3.1312 \(\int \frac{(A+B x) (a+c x^2)^2}{(d+e x)^{10}} \, dx\)

Optimal. Leaf size=206 \[ -\frac{c \left (a B e^2-2 A c d e+5 B c d^2\right )}{3 e^6 (d+e x)^6}+\frac{2 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{7 e^6 (d+e x)^7}-\frac{\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{8 e^6 (d+e x)^8}+\frac{\left (a e^2+c d^2\right )^2 (B d-A e)}{9 e^6 (d+e x)^9}+\frac{c^2 (5 B d-A e)}{5 e^6 (d+e x)^5}-\frac{B c^2}{4 e^6 (d+e x)^4} \]

[Out]

((B*d - A*e)*(c*d^2 + a*e^2)^2)/(9*e^6*(d + e*x)^9) - ((c*d^2 + a*e^2)*(5*B*c*d^2 - 4*A*c*d*e + a*B*e^2))/(8*e
^6*(d + e*x)^8) + (2*c*(5*B*c*d^3 - 3*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(7*e^6*(d + e*x)^7) - (c*(5*B*c*d^2
- 2*A*c*d*e + a*B*e^2))/(3*e^6*(d + e*x)^6) + (c^2*(5*B*d - A*e))/(5*e^6*(d + e*x)^5) - (B*c^2)/(4*e^6*(d + e*
x)^4)

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Rubi [A]  time = 0.136584, antiderivative size = 206, 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} \[ -\frac{c \left (a B e^2-2 A c d e+5 B c d^2\right )}{3 e^6 (d+e x)^6}+\frac{2 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{7 e^6 (d+e x)^7}-\frac{\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{8 e^6 (d+e x)^8}+\frac{\left (a e^2+c d^2\right )^2 (B d-A e)}{9 e^6 (d+e x)^9}+\frac{c^2 (5 B d-A e)}{5 e^6 (d+e x)^5}-\frac{B c^2}{4 e^6 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^10,x]

[Out]

((B*d - A*e)*(c*d^2 + a*e^2)^2)/(9*e^6*(d + e*x)^9) - ((c*d^2 + a*e^2)*(5*B*c*d^2 - 4*A*c*d*e + a*B*e^2))/(8*e
^6*(d + e*x)^8) + (2*c*(5*B*c*d^3 - 3*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(7*e^6*(d + e*x)^7) - (c*(5*B*c*d^2
- 2*A*c*d*e + a*B*e^2))/(3*e^6*(d + e*x)^6) + (c^2*(5*B*d - A*e))/(5*e^6*(d + e*x)^5) - (B*c^2)/(4*e^6*(d + e*
x)^4)

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 )^2}{(d+e x)^{10}} \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2+a e^2\right )^2}{e^5 (d+e x)^{10}}+\frac{\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{e^5 (d+e x)^9}+\frac{2 c \left (-5 B c d^3+3 A c d^2 e-3 a B d e^2+a A e^3\right )}{e^5 (d+e x)^8}-\frac{2 c \left (-5 B c d^2+2 A c d e-a B e^2\right )}{e^5 (d+e x)^7}+\frac{c^2 (-5 B d+A e)}{e^5 (d+e x)^6}+\frac{B c^2}{e^5 (d+e x)^5}\right ) \, dx\\ &=\frac{(B d-A e) \left (c d^2+a e^2\right )^2}{9 e^6 (d+e x)^9}-\frac{\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{8 e^6 (d+e x)^8}+\frac{2 c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right )}{7 e^6 (d+e x)^7}-\frac{c \left (5 B c d^2-2 A c d e+a B e^2\right )}{3 e^6 (d+e x)^6}+\frac{c^2 (5 B d-A e)}{5 e^6 (d+e x)^5}-\frac{B c^2}{4 e^6 (d+e x)^4}\\ \end{align*}

Mathematica [A]  time = 0.0990557, size = 202, normalized size = 0.98 \[ -\frac{4 A e \left (70 a^2 e^4+5 a c e^2 \left (d^2+9 d e x+36 e^2 x^2\right )+c^2 \left (36 d^2 e^2 x^2+9 d^3 e x+d^4+84 d e^3 x^3+126 e^4 x^4\right )\right )+5 B \left (7 a^2 e^4 (d+9 e x)+2 a c e^2 \left (9 d^2 e x+d^3+36 d e^2 x^2+84 e^3 x^3\right )+c^2 \left (36 d^3 e^2 x^2+84 d^2 e^3 x^3+9 d^4 e x+d^5+126 d e^4 x^4+126 e^5 x^5\right )\right )}{2520 e^6 (d+e x)^9} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^10,x]

[Out]

-(4*A*e*(70*a^2*e^4 + 5*a*c*e^2*(d^2 + 9*d*e*x + 36*e^2*x^2) + c^2*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^
3*x^3 + 126*e^4*x^4)) + 5*B*(7*a^2*e^4*(d + 9*e*x) + 2*a*c*e^2*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) +
c^2*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5)))/(2520*e^6*(d + e*x)^9
)

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Maple [A]  time = 0.009, size = 249, normalized size = 1.2 \begin{align*} -{\frac{2\,c \left ( aA{e}^{3}+3\,Ac{d}^{2}e-3\,aBd{e}^{2}-5\,Bc{d}^{3} \right ) }{7\,{e}^{6} \left ( ex+d \right ) ^{7}}}-{\frac{B{c}^{2}}{4\,{e}^{6} \left ( ex+d \right ) ^{4}}}-{\frac{-4\,Adac{e}^{3}-4\,A{c}^{2}{d}^{3}e+B{e}^{4}{a}^{2}+6\,aBc{d}^{2}{e}^{2}+5\,B{c}^{2}{d}^{4}}{8\,{e}^{6} \left ( ex+d \right ) ^{8}}}-{\frac{{c}^{2} \left ( Ae-5\,Bd \right ) }{5\,{e}^{6} \left ( ex+d \right ) ^{5}}}-{\frac{A{a}^{2}{e}^{5}+2\,A{d}^{2}ac{e}^{3}+A{d}^{4}{c}^{2}e-B{a}^{2}d{e}^{4}-2\,aBc{d}^{3}{e}^{2}-B{c}^{2}{d}^{5}}{9\,{e}^{6} \left ( ex+d \right ) ^{9}}}+{\frac{c \left ( 2\,Acde-aB{e}^{2}-5\,Bc{d}^{2} \right ) }{3\,{e}^{6} \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)^2/(e*x+d)^10,x)

[Out]

-2/7*c*(A*a*e^3+3*A*c*d^2*e-3*B*a*d*e^2-5*B*c*d^3)/e^6/(e*x+d)^7-1/4*B*c^2/e^6/(e*x+d)^4-1/8*(-4*A*a*c*d*e^3-4
*A*c^2*d^3*e+B*a^2*e^4+6*B*a*c*d^2*e^2+5*B*c^2*d^4)/e^6/(e*x+d)^8-1/5*c^2*(A*e-5*B*d)/e^6/(e*x+d)^5-1/9*(A*a^2
*e^5+2*A*a*c*d^2*e^3+A*c^2*d^4*e-B*a^2*d*e^4-2*B*a*c*d^3*e^2-B*c^2*d^5)/e^6/(e*x+d)^9+1/3*c*(2*A*c*d*e-B*a*e^2
-5*B*c*d^2)/e^6/(e*x+d)^6

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Maxima [A]  time = 1.11664, size = 458, normalized size = 2.22 \begin{align*} -\frac{630 \, B c^{2} e^{5} x^{5} + 5 \, B c^{2} d^{5} + 4 \, A c^{2} d^{4} e + 10 \, B a c d^{3} e^{2} + 20 \, A a c d^{2} e^{3} + 35 \, B a^{2} d e^{4} + 280 \, A a^{2} e^{5} + 126 \,{\left (5 \, B c^{2} d e^{4} + 4 \, A c^{2} e^{5}\right )} x^{4} + 84 \,{\left (5 \, B c^{2} d^{2} e^{3} + 4 \, A c^{2} d e^{4} + 10 \, B a c e^{5}\right )} x^{3} + 36 \,{\left (5 \, B c^{2} d^{3} e^{2} + 4 \, A c^{2} d^{2} e^{3} + 10 \, B a c d e^{4} + 20 \, A a c e^{5}\right )} x^{2} + 9 \,{\left (5 \, B c^{2} d^{4} e + 4 \, A c^{2} d^{3} e^{2} + 10 \, B a c d^{2} e^{3} + 20 \, A a c d e^{4} + 35 \, B a^{2} e^{5}\right )} x}{2520 \,{\left (e^{15} x^{9} + 9 \, d e^{14} x^{8} + 36 \, d^{2} e^{13} x^{7} + 84 \, d^{3} e^{12} x^{6} + 126 \, d^{4} e^{11} x^{5} + 126 \, d^{5} e^{10} x^{4} + 84 \, d^{6} e^{9} x^{3} + 36 \, d^{7} e^{8} x^{2} + 9 \, d^{8} e^{7} x + d^{9} e^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^10,x, algorithm="maxima")

[Out]

-1/2520*(630*B*c^2*e^5*x^5 + 5*B*c^2*d^5 + 4*A*c^2*d^4*e + 10*B*a*c*d^3*e^2 + 20*A*a*c*d^2*e^3 + 35*B*a^2*d*e^
4 + 280*A*a^2*e^5 + 126*(5*B*c^2*d*e^4 + 4*A*c^2*e^5)*x^4 + 84*(5*B*c^2*d^2*e^3 + 4*A*c^2*d*e^4 + 10*B*a*c*e^5
)*x^3 + 36*(5*B*c^2*d^3*e^2 + 4*A*c^2*d^2*e^3 + 10*B*a*c*d*e^4 + 20*A*a*c*e^5)*x^2 + 9*(5*B*c^2*d^4*e + 4*A*c^
2*d^3*e^2 + 10*B*a*c*d^2*e^3 + 20*A*a*c*d*e^4 + 35*B*a^2*e^5)*x)/(e^15*x^9 + 9*d*e^14*x^8 + 36*d^2*e^13*x^7 +
84*d^3*e^12*x^6 + 126*d^4*e^11*x^5 + 126*d^5*e^10*x^4 + 84*d^6*e^9*x^3 + 36*d^7*e^8*x^2 + 9*d^8*e^7*x + d^9*e^
6)

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Fricas [A]  time = 1.65422, size = 748, normalized size = 3.63 \begin{align*} -\frac{630 \, B c^{2} e^{5} x^{5} + 5 \, B c^{2} d^{5} + 4 \, A c^{2} d^{4} e + 10 \, B a c d^{3} e^{2} + 20 \, A a c d^{2} e^{3} + 35 \, B a^{2} d e^{4} + 280 \, A a^{2} e^{5} + 126 \,{\left (5 \, B c^{2} d e^{4} + 4 \, A c^{2} e^{5}\right )} x^{4} + 84 \,{\left (5 \, B c^{2} d^{2} e^{3} + 4 \, A c^{2} d e^{4} + 10 \, B a c e^{5}\right )} x^{3} + 36 \,{\left (5 \, B c^{2} d^{3} e^{2} + 4 \, A c^{2} d^{2} e^{3} + 10 \, B a c d e^{4} + 20 \, A a c e^{5}\right )} x^{2} + 9 \,{\left (5 \, B c^{2} d^{4} e + 4 \, A c^{2} d^{3} e^{2} + 10 \, B a c d^{2} e^{3} + 20 \, A a c d e^{4} + 35 \, B a^{2} e^{5}\right )} x}{2520 \,{\left (e^{15} x^{9} + 9 \, d e^{14} x^{8} + 36 \, d^{2} e^{13} x^{7} + 84 \, d^{3} e^{12} x^{6} + 126 \, d^{4} e^{11} x^{5} + 126 \, d^{5} e^{10} x^{4} + 84 \, d^{6} e^{9} x^{3} + 36 \, d^{7} e^{8} x^{2} + 9 \, d^{8} e^{7} x + d^{9} e^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^10,x, algorithm="fricas")

[Out]

-1/2520*(630*B*c^2*e^5*x^5 + 5*B*c^2*d^5 + 4*A*c^2*d^4*e + 10*B*a*c*d^3*e^2 + 20*A*a*c*d^2*e^3 + 35*B*a^2*d*e^
4 + 280*A*a^2*e^5 + 126*(5*B*c^2*d*e^4 + 4*A*c^2*e^5)*x^4 + 84*(5*B*c^2*d^2*e^3 + 4*A*c^2*d*e^4 + 10*B*a*c*e^5
)*x^3 + 36*(5*B*c^2*d^3*e^2 + 4*A*c^2*d^2*e^3 + 10*B*a*c*d*e^4 + 20*A*a*c*e^5)*x^2 + 9*(5*B*c^2*d^4*e + 4*A*c^
2*d^3*e^2 + 10*B*a*c*d^2*e^3 + 20*A*a*c*d*e^4 + 35*B*a^2*e^5)*x)/(e^15*x^9 + 9*d*e^14*x^8 + 36*d^2*e^13*x^7 +
84*d^3*e^12*x^6 + 126*d^4*e^11*x^5 + 126*d^5*e^10*x^4 + 84*d^6*e^9*x^3 + 36*d^7*e^8*x^2 + 9*d^8*e^7*x + d^9*e^
6)

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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)**2/(e*x+d)**10,x)

[Out]

Timed out

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Giac [A]  time = 1.14045, size = 327, normalized size = 1.59 \begin{align*} -\frac{{\left (630 \, B c^{2} x^{5} e^{5} + 630 \, B c^{2} d x^{4} e^{4} + 420 \, B c^{2} d^{2} x^{3} e^{3} + 180 \, B c^{2} d^{3} x^{2} e^{2} + 45 \, B c^{2} d^{4} x e + 5 \, B c^{2} d^{5} + 504 \, A c^{2} x^{4} e^{5} + 336 \, A c^{2} d x^{3} e^{4} + 144 \, A c^{2} d^{2} x^{2} e^{3} + 36 \, A c^{2} d^{3} x e^{2} + 4 \, A c^{2} d^{4} e + 840 \, B a c x^{3} e^{5} + 360 \, B a c d x^{2} e^{4} + 90 \, B a c d^{2} x e^{3} + 10 \, B a c d^{3} e^{2} + 720 \, A a c x^{2} e^{5} + 180 \, A a c d x e^{4} + 20 \, A a c d^{2} e^{3} + 315 \, B a^{2} x e^{5} + 35 \, B a^{2} d e^{4} + 280 \, A a^{2} e^{5}\right )} e^{\left (-6\right )}}{2520 \,{\left (x e + d\right )}^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^10,x, algorithm="giac")

[Out]

-1/2520*(630*B*c^2*x^5*e^5 + 630*B*c^2*d*x^4*e^4 + 420*B*c^2*d^2*x^3*e^3 + 180*B*c^2*d^3*x^2*e^2 + 45*B*c^2*d^
4*x*e + 5*B*c^2*d^5 + 504*A*c^2*x^4*e^5 + 336*A*c^2*d*x^3*e^4 + 144*A*c^2*d^2*x^2*e^3 + 36*A*c^2*d^3*x*e^2 + 4
*A*c^2*d^4*e + 840*B*a*c*x^3*e^5 + 360*B*a*c*d*x^2*e^4 + 90*B*a*c*d^2*x*e^3 + 10*B*a*c*d^3*e^2 + 720*A*a*c*x^2
*e^5 + 180*A*a*c*d*x*e^4 + 20*A*a*c*d^2*e^3 + 315*B*a^2*x*e^5 + 35*B*a^2*d*e^4 + 280*A*a^2*e^5)*e^(-6)/(x*e +
d)^9