3.2346 \(\int \frac{(A+B x) (a+b x+c x^2)^3}{(d+e x)^8} \, dx\)

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

[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 = 0.767961, 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, Rules used = {771} \[ \frac{A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )}{4 e^8 (d+e x)^4}+\frac{3 c \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{2 e^8 (d+e x)^2}+\frac{B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^8 (d+e x)^3}+\frac{3 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{5 e^8 (d+e x)^5}-\frac{\left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{6 e^8 (d+e x)^6}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{7 e^8 (d+e x)^7}+\frac{c^2 (-A c e-3 b B e+7 B c d)}{e^8 (d+e x)}+\frac{B c^3 \log (d+e x)}{e^8} \]

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

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^8} \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2-b d e+a e^2\right )^3}{e^7 (d+e x)^8}+\frac{\left (c d^2-b d e+a e^2\right )^2 \left (7 B c d^2-B e (4 b d-a e)-3 A e (2 c d-b e)\right )}{e^7 (d+e x)^7}+\frac{3 \left (c d^2-b d e+a e^2\right ) \left (-B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )+A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right )}{e^7 (d+e x)^6}+\frac{-A e (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )+B \left (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 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )}{e^7 (d+e x)^5}+\frac{-B \left (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)\right )+3 A c e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^7 (d+e x)^4}+\frac{3 c \left (-A c e (2 c d-b e)+B \left (7 c^2 d^2+b^2 e^2-c e (6 b d-a e)\right )\right )}{e^7 (d+e x)^3}+\frac{c^2 (-7 B c d+3 b B e+A c e)}{e^7 (d+e x)^2}+\frac{B c^3}{e^7 (d+e x)}\right ) \, dx\\ &=\frac{(B d-A e) \left (c d^2-b d e+a e^2\right )^3}{7 e^8 (d+e x)^7}-\frac{\left (c d^2-b d e+a e^2\right )^2 \left (7 B c d^2-B e (4 b d-a e)-3 A e (2 c d-b e)\right )}{6 e^8 (d+e x)^6}+\frac{3 \left (c d^2-b d e+a e^2\right ) \left (B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )-A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right )}{5 e^8 (d+e x)^5}+\frac{A e (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )-B \left (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 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )}{4 e^8 (d+e x)^4}+\frac{B \left (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)\right )-3 A c e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{3 e^8 (d+e x)^3}+\frac{3 c \left (A c e (2 c d-b e)-B \left (7 c^2 d^2+b^2 e^2-c e (6 b d-a e)\right )\right )}{2 e^8 (d+e x)^2}+\frac{c^2 (7 B c d-3 b B e-A c e)}{e^8 (d+e x)}+\frac{B c^3 \log (d+e x)}{e^8}\\ \end{align*}

Mathematica [A]  time = 0.567264, 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 + 2
1*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 = 0.013, size = 1661, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

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]

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

________________________________________________________________________________________

Maxima [A]  time = 1.17021, size = 1250, normalized size = 2.28 \begin{align*} \text{result too large to display} \end{align*}

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, 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 = 1.10028, size = 2209, normalized size = 4.03 \begin{align*} \text{result too large to display} \end{align*}

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, 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. \begin{align*} \text{Timed out} \end{align*}

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 [A]  time = 1.18371, size = 1351, normalized size = 2.47 \begin{align*} \text{result too large to display} \end{align*}

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, algorithm="giac")

[Out]

B*c^3*e^(-8)*log(abs(x*e + d)) + 1/420*(420*(7*B*c^3*d*e^5 - 3*B*b*c^2*e^6 - A*c^3*e^6)*x^6 + 630*(21*B*c^3*d^
2*e^4 - 6*B*b*c^2*d*e^5 - 2*A*c^3*d*e^5 - B*b^2*c*e^6 - B*a*c^2*e^6 - A*b*c^2*e^6)*x^5 + 70*(385*B*c^3*d^3*e^3
- 90*B*b*c^2*d^2*e^4 - 30*A*c^3*d^2*e^4 - 15*B*b^2*c*d*e^5 - 15*B*a*c^2*d*e^5 - 15*A*b*c^2*d*e^5 - 2*B*b^3*e^
6 - 12*B*a*b*c*e^6 - 6*A*b^2*c*e^6 - 6*A*a*c^2*e^6)*x^4 + 35*(875*B*c^3*d^4*e^2 - 180*B*b*c^2*d^3*e^3 - 60*A*c
^3*d^3*e^3 - 30*B*b^2*c*d^2*e^4 - 30*B*a*c^2*d^2*e^4 - 30*A*b*c^2*d^2*e^4 - 4*B*b^3*d*e^5 - 24*B*a*b*c*d*e^5 -
12*A*b^2*c*d*e^5 - 12*A*a*c^2*d*e^5 - 9*B*a*b^2*e^6 - 3*A*b^3*e^6 - 9*B*a^2*c*e^6 - 18*A*a*b*c*e^6)*x^3 + 21*
(959*B*c^3*d^5*e - 180*B*b*c^2*d^4*e^2 - 60*A*c^3*d^4*e^2 - 30*B*b^2*c*d^3*e^3 - 30*B*a*c^2*d^3*e^3 - 30*A*b*c
^2*d^3*e^3 - 4*B*b^3*d^2*e^4 - 24*B*a*b*c*d^2*e^4 - 12*A*b^2*c*d^2*e^4 - 12*A*a*c^2*d^2*e^4 - 9*B*a*b^2*d*e^5
- 3*A*b^3*d*e^5 - 9*B*a^2*c*d*e^5 - 18*A*a*b*c*d*e^5 - 12*B*a^2*b*e^6 - 12*A*a*b^2*e^6 - 12*A*a^2*c*e^6)*x^2 +
7*(1029*B*c^3*d^6 - 180*B*b*c^2*d^5*e - 60*A*c^3*d^5*e - 30*B*b^2*c*d^4*e^2 - 30*B*a*c^2*d^4*e^2 - 30*A*b*c^2
*d^4*e^2 - 4*B*b^3*d^3*e^3 - 24*B*a*b*c*d^3*e^3 - 12*A*b^2*c*d^3*e^3 - 12*A*a*c^2*d^3*e^3 - 9*B*a*b^2*d^2*e^4
- 3*A*b^3*d^2*e^4 - 9*B*a^2*c*d^2*e^4 - 18*A*a*b*c*d^2*e^4 - 12*B*a^2*b*d*e^5 - 12*A*a*b^2*d*e^5 - 12*A*a^2*c*
d*e^5 - 10*B*a^3*e^6 - 30*A*a^2*b*e^6)*x + (1089*B*c^3*d^7 - 180*B*b*c^2*d^6*e - 60*A*c^3*d^6*e - 30*B*b^2*c*d
^5*e^2 - 30*B*a*c^2*d^5*e^2 - 30*A*b*c^2*d^5*e^2 - 4*B*b^3*d^4*e^3 - 24*B*a*b*c*d^4*e^3 - 12*A*b^2*c*d^4*e^3 -
12*A*a*c^2*d^4*e^3 - 9*B*a*b^2*d^3*e^4 - 3*A*b^3*d^3*e^4 - 9*B*a^2*c*d^3*e^4 - 18*A*a*b*c*d^3*e^4 - 12*B*a^2*
b*d^2*e^5 - 12*A*a*b^2*d^2*e^5 - 12*A*a^2*c*d^2*e^5 - 10*B*a^3*d*e^6 - 30*A*a^2*b*d*e^6 - 60*A*a^3*e^7)*e^(-1)
)*e^(-7)/(x*e + d)^7