3.22.6 \(\int \frac {(A+B x) (a+b x+c x^2)^3}{(d+e x)^9} \, dx\)

Optimal. Leaf size=550 \[ \frac {A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )}{5 e^8 (d+e x)^5}+\frac {c \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{e^8 (d+e x)^3}+\frac {\left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{2 e^8 (d+e x)^6}+\frac {B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{4 e^8 (d+e x)^4}+\frac {\left (a e^2-b d e+c d^2\right )^2 \left (3 A e (2 c d-b e)-B \left (7 c d^2-e (4 b d-a e)\right )\right )}{7 e^8 (d+e x)^7}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{8 e^8 (d+e x)^8}+\frac {c^2 (-A c e-3 b B e+7 B c d)}{2 e^8 (d+e x)^2}-\frac {B c^3}{e^8 (d+e x)} \]

________________________________________________________________________________________

Rubi [A]  time = 0.71, antiderivative size = 548, 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 )}{5 e^8 (d+e x)^5}+\frac {c \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{e^8 (d+e x)^3}+\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 )}{4 e^8 (d+e x)^4}+\frac {\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 )}{2 e^8 (d+e x)^6}-\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 )}{7 e^8 (d+e x)^7}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{8 e^8 (d+e x)^8}+\frac {c^2 (-A c e-3 b B e+7 B c d)}{2 e^8 (d+e x)^2}-\frac {B c^3}{e^8 (d+e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^9,x]

[Out]

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

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)^9} \, 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)^9}+\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)^8}+\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)^7}+\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)^6}+\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)^5}+\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)^4}+\frac {c^2 (-7 B c d+3 b B e+A c e)}{e^7 (d+e x)^3}+\frac {B c^3}{e^7 (d+e x)^2}\right ) \, dx\\ &=\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^3}{8 e^8 (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 )}{7 e^8 (d+e x)^7}+\frac {\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 )}{2 e^8 (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 )}{5 e^8 (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 )}{4 e^8 (d+e x)^4}+\frac {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^8 (d+e x)^3}+\frac {c^2 (7 B c d-3 b B e-A c e)}{2 e^8 (d+e x)^2}-\frac {B c^3}{e^8 (d+e x)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.47, size = 847, normalized size = 1.54 \begin {gather*} -\frac {A e \left (5 \left (d^6+8 e x d^5+28 e^2 x^2 d^4+56 e^3 x^3 d^3+70 e^4 x^4 d^2+56 e^5 x^5 d+28 e^6 x^6\right ) c^3+e \left (3 a e \left (d^4+8 e x d^3+28 e^2 x^2 d^2+56 e^3 x^3 d+70 e^4 x^4\right )+5 b \left (d^5+8 e x d^4+28 e^2 x^2 d^3+56 e^3 x^3 d^2+70 e^4 x^4 d+56 e^5 x^5\right )\right ) c^2+e^2 \left (3 \left (d^4+8 e x d^3+28 e^2 x^2 d^2+56 e^3 x^3 d+70 e^4 x^4\right ) b^2+6 a e \left (d^3+8 e x d^2+28 e^2 x^2 d+56 e^3 x^3\right ) b+5 a^2 e^2 \left (d^2+8 e x d+28 e^2 x^2\right )\right ) c+e^3 \left (\left (d^3+8 e x d^2+28 e^2 x^2 d+56 e^3 x^3\right ) b^3+5 a e \left (d^2+8 e x d+28 e^2 x^2\right ) b^2+15 a^2 e^2 (d+8 e x) b+35 a^3 e^3\right )\right )+B \left (35 \left (d^7+8 e x d^6+28 e^2 x^2 d^5+56 e^3 x^3 d^4+70 e^4 x^4 d^3+56 e^5 x^5 d^2+28 e^6 x^6 d+8 e^7 x^7\right ) c^3+5 e \left (a e \left (d^5+8 e x d^4+28 e^2 x^2 d^3+56 e^3 x^3 d^2+70 e^4 x^4 d+56 e^5 x^5\right )+3 b \left (d^6+8 e x d^5+28 e^2 x^2 d^4+56 e^3 x^3 d^3+70 e^4 x^4 d^2+56 e^5 x^5 d+28 e^6 x^6\right )\right ) c^2+e^2 \left (5 \left (d^5+8 e x d^4+28 e^2 x^2 d^3+56 e^3 x^3 d^2+70 e^4 x^4 d+56 e^5 x^5\right ) b^2+6 a e \left (d^4+8 e x d^3+28 e^2 x^2 d^2+56 e^3 x^3 d+70 e^4 x^4\right ) b+3 a^2 e^2 \left (d^3+8 e x d^2+28 e^2 x^2 d+56 e^3 x^3\right )\right ) c+e^3 \left (\left (d^4+8 e x d^3+28 e^2 x^2 d^2+56 e^3 x^3 d+70 e^4 x^4\right ) b^3+3 a e \left (d^3+8 e x d^2+28 e^2 x^2 d+56 e^3 x^3\right ) b^2+5 a^2 e^2 \left (d^2+8 e x d+28 e^2 x^2\right ) b+5 a^3 e^3 (d+8 e x)\right )\right )}{280 e^8 (d+e x)^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^9,x]

[Out]

-1/280*(A*e*(5*c^3*(d^6 + 8*d^5*e*x + 28*d^4*e^2*x^2 + 56*d^3*e^3*x^3 + 70*d^2*e^4*x^4 + 56*d*e^5*x^5 + 28*e^6
*x^6) + e^3*(35*a^3*e^3 + 15*a^2*b*e^2*(d + 8*e*x) + 5*a*b^2*e*(d^2 + 8*d*e*x + 28*e^2*x^2) + b^3*(d^3 + 8*d^2
*e*x + 28*d*e^2*x^2 + 56*e^3*x^3)) + c*e^2*(5*a^2*e^2*(d^2 + 8*d*e*x + 28*e^2*x^2) + 6*a*b*e*(d^3 + 8*d^2*e*x
+ 28*d*e^2*x^2 + 56*e^3*x^3) + 3*b^2*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4)) + c^2*e*(
3*a*e*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4) + 5*b*(d^5 + 8*d^4*e*x + 28*d^3*e^2*x^2 +
56*d^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e^5*x^5))) + B*(35*c^3*(d^7 + 8*d^6*e*x + 28*d^5*e^2*x^2 + 56*d^4*e^3*x^3
+ 70*d^3*e^4*x^4 + 56*d^2*e^5*x^5 + 28*d*e^6*x^6 + 8*e^7*x^7) + e^3*(5*a^3*e^3*(d + 8*e*x) + 5*a^2*b*e^2*(d^2
+ 8*d*e*x + 28*e^2*x^2) + 3*a*b^2*e*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + b^3*(d^4 + 8*d^3*e*x + 28*
d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4)) + c*e^2*(3*a^2*e^2*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + 6
*a*b*e*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4) + 5*b^2*(d^5 + 8*d^4*e*x + 28*d^3*e^2*x^
2 + 56*d^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e^5*x^5)) + 5*c^2*e*(a*e*(d^5 + 8*d^4*e*x + 28*d^3*e^2*x^2 + 56*d^2*e^3
*x^3 + 70*d*e^4*x^4 + 56*e^5*x^5) + 3*b*(d^6 + 8*d^5*e*x + 28*d^4*e^2*x^2 + 56*d^3*e^3*x^3 + 70*d^2*e^4*x^4 +
56*d*e^5*x^5 + 28*e^6*x^6))))/(e^8*(d + e*x)^8)

________________________________________________________________________________________

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)^9} \, 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)^9,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^9, x]

________________________________________________________________________________________

fricas [A]  time = 0.38, size = 922, normalized size = 1.68 \begin {gather*} -\frac {280 \, B c^{3} e^{7} x^{7} + 35 \, B c^{3} d^{7} + 35 \, A a^{3} e^{7} + 5 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 5 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{5} e^{2} + {\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} + {\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} + 5 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{2} e^{5} + 5 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{6} + 140 \, {\left (7 \, B c^{3} d e^{6} + {\left (3 \, B b c^{2} + A c^{3}\right )} e^{7}\right )} x^{6} + 280 \, {\left (7 \, B c^{3} d^{2} e^{5} + {\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 (35 \, B c^{3} d^{3} e^{4} + 5 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{5} + 5 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d e^{6} + {\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} + 56 \, {\left (35 \, B c^{3} d^{4} e^{3} + 5 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e^{4} + 5 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{2} e^{5} + {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d e^{6} + {\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} + 28 \, {\left (35 \, B c^{3} d^{5} e^{2} + 5 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{3} + 5 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{3} e^{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} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d e^{6} + 5 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e^{7}\right )} x^{2} + 8 \, {\left (35 \, B c^{3} d^{6} e + 5 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} + 5 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{4} e^{3} + {\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} + {\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} + 5 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d e^{6} + 5 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{7}\right )} x}{280 \, {\left (e^{16} x^{8} + 8 \, d e^{15} x^{7} + 28 \, d^{2} e^{14} x^{6} + 56 \, d^{3} e^{13} x^{5} + 70 \, d^{4} e^{12} x^{4} + 56 \, d^{5} e^{11} x^{3} + 28 \, d^{6} e^{10} x^{2} + 8 \, d^{7} e^{9} x + d^{8} 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)^9,x, algorithm="fricas")

[Out]

-1/280*(280*B*c^3*e^7*x^7 + 35*B*c^3*d^7 + 35*A*a^3*e^7 + 5*(3*B*b*c^2 + A*c^3)*d^6*e + 5*(B*b^2*c + (B*a + A*
b)*c^2)*d^5*e^2 + (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^4*e^3 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*
b)*c)*d^3*e^4 + 5*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^2*e^5 + 5*(B*a^3 + 3*A*a^2*b)*d*e^6 + 140*(7*B*c^3*d*e^6 + (
3*B*b*c^2 + A*c^3)*e^7)*x^6 + 280*(7*B*c^3*d^2*e^5 + (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*(35*B*c^3*d^3*e^4 + 5*(3*B*b*c^2 + A*c^3)*d^2*e^5 + 5*(B*b^2*c + (B*a + A*b)*c^2)*d*e^6 + (B*b^3
+ 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*e^7)*x^4 + 56*(35*B*c^3*d^4*e^3 + 5*(3*B*b*c^2 + A*c^3)*d^3*e^4 + 5*(B*b^
2*c + (B*a + A*b)*c^2)*d^2*e^5 + (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d*e^6 + (3*B*a*b^2 + A*b^3 + 3*(B
*a^2 + 2*A*a*b)*c)*e^7)*x^3 + 28*(35*B*c^3*d^5*e^2 + 5*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 5*(B*b^2*c + (B*a + A*b)*
c^2)*d^3*e^4 + (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^2*e^5 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*
c)*d*e^6 + 5*(B*a^2*b + A*a*b^2 + A*a^2*c)*e^7)*x^2 + 8*(35*B*c^3*d^6*e + 5*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 5*(B
*b^2*c + (B*a + A*b)*c^2)*d^4*e^3 + (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^3*e^4 + (3*B*a*b^2 + A*b^3 +
3*(B*a^2 + 2*A*a*b)*c)*d^2*e^5 + 5*(B*a^2*b + A*a*b^2 + A*a^2*c)*d*e^6 + 5*(B*a^3 + 3*A*a^2*b)*e^7)*x)/(e^16*
x^8 + 8*d*e^15*x^7 + 28*d^2*e^14*x^6 + 56*d^3*e^13*x^5 + 70*d^4*e^12*x^4 + 56*d^5*e^11*x^3 + 28*d^6*e^10*x^2 +
8*d^7*e^9*x + d^8*e^8)

________________________________________________________________________________________

giac [B]  time = 0.18, size = 1127, normalized size = 2.05 \begin {gather*} -\frac {{\left (280 \, B c^{3} x^{7} e^{7} + 980 \, B c^{3} d x^{6} e^{6} + 1960 \, B c^{3} d^{2} x^{5} e^{5} + 2450 \, B c^{3} d^{3} x^{4} e^{4} + 1960 \, B c^{3} d^{4} x^{3} e^{3} + 980 \, B c^{3} d^{5} x^{2} e^{2} + 280 \, B c^{3} d^{6} x e + 35 \, B c^{3} d^{7} + 420 \, B b c^{2} x^{6} e^{7} + 140 \, A c^{3} x^{6} e^{7} + 840 \, B b c^{2} d x^{5} e^{6} + 280 \, A c^{3} d x^{5} e^{6} + 1050 \, B b c^{2} d^{2} x^{4} e^{5} + 350 \, A c^{3} d^{2} x^{4} e^{5} + 840 \, B b c^{2} d^{3} x^{3} e^{4} + 280 \, A c^{3} d^{3} x^{3} e^{4} + 420 \, B b c^{2} d^{4} x^{2} e^{3} + 140 \, A c^{3} d^{4} x^{2} e^{3} + 120 \, B b c^{2} d^{5} x e^{2} + 40 \, A c^{3} d^{5} x e^{2} + 15 \, B b c^{2} d^{6} e + 5 \, A c^{3} d^{6} e + 280 \, B b^{2} c x^{5} e^{7} + 280 \, B a c^{2} x^{5} e^{7} + 280 \, A b c^{2} x^{5} e^{7} + 350 \, B b^{2} c d x^{4} e^{6} + 350 \, B a c^{2} d x^{4} e^{6} + 350 \, A b c^{2} d x^{4} e^{6} + 280 \, B b^{2} c d^{2} x^{3} e^{5} + 280 \, B a c^{2} d^{2} x^{3} e^{5} + 280 \, A b c^{2} d^{2} x^{3} e^{5} + 140 \, B b^{2} c d^{3} x^{2} e^{4} + 140 \, B a c^{2} d^{3} x^{2} e^{4} + 140 \, A b c^{2} d^{3} x^{2} e^{4} + 40 \, B b^{2} c d^{4} x e^{3} + 40 \, B a c^{2} d^{4} x e^{3} + 40 \, A b c^{2} d^{4} x e^{3} + 5 \, B b^{2} c d^{5} e^{2} + 5 \, B a c^{2} d^{5} e^{2} + 5 \, A b c^{2} d^{5} e^{2} + 70 \, B b^{3} x^{4} e^{7} + 420 \, B a b c x^{4} e^{7} + 210 \, A b^{2} c x^{4} e^{7} + 210 \, A a c^{2} x^{4} e^{7} + 56 \, B b^{3} d x^{3} e^{6} + 336 \, B a b c d x^{3} e^{6} + 168 \, A b^{2} c d x^{3} e^{6} + 168 \, A a c^{2} d x^{3} e^{6} + 28 \, B b^{3} d^{2} x^{2} e^{5} + 168 \, B a b c d^{2} x^{2} e^{5} + 84 \, A b^{2} c d^{2} x^{2} e^{5} + 84 \, A a c^{2} d^{2} x^{2} e^{5} + 8 \, B b^{3} d^{3} x e^{4} + 48 \, B a b c d^{3} x e^{4} + 24 \, A b^{2} c d^{3} x e^{4} + 24 \, A a c^{2} d^{3} x e^{4} + B b^{3} d^{4} e^{3} + 6 \, B a b c d^{4} e^{3} + 3 \, A b^{2} c d^{4} e^{3} + 3 \, A a c^{2} d^{4} e^{3} + 168 \, B a b^{2} x^{3} e^{7} + 56 \, A b^{3} x^{3} e^{7} + 168 \, B a^{2} c x^{3} e^{7} + 336 \, A a b c x^{3} e^{7} + 84 \, B a b^{2} d x^{2} e^{6} + 28 \, A b^{3} d x^{2} e^{6} + 84 \, B a^{2} c d x^{2} e^{6} + 168 \, A a b c d x^{2} e^{6} + 24 \, B a b^{2} d^{2} x e^{5} + 8 \, A b^{3} d^{2} x e^{5} + 24 \, B a^{2} c d^{2} x e^{5} + 48 \, A a b c d^{2} x e^{5} + 3 \, B a b^{2} d^{3} e^{4} + A b^{3} d^{3} e^{4} + 3 \, B a^{2} c d^{3} e^{4} + 6 \, A a b c d^{3} e^{4} + 140 \, B a^{2} b x^{2} e^{7} + 140 \, A a b^{2} x^{2} e^{7} + 140 \, A a^{2} c x^{2} e^{7} + 40 \, B a^{2} b d x e^{6} + 40 \, A a b^{2} d x e^{6} + 40 \, A a^{2} c d x e^{6} + 5 \, B a^{2} b d^{2} e^{5} + 5 \, A a b^{2} d^{2} e^{5} + 5 \, A a^{2} c d^{2} e^{5} + 40 \, B a^{3} x e^{7} + 120 \, A a^{2} b x e^{7} + 5 \, B a^{3} d e^{6} + 15 \, A a^{2} b d e^{6} + 35 \, A a^{3} e^{7}\right )} e^{\left (-8\right )}}{280 \, {\left (x e + d\right )}^{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)^9,x, algorithm="giac")

[Out]

-1/280*(280*B*c^3*x^7*e^7 + 980*B*c^3*d*x^6*e^6 + 1960*B*c^3*d^2*x^5*e^5 + 2450*B*c^3*d^3*x^4*e^4 + 1960*B*c^3
*d^4*x^3*e^3 + 980*B*c^3*d^5*x^2*e^2 + 280*B*c^3*d^6*x*e + 35*B*c^3*d^7 + 420*B*b*c^2*x^6*e^7 + 140*A*c^3*x^6*
e^7 + 840*B*b*c^2*d*x^5*e^6 + 280*A*c^3*d*x^5*e^6 + 1050*B*b*c^2*d^2*x^4*e^5 + 350*A*c^3*d^2*x^4*e^5 + 840*B*b
*c^2*d^3*x^3*e^4 + 280*A*c^3*d^3*x^3*e^4 + 420*B*b*c^2*d^4*x^2*e^3 + 140*A*c^3*d^4*x^2*e^3 + 120*B*b*c^2*d^5*x
*e^2 + 40*A*c^3*d^5*x*e^2 + 15*B*b*c^2*d^6*e + 5*A*c^3*d^6*e + 280*B*b^2*c*x^5*e^7 + 280*B*a*c^2*x^5*e^7 + 280
*A*b*c^2*x^5*e^7 + 350*B*b^2*c*d*x^4*e^6 + 350*B*a*c^2*d*x^4*e^6 + 350*A*b*c^2*d*x^4*e^6 + 280*B*b^2*c*d^2*x^3
*e^5 + 280*B*a*c^2*d^2*x^3*e^5 + 280*A*b*c^2*d^2*x^3*e^5 + 140*B*b^2*c*d^3*x^2*e^4 + 140*B*a*c^2*d^3*x^2*e^4 +
140*A*b*c^2*d^3*x^2*e^4 + 40*B*b^2*c*d^4*x*e^3 + 40*B*a*c^2*d^4*x*e^3 + 40*A*b*c^2*d^4*x*e^3 + 5*B*b^2*c*d^5*
e^2 + 5*B*a*c^2*d^5*e^2 + 5*A*b*c^2*d^5*e^2 + 70*B*b^3*x^4*e^7 + 420*B*a*b*c*x^4*e^7 + 210*A*b^2*c*x^4*e^7 + 2
10*A*a*c^2*x^4*e^7 + 56*B*b^3*d*x^3*e^6 + 336*B*a*b*c*d*x^3*e^6 + 168*A*b^2*c*d*x^3*e^6 + 168*A*a*c^2*d*x^3*e^
6 + 28*B*b^3*d^2*x^2*e^5 + 168*B*a*b*c*d^2*x^2*e^5 + 84*A*b^2*c*d^2*x^2*e^5 + 84*A*a*c^2*d^2*x^2*e^5 + 8*B*b^3
*d^3*x*e^4 + 48*B*a*b*c*d^3*x*e^4 + 24*A*b^2*c*d^3*x*e^4 + 24*A*a*c^2*d^3*x*e^4 + B*b^3*d^4*e^3 + 6*B*a*b*c*d^
4*e^3 + 3*A*b^2*c*d^4*e^3 + 3*A*a*c^2*d^4*e^3 + 168*B*a*b^2*x^3*e^7 + 56*A*b^3*x^3*e^7 + 168*B*a^2*c*x^3*e^7 +
336*A*a*b*c*x^3*e^7 + 84*B*a*b^2*d*x^2*e^6 + 28*A*b^3*d*x^2*e^6 + 84*B*a^2*c*d*x^2*e^6 + 168*A*a*b*c*d*x^2*e^
6 + 24*B*a*b^2*d^2*x*e^5 + 8*A*b^3*d^2*x*e^5 + 24*B*a^2*c*d^2*x*e^5 + 48*A*a*b*c*d^2*x*e^5 + 3*B*a*b^2*d^3*e^4
+ A*b^3*d^3*e^4 + 3*B*a^2*c*d^3*e^4 + 6*A*a*b*c*d^3*e^4 + 140*B*a^2*b*x^2*e^7 + 140*A*a*b^2*x^2*e^7 + 140*A*a
^2*c*x^2*e^7 + 40*B*a^2*b*d*x*e^6 + 40*A*a*b^2*d*x*e^6 + 40*A*a^2*c*d*x*e^6 + 5*B*a^2*b*d^2*e^5 + 5*A*a*b^2*d^
2*e^5 + 5*A*a^2*c*d^2*e^5 + 40*B*a^3*x*e^7 + 120*A*a^2*b*x*e^7 + 5*B*a^3*d*e^6 + 15*A*a^2*b*d*e^6 + 35*A*a^3*e
^7)*e^(-8)/(x*e + d)^8

________________________________________________________________________________________

maple [A]  time = 0.05, size = 1067, normalized size = 1.94 \begin {gather*} -\frac {B \,c^{3}}{\left (e x +d \right ) e^{8}}-\frac {\left (A c e +3 B b e -7 B c d \right ) c^{2}}{2 \left (e x +d \right )^{2} e^{8}}-\frac {\left (A b c \,e^{2}-2 A \,c^{2} d e +a B c \,e^{2}+b^{2} B \,e^{2}-6 B b c d e +7 B \,c^{2} d^{2}\right ) c}{\left (e x +d \right )^{3} e^{8}}-\frac {A \,a^{3} e^{7}-3 A d \,a^{2} b \,e^{6}+3 A \,d^{2} a^{2} c \,e^{5}+3 A \,d^{2} a \,b^{2} e^{5}-6 A \,d^{3} a b c \,e^{4}+3 A a \,c^{2} d^{4} e^{3}-A \,b^{3} d^{3} e^{4}+3 A \,d^{4} b^{2} c \,e^{3}-3 A b \,c^{2} d^{5} e^{2}+A \,d^{6} c^{3} e -B \,a^{3} d \,e^{6}+3 B \,d^{2} a^{2} b \,e^{5}-3 B \,d^{3} a^{2} c \,e^{4}-3 B a \,b^{2} d^{3} e^{4}+6 B \,d^{4} a b c \,e^{3}-3 B a \,c^{2} d^{5} e^{2}+B \,d^{4} b^{3} e^{3}-3 B \,d^{5} b^{2} c \,e^{2}+3 B \,d^{6} b \,c^{2} e -B \,d^{7} c^{3}}{8 \left (e x +d \right )^{8} e^{8}}-\frac {3 A a \,c^{2} e^{3}+3 A \,b^{2} c \,e^{3}-15 A b \,c^{2} d \,e^{2}+15 A \,c^{3} d^{2} e +6 a b B c \,e^{3}-15 B d a \,c^{2} e^{2}+b^{3} B \,e^{3}-15 B d \,b^{2} c \,e^{2}+45 B b \,c^{2} d^{2} e -35 B \,d^{3} c^{3}}{4 \left (e x +d \right )^{4} e^{8}}-\frac {3 A \,a^{2} c \,e^{5}+3 A a \,b^{2} e^{5}-18 A d a b c \,e^{4}+18 A a \,c^{2} d^{2} e^{3}-3 A d \,b^{3} e^{4}+18 A \,b^{2} c \,d^{2} e^{3}-30 A b \,c^{2} d^{3} e^{2}+15 A \,c^{3} d^{4} e +3 B \,a^{2} b \,e^{5}-9 B \,a^{2} c d \,e^{4}-9 B d a \,b^{2} e^{4}+36 B \,d^{2} a b c \,e^{3}-30 B a \,c^{2} d^{3} e^{2}+6 B \,d^{2} b^{3} e^{3}-30 B \,d^{3} b^{2} c \,e^{2}+45 B \,d^{4} b \,c^{2} e -21 B \,d^{5} c^{3}}{6 \left (e x +d \right )^{6} e^{8}}-\frac {3 A \,a^{2} b \,e^{6}-6 A \,a^{2} c d \,e^{5}-6 A d a \,b^{2} e^{5}+18 A \,d^{2} a b c \,e^{4}-12 A \,d^{3} a \,c^{2} e^{3}+3 A \,d^{2} b^{3} e^{4}-12 A \,d^{3} b^{2} c \,e^{3}+15 A b \,c^{2} d^{4} e^{2}-6 A \,c^{3} d^{5} e +B \,a^{3} e^{6}-6 B d \,a^{2} b \,e^{5}+9 B \,a^{2} c \,d^{2} e^{4}+9 B \,d^{2} a \,b^{2} e^{4}-24 B \,d^{3} a b c \,e^{3}+15 B \,d^{4} a \,c^{2} e^{2}-4 B \,b^{3} d^{3} e^{3}+15 B \,d^{4} b^{2} c \,e^{2}-18 B \,d^{5} b \,c^{2} e +7 B \,d^{6} c^{3}}{7 \left (e x +d \right )^{7} e^{8}}-\frac {6 A a b c \,e^{4}-12 A d a \,c^{2} e^{3}+A \,b^{3} e^{4}-12 A d \,b^{2} c \,e^{3}+30 A b \,c^{2} d^{2} e^{2}-20 A \,c^{3} d^{3} e +3 B \,a^{2} c \,e^{4}+3 B a \,b^{2} e^{4}-24 B d a b c \,e^{3}+30 B \,d^{2} a \,c^{2} e^{2}-4 B \,b^{3} d \,e^{3}+30 B \,d^{2} b^{2} c \,e^{2}-60 B \,d^{3} b \,c^{2} e +35 B \,c^{3} d^{4}}{5 \left (e x +d \right )^{5} e^{8}} \end {gather*}

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)^9,x)

[Out]

-1/(e*x+d)*B*c^3/e^8-1/8*(A*a^3*e^7-3*A*a^2*b*d*e^6+3*A*a^2*c*d^2*e^5+3*A*a*b^2*d^2*e^5-6*A*a*b*c*d^3*e^4+3*A*
a*c^2*d^4*e^3-A*b^3*d^3*e^4+3*A*b^2*c*d^4*e^3-3*A*b*c^2*d^5*e^2+A*c^3*d^6*e-B*a^3*d*e^6+3*B*a^2*b*d^2*e^5-3*B*
a^2*c*d^3*e^4-3*B*a*b^2*d^3*e^4+6*B*a*b*c*d^4*e^3-3*B*a*c^2*d^5*e^2+B*b^3*d^4*e^3-3*B*b^2*c*d^5*e^2+3*B*b*c^2*
d^6*e-B*c^3*d^7)/e^8/(e*x+d)^8-1/2*c^2*(A*c*e+3*B*b*e-7*B*c*d)/e^8/(e*x+d)^2-1/4*(3*A*a*c^2*e^3+3*A*b^2*c*e^3-
15*A*b*c^2*d*e^2+15*A*c^3*d^2*e+6*B*a*b*c*e^3-15*B*a*c^2*d*e^2+B*b^3*e^3-15*B*b^2*c*d*e^2+45*B*b*c^2*d^2*e-35*
B*c^3*d^3)/e^8/(e*x+d)^4-1/6*(3*A*a^2*c*e^5+3*A*a*b^2*e^5-18*A*a*b*c*d*e^4+18*A*a*c^2*d^2*e^3-3*A*b^3*d*e^4+18
*A*b^2*c*d^2*e^3-30*A*b*c^2*d^3*e^2+15*A*c^3*d^4*e+3*B*a^2*b*e^5-9*B*a^2*c*d*e^4-9*B*a*b^2*d*e^4+36*B*a*b*c*d^
2*e^3-30*B*a*c^2*d^3*e^2+6*B*b^3*d^2*e^3-30*B*b^2*c*d^3*e^2+45*B*b*c^2*d^4*e-21*B*c^3*d^5)/e^8/(e*x+d)^6-1/7*(
3*A*a^2*b*e^6-6*A*a^2*c*d*e^5-6*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+15*A*b*c^2*d^4*e^2-6*A*c^3*d^5*e+B*a^3*e^6-6*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+15*B*a*c^2*d^4*e^2-4*B*b^3*d^3*e^3+15*B*b^2*c*d^4*e^2-18*B*b*c^2*d^5*e+7*B*c^3*d^6)/e^8/(e*x+d)^7
-1/5*(6*A*a*b*c*e^4-12*A*a*c^2*d*e^3+A*b^3*e^4-12*A*b^2*c*d*e^3+30*A*b*c^2*d^2*e^2-20*A*c^3*d^3*e+3*B*a^2*c*e^
4+3*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^2-60*B*b*c^2*d^3*e+35*B*c^3
*d^4)/e^8/(e*x+d)^5-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+7*B*c^2*d^2)/e^8/(e*x+d)^3

________________________________________________________________________________________

maxima [A]  time = 1.06, size = 922, normalized size = 1.68 \begin {gather*} -\frac {280 \, B c^{3} e^{7} x^{7} + 35 \, B c^{3} d^{7} + 35 \, A a^{3} e^{7} + 5 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 5 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{5} e^{2} + {\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} + {\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} + 5 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{2} e^{5} + 5 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{6} + 140 \, {\left (7 \, B c^{3} d e^{6} + {\left (3 \, B b c^{2} + A c^{3}\right )} e^{7}\right )} x^{6} + 280 \, {\left (7 \, B c^{3} d^{2} e^{5} + {\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 (35 \, B c^{3} d^{3} e^{4} + 5 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{5} + 5 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d e^{6} + {\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} + 56 \, {\left (35 \, B c^{3} d^{4} e^{3} + 5 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e^{4} + 5 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{2} e^{5} + {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d e^{6} + {\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} + 28 \, {\left (35 \, B c^{3} d^{5} e^{2} + 5 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{3} + 5 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{3} e^{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} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d e^{6} + 5 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e^{7}\right )} x^{2} + 8 \, {\left (35 \, B c^{3} d^{6} e + 5 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} + 5 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{4} e^{3} + {\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} + {\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} + 5 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d e^{6} + 5 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{7}\right )} x}{280 \, {\left (e^{16} x^{8} + 8 \, d e^{15} x^{7} + 28 \, d^{2} e^{14} x^{6} + 56 \, d^{3} e^{13} x^{5} + 70 \, d^{4} e^{12} x^{4} + 56 \, d^{5} e^{11} x^{3} + 28 \, d^{6} e^{10} x^{2} + 8 \, d^{7} e^{9} x + d^{8} 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)^9,x, algorithm="maxima")

[Out]

-1/280*(280*B*c^3*e^7*x^7 + 35*B*c^3*d^7 + 35*A*a^3*e^7 + 5*(3*B*b*c^2 + A*c^3)*d^6*e + 5*(B*b^2*c + (B*a + A*
b)*c^2)*d^5*e^2 + (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^4*e^3 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*
b)*c)*d^3*e^4 + 5*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^2*e^5 + 5*(B*a^3 + 3*A*a^2*b)*d*e^6 + 140*(7*B*c^3*d*e^6 + (
3*B*b*c^2 + A*c^3)*e^7)*x^6 + 280*(7*B*c^3*d^2*e^5 + (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*(35*B*c^3*d^3*e^4 + 5*(3*B*b*c^2 + A*c^3)*d^2*e^5 + 5*(B*b^2*c + (B*a + A*b)*c^2)*d*e^6 + (B*b^3
+ 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*e^7)*x^4 + 56*(35*B*c^3*d^4*e^3 + 5*(3*B*b*c^2 + A*c^3)*d^3*e^4 + 5*(B*b^
2*c + (B*a + A*b)*c^2)*d^2*e^5 + (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d*e^6 + (3*B*a*b^2 + A*b^3 + 3*(B
*a^2 + 2*A*a*b)*c)*e^7)*x^3 + 28*(35*B*c^3*d^5*e^2 + 5*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 5*(B*b^2*c + (B*a + A*b)*
c^2)*d^3*e^4 + (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^2*e^5 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*
c)*d*e^6 + 5*(B*a^2*b + A*a*b^2 + A*a^2*c)*e^7)*x^2 + 8*(35*B*c^3*d^6*e + 5*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 5*(B
*b^2*c + (B*a + A*b)*c^2)*d^4*e^3 + (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^3*e^4 + (3*B*a*b^2 + A*b^3 +
3*(B*a^2 + 2*A*a*b)*c)*d^2*e^5 + 5*(B*a^2*b + A*a*b^2 + A*a^2*c)*d*e^6 + 5*(B*a^3 + 3*A*a^2*b)*e^7)*x)/(e^16*
x^8 + 8*d*e^15*x^7 + 28*d^2*e^14*x^6 + 56*d^3*e^13*x^5 + 70*d^4*e^12*x^4 + 56*d^5*e^11*x^3 + 28*d^6*e^10*x^2 +
8*d^7*e^9*x + d^8*e^8)

________________________________________________________________________________________

mupad [B]  time = 0.28, size = 1115, normalized size = 2.03 \begin {gather*} -\frac {\frac {5\,B\,a^3\,d\,e^6+35\,A\,a^3\,e^7+5\,B\,a^2\,b\,d^2\,e^5+15\,A\,a^2\,b\,d\,e^6+3\,B\,a^2\,c\,d^3\,e^4+5\,A\,a^2\,c\,d^2\,e^5+3\,B\,a\,b^2\,d^3\,e^4+5\,A\,a\,b^2\,d^2\,e^5+6\,B\,a\,b\,c\,d^4\,e^3+6\,A\,a\,b\,c\,d^3\,e^4+5\,B\,a\,c^2\,d^5\,e^2+3\,A\,a\,c^2\,d^4\,e^3+B\,b^3\,d^4\,e^3+A\,b^3\,d^3\,e^4+5\,B\,b^2\,c\,d^5\,e^2+3\,A\,b^2\,c\,d^4\,e^3+15\,B\,b\,c^2\,d^6\,e+5\,A\,b\,c^2\,d^5\,e^2+35\,B\,c^3\,d^7+5\,A\,c^3\,d^6\,e}{280\,e^8}+\frac {x^4\,\left (B\,b^3\,e^3+5\,B\,b^2\,c\,d\,e^2+3\,A\,b^2\,c\,e^3+15\,B\,b\,c^2\,d^2\,e+5\,A\,b\,c^2\,d\,e^2+6\,B\,a\,b\,c\,e^3+35\,B\,c^3\,d^3+5\,A\,c^3\,d^2\,e+5\,B\,a\,c^2\,d\,e^2+3\,A\,a\,c^2\,e^3\right )}{4\,e^4}+\frac {x\,\left (5\,B\,a^3\,e^6+5\,B\,a^2\,b\,d\,e^5+15\,A\,a^2\,b\,e^6+3\,B\,a^2\,c\,d^2\,e^4+5\,A\,a^2\,c\,d\,e^5+3\,B\,a\,b^2\,d^2\,e^4+5\,A\,a\,b^2\,d\,e^5+6\,B\,a\,b\,c\,d^3\,e^3+6\,A\,a\,b\,c\,d^2\,e^4+5\,B\,a\,c^2\,d^4\,e^2+3\,A\,a\,c^2\,d^3\,e^3+B\,b^3\,d^3\,e^3+A\,b^3\,d^2\,e^4+5\,B\,b^2\,c\,d^4\,e^2+3\,A\,b^2\,c\,d^3\,e^3+15\,B\,b\,c^2\,d^5\,e+5\,A\,b\,c^2\,d^4\,e^2+35\,B\,c^3\,d^6+5\,A\,c^3\,d^5\,e\right )}{35\,e^7}+\frac {x^2\,\left (5\,B\,a^2\,b\,e^5+3\,B\,a^2\,c\,d\,e^4+5\,A\,a^2\,c\,e^5+3\,B\,a\,b^2\,d\,e^4+5\,A\,a\,b^2\,e^5+6\,B\,a\,b\,c\,d^2\,e^3+6\,A\,a\,b\,c\,d\,e^4+5\,B\,a\,c^2\,d^3\,e^2+3\,A\,a\,c^2\,d^2\,e^3+B\,b^3\,d^2\,e^3+A\,b^3\,d\,e^4+5\,B\,b^2\,c\,d^3\,e^2+3\,A\,b^2\,c\,d^2\,e^3+15\,B\,b\,c^2\,d^4\,e+5\,A\,b\,c^2\,d^3\,e^2+35\,B\,c^3\,d^5+5\,A\,c^3\,d^4\,e\right )}{10\,e^6}+\frac {x^5\,\left (B\,b^2\,c\,e^2+3\,B\,b\,c^2\,d\,e+A\,b\,c^2\,e^2+7\,B\,c^3\,d^2+A\,c^3\,d\,e+B\,a\,c^2\,e^2\right )}{e^3}+\frac {x^3\,\left (3\,B\,a^2\,c\,e^4+3\,B\,a\,b^2\,e^4+6\,B\,a\,b\,c\,d\,e^3+6\,A\,a\,b\,c\,e^4+5\,B\,a\,c^2\,d^2\,e^2+3\,A\,a\,c^2\,d\,e^3+B\,b^3\,d\,e^3+A\,b^3\,e^4+5\,B\,b^2\,c\,d^2\,e^2+3\,A\,b^2\,c\,d\,e^3+15\,B\,b\,c^2\,d^3\,e+5\,A\,b\,c^2\,d^2\,e^2+35\,B\,c^3\,d^4+5\,A\,c^3\,d^3\,e\right )}{5\,e^5}+\frac {c^2\,x^6\,\left (A\,c\,e+3\,B\,b\,e+7\,B\,c\,d\right )}{2\,e^2}+\frac {B\,c^3\,x^7}{e}}{d^8+8\,d^7\,e\,x+28\,d^6\,e^2\,x^2+56\,d^5\,e^3\,x^3+70\,d^4\,e^4\,x^4+56\,d^3\,e^5\,x^5+28\,d^2\,e^6\,x^6+8\,d\,e^7\,x^7+e^8\,x^8} \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)^9,x)

[Out]

-((35*A*a^3*e^7 + 35*B*c^3*d^7 + 5*B*a^3*d*e^6 + 5*A*c^3*d^6*e + A*b^3*d^3*e^4 + B*b^3*d^4*e^3 + 5*A*a*b^2*d^2
*e^5 + 3*A*a*c^2*d^4*e^3 + 5*A*a^2*c*d^2*e^5 + 3*B*a*b^2*d^3*e^4 + 5*B*a^2*b*d^2*e^5 + 5*A*b*c^2*d^5*e^2 + 3*A
*b^2*c*d^4*e^3 + 5*B*a*c^2*d^5*e^2 + 3*B*a^2*c*d^3*e^4 + 5*B*b^2*c*d^5*e^2 + 15*A*a^2*b*d*e^6 + 15*B*b*c^2*d^6
*e + 6*A*a*b*c*d^3*e^4 + 6*B*a*b*c*d^4*e^3)/(280*e^8) + (x^4*(B*b^3*e^3 + 35*B*c^3*d^3 + 3*A*a*c^2*e^3 + 3*A*b
^2*c*e^3 + 5*A*c^3*d^2*e + 6*B*a*b*c*e^3 + 5*A*b*c^2*d*e^2 + 5*B*a*c^2*d*e^2 + 15*B*b*c^2*d^2*e + 5*B*b^2*c*d*
e^2))/(4*e^4) + (x*(5*B*a^3*e^6 + 35*B*c^3*d^6 + 15*A*a^2*b*e^6 + 5*A*c^3*d^5*e + A*b^3*d^2*e^4 + B*b^3*d^3*e^
3 + 3*A*a*c^2*d^3*e^3 + 3*B*a*b^2*d^2*e^4 + 5*A*b*c^2*d^4*e^2 + 3*A*b^2*c*d^3*e^3 + 5*B*a*c^2*d^4*e^2 + 3*B*a^
2*c*d^2*e^4 + 5*B*b^2*c*d^4*e^2 + 5*A*a*b^2*d*e^5 + 5*A*a^2*c*d*e^5 + 5*B*a^2*b*d*e^5 + 15*B*b*c^2*d^5*e + 6*A
*a*b*c*d^2*e^4 + 6*B*a*b*c*d^3*e^3))/(35*e^7) + (x^2*(35*B*c^3*d^5 + 5*A*a*b^2*e^5 + 5*A*a^2*c*e^5 + 5*B*a^2*b
*e^5 + A*b^3*d*e^4 + 5*A*c^3*d^4*e + B*b^3*d^2*e^3 + 3*A*a*c^2*d^2*e^3 + 5*A*b*c^2*d^3*e^2 + 3*A*b^2*c*d^2*e^3
+ 5*B*a*c^2*d^3*e^2 + 5*B*b^2*c*d^3*e^2 + 3*B*a*b^2*d*e^4 + 3*B*a^2*c*d*e^4 + 15*B*b*c^2*d^4*e + 6*B*a*b*c*d^
2*e^3 + 6*A*a*b*c*d*e^4))/(10*e^6) + (x^5*(7*B*c^3*d^2 + A*c^3*d*e + A*b*c^2*e^2 + B*a*c^2*e^2 + B*b^2*c*e^2 +
3*B*b*c^2*d*e))/e^3 + (x^3*(A*b^3*e^4 + 35*B*c^3*d^4 + 3*B*a*b^2*e^4 + 3*B*a^2*c*e^4 + 5*A*c^3*d^3*e + B*b^3*
d*e^3 + 5*A*b*c^2*d^2*e^2 + 5*B*a*c^2*d^2*e^2 + 5*B*b^2*c*d^2*e^2 + 6*A*a*b*c*e^4 + 3*A*a*c^2*d*e^3 + 3*A*b^2*
c*d*e^3 + 15*B*b*c^2*d^3*e + 6*B*a*b*c*d*e^3))/(5*e^5) + (c^2*x^6*(A*c*e + 3*B*b*e + 7*B*c*d))/(2*e^2) + (B*c^
3*x^7)/e)/(d^8 + e^8*x^8 + 8*d*e^7*x^7 + 28*d^6*e^2*x^2 + 56*d^5*e^3*x^3 + 70*d^4*e^4*x^4 + 56*d^3*e^5*x^5 + 2
8*d^2*e^6*x^6 + 8*d^7*e*x)

________________________________________________________________________________________

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)**9,x)

[Out]

Timed out

________________________________________________________________________________________