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

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

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Rubi [A]  time = 0.77, antiderivative size = 546, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {771} \begin {gather*} \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} \end {gather*}

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*}

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Mathematica [A]  time = 0.49, size = 863, normalized size = 1.57 \begin {gather*} \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} \end {gather*}

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)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 1023, normalized size = 1.87 \begin {gather*} \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 )}} \end {gather*}

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)

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giac [A]  time = 0.18, size = 1001, normalized size = 1.83 \begin {gather*} B c^{3} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {{\left (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 + {\left (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}\right )} e^{\left (-1\right )}\right )} e^{\left (-7\right )}}{420 \, {\left (x e + d\right )}^{7}} \end {gather*}

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

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maple [B]  time = 0.06, size = 1661, normalized size = 3.03

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]

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

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maxima [A]  time = 1.01, size = 926, normalized size = 1.69 \begin {gather*} \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}} \end {gather*}

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

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mupad [B]  time = 2.58, size = 1353, normalized size = 2.47 \begin {gather*} -\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

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