3.2344 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^7} \, dx\)

Optimal. Leaf size=541 \[ \frac{A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )}{3 e^8 (d+e x)^3}+\frac{3 c \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{e^8 (d+e x)}+\frac{3 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{4 e^8 (d+e x)^4}+\frac{B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^8 (d+e x)^2}+\frac{\left (a e^2-b d e+c d^2\right )^2 \left (3 A e (2 c d-b e)-B \left (7 c d^2-e (4 b d-a e)\right )\right )}{5 e^8 (d+e x)^5}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{6 e^8 (d+e x)^6}-\frac{c^2 \log (d+e x) (-A c e-3 b B e+7 B c d)}{e^8}+\frac{B c^3 x}{e^7} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 3.65568, antiderivative size = 539, 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 \[ \frac{A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )}{3 e^8 (d+e x)^3}+\frac{3 c \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{e^8 (d+e x)}+\frac{3 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{4 e^8 (d+e x)^4}+\frac{B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^8 (d+e x)^2}-\frac{\left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{5 e^8 (d+e x)^5}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{6 e^8 (d+e x)^6}-\frac{c^2 \log (d+e x) (-A c e-3 b B e+7 B c d)}{e^8}+\frac{B c^3 x}{e^7} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^7,x]

[Out]

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

_______________________________________________________________________________________

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)**7,x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 6.02211, size = 868, normalized size = 1.6 \[ -\frac{60 c^2 (7 B c d-3 b B e-A c e) \log (d+e x) (d+e x)^6+A e \left (-d \left (147 d^5+822 e x d^4+1875 e^2 x^2 d^3+2200 e^3 x^3 d^2+1350 e^4 x^4 d+360 e^5 x^5\right ) c^3+6 e \left (a e \left (d^4+6 e x d^3+15 e^2 x^2 d^2+20 e^3 x^3 d+15 e^4 x^4\right )+5 b \left (d^5+6 e x d^4+15 e^2 x^2 d^3+20 e^3 x^3 d^2+15 e^4 x^4 d+6 e^5 x^5\right )\right ) c^2+3 e^2 \left (2 \left (d^4+6 e x d^3+15 e^2 x^2 d^2+20 e^3 x^3 d+15 e^4 x^4\right ) b^2+2 a e \left (d^3+6 e x d^2+15 e^2 x^2 d+20 e^3 x^3\right ) b+a^2 e^2 \left (d^2+6 e x d+15 e^2 x^2\right )\right ) c+e^3 \left (\left (d^3+6 e x d^2+15 e^2 x^2 d+20 e^3 x^3\right ) b^3+3 a e \left (d^2+6 e x d+15 e^2 x^2\right ) b^2+6 a^2 e^2 (d+6 e x) b+10 a^3 e^3\right )\right )+B \left (\left (669 d^7+3594 e x d^6+7725 e^2 x^2 d^5+8200 e^3 x^3 d^4+4050 e^4 x^4 d^3+360 e^5 x^5 d^2-360 e^6 x^6 d-60 e^7 x^7\right ) c^3+3 e \left (10 a e \left (d^5+6 e x d^4+15 e^2 x^2 d^3+20 e^3 x^3 d^2+15 e^4 x^4 d+6 e^5 x^5\right )-b d \left (147 d^5+822 e x d^4+1875 e^2 x^2 d^3+2200 e^3 x^3 d^2+1350 e^4 x^4 d+360 e^5 x^5\right )\right ) c^2+3 e^2 \left (10 \left (d^5+6 e x d^4+15 e^2 x^2 d^3+20 e^3 x^3 d^2+15 e^4 x^4 d+6 e^5 x^5\right ) b^2+4 a e \left (d^4+6 e x d^3+15 e^2 x^2 d^2+20 e^3 x^3 d+15 e^4 x^4\right ) b+a^2 e^2 \left (d^3+6 e x d^2+15 e^2 x^2 d+20 e^3 x^3\right )\right ) c+e^3 \left (2 \left (d^4+6 e x d^3+15 e^2 x^2 d^2+20 e^3 x^3 d+15 e^4 x^4\right ) b^3+3 a e \left (d^3+6 e x d^2+15 e^2 x^2 d+20 e^3 x^3\right ) b^2+3 a^2 e^2 \left (d^2+6 e x d+15 e^2 x^2\right ) b+2 a^3 e^3 (d+6 e x)\right )\right )}{60 e^8 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^7,x]

[Out]

-(A*e*(-(c^3*d*(147*d^5 + 822*d^4*e*x + 1875*d^3*e^2*x^2 + 2200*d^2*e^3*x^3 + 13
50*d*e^4*x^4 + 360*e^5*x^5)) + e^3*(10*a^3*e^3 + 6*a^2*b*e^2*(d + 6*e*x) + 3*a*b
^2*e*(d^2 + 6*d*e*x + 15*e^2*x^2) + b^3*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3
*x^3)) + 3*c*e^2*(a^2*e^2*(d^2 + 6*d*e*x + 15*e^2*x^2) + 2*a*b*e*(d^3 + 6*d^2*e*
x + 15*d*e^2*x^2 + 20*e^3*x^3) + 2*b^2*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*
e^3*x^3 + 15*e^4*x^4)) + 6*c^2*e*(a*e*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e
^3*x^3 + 15*e^4*x^4) + 5*b*(d^5 + 6*d^4*e*x + 15*d^3*e^2*x^2 + 20*d^2*e^3*x^3 +
15*d*e^4*x^4 + 6*e^5*x^5))) + B*(c^3*(669*d^7 + 3594*d^6*e*x + 7725*d^5*e^2*x^2
+ 8200*d^4*e^3*x^3 + 4050*d^3*e^4*x^4 + 360*d^2*e^5*x^5 - 360*d*e^6*x^6 - 60*e^7
*x^7) + e^3*(2*a^3*e^3*(d + 6*e*x) + 3*a^2*b*e^2*(d^2 + 6*d*e*x + 15*e^2*x^2) +
3*a*b^2*e*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + 2*b^3*(d^4 + 6*d^3*e*x
+ 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4)) + 3*c*e^2*(a^2*e^2*(d^3 + 6*d^2*
e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + 4*a*b*e*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 2
0*d*e^3*x^3 + 15*e^4*x^4) + 10*b^2*(d^5 + 6*d^4*e*x + 15*d^3*e^2*x^2 + 20*d^2*e^
3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5)) + 3*c^2*e*(10*a*e*(d^5 + 6*d^4*e*x + 15*d^3*e
^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5) - b*d*(147*d^5 + 822*d^4*e*x
+ 1875*d^3*e^2*x^2 + 2200*d^2*e^3*x^3 + 1350*d*e^4*x^4 + 360*e^5*x^5))) + 60*c^
2*(7*B*c*d - 3*b*B*e - A*c*e)*(d + e*x)^6*Log[d + e*x])/(60*e^8*(d + e*x)^6)

_______________________________________________________________________________________

Maple [B]  time = 0.023, size = 1656, 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)^7,x)

[Out]

9/2/e^4/(e*x+d)^4*A*a*b*c*d-9/e^5/(e*x+d)^4*B*a*b*c*d^2-18/5/e^4/(e*x+d)^5*A*a*b
*c*d^2+24/5/e^5/(e*x+d)^5*B*a*b*c*d^3+1/e^4/(e*x+d)^6*A*d^3*a*b*c-1/e^5/(e*x+d)^
6*B*d^4*a*b*c+8/e^5/(e*x+d)^3*B*a*b*c*d+18*c^2/e^7/(e*x+d)*B*b*d-9/2/e^5/(e*x+d)
^4*A*a*c^2*d^2-9/2/e^5/(e*x+d)^4*A*b^2*c*d^2+4/e^5/(e*x+d)^3*A*a*c^2*d+4/e^5/(e*
x+d)^3*A*b^2*c*d-2/e^4/(e*x+d)^3*A*a*b*c+B*c^3*x/e^7-10/e^6/(e*x+d)^3*B*a*c^2*d^
2-10/e^6/(e*x+d)^3*B*b^2*c*d^2+20/e^7/(e*x+d)^3*B*b*c^2*d^3-9/5/e^4/(e*x+d)^5*B*
a^2*c*d^2-9/5/e^4/(e*x+d)^5*B*a*b^2*d^2-3/e^6/(e*x+d)^5*B*a*c^2*d^4-3/e^6/(e*x+d
)^5*B*b^2*c*d^4+18/5/e^7/(e*x+d)^5*B*b*c^2*d^5+12/5/e^5/(e*x+d)^5*A*b^2*c*d^3-3/
e^6/(e*x+d)^5*A*b*c^2*d^4+6/5/e^3/(e*x+d)^5*A*a^2*c*d+6/5/e^3/(e*x+d)^5*A*a*b^2*
d+12/5/e^5/(e*x+d)^5*A*a*c^2*d^3+15/2/e^6/(e*x+d)^2*B*a*c^2*d+15/2/e^6/(e*x+d)^2
*B*b^2*c*d-45/2/e^7/(e*x+d)^2*B*b*c^2*d^2+9/4/e^4/(e*x+d)^4*B*a*b^2*d+15/2/e^6/(
e*x+d)^4*B*a*c^2*d^3+15/2/e^6/(e*x+d)^4*B*b^2*c*d^3-45/4/e^7/(e*x+d)^4*B*b*c^2*d
^4+15/2/e^6/(e*x+d)^2*A*b*c^2*d-3/e^5/(e*x+d)^2*B*a*b*c+15/2/e^6/(e*x+d)^4*A*b*c
^2*d^3+9/4/e^4/(e*x+d)^4*B*a^2*c*d-10/e^6/(e*x+d)^3*A*b*c^2*d^2+1/2/e^4/(e*x+d)^
6*B*d^3*a^2*c+1/2/e^4/(e*x+d)^6*B*d^3*a*b^2+1/2/e^6/(e*x+d)^6*B*d^5*a*c^2+6/5/e^
3/(e*x+d)^5*B*a^2*b*d+1/2/e^6/(e*x+d)^6*B*d^5*b^2*c-1/2/e^7/(e*x+d)^6*B*d^6*b*c^
2+1/2/e^2/(e*x+d)^6*A*d*a^2*b-1/2/e^3/(e*x+d)^6*A*d^2*a^2*c-1/2/e^3/(e*x+d)^6*A*
d^2*a*b^2-1/2/e^5/(e*x+d)^6*A*d^4*a*c^2-1/2/e^5/(e*x+d)^6*A*d^4*b^2*c+1/2/e^6/(e
*x+d)^6*A*d^5*b*c^2-1/2/e^3/(e*x+d)^6*B*d^2*a^2*b-1/6/e/(e*x+d)^6*A*a^3-1/3/e^4/
(e*x+d)^3*A*b^3+c^3/e^7*ln(e*x+d)*A-1/2/e^5/(e*x+d)^2*B*b^3-1/5/e^2/(e*x+d)^5*B*
a^3-3/4/e^3/(e*x+d)^4*A*a*b^2+3/4/e^4/(e*x+d)^4*A*b^3*d-15/4/e^7/(e*x+d)^4*A*c^3
*d^4-3/4/e^3/(e*x+d)^4*B*a^2*b-3/2/e^5/(e*x+d)^4*B*b^3*d^2+21/4/e^8/(e*x+d)^4*B*
c^3*d^5-3/2/e^5/(e*x+d)^2*A*a*c^2-3/2/e^5/(e*x+d)^2*A*b^2*c+20/3/e^7/(e*x+d)^3*A
*c^3*d^3-1/e^4/(e*x+d)^3*B*a^2*c-1/e^4/(e*x+d)^3*B*a*b^2+4/3/e^5/(e*x+d)^3*B*b^3
*d-35/3/e^8/(e*x+d)^3*B*c^3*d^4+3*c^2/e^7*ln(e*x+d)*b*B-7*c^3/e^8*ln(e*x+d)*B*d-
3*c/e^6/(e*x+d)*B*b^2-21*c^3/e^8/(e*x+d)*B*d^2-3/4/e^3/(e*x+d)^4*A*a^2*c+6/5/e^7
/(e*x+d)^5*A*c^3*d^5+4/5/e^5/(e*x+d)^5*B*b^3*d^3-7/5/e^8/(e*x+d)^5*B*c^3*d^6+1/6
/e^4/(e*x+d)^6*A*d^3*b^3-1/6/e^7/(e*x+d)^6*A*d^6*c^3+1/6/e^2/(e*x+d)^6*B*d*a^3-1
/6/e^5/(e*x+d)^6*B*d^4*b^3+1/6/e^8/(e*x+d)^6*B*c^3*d^7-15/2/e^7/(e*x+d)^2*A*c^3*
d^2+35/2/e^8/(e*x+d)^2*B*c^3*d^3-3*c^2/e^6/(e*x+d)*A*b+6*c^3/e^7/(e*x+d)*A*d-3*c
^2/e^6/(e*x+d)*B*a-3/5/e^2/(e*x+d)^5*A*a^2*b-3/5/e^4/(e*x+d)^5*A*b^3*d^2

_______________________________________________________________________________________

Maxima [A]  time = 0.743603, size = 1222, normalized size = 2.26 \[ \text{result too large to display} \]

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)^7,x, algorithm="maxima")

[Out]

-1/60*(669*B*c^3*d^7 + 10*A*a^3*e^7 - 147*(3*B*b*c^2 + A*c^3)*d^6*e + 30*(B*b^2*
c + (B*a + A*b)*c^2)*d^5*e^2 + 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 + 2*(B*a^3 + 3*A*a^2*b)*d*e^6 + 180*(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 + 30*(175*B*c^3*d
^3*e^4 - 45*(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 + 20*(455*B*c^3*d^4*e^3 -
110*(3*B*b*c^2 + A*c^3)*d^3*e^4 + 30*(B*b^2*c + (B*a + A*b)*c^2)*d^2*e^5 + 2*(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 + 15*(539*B*c^3*d^5*e^2 - 125*(3*B*b*c^2 + A*c^3)*d^4*e^3
+ 30*(B*b^2*c + (B*a + A*b)*c^2)*d^3*e^4 + 2*(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 + 6*(609*B*c^3*d^6*e - 137*(3*B*b*c^2 + A*c^3)*d
^5*e^2 + 30*(B*b^2*c + (B*a + A*b)*c^2)*d^4*e^3 + 2*(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 + 2*(B*a^3 + 3*A*a^2*b)*e^7)*x)/(e^14*x^6
+ 6*d*e^13*x^5 + 15*d^2*e^12*x^4 + 20*d^3*e^11*x^3 + 15*d^4*e^10*x^2 + 6*d^5*e^9
*x + d^6*e^8) + B*c^3*x/e^7 - (7*B*c^3*d - (3*B*b*c^2 + A*c^3)*e)*log(e*x + d)/e
^8

_______________________________________________________________________________________

Fricas [A]  time = 0.265405, size = 1546, normalized size = 2.86 \[ \text{result too large to display} \]

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)^7,x, algorithm="fricas")

[Out]

1/60*(60*B*c^3*e^7*x^7 + 360*B*c^3*d*e^6*x^6 - 669*B*c^3*d^7 - 10*A*a^3*e^7 + 14
7*(3*B*b*c^2 + A*c^3)*d^6*e - 30*(B*b^2*c + (B*a + A*b)*c^2)*d^5*e^2 - 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 - 2*(B*a^3 + 3*A*a^
2*b)*d*e^6 - 180*(2*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 - 30*(135*B*c^3*d^3*e^4 - 45*(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 - 20*(410*B*c^3*d^4*e^3 - 110*(3*B*b*c^2 + A*c^3)*d^3*e^4 + 30*(B*
b^2*c + (B*a + A*b)*c^2)*d^2*e^5 + 2*(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 - 15*(515*B*c^3*d^
5*e^2 - 125*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 30*(B*b^2*c + (B*a + A*b)*c^2)*d^3*e^4
+ 2*(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 - 6*(599
*B*c^3*d^6*e - 137*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 30*(B*b^2*c + (B*a + A*b)*c^2)*
d^4*e^3 + 2*(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 +
2*(B*a^3 + 3*A*a^2*b)*e^7)*x - 60*(7*B*c^3*d^7 - (3*B*b*c^2 + A*c^3)*d^6*e + (7*
B*c^3*d*e^6 - (3*B*b*c^2 + A*c^3)*e^7)*x^6 + 6*(7*B*c^3*d^2*e^5 - (3*B*b*c^2 + A
*c^3)*d*e^6)*x^5 + 15*(7*B*c^3*d^3*e^4 - (3*B*b*c^2 + A*c^3)*d^2*e^5)*x^4 + 20*(
7*B*c^3*d^4*e^3 - (3*B*b*c^2 + A*c^3)*d^3*e^4)*x^3 + 15*(7*B*c^3*d^5*e^2 - (3*B*
b*c^2 + A*c^3)*d^4*e^3)*x^2 + 6*(7*B*c^3*d^6*e - (3*B*b*c^2 + A*c^3)*d^5*e^2)*x)
*log(e*x + d))/(e^14*x^6 + 6*d*e^13*x^5 + 15*d^2*e^12*x^4 + 20*d^3*e^11*x^3 + 15
*d^4*e^10*x^2 + 6*d^5*e^9*x + d^6*e^8)

_______________________________________________________________________________________

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)**7,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.260533, 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)^7,x, algorithm="giac")

[Out]

Done