\(\int \frac {(A+B x) (a+b x+c x^2)^3}{(d+e x)^8} \, dx\) [2346]

Optimal result

Integrand size = 25, antiderivative size = 548 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^8} \, dx=\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^3}{7 e^8 (d+e x)^7}+\frac {\left (c d^2-b d e+a e^2\right )^2 \left (3 A e (2 c d-b e)-B \left (7 c d^2-e (4 b d-a e)\right )\right )}{6 e^8 (d+e x)^6}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )-A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right )}{5 e^8 (d+e x)^5}+\frac {A e (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )-B \left (35 c^3 d^4-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )}{4 e^8 (d+e x)^4}+\frac {B \left (35 c^3 d^3-b^3 e^3+3 b c e^2 (5 b d-2 a e)-15 c^2 d e (3 b d-a e)\right )-3 A c e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{3 e^8 (d+e x)^3}+\frac {3 c \left (A c e (2 c d-b e)-B \left (7 c^2 d^2+b^2 e^2-c e (6 b d-a e)\right )\right )}{2 e^8 (d+e x)^2}+\frac {c^2 (7 B c d-3 b B e-A c e)}{e^8 (d+e x)}+\frac {B c^3 \log (d+e x)}{e^8} \]

[Out]

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

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 546, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {785} \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^8} \, dx=\frac {A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )}{4 e^8 (d+e x)^4}+\frac {3 c \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{2 e^8 (d+e x)^2}+\frac {3 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{5 e^8 (d+e x)^5}+\frac {B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^8 (d+e x)^3}-\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{6 e^8 (d+e x)^6}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{7 e^8 (d+e x)^7}+\frac {c^2 (-A c e-3 b B e+7 B c d)}{e^8 (d+e x)}+\frac {B c^3 \log (d+e x)}{e^8} \]

[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 785

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*} \text {integral}& = \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] (verified)

Time = 0.32 (sec) , antiderivative size = 863, normalized size of antiderivative = 1.57 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^8} \, dx=\frac {-3 A e \left (20 c^3 \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )+e^3 \left (20 a^3 e^3+10 a^2 b e^2 (d+7 e x)+4 a b^2 e \left (d^2+7 d e x+21 e^2 x^2\right )+b^3 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )\right )+2 c e^2 \left (2 a^2 e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+3 a b e \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+2 b^2 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )\right )+2 c^2 e \left (2 a e \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+5 b \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )\right )\right )+B \left (c^3 d \left (1089 d^6+7203 d^5 e x+20139 d^4 e^2 x^2+30625 d^3 e^3 x^3+26950 d^2 e^4 x^4+13230 d e^5 x^5+2940 e^6 x^6\right )-e^3 \left (10 a^3 e^3 (d+7 e x)+12 a^2 b e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+9 a b^2 e \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+4 b^3 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )\right )-3 c e^2 \left (3 a^2 e^2 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+8 a b e \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+10 b^2 \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )\right )-30 c^2 e \left (a e \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )+6 b \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )\right )\right )+420 B c^3 (d+e x)^7 \log (d+e x)}{420 e^8 (d+e x)^7} \]

[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 [A] (verified)

Time = 0.39 (sec) , antiderivative size = 1047, normalized size of antiderivative = 1.91

method	result	size
risch	\(\text {Expression too large to display}\)	\(1047\)
norman	\(\text {Expression too large to display}\)	\(1057\)
default	\(\text {Expression too large to display}\)	\(1065\)
parallelrisch	\(\text {Expression too large to display}\)	\(1354\)

[In]

int((B*x+A)*(c*x^2+b*x+a)^3/(e*x+d)^8,x,method=_RETURNVERBOSE)

[Out]

(-c^2*(A*c*e+3*B*b*e-7*B*c*d)/e^2*x^6-3/2*c*(A*b*c*e^2+2*A*c^2*d*e+B*a*c*e^2+B*b^2*e^2+6*B*b*c*d*e-21*B*c^2*d^
2)/e^3*x^5-1/6*(6*A*a*c^2*e^3+6*A*b^2*c*e^3+15*A*b*c^2*d*e^2+30*A*c^3*d^2*e+12*B*a*b*c*e^3+15*B*a*c^2*d*e^2+2*
B*b^3*e^3+15*B*b^2*c*d*e^2+90*B*b*c^2*d^2*e-385*B*c^3*d^3)/e^4*x^4-1/12*(18*A*a*b*c*e^4+12*A*a*c^2*d*e^3+3*A*b
^3*e^4+12*A*b^2*c*d*e^3+30*A*b*c^2*d^2*e^2+60*A*c^3*d^3*e+9*B*a^2*c*e^4+9*B*a*b^2*e^4+24*B*a*b*c*d*e^3+30*B*a*
c^2*d^2*e^2+4*B*b^3*d*e^3+30*B*b^2*c*d^2*e^2+180*B*b*c^2*d^3*e-875*B*c^3*d^4)/e^5*x^3-1/20*(12*A*a^2*c*e^5+12*
A*a*b^2*e^5+18*A*a*b*c*d*e^4+12*A*a*c^2*d^2*e^3+3*A*b^3*d*e^4+12*A*b^2*c*d^2*e^3+30*A*b*c^2*d^3*e^2+60*A*c^3*d
^4*e+12*B*a^2*b*e^5+9*B*a^2*c*d*e^4+9*B*a*b^2*d*e^4+24*B*a*b*c*d^2*e^3+30*B*a*c^2*d^3*e^2+4*B*b^3*d^2*e^3+30*B
*b^2*c*d^3*e^2+180*B*b*c^2*d^4*e-959*B*c^3*d^5)/e^6*x^2-1/60*(30*A*a^2*b*e^6+12*A*a^2*c*d*e^5+12*A*a*b^2*d*e^5
+18*A*a*b*c*d^2*e^4+12*A*a*c^2*d^3*e^3+3*A*b^3*d^2*e^4+12*A*b^2*c*d^3*e^3+30*A*b*c^2*d^4*e^2+60*A*c^3*d^5*e+10
*B*a^3*e^6+12*B*a^2*b*d*e^5+9*B*a^2*c*d^2*e^4+9*B*a*b^2*d^2*e^4+24*B*a*b*c*d^3*e^3+30*B*a*c^2*d^4*e^2+4*B*b^3*
d^3*e^3+30*B*b^2*c*d^4*e^2+180*B*b*c^2*d^5*e-1029*B*c^3*d^6)/e^7*x-1/420*(60*A*a^3*e^7+30*A*a^2*b*d*e^6+12*A*a
^2*c*d^2*e^5+12*A*a*b^2*d^2*e^5+18*A*a*b*c*d^3*e^4+12*A*a*c^2*d^4*e^3+3*A*b^3*d^3*e^4+12*A*b^2*c*d^4*e^3+30*A*
b*c^2*d^5*e^2+60*A*c^3*d^6*e+10*B*a^3*d*e^6+12*B*a^2*b*d^2*e^5+9*B*a^2*c*d^3*e^4+9*B*a*b^2*d^3*e^4+24*B*a*b*c*
d^4*e^3+30*B*a*c^2*d^5*e^2+4*B*b^3*d^4*e^3+30*B*b^2*c*d^5*e^2+180*B*b*c^2*d^6*e-1089*B*c^3*d^7)/e^8)/(e*x+d)^7
+B*c^3*ln(e*x+d)/e^8

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 1023, normalized size of antiderivative = 1.87 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^8} \, dx=\frac {1089 \, B c^{3} d^{7} - 60 \, A a^{3} e^{7} - 60 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e - 30 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{5} e^{2} - 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{4} e^{3} - 3 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{3} e^{4} - 12 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{2} e^{5} - 10 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{6} + 420 \, {\left (7 \, B c^{3} d e^{6} - {\left (3 \, B b c^{2} + A c^{3}\right )} e^{7}\right )} x^{6} + 630 \, {\left (21 \, B c^{3} d^{2} e^{5} - 2 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d e^{6} - {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} e^{7}\right )} x^{5} + 70 \, {\left (385 \, B c^{3} d^{3} e^{4} - 30 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{5} - 15 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d e^{6} - 2 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} e^{7}\right )} x^{4} + 35 \, {\left (875 \, B c^{3} d^{4} e^{3} - 60 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e^{4} - 30 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{2} e^{5} - 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d e^{6} - 3 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} e^{7}\right )} x^{3} + 21 \, {\left (959 \, B c^{3} d^{5} e^{2} - 60 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{3} - 30 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{3} e^{4} - 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{2} e^{5} - 3 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d e^{6} - 12 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e^{7}\right )} x^{2} + 7 \, {\left (1029 \, B c^{3} d^{6} e - 60 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} - 30 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{4} e^{3} - 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{3} e^{4} - 3 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{2} e^{5} - 12 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d e^{6} - 10 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{7}\right )} x + 420 \, {\left (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}\right )} \log \left (e x + d\right )}{420 \, {\left (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}\right )}} \]

[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)]

Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^8} \, dx=\text {Timed out} \]

[In]

integrate((B*x+A)*(c*x**2+b*x+a)**3/(e*x+d)**8,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 926, normalized size of antiderivative = 1.69 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^8} \, dx=\frac {1089 \, B c^{3} d^{7} - 60 \, A a^{3} e^{7} - 60 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e - 30 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{5} e^{2} - 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{4} e^{3} - 3 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{3} e^{4} - 12 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{2} e^{5} - 10 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{6} + 420 \, {\left (7 \, B c^{3} d e^{6} - {\left (3 \, B b c^{2} + A c^{3}\right )} e^{7}\right )} x^{6} + 630 \, {\left (21 \, B c^{3} d^{2} e^{5} - 2 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d e^{6} - {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} e^{7}\right )} x^{5} + 70 \, {\left (385 \, B c^{3} d^{3} e^{4} - 30 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{5} - 15 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d e^{6} - 2 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} e^{7}\right )} x^{4} + 35 \, {\left (875 \, B c^{3} d^{4} e^{3} - 60 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e^{4} - 30 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{2} e^{5} - 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d e^{6} - 3 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} e^{7}\right )} x^{3} + 21 \, {\left (959 \, B c^{3} d^{5} e^{2} - 60 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{3} - 30 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{3} e^{4} - 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{2} e^{5} - 3 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d e^{6} - 12 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e^{7}\right )} x^{2} + 7 \, {\left (1029 \, B c^{3} d^{6} e - 60 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} - 30 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{4} e^{3} - 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{3} e^{4} - 3 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{2} e^{5} - 12 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d e^{6} - 10 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{7}\right )} x}{420 \, {\left (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}\right )}} + \frac {B c^{3} \log \left (e x + d\right )}{e^{8}} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1079 vs. \(2 (537) = 1074\).

Time = 0.28 (sec) , antiderivative size = 1079, normalized size of antiderivative = 1.97 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^8} \, dx=\frac {B c^{3} \log \left ({\left | e x + d \right |}\right )}{e^{8}} + \frac {420 \, {\left (7 \, B c^{3} d e^{5} - 3 \, B b c^{2} e^{6} - A c^{3} e^{6}\right )} x^{6} + 630 \, {\left (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}\right )} x^{5} + 70 \, {\left (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}\right )} x^{4} + 35 \, {\left (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}\right )} x^{3} + 21 \, {\left (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}\right )} x^{2} + 7 \, {\left (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}\right )} x + \frac {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}}{420 \, {\left (e x + d\right )}^{7} e^{7}} \]

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^3/(e*x+d)^8,x, algorithm="giac")

[Out]

B*c^3*log(abs(e*x + d))/e^8 + 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*(95
9*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)/((e*x
+ d)^7*e^7)

Mupad [B] (verification not implemented)

Time = 11.42 (sec) , antiderivative size = 1353, normalized size of antiderivative = 2.47 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^8} \, dx=-\frac {60\,A\,a^3\,e^7-1089\,B\,c^3\,d^7+10\,B\,a^3\,d\,e^6+60\,A\,c^3\,d^6\,e-420\,B\,c^3\,d^7\,\ln \left (d+e\,x\right )+70\,B\,a^3\,e^7\,x+3\,A\,b^3\,d^3\,e^4+4\,B\,b^3\,d^4\,e^3+105\,A\,b^3\,e^7\,x^3+140\,B\,b^3\,e^7\,x^4+420\,A\,c^3\,e^7\,x^6-7203\,B\,c^3\,d^6\,e\,x+12\,A\,a\,b^2\,d^2\,e^5+12\,A\,a\,c^2\,d^4\,e^3+12\,A\,a^2\,c\,d^2\,e^5+9\,B\,a\,b^2\,d^3\,e^4+12\,B\,a^2\,b\,d^2\,e^5+30\,A\,b\,c^2\,d^5\,e^2+12\,A\,b^2\,c\,d^4\,e^3+30\,B\,a\,c^2\,d^5\,e^2+9\,B\,a^2\,c\,d^3\,e^4+30\,B\,b^2\,c\,d^5\,e^2+252\,A\,a\,b^2\,e^7\,x^2+252\,A\,a^2\,c\,e^7\,x^2+252\,B\,a^2\,b\,e^7\,x^2+315\,B\,a\,b^2\,e^7\,x^3+420\,A\,a\,c^2\,e^7\,x^4+315\,B\,a^2\,c\,e^7\,x^3+420\,A\,b^2\,c\,e^7\,x^4+630\,A\,b\,c^2\,e^7\,x^5+630\,B\,a\,c^2\,e^7\,x^5+21\,A\,b^3\,d^2\,e^5\,x+63\,A\,b^3\,d\,e^6\,x^2+630\,B\,b^2\,c\,e^7\,x^5+1260\,B\,b\,c^2\,e^7\,x^6+420\,A\,c^3\,d^5\,e^2\,x+28\,B\,b^3\,d^3\,e^4\,x+140\,B\,b^3\,d\,e^6\,x^3+1260\,A\,c^3\,d\,e^6\,x^5-2940\,B\,c^3\,d\,e^6\,x^6-420\,B\,c^3\,e^7\,x^7\,\ln \left (d+e\,x\right )+1260\,A\,c^3\,d^4\,e^3\,x^2+84\,B\,b^3\,d^2\,e^5\,x^2+2100\,A\,c^3\,d^3\,e^4\,x^3+2100\,A\,c^3\,d^2\,e^5\,x^4-20139\,B\,c^3\,d^5\,e^2\,x^2-30625\,B\,c^3\,d^4\,e^3\,x^3-26950\,B\,c^3\,d^3\,e^4\,x^4-13230\,B\,c^3\,d^2\,e^5\,x^5+30\,A\,a^2\,b\,d\,e^6+180\,B\,b\,c^2\,d^6\,e+210\,A\,a^2\,b\,e^7\,x+252\,A\,a\,c^2\,d^2\,e^5\,x^2+630\,A\,b\,c^2\,d^3\,e^4\,x^2+252\,A\,b^2\,c\,d^2\,e^5\,x^2+630\,B\,a\,c^2\,d^3\,e^4\,x^2+1050\,A\,b\,c^2\,d^2\,e^5\,x^3+1050\,B\,a\,c^2\,d^2\,e^5\,x^3+3780\,B\,b\,c^2\,d^4\,e^3\,x^2+630\,B\,b^2\,c\,d^3\,e^4\,x^2+6300\,B\,b\,c^2\,d^3\,e^4\,x^3+1050\,B\,b^2\,c\,d^2\,e^5\,x^3+6300\,B\,b\,c^2\,d^2\,e^5\,x^4-8820\,B\,c^3\,d^5\,e^2\,x^2\,\ln \left (d+e\,x\right )-14700\,B\,c^3\,d^4\,e^3\,x^3\,\ln \left (d+e\,x\right )-14700\,B\,c^3\,d^3\,e^4\,x^4\,\ln \left (d+e\,x\right )-8820\,B\,c^3\,d^2\,e^5\,x^5\,\ln \left (d+e\,x\right )+18\,A\,a\,b\,c\,d^3\,e^4+24\,B\,a\,b\,c\,d^4\,e^3+630\,A\,a\,b\,c\,e^7\,x^3+84\,A\,a\,b^2\,d\,e^6\,x+840\,B\,a\,b\,c\,e^7\,x^4+84\,A\,a^2\,c\,d\,e^6\,x+84\,B\,a^2\,b\,d\,e^6\,x-2940\,B\,c^3\,d^6\,e\,x\,\ln \left (d+e\,x\right )+84\,A\,a\,c^2\,d^3\,e^4\,x+63\,B\,a\,b^2\,d^2\,e^5\,x+189\,B\,a\,b^2\,d\,e^6\,x^2+420\,A\,a\,c^2\,d\,e^6\,x^3+210\,A\,b\,c^2\,d^4\,e^3\,x+84\,A\,b^2\,c\,d^3\,e^4\,x+210\,B\,a\,c^2\,d^4\,e^3\,x+63\,B\,a^2\,c\,d^2\,e^5\,x+189\,B\,a^2\,c\,d\,e^6\,x^2+420\,A\,b^2\,c\,d\,e^6\,x^3+1050\,A\,b\,c^2\,d\,e^6\,x^4+1050\,B\,a\,c^2\,d\,e^6\,x^4+1260\,B\,b\,c^2\,d^5\,e^2\,x+210\,B\,b^2\,c\,d^4\,e^3\,x+1050\,B\,b^2\,c\,d\,e^6\,x^4+3780\,B\,b\,c^2\,d\,e^6\,x^5-2940\,B\,c^3\,d\,e^6\,x^6\,\ln \left (d+e\,x\right )+504\,B\,a\,b\,c\,d^2\,e^5\,x^2+126\,A\,a\,b\,c\,d^2\,e^5\,x+378\,A\,a\,b\,c\,d\,e^6\,x^2+168\,B\,a\,b\,c\,d^3\,e^4\,x+840\,B\,a\,b\,c\,d\,e^6\,x^3}{420\,e^8\,{\left (d+e\,x\right )}^7} \]

[In]

int(((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^8,x)

[Out]

-(60*A*a^3*e^7 - 1089*B*c^3*d^7 + 10*B*a^3*d*e^6 + 60*A*c^3*d^6*e - 420*B*c^3*d^7*log(d + e*x) + 70*B*a^3*e^7*
x + 3*A*b^3*d^3*e^4 + 4*B*b^3*d^4*e^3 + 105*A*b^3*e^7*x^3 + 140*B*b^3*e^7*x^4 + 420*A*c^3*e^7*x^6 - 7203*B*c^3
*d^6*e*x + 12*A*a*b^2*d^2*e^5 + 12*A*a*c^2*d^4*e^3 + 12*A*a^2*c*d^2*e^5 + 9*B*a*b^2*d^3*e^4 + 12*B*a^2*b*d^2*e
^5 + 30*A*b*c^2*d^5*e^2 + 12*A*b^2*c*d^4*e^3 + 30*B*a*c^2*d^5*e^2 + 9*B*a^2*c*d^3*e^4 + 30*B*b^2*c*d^5*e^2 + 2
52*A*a*b^2*e^7*x^2 + 252*A*a^2*c*e^7*x^2 + 252*B*a^2*b*e^7*x^2 + 315*B*a*b^2*e^7*x^3 + 420*A*a*c^2*e^7*x^4 + 3
15*B*a^2*c*e^7*x^3 + 420*A*b^2*c*e^7*x^4 + 630*A*b*c^2*e^7*x^5 + 630*B*a*c^2*e^7*x^5 + 21*A*b^3*d^2*e^5*x + 63
*A*b^3*d*e^6*x^2 + 630*B*b^2*c*e^7*x^5 + 1260*B*b*c^2*e^7*x^6 + 420*A*c^3*d^5*e^2*x + 28*B*b^3*d^3*e^4*x + 140
*B*b^3*d*e^6*x^3 + 1260*A*c^3*d*e^6*x^5 - 2940*B*c^3*d*e^6*x^6 - 420*B*c^3*e^7*x^7*log(d + e*x) + 1260*A*c^3*d
^4*e^3*x^2 + 84*B*b^3*d^2*e^5*x^2 + 2100*A*c^3*d^3*e^4*x^3 + 2100*A*c^3*d^2*e^5*x^4 - 20139*B*c^3*d^5*e^2*x^2
- 30625*B*c^3*d^4*e^3*x^3 - 26950*B*c^3*d^3*e^4*x^4 - 13230*B*c^3*d^2*e^5*x^5 + 30*A*a^2*b*d*e^6 + 180*B*b*c^2
*d^6*e + 210*A*a^2*b*e^7*x + 252*A*a*c^2*d^2*e^5*x^2 + 630*A*b*c^2*d^3*e^4*x^2 + 252*A*b^2*c*d^2*e^5*x^2 + 630
*B*a*c^2*d^3*e^4*x^2 + 1050*A*b*c^2*d^2*e^5*x^3 + 1050*B*a*c^2*d^2*e^5*x^3 + 3780*B*b*c^2*d^4*e^3*x^2 + 630*B*
b^2*c*d^3*e^4*x^2 + 6300*B*b*c^2*d^3*e^4*x^3 + 1050*B*b^2*c*d^2*e^5*x^3 + 6300*B*b*c^2*d^2*e^5*x^4 - 8820*B*c^
3*d^5*e^2*x^2*log(d + e*x) - 14700*B*c^3*d^4*e^3*x^3*log(d + e*x) - 14700*B*c^3*d^3*e^4*x^4*log(d + e*x) - 882
0*B*c^3*d^2*e^5*x^5*log(d + e*x) + 18*A*a*b*c*d^3*e^4 + 24*B*a*b*c*d^4*e^3 + 630*A*a*b*c*e^7*x^3 + 84*A*a*b^2*
d*e^6*x + 840*B*a*b*c*e^7*x^4 + 84*A*a^2*c*d*e^6*x + 84*B*a^2*b*d*e^6*x - 2940*B*c^3*d^6*e*x*log(d + e*x) + 84
*A*a*c^2*d^3*e^4*x + 63*B*a*b^2*d^2*e^5*x + 189*B*a*b^2*d*e^6*x^2 + 420*A*a*c^2*d*e^6*x^3 + 210*A*b*c^2*d^4*e^
3*x + 84*A*b^2*c*d^3*e^4*x + 210*B*a*c^2*d^4*e^3*x + 63*B*a^2*c*d^2*e^5*x + 189*B*a^2*c*d*e^6*x^2 + 420*A*b^2*
c*d*e^6*x^3 + 1050*A*b*c^2*d*e^6*x^4 + 1050*B*a*c^2*d*e^6*x^4 + 1260*B*b*c^2*d^5*e^2*x + 210*B*b^2*c*d^4*e^3*x
+ 1050*B*b^2*c*d*e^6*x^4 + 3780*B*b*c^2*d*e^6*x^5 - 2940*B*c^3*d*e^6*x^6*log(d + e*x) + 504*B*a*b*c*d^2*e^5*x
^2 + 126*A*a*b*c*d^2*e^5*x + 378*A*a*b*c*d*e^6*x^2 + 168*B*a*b*c*d^3*e^4*x + 840*B*a*b*c*d*e^6*x^3)/(420*e^8*(
d + e*x)^7)