3.2345 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^8} \, dx\)

Optimal. Leaf size=548 \[ \frac{A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )}{4 e^8 (d+e x)^4}+\frac{3 c \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{2 e^8 (d+e x)^2}+\frac{3 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{5 e^8 (d+e x)^5}+\frac{B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^8 (d+e x)^3}+\frac{\left (a e^2-b d e+c d^2\right )^2 \left (3 A e (2 c d-b e)-B \left (7 c d^2-e (4 b d-a e)\right )\right )}{6 e^8 (d+e x)^6}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{7 e^8 (d+e x)^7}+\frac{c^2 (-A c e-3 b B e+7 B c d)}{e^8 (d+e x)}+\frac{B c^3 \log (d+e x)}{e^8} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 3.52566, antiderivative size = 546, 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 \[ \frac{A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )}{4 e^8 (d+e x)^4}+\frac{3 c \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{2 e^8 (d+e x)^2}+\frac{3 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{5 e^8 (d+e x)^5}+\frac{B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^8 (d+e x)^3}-\frac{\left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{6 e^8 (d+e x)^6}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{7 e^8 (d+e x)^7}+\frac{c^2 (-A c e-3 b B e+7 B c d)}{e^8 (d+e x)}+\frac{B c^3 \log (d+e x)}{e^8} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^8,x]

[Out]

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

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 5.08611, size = 863, normalized size = 1.57 \[ \frac{420 B c^3 \log (d+e x) (d+e x)^7-3 A e \left (20 \left (d^6+7 e x d^5+21 e^2 x^2 d^4+35 e^3 x^3 d^3+35 e^4 x^4 d^2+21 e^5 x^5 d+7 e^6 x^6\right ) c^3+2 e \left (2 a e \left (d^4+7 e x d^3+21 e^2 x^2 d^2+35 e^3 x^3 d+35 e^4 x^4\right )+5 b \left (d^5+7 e x d^4+21 e^2 x^2 d^3+35 e^3 x^3 d^2+35 e^4 x^4 d+21 e^5 x^5\right )\right ) c^2+2 e^2 \left (2 \left (d^4+7 e x d^3+21 e^2 x^2 d^2+35 e^3 x^3 d+35 e^4 x^4\right ) b^2+3 a e \left (d^3+7 e x d^2+21 e^2 x^2 d+35 e^3 x^3\right ) b+2 a^2 e^2 \left (d^2+7 e x d+21 e^2 x^2\right )\right ) c+e^3 \left (\left (d^3+7 e x d^2+21 e^2 x^2 d+35 e^3 x^3\right ) b^3+4 a e \left (d^2+7 e x d+21 e^2 x^2\right ) b^2+10 a^2 e^2 (d+7 e x) b+20 a^3 e^3\right )\right )+B \left (d \left (1089 d^6+7203 e x d^5+20139 e^2 x^2 d^4+30625 e^3 x^3 d^3+26950 e^4 x^4 d^2+13230 e^5 x^5 d+2940 e^6 x^6\right ) c^3-30 e \left (a e \left (d^5+7 e x d^4+21 e^2 x^2 d^3+35 e^3 x^3 d^2+35 e^4 x^4 d+21 e^5 x^5\right )+6 b \left (d^6+7 e x d^5+21 e^2 x^2 d^4+35 e^3 x^3 d^3+35 e^4 x^4 d^2+21 e^5 x^5 d+7 e^6 x^6\right )\right ) c^2-3 e^2 \left (10 \left (d^5+7 e x d^4+21 e^2 x^2 d^3+35 e^3 x^3 d^2+35 e^4 x^4 d+21 e^5 x^5\right ) b^2+8 a e \left (d^4+7 e x d^3+21 e^2 x^2 d^2+35 e^3 x^3 d+35 e^4 x^4\right ) b+3 a^2 e^2 \left (d^3+7 e x d^2+21 e^2 x^2 d+35 e^3 x^3\right )\right ) c-e^3 \left (4 \left (d^4+7 e x d^3+21 e^2 x^2 d^2+35 e^3 x^3 d+35 e^4 x^4\right ) b^3+9 a e \left (d^3+7 e x d^2+21 e^2 x^2 d+35 e^3 x^3\right ) b^2+12 a^2 e^2 \left (d^2+7 e x d+21 e^2 x^2\right ) b+10 a^3 e^3 (d+7 e x)\right )\right )}{420 e^8 (d+e x)^7} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^8,x]

[Out]

(-3*A*e*(20*c^3*(d^6 + 7*d^5*e*x + 21*d^4*e^2*x^2 + 35*d^3*e^3*x^3 + 35*d^2*e^4*
x^4 + 21*d*e^5*x^5 + 7*e^6*x^6) + e^3*(20*a^3*e^3 + 10*a^2*b*e^2*(d + 7*e*x) + 4
*a*b^2*e*(d^2 + 7*d*e*x + 21*e^2*x^2) + b^3*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35
*e^3*x^3)) + 2*c*e^2*(2*a^2*e^2*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3*a*b*e*(d^3 + 7*
d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + 2*b^2*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 +
35*d*e^3*x^3 + 35*e^4*x^4)) + 2*c^2*e*(2*a*e*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2
+ 35*d*e^3*x^3 + 35*e^4*x^4) + 5*b*(d^5 + 7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^
3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5))) + B*(c^3*d*(1089*d^6 + 7203*d^5*e*x + 20139
*d^4*e^2*x^2 + 30625*d^3*e^3*x^3 + 26950*d^2*e^4*x^4 + 13230*d*e^5*x^5 + 2940*e^
6*x^6) - e^3*(10*a^3*e^3*(d + 7*e*x) + 12*a^2*b*e^2*(d^2 + 7*d*e*x + 21*e^2*x^2)
+ 9*a*b^2*e*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + 4*b^3*(d^4 + 7*d^3*
e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4)) - 3*c*e^2*(3*a^2*e^2*(d^3 + 7
*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + 8*a*b*e*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^
2 + 35*d*e^3*x^3 + 35*e^4*x^4) + 10*b^2*(d^5 + 7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d
^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5)) - 30*c^2*e*(a*e*(d^5 + 7*d^4*e*x + 21*d
^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5) + 6*b*(d^6 + 7*d^5*e*x
+ 21*d^4*e^2*x^2 + 35*d^3*e^3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*x^5 + 7*e^6*x^6)))
+ 420*B*c^3*(d + e*x)^7*Log[d + e*x])/(420*e^8*(d + e*x)^7)

_______________________________________________________________________________________

Maple [B]  time = 0.018, size = 1661, normalized size = 3. \[ \text{result too large to display} \]

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)^8,x)

[Out]

1/e^3/(e*x+d)^6*A*d*a^2*c+3/e^5/(e*x+d)^4*A*d*a*c^2+3/e^5/(e*x+d)^4*A*d*b^2*c-15
/2/e^6/(e*x+d)^4*A*d^2*b*c^2-15/2/e^6/(e*x+d)^4*B*d^2*a*c^2-15/2/e^6/(e*x+d)^4*B
*d^2*b^2*c+15/e^7/(e*x+d)^4*B*d^3*b*c^2+9*c^2/e^7/(e*x+d)^2*B*b*d-18/5/e^5/(e*x+
d)^5*A*d^2*a*c^2+5/e^6/(e*x+d)^3*B*d*b^2*c-15/e^7/(e*x+d)^3*B*d^2*b*c^2-3/2/e^4/
(e*x+d)^4*a*A*b*c+5/e^6/(e*x+d)^3*B*d*a*c^2-2/e^5/(e*x+d)^3*a*b*B*c+5/e^6/(e*x+d
)^3*A*d*b*c^2-3/7/e^5/(e*x+d)^7*A*d^4*a*c^2-3/7/e^5/(e*x+d)^7*A*d^4*b^2*c+3/7/e^
6/(e*x+d)^7*A*d^5*b*c^2-3/7/e^3/(e*x+d)^7*B*d^2*a^2*b+3/7/e^4/(e*x+d)^7*B*d^3*a^
2*c+3/7/e^4/(e*x+d)^7*B*d^3*a*b^2+3/7/e^6/(e*x+d)^7*B*d^5*a*c^2+3/7/e^6/(e*x+d)^
7*B*d^5*b^2*c-3/7/e^7/(e*x+d)^7*B*d^6*b*c^2-3/7/e^3/(e*x+d)^7*A*d^2*a^2*c-3/7/e^
3/(e*x+d)^7*A*d^2*a*b^2+3/e^7/(e*x+d)^6*B*d^5*b*c^2+3/7/e^2/(e*x+d)^7*A*d*a^2*b+
2/e^5/(e*x+d)^6*A*d^3*a*c^2+2/e^5/(e*x+d)^6*A*d^3*b^2*c-5/2/e^6/(e*x+d)^6*A*d^4*
b*c^2+1/e^3/(e*x+d)^6*B*d*a^2*b-3/2/e^4/(e*x+d)^6*B*d^2*a^2*c-3/2/e^4/(e*x+d)^6*
B*d^2*a*b^2-5/2/e^6/(e*x+d)^6*B*d^4*a*c^2-5/2/e^6/(e*x+d)^6*B*d^4*b^2*c+1/e^3/(e
*x+d)^6*A*d*a*b^2-18/5/e^5/(e*x+d)^5*A*d^2*b^2*c+6/e^6/(e*x+d)^5*A*d^3*b*c^2+9/5
/e^4/(e*x+d)^5*B*d*a^2*c+9/5/e^4/(e*x+d)^5*B*d*a*b^2+6/e^6/(e*x+d)^5*B*d^3*a*c^2
+6/e^6/(e*x+d)^5*B*d^3*b^2*c-9/e^7/(e*x+d)^5*B*d^4*b*c^2+B*c^3*ln(e*x+d)/e^8-1/3
/e^5/(e*x+d)^3*b^3*B-c^3/e^7/(e*x+d)*A-1/4/e^4/(e*x+d)^4*A*b^3-1/6/e^2/(e*x+d)^6
*B*a^3-1/7/e/(e*x+d)^7*A*a^3+18/5/e^4/(e*x+d)^5*A*d*a*b*c-36/5/e^5/(e*x+d)^5*B*d
^2*a*b*c-3/e^4/(e*x+d)^6*A*d^2*a*b*c+4/e^5/(e*x+d)^6*B*d^3*a*b*c+6/7/e^4/(e*x+d)
^7*A*d^3*a*b*c-6/7/e^5/(e*x+d)^7*B*d^4*a*b*c+6/e^5/(e*x+d)^4*B*d*a*b*c-3/5/e^3/(
e*x+d)^5*B*a^2*b-6/5/e^5/(e*x+d)^5*B*d^2*b^3+21/5/e^8/(e*x+d)^5*B*c^3*d^5-1/2/e^
2/(e*x+d)^6*A*a^2*b-1/2/e^4/(e*x+d)^6*A*d^2*b^3+1/e^7/(e*x+d)^6*A*d^5*c^3+2/3/e^
5/(e*x+d)^6*B*d^3*b^3-7/6/e^8/(e*x+d)^6*B*c^3*d^6+1/7/e^4/(e*x+d)^7*A*d^3*b^3-1/
7/e^7/(e*x+d)^7*A*d^6*c^3+1/7/e^2/(e*x+d)^7*B*d*a^3-1/7/e^5/(e*x+d)^7*B*d^4*b^3+
1/7/e^8/(e*x+d)^7*B*c^3*d^7-1/e^5/(e*x+d)^3*a*A*c^2-1/e^5/(e*x+d)^3*A*b^2*c-5/e^
7/(e*x+d)^3*A*c^3*d^2+35/3/e^8/(e*x+d)^3*B*c^3*d^3-3*c^2/e^7/(e*x+d)*b*B+7*c^3/e
^8/(e*x+d)*B*d+5/e^7/(e*x+d)^4*A*c^3*d^3-3/4/e^4/(e*x+d)^4*B*a^2*c-3/4/e^4/(e*x+
d)^4*a*B*b^2+1/e^5/(e*x+d)^4*B*d*b^3-35/4/e^8/(e*x+d)^4*B*c^3*d^4-3/2*c^2/e^6/(e
*x+d)^2*B*a-3/2*c/e^6/(e*x+d)^2*B*b^2-21/2*c^3/e^8/(e*x+d)^2*B*d^2-3/5/e^3/(e*x+
d)^5*A*a^2*c-3/5/e^3/(e*x+d)^5*a*A*b^2+3/5/e^4/(e*x+d)^5*A*d*b^3-3/e^7/(e*x+d)^5
*A*d^4*c^3-3/2*c^2/e^6/(e*x+d)^2*A*b+3*c^3/e^7/(e*x+d)^2*A*d

_______________________________________________________________________________________

Maxima [A]  time = 0.739, size = 1250, normalized size = 2.28 \[ \text{result too large to display} \]

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)^8,x, algorithm="maxima")

[Out]

1/420*(1089*B*c^3*d^7 - 60*A*a^3*e^7 - 60*(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 - 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^4
*e^3 - 3*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^3*e^4 - 12*(B*a^2*b + A*a
*b^2 + A*a^2*c)*d^2*e^5 - 10*(B*a^3 + 3*A*a^2*b)*d*e^6 + 420*(7*B*c^3*d*e^6 - (3
*B*b*c^2 + A*c^3)*e^7)*x^6 + 630*(21*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 + 70*(385*B*c^3*d^3*e^4 - 30*(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 - 2*(B*b^3 + 3*A*a*c^2 +
3*(2*B*a*b + A*b^2)*c)*e^7)*x^4 + 35*(875*B*c^3*d^4*e^3 - 60*(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 - 4*(B*b^3 + 3*A*a*c^2 + 3*(2
*B*a*b + A*b^2)*c)*d*e^6 - 3*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*e^7)*x^
3 + 21*(959*B*c^3*d^5*e^2 - 60*(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 - 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^2*e^5 - 3*
(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d*e^6 - 12*(B*a^2*b + A*a*b^2 + A*a^
2*c)*e^7)*x^2 + 7*(1029*B*c^3*d^6*e - 60*(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 - 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^
3*e^4 - 3*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^2*e^5 - 12*(B*a^2*b + A*
a*b^2 + A*a^2*c)*d*e^6 - 10*(B*a^3 + 3*A*a^2*b)*e^7)*x)/(e^15*x^7 + 7*d*e^14*x^6
+ 21*d^2*e^13*x^5 + 35*d^3*e^12*x^4 + 35*d^4*e^11*x^3 + 21*d^5*e^10*x^2 + 7*d^6
*e^9*x + d^7*e^8) + B*c^3*log(e*x + d)/e^8

_______________________________________________________________________________________

Fricas [A]  time = 0.264431, size = 1381, normalized size = 2.52 \[ \text{result too large to display} \]

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)^8,x, algorithm="fricas")

[Out]

1/420*(1089*B*c^3*d^7 - 60*A*a^3*e^7 - 60*(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 - 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^4
*e^3 - 3*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^3*e^4 - 12*(B*a^2*b + A*a
*b^2 + A*a^2*c)*d^2*e^5 - 10*(B*a^3 + 3*A*a^2*b)*d*e^6 + 420*(7*B*c^3*d*e^6 - (3
*B*b*c^2 + A*c^3)*e^7)*x^6 + 630*(21*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 + 70*(385*B*c^3*d^3*e^4 - 30*(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 - 2*(B*b^3 + 3*A*a*c^2 +
3*(2*B*a*b + A*b^2)*c)*e^7)*x^4 + 35*(875*B*c^3*d^4*e^3 - 60*(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 - 4*(B*b^3 + 3*A*a*c^2 + 3*(2
*B*a*b + A*b^2)*c)*d*e^6 - 3*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*e^7)*x^
3 + 21*(959*B*c^3*d^5*e^2 - 60*(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 - 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^2*e^5 - 3*
(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d*e^6 - 12*(B*a^2*b + A*a*b^2 + A*a^
2*c)*e^7)*x^2 + 7*(1029*B*c^3*d^6*e - 60*(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 - 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^
3*e^4 - 3*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^2*e^5 - 12*(B*a^2*b + A*
a*b^2 + A*a^2*c)*d*e^6 - 10*(B*a^3 + 3*A*a^2*b)*e^7)*x + 420*(B*c^3*e^7*x^7 + 7*
B*c^3*d*e^6*x^6 + 21*B*c^3*d^2*e^5*x^5 + 35*B*c^3*d^3*e^4*x^4 + 35*B*c^3*d^4*e^3
*x^3 + 21*B*c^3*d^5*e^2*x^2 + 7*B*c^3*d^6*e*x + B*c^3*d^7)*log(e*x + d))/(e^15*x
^7 + 7*d*e^14*x^6 + 21*d^2*e^13*x^5 + 35*d^3*e^12*x^4 + 35*d^4*e^11*x^3 + 21*d^5
*e^10*x^2 + 7*d^6*e^9*x + d^7*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)**8,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.260448, 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)^8,x, algorithm="giac")

[Out]

Done