4 - 50*d*e^5*x^5 - 10*e^6*x^6))) + 60*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*Log[d
+ e*x])/(20*e^8*(d + e*x)^5)

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Maple [B]  time = 0.026, size = 1637, normalized size = 3.1 \[ \text{result too large to display} \]

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)^6,x)

[Out]

1/2*B*c^3*x^2/e^6-6/5/e^5/(e*x+d)^5*B*d^4*a*b*c+6/e^4/(e*x+d)^3*A*a*b*c*d-12/e^5
/(e*x+d)^3*B*a*b*c*d^2-9/2/e^4/(e*x+d)^4*A*a*b*c*d^2+6/e^5/(e*x+d)^4*B*a*b*c*d^3
+12/e^5/(e*x+d)^2*B*a*b*c*d+6/5/e^4/(e*x+d)^5*A*d^3*a*b*c-6/e^5/(e*x+d)^3*A*a*c^
2*d^2-6/e^5/(e*x+d)^3*A*b^2*c*d^2+10/e^6/(e*x+d)^3*A*b*c^2*d^3+3/e^4/(e*x+d)^3*B
*a^2*c*d+3/e^4/(e*x+d)^3*B*a*b^2*d+10/e^6/(e*x+d)^3*B*a*c^2*d^3+10/e^6/(e*x+d)^3
*B*b^2*c*d^3-15/e^7/(e*x+d)^3*B*b*c^2*d^4-18*c^2/e^7*ln(e*x+d)*B*b*d+15/e^6/(e*x
+d)*A*b*c^2*d-6/e^5/(e*x+d)*B*a*b*c+15/e^6/(e*x+d)*B*a*c^2*d+15/e^6/(e*x+d)*B*b^
2*c*d+6/e^5/(e*x+d)^2*A*b^2*c*d-15/e^6/(e*x+d)^2*A*b*c^2*d^2+6/e^5/(e*x+d)^2*A*a
*c^2*d-9/4/e^4/(e*x+d)^4*B*a*b^2*d^2-15/4/e^6/(e*x+d)^4*B*a*c^2*d^4-15/4/e^6/(e*
x+d)^4*B*b^2*c*d^4+9/2/e^7/(e*x+d)^4*B*b*c^2*d^5-3/e^4/(e*x+d)^2*A*a*b*c+3/2/e^3
/(e*x+d)^4*B*a^2*b*d-9/4/e^4/(e*x+d)^4*B*a^2*c*d^2+3/5/e^4/(e*x+d)^5*B*d^3*a^2*c
-45/e^7/(e*x+d)*B*b*c^2*d^2+3/2/e^3/(e*x+d)^4*A*a^2*c*d+3/2/e^3/(e*x+d)^4*A*a*b^
2*d+3/e^5/(e*x+d)^4*A*a*c^2*d^3+3/e^5/(e*x+d)^4*A*b^2*c*d^3-15/4/e^6/(e*x+d)^4*A
*b*c^2*d^4-3/5/e^3/(e*x+d)^5*A*d^2*a^2*c-3/5/e^3/(e*x+d)^5*A*d^2*a*b^2-3/5/e^5/(
e*x+d)^5*A*d^4*a*c^2-3/5/e^5/(e*x+d)^5*A*d^4*b^2*c+3/5/e^6/(e*x+d)^5*A*d^5*b*c^2
-3/5/e^3/(e*x+d)^5*B*d^2*a^2*b+3/5/e^6/(e*x+d)^5*B*d^5*a*c^2+3/5/e^6/(e*x+d)^5*B
*d^5*b^2*c-3/5/e^7/(e*x+d)^5*B*b*c^2*d^6-15/e^6/(e*x+d)^2*B*a*c^2*d^2-15/e^6/(e*
x+d)^2*B*b^2*c*d^2+30/e^7/(e*x+d)^2*B*b*c^2*d^3+3/5/e^2/(e*x+d)^5*A*d*a^2*b+3/5/
e^4/(e*x+d)^5*B*d^3*a*b^2-1/5/e/(e*x+d)^5*A*a^3-1/e^5/(e*x+d)*B*b^3-1/4/e^2/(e*x
+d)^4*B*a^3-1/2/e^4/(e*x+d)^2*A*b^3+c^3/e^6*A*x-15/e^7/(e*x+d)*A*c^3*d^2+35/e^8/
(e*x+d)*B*c^3*d^3-3/4/e^2/(e*x+d)^4*A*a^2*b-3/4/e^4/(e*x+d)^4*A*b^3*d^2+3/2/e^7/
(e*x+d)^4*A*c^3*d^5+1/e^5/(e*x+d)^4*B*b^3*d^3-7/4/e^8/(e*x+d)^4*B*c^3*d^6+3*c^2/
e^6*b*B*x-6*c^3/e^7*B*d*x-1/e^3/(e*x+d)^3*A*a^2*c-1/e^3/(e*x+d)^3*A*a*b^2+1/e^4/
(e*x+d)^3*A*b^3*d-5/e^7/(e*x+d)^3*A*c^3*d^4-3/2/e^4/(e*x+d)^2*B*a*b^2+2/e^5/(e*x
+d)^2*B*b^3*d-35/2/e^8/(e*x+d)^2*B*c^3*d^4+1/5/e^4/(e*x+d)^5*A*d^3*b^3-1/5/e^7/(
e*x+d)^5*A*c^3*d^6+1/5/e^2/(e*x+d)^5*B*d*a^3-1/5/e^5/(e*x+d)^5*B*d^4*b^3+1/5/e^8
/(e*x+d)^5*B*c^3*d^7-1/e^3/(e*x+d)^3*B*a^2*b-2/e^5/(e*x+d)^3*B*b^3*d^2+7/e^8/(e*
x+d)^3*B*c^3*d^5+3*c^2/e^6*ln(e*x+d)*A*b-6*c^3/e^7*ln(e*x+d)*A*d+3*c^2/e^6*ln(e*
x+d)*B*a+3*c/e^6*ln(e*x+d)*B*b^2+21*c^3/e^8*ln(e*x+d)*B*d^2-3/e^5/(e*x+d)*A*a*c^
2-3/e^5/(e*x+d)*A*b^2*c+10/e^7/(e*x+d)^2*A*c^3*d^3-3/2/e^4/(e*x+d)^2*B*a^2*c

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Maxima [A]  time = 0.729312, size = 1212, normalized size = 2.27 \[ \text{result too large to display} \]

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)^6,x, algorithm="maxima")

[Out]

1/20*(459*B*c^3*d^7 - 4*A*a^3*e^7 - 174*(3*B*b*c^2 + A*c^3)*d^6*e + 137*(B*b^2*c
+ (B*a + A*b)*c^2)*d^5*e^2 - 4*(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 - 2*(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 + 20*(35*B*c^3*d^3*e^4 - 15*(3*B
*b*c^2 + A*c^3)*d^2*e^5 + 15*(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 + 10*(245*B*c^3*d^4*e^3 - 100*(3*B*b*c^2 +
A*c^3)*d^3*e^4 + 90*(B*b^2*c + (B*a + A*b)*c^2)*d^2*e^5 - 4*(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 + 10*(329*B*c^3*d^5*e^2 - 130*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 110*(B*b^2*c +
(B*a + A*b)*c^2)*d^3*e^4 - 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 - 2*(B*a^2*b + A*a*b^2 + A
*a^2*c)*e^7)*x^2 + 5*(399*B*c^3*d^6*e - 154*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 125*(B
*b^2*c + (B*a + A*b)*c^2)*d^4*e^3 - 4*(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 - 2*(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)/(e^13*x^5 + 5*d*e^12*x^4 +
10*d^2*e^11*x^3 + 10*d^3*e^10*x^2 + 5*d^4*e^9*x + d^5*e^8) + 1/2*(B*c^3*e*x^2 -
2*(6*B*c^3*d - (3*B*b*c^2 + A*c^3)*e)*x)/e^7 + 3*(7*B*c^3*d^2 - 2*(3*B*b*c^2 +
A*c^3)*d*e + (B*b^2*c + (B*a + A*b)*c^2)*e^2)*log(e*x + d)/e^8

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Fricas [A]  time = 0.268766, size = 1683, normalized size = 3.15 \[ \text{result too large to display} \]

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)^6,x, algorithm="fricas")

[Out]

1/20*(10*B*c^3*e^7*x^7 + 459*B*c^3*d^7 - 4*A*a^3*e^7 - 174*(3*B*b*c^2 + A*c^3)*d
^6*e + 137*(B*b^2*c + (B*a + A*b)*c^2)*d^5*e^2 - 4*(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 - 2
*(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 - 10*(7*B*c^3
*d*e^6 - 2*(3*B*b*c^2 + A*c^3)*e^7)*x^6 - 100*(5*B*c^3*d^2*e^5 - (3*B*b*c^2 + A*
c^3)*d*e^6)*x^5 - 20*(20*B*c^3*d^3*e^4 + 5*(3*B*b*c^2 + A*c^3)*d^2*e^5 - 15*(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 + 10*(130*B*c^3*d^4*e^3 - 80*(3*B*b*c^2 + A*c^3)*d^3*e^4 + 90*(B*b^2*c + (B
*a + A*b)*c^2)*d^2*e^5 - 4*(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 + 10*(270*B*c^3*d^5*e^2 - 12
0*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 110*(B*b^2*c + (B*a + A*b)*c^2)*d^3*e^4 - 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 - 2*(B*a^2*b + A*a*b^2 + A*a^2*c)*e^7)*x^2 + 5*(375*B*c^3*d^
6*e - 150*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 125*(B*b^2*c + (B*a + A*b)*c^2)*d^4*e^3
- 4*(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 - 2*(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 + 60*(7*B*c^3*d^7 - 2*(3*B*b*c^2 + A*c^3)*d^6*e + (B*b^2*c +
(B*a + A*b)*c^2)*d^5*e^2 + (7*B*c^3*d^2*e^5 - 2*(3*B*b*c^2 + A*c^3)*d*e^6 + (B*b
^2*c + (B*a + A*b)*c^2)*e^7)*x^5 + 5*(7*B*c^3*d^3*e^4 - 2*(3*B*b*c^2 + A*c^3)*d^
2*e^5 + (B*b^2*c + (B*a + A*b)*c^2)*d*e^6)*x^4 + 10*(7*B*c^3*d^4*e^3 - 2*(3*B*b*
c^2 + A*c^3)*d^3*e^4 + (B*b^2*c + (B*a + A*b)*c^2)*d^2*e^5)*x^3 + 10*(7*B*c^3*d^
5*e^2 - 2*(3*B*b*c^2 + A*c^3)*d^4*e^3 + (B*b^2*c + (B*a + A*b)*c^2)*d^3*e^4)*x^2
+ 5*(7*B*c^3*d^6*e - 2*(3*B*b*c^2 + A*c^3)*d^5*e^2 + (B*b^2*c + (B*a + A*b)*c^2
)*d^4*e^3)*x)*log(e*x + d))/(e^13*x^5 + 5*d*e^12*x^4 + 10*d^2*e^11*x^3 + 10*d^3*
e^10*x^2 + 5*d^4*e^9*x + d^5*e^8)

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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((B*x+A)*(c*x**2+b*x+a)**3/(e*x+d)**6,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.257057, 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)^6,x, algorithm="giac")

[Out]

Done