3.52 \(\int \frac{(a+b x^2)^5 (A+B x^2)}{x^{20}} \, dx\)

Optimal. Leaf size=117 \[ -\frac{10 a^2 b^2 (a B+A b)}{13 x^{13}}-\frac{a^4 (a B+5 A b)}{17 x^{17}}-\frac{a^3 b (a B+2 A b)}{3 x^{15}}-\frac{a^5 A}{19 x^{19}}-\frac{5 a b^3 (2 a B+A b)}{11 x^{11}}-\frac{b^4 (5 a B+A b)}{9 x^9}-\frac{b^5 B}{7 x^7} \]

[Out]

-(a^5*A)/(19*x^19) - (a^4*(5*A*b + a*B))/(17*x^17) - (a^3*b*(2*A*b + a*B))/(3*x^15) - (10*a^2*b^2*(A*b + a*B))
/(13*x^13) - (5*a*b^3*(A*b + 2*a*B))/(11*x^11) - (b^4*(A*b + 5*a*B))/(9*x^9) - (b^5*B)/(7*x^7)

________________________________________________________________________________________

Rubi [A]  time = 0.0586581, 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} \[ -\frac{10 a^2 b^2 (a B+A b)}{13 x^{13}}-\frac{a^4 (a B+5 A b)}{17 x^{17}}-\frac{a^3 b (a B+2 A b)}{3 x^{15}}-\frac{a^5 A}{19 x^{19}}-\frac{5 a b^3 (2 a B+A b)}{11 x^{11}}-\frac{b^4 (5 a B+A b)}{9 x^9}-\frac{b^5 B}{7 x^7} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^5*(A + B*x^2))/x^20,x]

[Out]

-(a^5*A)/(19*x^19) - (a^4*(5*A*b + a*B))/(17*x^17) - (a^3*b*(2*A*b + a*B))/(3*x^15) - (10*a^2*b^2*(A*b + a*B))
/(13*x^13) - (5*a*b^3*(A*b + 2*a*B))/(11*x^11) - (b^4*(A*b + 5*a*B))/(9*x^9) - (b^5*B)/(7*x^7)

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^{20}} \, dx &=\int \left (\frac{a^5 A}{x^{20}}+\frac{a^4 (5 A b+a B)}{x^{18}}+\frac{5 a^3 b (2 A b+a B)}{x^{16}}+\frac{10 a^2 b^2 (A b+a B)}{x^{14}}+\frac{5 a b^3 (A b+2 a B)}{x^{12}}+\frac{b^4 (A b+5 a B)}{x^{10}}+\frac{b^5 B}{x^8}\right ) \, dx\\ &=-\frac{a^5 A}{19 x^{19}}-\frac{a^4 (5 A b+a B)}{17 x^{17}}-\frac{a^3 b (2 A b+a B)}{3 x^{15}}-\frac{10 a^2 b^2 (A b+a B)}{13 x^{13}}-\frac{5 a b^3 (A b+2 a B)}{11 x^{11}}-\frac{b^4 (A b+5 a B)}{9 x^9}-\frac{b^5 B}{7 x^7}\\ \end{align*}

Mathematica [A]  time = 0.0425499, size = 117, normalized size = 1. \[ -\frac{10 a^2 b^2 (a B+A b)}{13 x^{13}}-\frac{a^4 (a B+5 A b)}{17 x^{17}}-\frac{a^3 b (a B+2 A b)}{3 x^{15}}-\frac{a^5 A}{19 x^{19}}-\frac{5 a b^3 (2 a B+A b)}{11 x^{11}}-\frac{b^4 (5 a B+A b)}{9 x^9}-\frac{b^5 B}{7 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^5*(A + B*x^2))/x^20,x]

[Out]

-(a^5*A)/(19*x^19) - (a^4*(5*A*b + a*B))/(17*x^17) - (a^3*b*(2*A*b + a*B))/(3*x^15) - (10*a^2*b^2*(A*b + a*B))
/(13*x^13) - (5*a*b^3*(A*b + 2*a*B))/(11*x^11) - (b^4*(A*b + 5*a*B))/(9*x^9) - (b^5*B)/(7*x^7)

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 104, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{5}}{19\,{x}^{19}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{17\,{x}^{17}}}-{\frac{{a}^{3}b \left ( 2\,Ab+Ba \right ) }{3\,{x}^{15}}}-{\frac{10\,{b}^{2}{a}^{2} \left ( Ab+Ba \right ) }{13\,{x}^{13}}}-{\frac{5\,a{b}^{3} \left ( Ab+2\,Ba \right ) }{11\,{x}^{11}}}-{\frac{{b}^{4} \left ( Ab+5\,Ba \right ) }{9\,{x}^{9}}}-{\frac{B{b}^{5}}{7\,{x}^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5*(B*x^2+A)/x^20,x)

[Out]

-1/19*a^5*A/x^19-1/17*a^4*(5*A*b+B*a)/x^17-1/3*a^3*b*(2*A*b+B*a)/x^15-10/13*a^2*b^2*(A*b+B*a)/x^13-5/11*a*b^3*
(A*b+2*B*a)/x^11-1/9*b^4*(A*b+5*B*a)/x^9-1/7*b^5*B/x^7

________________________________________________________________________________________

Maxima [A]  time = 1.00315, size = 163, normalized size = 1.39 \begin{align*} -\frac{415701 \, B b^{5} x^{12} + 323323 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 1322685 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 2238390 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 153153 \, A a^{5} + 969969 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 171171 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{2909907 \, x^{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^20,x, algorithm="maxima")

[Out]

-1/2909907*(415701*B*b^5*x^12 + 323323*(5*B*a*b^4 + A*b^5)*x^10 + 1322685*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 223839
0*(B*a^3*b^2 + A*a^2*b^3)*x^6 + 153153*A*a^5 + 969969*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 171171*(B*a^5 + 5*A*a^4*b)
*x^2)/x^19

________________________________________________________________________________________

Fricas [A]  time = 1.38052, size = 313, normalized size = 2.68 \begin{align*} -\frac{415701 \, B b^{5} x^{12} + 323323 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 1322685 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 2238390 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 153153 \, A a^{5} + 969969 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 171171 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{2909907 \, x^{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^20,x, algorithm="fricas")

[Out]

-1/2909907*(415701*B*b^5*x^12 + 323323*(5*B*a*b^4 + A*b^5)*x^10 + 1322685*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 223839
0*(B*a^3*b^2 + A*a^2*b^3)*x^6 + 153153*A*a^5 + 969969*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 171171*(B*a^5 + 5*A*a^4*b)
*x^2)/x^19

________________________________________________________________________________________

Sympy [A]  time = 176.26, size = 128, normalized size = 1.09 \begin{align*} - \frac{153153 A a^{5} + 415701 B b^{5} x^{12} + x^{10} \left (323323 A b^{5} + 1616615 B a b^{4}\right ) + x^{8} \left (1322685 A a b^{4} + 2645370 B a^{2} b^{3}\right ) + x^{6} \left (2238390 A a^{2} b^{3} + 2238390 B a^{3} b^{2}\right ) + x^{4} \left (1939938 A a^{3} b^{2} + 969969 B a^{4} b\right ) + x^{2} \left (855855 A a^{4} b + 171171 B a^{5}\right )}{2909907 x^{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5*(B*x**2+A)/x**20,x)

[Out]

-(153153*A*a**5 + 415701*B*b**5*x**12 + x**10*(323323*A*b**5 + 1616615*B*a*b**4) + x**8*(1322685*A*a*b**4 + 26
45370*B*a**2*b**3) + x**6*(2238390*A*a**2*b**3 + 2238390*B*a**3*b**2) + x**4*(1939938*A*a**3*b**2 + 969969*B*a
**4*b) + x**2*(855855*A*a**4*b + 171171*B*a**5))/(2909907*x**19)

________________________________________________________________________________________

Giac [A]  time = 1.18392, size = 171, normalized size = 1.46 \begin{align*} -\frac{415701 \, B b^{5} x^{12} + 1616615 \, B a b^{4} x^{10} + 323323 \, A b^{5} x^{10} + 2645370 \, B a^{2} b^{3} x^{8} + 1322685 \, A a b^{4} x^{8} + 2238390 \, B a^{3} b^{2} x^{6} + 2238390 \, A a^{2} b^{3} x^{6} + 969969 \, B a^{4} b x^{4} + 1939938 \, A a^{3} b^{2} x^{4} + 171171 \, B a^{5} x^{2} + 855855 \, A a^{4} b x^{2} + 153153 \, A a^{5}}{2909907 \, x^{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^20,x, algorithm="giac")

[Out]

-1/2909907*(415701*B*b^5*x^12 + 1616615*B*a*b^4*x^10 + 323323*A*b^5*x^10 + 2645370*B*a^2*b^3*x^8 + 1322685*A*a
*b^4*x^8 + 2238390*B*a^3*b^2*x^6 + 2238390*A*a^2*b^3*x^6 + 969969*B*a^4*b*x^4 + 1939938*A*a^3*b^2*x^4 + 171171
*B*a^5*x^2 + 855855*A*a^4*b*x^2 + 153153*A*a^5)/x^19