3.54 \(\int \frac{(a+b x^2)^5 (A+B x^2)}{x^{22}} \, dx\)

Optimal. Leaf size=117 \[ -\frac{2 a^2 b^2 (a B+A b)}{3 x^{15}}-\frac{a^4 (a B+5 A b)}{19 x^{19}}-\frac{5 a^3 b (a B+2 A b)}{17 x^{17}}-\frac{a^5 A}{21 x^{21}}-\frac{5 a b^3 (2 a B+A b)}{13 x^{13}}-\frac{b^4 (5 a B+A b)}{11 x^{11}}-\frac{b^5 B}{9 x^9} \]

[Out]

-(a^5*A)/(21*x^21) - (a^4*(5*A*b + a*B))/(19*x^19) - (5*a^3*b*(2*A*b + a*B))/(17*x^17) - (2*a^2*b^2*(A*b + a*B
))/(3*x^15) - (5*a*b^3*(A*b + 2*a*B))/(13*x^13) - (b^4*(A*b + 5*a*B))/(11*x^11) - (b^5*B)/(9*x^9)

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Rubi [A]  time = 0.0620889, 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} \[ -\frac{2 a^2 b^2 (a B+A b)}{3 x^{15}}-\frac{a^4 (a B+5 A b)}{19 x^{19}}-\frac{5 a^3 b (a B+2 A b)}{17 x^{17}}-\frac{a^5 A}{21 x^{21}}-\frac{5 a b^3 (2 a B+A b)}{13 x^{13}}-\frac{b^4 (5 a B+A b)}{11 x^{11}}-\frac{b^5 B}{9 x^9} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^5*(A + B*x^2))/x^22,x]

[Out]

-(a^5*A)/(21*x^21) - (a^4*(5*A*b + a*B))/(19*x^19) - (5*a^3*b*(2*A*b + a*B))/(17*x^17) - (2*a^2*b^2*(A*b + a*B
))/(3*x^15) - (5*a*b^3*(A*b + 2*a*B))/(13*x^13) - (b^4*(A*b + 5*a*B))/(11*x^11) - (b^5*B)/(9*x^9)

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^{22}} \, dx &=\int \left (\frac{a^5 A}{x^{22}}+\frac{a^4 (5 A b+a B)}{x^{20}}+\frac{5 a^3 b (2 A b+a B)}{x^{18}}+\frac{10 a^2 b^2 (A b+a B)}{x^{16}}+\frac{5 a b^3 (A b+2 a B)}{x^{14}}+\frac{b^4 (A b+5 a B)}{x^{12}}+\frac{b^5 B}{x^{10}}\right ) \, dx\\ &=-\frac{a^5 A}{21 x^{21}}-\frac{a^4 (5 A b+a B)}{19 x^{19}}-\frac{5 a^3 b (2 A b+a B)}{17 x^{17}}-\frac{2 a^2 b^2 (A b+a B)}{3 x^{15}}-\frac{5 a b^3 (A b+2 a B)}{13 x^{13}}-\frac{b^4 (A b+5 a B)}{11 x^{11}}-\frac{b^5 B}{9 x^9}\\ \end{align*}

Mathematica [A]  time = 0.0422499, size = 117, normalized size = 1. \[ -\frac{2 a^2 b^2 (a B+A b)}{3 x^{15}}-\frac{a^4 (a B+5 A b)}{19 x^{19}}-\frac{5 a^3 b (a B+2 A b)}{17 x^{17}}-\frac{a^5 A}{21 x^{21}}-\frac{5 a b^3 (2 a B+A b)}{13 x^{13}}-\frac{b^4 (5 a B+A b)}{11 x^{11}}-\frac{b^5 B}{9 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^5*(A + B*x^2))/x^22,x]

[Out]

-(a^5*A)/(21*x^21) - (a^4*(5*A*b + a*B))/(19*x^19) - (5*a^3*b*(2*A*b + a*B))/(17*x^17) - (2*a^2*b^2*(A*b + a*B
))/(3*x^15) - (5*a*b^3*(A*b + 2*a*B))/(13*x^13) - (b^4*(A*b + 5*a*B))/(11*x^11) - (b^5*B)/(9*x^9)

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Maple [A]  time = 0.007, size = 104, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{5}}{21\,{x}^{21}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{19\,{x}^{19}}}-{\frac{5\,{a}^{3}b \left ( 2\,Ab+Ba \right ) }{17\,{x}^{17}}}-{\frac{2\,{b}^{2}{a}^{2} \left ( Ab+Ba \right ) }{3\,{x}^{15}}}-{\frac{5\,a{b}^{3} \left ( Ab+2\,Ba \right ) }{13\,{x}^{13}}}-{\frac{{b}^{4} \left ( Ab+5\,Ba \right ) }{11\,{x}^{11}}}-{\frac{B{b}^{5}}{9\,{x}^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5*(B*x^2+A)/x^22,x)

[Out]

-1/21*a^5*A/x^21-1/19*a^4*(5*A*b+B*a)/x^19-5/17*a^3*b*(2*A*b+B*a)/x^17-2/3*a^2*b^2*(A*b+B*a)/x^15-5/13*a*b^3*(
A*b+2*B*a)/x^13-1/11*b^4*(A*b+5*B*a)/x^11-1/9*b^5*B/x^9

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Maxima [A]  time = 1.02141, size = 163, normalized size = 1.39 \begin{align*} -\frac{323323 \, B b^{5} x^{12} + 264537 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 1119195 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 1939938 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 138567 \, A a^{5} + 855855 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 153153 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{2909907 \, x^{21}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^22,x, algorithm="maxima")

[Out]

-1/2909907*(323323*B*b^5*x^12 + 264537*(5*B*a*b^4 + A*b^5)*x^10 + 1119195*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 193993
8*(B*a^3*b^2 + A*a^2*b^3)*x^6 + 138567*A*a^5 + 855855*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 153153*(B*a^5 + 5*A*a^4*b)
*x^2)/x^21

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Fricas [A]  time = 1.36854, size = 313, normalized size = 2.68 \begin{align*} -\frac{323323 \, B b^{5} x^{12} + 264537 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 1119195 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 1939938 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 138567 \, A a^{5} + 855855 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 153153 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{2909907 \, x^{21}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^22,x, algorithm="fricas")

[Out]

-1/2909907*(323323*B*b^5*x^12 + 264537*(5*B*a*b^4 + A*b^5)*x^10 + 1119195*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 193993
8*(B*a^3*b^2 + A*a^2*b^3)*x^6 + 138567*A*a^5 + 855855*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 153153*(B*a^5 + 5*A*a^4*b)
*x^2)/x^21

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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**2+a)**5*(B*x**2+A)/x**22,x)

[Out]

Timed out

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Giac [A]  time = 1.11926, size = 171, normalized size = 1.46 \begin{align*} -\frac{323323 \, B b^{5} x^{12} + 1322685 \, B a b^{4} x^{10} + 264537 \, A b^{5} x^{10} + 2238390 \, B a^{2} b^{3} x^{8} + 1119195 \, A a b^{4} x^{8} + 1939938 \, B a^{3} b^{2} x^{6} + 1939938 \, A a^{2} b^{3} x^{6} + 855855 \, B a^{4} b x^{4} + 1711710 \, A a^{3} b^{2} x^{4} + 153153 \, B a^{5} x^{2} + 765765 \, A a^{4} b x^{2} + 138567 \, A a^{5}}{2909907 \, x^{21}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^22,x, algorithm="giac")

[Out]

-1/2909907*(323323*B*b^5*x^12 + 1322685*B*a*b^4*x^10 + 264537*A*b^5*x^10 + 2238390*B*a^2*b^3*x^8 + 1119195*A*a
*b^4*x^8 + 1939938*B*a^3*b^2*x^6 + 1939938*A*a^2*b^3*x^6 + 855855*B*a^4*b*x^4 + 1711710*A*a^3*b^2*x^4 + 153153
*B*a^5*x^2 + 765765*A*a^4*b*x^2 + 138567*A*a^5)/x^21