3.1327 \(\int \frac{(A+B x) (a+c x^2)^3}{(d+e x)^9} \, dx\)

Optimal. Leaf size=330 \[ \frac{c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{5 e^8 (d+e x)^5}-\frac{c^2 \left (a B e^2-2 A c d e+7 B c d^2\right )}{e^8 (d+e x)^3}+\frac{c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{4 e^8 (d+e x)^4}+\frac{c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{2 e^8 (d+e x)^6}-\frac{\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{7 e^8 (d+e x)^7}+\frac{\left (a e^2+c d^2\right )^3 (B d-A e)}{8 e^8 (d+e x)^8}+\frac{c^3 (7 B d-A e)}{2 e^8 (d+e x)^2}-\frac{B c^3}{e^8 (d+e x)} \]

[Out]

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

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Rubi [A]  time = 0.273676, antiderivative size = 330, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {772} \[ \frac{c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{5 e^8 (d+e x)^5}-\frac{c^2 \left (a B e^2-2 A c d e+7 B c d^2\right )}{e^8 (d+e x)^3}+\frac{c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{4 e^8 (d+e x)^4}+\frac{c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{2 e^8 (d+e x)^6}-\frac{\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{7 e^8 (d+e x)^7}+\frac{\left (a e^2+c d^2\right )^3 (B d-A e)}{8 e^8 (d+e x)^8}+\frac{c^3 (7 B d-A e)}{2 e^8 (d+e x)^2}-\frac{B c^3}{e^8 (d+e x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (a+c x^2\right )^3}{(d+e x)^9} \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2+a e^2\right )^3}{e^7 (d+e x)^9}+\frac{\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right )}{e^7 (d+e x)^8}+\frac{3 c \left (c d^2+a e^2\right ) \left (-7 B c d^3+5 A c d^2 e-3 a B d e^2+a A e^3\right )}{e^7 (d+e x)^7}-\frac{c \left (-35 B c^2 d^4+20 A c^2 d^3 e-30 a B c d^2 e^2+12 a A c d e^3-3 a^2 B e^4\right )}{e^7 (d+e x)^6}+\frac{c^2 \left (-35 B c d^3+15 A c d^2 e-15 a B d e^2+3 a A e^3\right )}{e^7 (d+e x)^5}-\frac{3 c^2 \left (-7 B c d^2+2 A c d e-a B e^2\right )}{e^7 (d+e x)^4}+\frac{c^3 (-7 B d+A 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+a e^2\right )^3}{8 e^8 (d+e x)^8}-\frac{\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right )}{7 e^8 (d+e x)^7}+\frac{c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right )}{2 e^8 (d+e x)^6}+\frac{c \left (4 A c d e \left (5 c d^2+3 a e^2\right )-B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right )}{5 e^8 (d+e x)^5}+\frac{c^2 \left (35 B c d^3-15 A c d^2 e+15 a B d e^2-3 a A e^3\right )}{4 e^8 (d+e x)^4}-\frac{c^2 \left (7 B c d^2-2 A c d e+a B e^2\right )}{e^8 (d+e x)^3}+\frac{c^3 (7 B d-A e)}{2 e^8 (d+e x)^2}-\frac{B c^3}{e^8 (d+e x)}\\ \end{align*}

Mathematica [A]  time = 0.181602, size = 357, normalized size = 1.08 \[ -\frac{A e \left (5 a^2 c e^4 \left (d^2+8 d e x+28 e^2 x^2\right )+35 a^3 e^6+3 a c^2 e^2 \left (28 d^2 e^2 x^2+8 d^3 e x+d^4+56 d e^3 x^3+70 e^4 x^4\right )+5 c^3 \left (28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+8 d^5 e x+d^6+56 d e^5 x^5+28 e^6 x^6\right )\right )+B \left (3 a^2 c e^4 \left (8 d^2 e x+d^3+28 d e^2 x^2+56 e^3 x^3\right )+5 a^3 e^6 (d+8 e x)+5 a c^2 e^2 \left (28 d^3 e^2 x^2+56 d^2 e^3 x^3+8 d^4 e x+d^5+70 d e^4 x^4+56 e^5 x^5\right )+35 c^3 \left (28 d^5 e^2 x^2+56 d^4 e^3 x^3+70 d^3 e^4 x^4+56 d^2 e^5 x^5+8 d^6 e x+d^7+28 d e^6 x^6+8 e^7 x^7\right )\right )}{280 e^8 (d+e x)^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.008, size = 448, normalized size = 1.4 \begin{align*} -{\frac{B{c}^{3}}{{e}^{8} \left ( ex+d \right ) }}-{\frac{-6\,A{a}^{2}cd{e}^{5}-12\,A{d}^{3}a{c}^{2}{e}^{3}-6\,A{d}^{5}{c}^{3}e+B{a}^{3}{e}^{6}+9\,B{a}^{2}c{d}^{2}{e}^{4}+15\,Ba{c}^{2}{d}^{4}{e}^{2}+7\,B{c}^{3}{d}^{6}}{7\,{e}^{8} \left ( ex+d \right ) ^{7}}}-{\frac{{c}^{3} \left ( Ae-7\,Bd \right ) }{2\,{e}^{8} \left ( ex+d \right ) ^{2}}}+{\frac{{c}^{2} \left ( 2\,Acde-aB{e}^{2}-7\,Bc{d}^{2} \right ) }{{e}^{8} \left ( ex+d \right ) ^{3}}}-{\frac{{c}^{2} \left ( 3\,aA{e}^{3}+15\,Ac{d}^{2}e-15\,aBd{e}^{2}-35\,Bc{d}^{3} \right ) }{4\,{e}^{8} \left ( ex+d \right ) ^{4}}}-{\frac{A{a}^{3}{e}^{7}+3\,A{d}^{2}{a}^{2}c{e}^{5}+3\,A{d}^{4}a{c}^{2}{e}^{3}+A{d}^{6}{c}^{3}e-B{a}^{3}d{e}^{6}-3\,B{a}^{2}c{d}^{3}{e}^{4}-3\,Ba{c}^{2}{d}^{5}{e}^{2}-B{c}^{3}{d}^{7}}{8\,{e}^{8} \left ( ex+d \right ) ^{8}}}+{\frac{c \left ( 12\,Adac{e}^{3}+20\,A{c}^{2}{d}^{3}e-3\,B{e}^{4}{a}^{2}-30\,Bac{d}^{2}{e}^{2}-35\,B{c}^{2}{d}^{4} \right ) }{5\,{e}^{8} \left ( ex+d \right ) ^{5}}}-{\frac{c \left ( A{a}^{2}{e}^{5}+6\,A{d}^{2}ac{e}^{3}+5\,A{d}^{4}{c}^{2}e-3\,B{a}^{2}d{e}^{4}-10\,aBc{d}^{3}{e}^{2}-7\,B{c}^{2}{d}^{5} \right ) }{2\,{e}^{8} \left ( ex+d \right ) ^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^3/(e*x+d)^9,x)

[Out]

-B*c^3/e^8/(e*x+d)-1/7*(-6*A*a^2*c*d*e^5-12*A*a*c^2*d^3*e^3-6*A*c^3*d^5*e+B*a^3*e^6+9*B*a^2*c*d^2*e^4+15*B*a*c
^2*d^4*e^2+7*B*c^3*d^6)/e^8/(e*x+d)^7-1/2*c^3*(A*e-7*B*d)/e^8/(e*x+d)^2+c^2*(2*A*c*d*e-B*a*e^2-7*B*c*d^2)/e^8/
(e*x+d)^3-1/4*c^2*(3*A*a*e^3+15*A*c*d^2*e-15*B*a*d*e^2-35*B*c*d^3)/e^8/(e*x+d)^4-1/8*(A*a^3*e^7+3*A*a^2*c*d^2*
e^5+3*A*a*c^2*d^4*e^3+A*c^3*d^6*e-B*a^3*d*e^6-3*B*a^2*c*d^3*e^4-3*B*a*c^2*d^5*e^2-B*c^3*d^7)/e^8/(e*x+d)^8+1/5
*c*(12*A*a*c*d*e^3+20*A*c^2*d^3*e-3*B*a^2*e^4-30*B*a*c*d^2*e^2-35*B*c^2*d^4)/e^8/(e*x+d)^5-1/2*c*(A*a^2*e^5+6*
A*a*c*d^2*e^3+5*A*c^2*d^4*e-3*B*a^2*d*e^4-10*B*a*c*d^3*e^2-7*B*c^2*d^5)/e^8/(e*x+d)^6

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Maxima [A]  time = 1.24498, size = 718, normalized size = 2.18 \begin{align*} -\frac{280 \, B c^{3} e^{7} x^{7} + 35 \, B c^{3} d^{7} + 5 \, A c^{3} d^{6} e + 5 \, B a c^{2} d^{5} e^{2} + 3 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} + 5 \, A a^{2} c d^{2} e^{5} + 5 \, B a^{3} d e^{6} + 35 \, A a^{3} e^{7} + 140 \,{\left (7 \, B c^{3} d e^{6} + A c^{3} e^{7}\right )} x^{6} + 280 \,{\left (7 \, B c^{3} d^{2} e^{5} + A c^{3} d e^{6} + B a c^{2} e^{7}\right )} x^{5} + 70 \,{\left (35 \, B c^{3} d^{3} e^{4} + 5 \, A c^{3} d^{2} e^{5} + 5 \, B a c^{2} d e^{6} + 3 \, A a c^{2} e^{7}\right )} x^{4} + 56 \,{\left (35 \, B c^{3} d^{4} e^{3} + 5 \, A c^{3} d^{3} e^{4} + 5 \, B a c^{2} d^{2} e^{5} + 3 \, A a c^{2} d e^{6} + 3 \, B a^{2} c e^{7}\right )} x^{3} + 28 \,{\left (35 \, B c^{3} d^{5} e^{2} + 5 \, A c^{3} d^{4} e^{3} + 5 \, B a c^{2} d^{3} e^{4} + 3 \, A a c^{2} d^{2} e^{5} + 3 \, B a^{2} c d e^{6} + 5 \, A a^{2} c e^{7}\right )} x^{2} + 8 \,{\left (35 \, B c^{3} d^{6} e + 5 \, A c^{3} d^{5} e^{2} + 5 \, B a c^{2} d^{4} e^{3} + 3 \, A a c^{2} d^{3} e^{4} + 3 \, B a^{2} c d^{2} e^{5} + 5 \, A a^{2} c d e^{6} + 5 \, B a^{3} 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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+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 + 5*A*c^3*d^6*e + 5*B*a*c^2*d^5*e^2 + 3*A*a*c^2*d^4*e^3 + 3*B*a^2*c*d
^3*e^4 + 5*A*a^2*c*d^2*e^5 + 5*B*a^3*d*e^6 + 35*A*a^3*e^7 + 140*(7*B*c^3*d*e^6 + A*c^3*e^7)*x^6 + 280*(7*B*c^3
*d^2*e^5 + A*c^3*d*e^6 + B*a*c^2*e^7)*x^5 + 70*(35*B*c^3*d^3*e^4 + 5*A*c^3*d^2*e^5 + 5*B*a*c^2*d*e^6 + 3*A*a*c
^2*e^7)*x^4 + 56*(35*B*c^3*d^4*e^3 + 5*A*c^3*d^3*e^4 + 5*B*a*c^2*d^2*e^5 + 3*A*a*c^2*d*e^6 + 3*B*a^2*c*e^7)*x^
3 + 28*(35*B*c^3*d^5*e^2 + 5*A*c^3*d^4*e^3 + 5*B*a*c^2*d^3*e^4 + 3*A*a*c^2*d^2*e^5 + 3*B*a^2*c*d*e^6 + 5*A*a^2
*c*e^7)*x^2 + 8*(35*B*c^3*d^6*e + 5*A*c^3*d^5*e^2 + 5*B*a*c^2*d^4*e^3 + 3*A*a*c^2*d^3*e^4 + 3*B*a^2*c*d^2*e^5
+ 5*A*a^2*c*d*e^6 + 5*B*a^3*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)

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Fricas [A]  time = 1.82934, size = 1130, normalized size = 3.42 \begin{align*} -\frac{280 \, B c^{3} e^{7} x^{7} + 35 \, B c^{3} d^{7} + 5 \, A c^{3} d^{6} e + 5 \, B a c^{2} d^{5} e^{2} + 3 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} + 5 \, A a^{2} c d^{2} e^{5} + 5 \, B a^{3} d e^{6} + 35 \, A a^{3} e^{7} + 140 \,{\left (7 \, B c^{3} d e^{6} + A c^{3} e^{7}\right )} x^{6} + 280 \,{\left (7 \, B c^{3} d^{2} e^{5} + A c^{3} d e^{6} + B a c^{2} e^{7}\right )} x^{5} + 70 \,{\left (35 \, B c^{3} d^{3} e^{4} + 5 \, A c^{3} d^{2} e^{5} + 5 \, B a c^{2} d e^{6} + 3 \, A a c^{2} e^{7}\right )} x^{4} + 56 \,{\left (35 \, B c^{3} d^{4} e^{3} + 5 \, A c^{3} d^{3} e^{4} + 5 \, B a c^{2} d^{2} e^{5} + 3 \, A a c^{2} d e^{6} + 3 \, B a^{2} c e^{7}\right )} x^{3} + 28 \,{\left (35 \, B c^{3} d^{5} e^{2} + 5 \, A c^{3} d^{4} e^{3} + 5 \, B a c^{2} d^{3} e^{4} + 3 \, A a c^{2} d^{2} e^{5} + 3 \, B a^{2} c d e^{6} + 5 \, A a^{2} c e^{7}\right )} x^{2} + 8 \,{\left (35 \, B c^{3} d^{6} e + 5 \, A c^{3} d^{5} e^{2} + 5 \, B a c^{2} d^{4} e^{3} + 3 \, A a c^{2} d^{3} e^{4} + 3 \, B a^{2} c d^{2} e^{5} + 5 \, A a^{2} c d e^{6} + 5 \, B a^{3} 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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+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 + 5*A*c^3*d^6*e + 5*B*a*c^2*d^5*e^2 + 3*A*a*c^2*d^4*e^3 + 3*B*a^2*c*d
^3*e^4 + 5*A*a^2*c*d^2*e^5 + 5*B*a^3*d*e^6 + 35*A*a^3*e^7 + 140*(7*B*c^3*d*e^6 + A*c^3*e^7)*x^6 + 280*(7*B*c^3
*d^2*e^5 + A*c^3*d*e^6 + B*a*c^2*e^7)*x^5 + 70*(35*B*c^3*d^3*e^4 + 5*A*c^3*d^2*e^5 + 5*B*a*c^2*d*e^6 + 3*A*a*c
^2*e^7)*x^4 + 56*(35*B*c^3*d^4*e^3 + 5*A*c^3*d^3*e^4 + 5*B*a*c^2*d^2*e^5 + 3*A*a*c^2*d*e^6 + 3*B*a^2*c*e^7)*x^
3 + 28*(35*B*c^3*d^5*e^2 + 5*A*c^3*d^4*e^3 + 5*B*a*c^2*d^3*e^4 + 3*A*a*c^2*d^2*e^5 + 3*B*a^2*c*d*e^6 + 5*A*a^2
*c*e^7)*x^2 + 8*(35*B*c^3*d^6*e + 5*A*c^3*d^5*e^2 + 5*B*a*c^2*d^4*e^3 + 3*A*a*c^2*d^3*e^4 + 3*B*a^2*c*d^2*e^5
+ 5*A*a^2*c*d*e^6 + 5*B*a^3*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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**3/(e*x+d)**9,x)

[Out]

Timed out

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Giac [A]  time = 1.16856, size = 617, normalized size = 1.87 \begin{align*} -\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} + 140 \, A c^{3} x^{6} e^{7} + 280 \, A c^{3} d x^{5} e^{6} + 350 \, A c^{3} d^{2} x^{4} e^{5} + 280 \, A c^{3} d^{3} x^{3} e^{4} + 140 \, A c^{3} d^{4} x^{2} e^{3} + 40 \, A c^{3} d^{5} x e^{2} + 5 \, A c^{3} d^{6} e + 280 \, B a c^{2} x^{5} e^{7} + 350 \, B a c^{2} d x^{4} e^{6} + 280 \, B a c^{2} d^{2} x^{3} e^{5} + 140 \, B a c^{2} d^{3} x^{2} e^{4} + 40 \, B a c^{2} d^{4} x e^{3} + 5 \, B a c^{2} d^{5} e^{2} + 210 \, A a c^{2} x^{4} e^{7} + 168 \, A a c^{2} d x^{3} e^{6} + 84 \, A a c^{2} d^{2} x^{2} e^{5} + 24 \, A a c^{2} d^{3} x e^{4} + 3 \, A a c^{2} d^{4} e^{3} + 168 \, B a^{2} c x^{3} e^{7} + 84 \, B a^{2} c d x^{2} e^{6} + 24 \, B a^{2} c d^{2} x e^{5} + 3 \, B a^{2} c d^{3} e^{4} + 140 \, A a^{2} c x^{2} e^{7} + 40 \, A a^{2} c d x e^{6} + 5 \, A a^{2} c d^{2} e^{5} + 40 \, B a^{3} x e^{7} + 5 \, B a^{3} d e^{6} + 35 \, A a^{3} e^{7}\right )} e^{\left (-8\right )}}{280 \,{\left (x e + d\right )}^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+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 + 140*A*c^3*x^6*e^7 + 280*A*c^3*d*x^5*
e^6 + 350*A*c^3*d^2*x^4*e^5 + 280*A*c^3*d^3*x^3*e^4 + 140*A*c^3*d^4*x^2*e^3 + 40*A*c^3*d^5*x*e^2 + 5*A*c^3*d^6
*e + 280*B*a*c^2*x^5*e^7 + 350*B*a*c^2*d*x^4*e^6 + 280*B*a*c^2*d^2*x^3*e^5 + 140*B*a*c^2*d^3*x^2*e^4 + 40*B*a*
c^2*d^4*x*e^3 + 5*B*a*c^2*d^5*e^2 + 210*A*a*c^2*x^4*e^7 + 168*A*a*c^2*d*x^3*e^6 + 84*A*a*c^2*d^2*x^2*e^5 + 24*
A*a*c^2*d^3*x*e^4 + 3*A*a*c^2*d^4*e^3 + 168*B*a^2*c*x^3*e^7 + 84*B*a^2*c*d*x^2*e^6 + 24*B*a^2*c*d^2*x*e^5 + 3*
B*a^2*c*d^3*e^4 + 140*A*a^2*c*x^2*e^7 + 40*A*a^2*c*d*x*e^6 + 5*A*a^2*c*d^2*e^5 + 40*B*a^3*x*e^7 + 5*B*a^3*d*e^
6 + 35*A*a^3*e^7)*e^(-8)/(x*e + d)^8