3.50 \(\int \frac{(a+b x^2)^5 (A+B x^2)}{x^{18}} \, dx\)

Optimal. Leaf size=117 \[ -\frac{10 a^2 b^2 (a B+A b)}{11 x^{11}}-\frac{a^4 (a B+5 A b)}{15 x^{15}}-\frac{5 a^3 b (a B+2 A b)}{13 x^{13}}-\frac{a^5 A}{17 x^{17}}-\frac{5 a b^3 (2 a B+A b)}{9 x^9}-\frac{b^4 (5 a B+A b)}{7 x^7}-\frac{b^5 B}{5 x^5} \]

[Out]

-(a^5*A)/(17*x^17) - (a^4*(5*A*b + a*B))/(15*x^15) - (5*a^3*b*(2*A*b + a*B))/(13*x^13) - (10*a^2*b^2*(A*b + a*
B))/(11*x^11) - (5*a*b^3*(A*b + 2*a*B))/(9*x^9) - (b^4*(A*b + 5*a*B))/(7*x^7) - (b^5*B)/(5*x^5)

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Rubi [A]  time = 0.0593107, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {448} \[ -\frac{10 a^2 b^2 (a B+A b)}{11 x^{11}}-\frac{a^4 (a B+5 A b)}{15 x^{15}}-\frac{5 a^3 b (a B+2 A b)}{13 x^{13}}-\frac{a^5 A}{17 x^{17}}-\frac{5 a b^3 (2 a B+A b)}{9 x^9}-\frac{b^4 (5 a B+A b)}{7 x^7}-\frac{b^5 B}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^5*(A + B*x^2))/x^18,x]

[Out]

-(a^5*A)/(17*x^17) - (a^4*(5*A*b + a*B))/(15*x^15) - (5*a^3*b*(2*A*b + a*B))/(13*x^13) - (10*a^2*b^2*(A*b + a*
B))/(11*x^11) - (5*a*b^3*(A*b + 2*a*B))/(9*x^9) - (b^4*(A*b + 5*a*B))/(7*x^7) - (b^5*B)/(5*x^5)

Rule 448

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{18}} \, dx &=\int \left (\frac{a^5 A}{x^{18}}+\frac{a^4 (5 A b+a B)}{x^{16}}+\frac{5 a^3 b (2 A b+a B)}{x^{14}}+\frac{10 a^2 b^2 (A b+a B)}{x^{12}}+\frac{5 a b^3 (A b+2 a B)}{x^{10}}+\frac{b^4 (A b+5 a B)}{x^8}+\frac{b^5 B}{x^6}\right ) \, dx\\ &=-\frac{a^5 A}{17 x^{17}}-\frac{a^4 (5 A b+a B)}{15 x^{15}}-\frac{5 a^3 b (2 A b+a B)}{13 x^{13}}-\frac{10 a^2 b^2 (A b+a B)}{11 x^{11}}-\frac{5 a b^3 (A b+2 a B)}{9 x^9}-\frac{b^4 (A b+5 a B)}{7 x^7}-\frac{b^5 B}{5 x^5}\\ \end{align*}

Mathematica [A]  time = 0.0470096, size = 117, normalized size = 1. \[ -\frac{10 a^2 b^2 (a B+A b)}{11 x^{11}}-\frac{a^4 (a B+5 A b)}{15 x^{15}}-\frac{5 a^3 b (a B+2 A b)}{13 x^{13}}-\frac{a^5 A}{17 x^{17}}-\frac{5 a b^3 (2 a B+A b)}{9 x^9}-\frac{b^4 (5 a B+A b)}{7 x^7}-\frac{b^5 B}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^5*(A + B*x^2))/x^18,x]

[Out]

-(a^5*A)/(17*x^17) - (a^4*(5*A*b + a*B))/(15*x^15) - (5*a^3*b*(2*A*b + a*B))/(13*x^13) - (10*a^2*b^2*(A*b + a*
B))/(11*x^11) - (5*a*b^3*(A*b + 2*a*B))/(9*x^9) - (b^4*(A*b + 5*a*B))/(7*x^7) - (b^5*B)/(5*x^5)

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Maple [A]  time = 0.005, size = 104, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{5}}{17\,{x}^{17}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{15\,{x}^{15}}}-{\frac{5\,{a}^{3}b \left ( 2\,Ab+Ba \right ) }{13\,{x}^{13}}}-{\frac{10\,{b}^{2}{a}^{2} \left ( Ab+Ba \right ) }{11\,{x}^{11}}}-{\frac{5\,a{b}^{3} \left ( Ab+2\,Ba \right ) }{9\,{x}^{9}}}-{\frac{{b}^{4} \left ( Ab+5\,Ba \right ) }{7\,{x}^{7}}}-{\frac{B{b}^{5}}{5\,{x}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5*(B*x^2+A)/x^18,x)

[Out]

-1/17*a^5*A/x^17-1/15*a^4*(5*A*b+B*a)/x^15-5/13*a^3*b*(2*A*b+B*a)/x^13-10/11*a^2*b^2*(A*b+B*a)/x^11-5/9*a*b^3*
(A*b+2*B*a)/x^9-1/7*b^4*(A*b+5*B*a)/x^7-1/5*b^5*B/x^5

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Maxima [A]  time = 1.02527, size = 163, normalized size = 1.39 \begin{align*} -\frac{153153 \, B b^{5} x^{12} + 109395 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 425425 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 696150 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 45045 \, A a^{5} + 294525 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 51051 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{765765 \, x^{17}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^18,x, algorithm="maxima")

[Out]

-1/765765*(153153*B*b^5*x^12 + 109395*(5*B*a*b^4 + A*b^5)*x^10 + 425425*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 696150*(
B*a^3*b^2 + A*a^2*b^3)*x^6 + 45045*A*a^5 + 294525*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 51051*(B*a^5 + 5*A*a^4*b)*x^2)
/x^17

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Fricas [A]  time = 1.43609, size = 306, normalized size = 2.62 \begin{align*} -\frac{153153 \, B b^{5} x^{12} + 109395 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 425425 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 696150 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 45045 \, A a^{5} + 294525 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 51051 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{765765 \, x^{17}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^18,x, algorithm="fricas")

[Out]

-1/765765*(153153*B*b^5*x^12 + 109395*(5*B*a*b^4 + A*b^5)*x^10 + 425425*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 696150*(
B*a^3*b^2 + A*a^2*b^3)*x^6 + 45045*A*a^5 + 294525*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 51051*(B*a^5 + 5*A*a^4*b)*x^2)
/x^17

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Sympy [A]  time = 86.1741, size = 128, normalized size = 1.09 \begin{align*} - \frac{45045 A a^{5} + 153153 B b^{5} x^{12} + x^{10} \left (109395 A b^{5} + 546975 B a b^{4}\right ) + x^{8} \left (425425 A a b^{4} + 850850 B a^{2} b^{3}\right ) + x^{6} \left (696150 A a^{2} b^{3} + 696150 B a^{3} b^{2}\right ) + x^{4} \left (589050 A a^{3} b^{2} + 294525 B a^{4} b\right ) + x^{2} \left (255255 A a^{4} b + 51051 B a^{5}\right )}{765765 x^{17}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5*(B*x**2+A)/x**18,x)

[Out]

-(45045*A*a**5 + 153153*B*b**5*x**12 + x**10*(109395*A*b**5 + 546975*B*a*b**4) + x**8*(425425*A*a*b**4 + 85085
0*B*a**2*b**3) + x**6*(696150*A*a**2*b**3 + 696150*B*a**3*b**2) + x**4*(589050*A*a**3*b**2 + 294525*B*a**4*b)
+ x**2*(255255*A*a**4*b + 51051*B*a**5))/(765765*x**17)

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Giac [A]  time = 1.1154, size = 171, normalized size = 1.46 \begin{align*} -\frac{153153 \, B b^{5} x^{12} + 546975 \, B a b^{4} x^{10} + 109395 \, A b^{5} x^{10} + 850850 \, B a^{2} b^{3} x^{8} + 425425 \, A a b^{4} x^{8} + 696150 \, B a^{3} b^{2} x^{6} + 696150 \, A a^{2} b^{3} x^{6} + 294525 \, B a^{4} b x^{4} + 589050 \, A a^{3} b^{2} x^{4} + 51051 \, B a^{5} x^{2} + 255255 \, A a^{4} b x^{2} + 45045 \, A a^{5}}{765765 \, x^{17}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^18,x, algorithm="giac")

[Out]

-1/765765*(153153*B*b^5*x^12 + 546975*B*a*b^4*x^10 + 109395*A*b^5*x^10 + 850850*B*a^2*b^3*x^8 + 425425*A*a*b^4
*x^8 + 696150*B*a^3*b^2*x^6 + 696150*A*a^2*b^3*x^6 + 294525*B*a^4*b*x^4 + 589050*A*a^3*b^2*x^4 + 51051*B*a^5*x
^2 + 255255*A*a^4*b*x^2 + 45045*A*a^5)/x^17