Optimal. Leaf size=91 \[ -\frac {a^5 B}{10 x^{10}}-\frac {5 a^4 b B}{8 x^8}-\frac {5 a^3 b^2 B}{3 x^6}-\frac {5 a^2 b^3 B}{2 x^4}-\frac {A \left (a+b x^2\right )^6}{12 a x^{12}}-\frac {5 a b^4 B}{2 x^2}+b^5 B \log (x) \]
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Rubi [A] time = 0.06, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {446, 78, 43} \[ -\frac {5 a^2 b^3 B}{2 x^4}-\frac {5 a^3 b^2 B}{3 x^6}-\frac {5 a^4 b B}{8 x^8}-\frac {a^5 B}{10 x^{10}}-\frac {A \left (a+b x^2\right )^6}{12 a x^{12}}-\frac {5 a b^4 B}{2 x^2}+b^5 B \log (x) \]
Antiderivative was successfully verified.
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Rule 43
Rule 78
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{13}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^5 (A+B x)}{x^7} \, dx,x,x^2\right )\\ &=-\frac {A \left (a+b x^2\right )^6}{12 a x^{12}}+\frac {1}{2} B \operatorname {Subst}\left (\int \frac {(a+b x)^5}{x^6} \, dx,x,x^2\right )\\ &=-\frac {A \left (a+b x^2\right )^6}{12 a x^{12}}+\frac {1}{2} B \operatorname {Subst}\left (\int \left (\frac {a^5}{x^6}+\frac {5 a^4 b}{x^5}+\frac {10 a^3 b^2}{x^4}+\frac {10 a^2 b^3}{x^3}+\frac {5 a b^4}{x^2}+\frac {b^5}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^5 B}{10 x^{10}}-\frac {5 a^4 b B}{8 x^8}-\frac {5 a^3 b^2 B}{3 x^6}-\frac {5 a^2 b^3 B}{2 x^4}-\frac {5 a b^4 B}{2 x^2}-\frac {A \left (a+b x^2\right )^6}{12 a x^{12}}+b^5 B \log (x)\\ \end {align*}
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Mathematica [A] time = 0.05, size = 118, normalized size = 1.30 \[ b^5 B \log (x)-\frac {2 a^5 \left (5 A+6 B x^2\right )+15 a^4 b x^2 \left (4 A+5 B x^2\right )+50 a^3 b^2 x^4 \left (3 A+4 B x^2\right )+100 a^2 b^3 x^6 \left (2 A+3 B x^2\right )+150 a b^4 x^8 \left (A+2 B x^2\right )+60 A b^5 x^{10}}{120 x^{12}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 123, normalized size = 1.35 \[ \frac {120 \, B b^{5} x^{12} \log \relax (x) - 60 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} - 150 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} - 200 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} - 10 \, A a^{5} - 75 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} - 12 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{120 \, x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 138, normalized size = 1.52 \[ \frac {1}{2} \, B b^{5} \log \left (x^{2}\right ) - \frac {147 \, B b^{5} x^{12} + 300 \, B a b^{4} x^{10} + 60 \, A b^{5} x^{10} + 300 \, B a^{2} b^{3} x^{8} + 150 \, A a b^{4} x^{8} + 200 \, B a^{3} b^{2} x^{6} + 200 \, A a^{2} b^{3} x^{6} + 75 \, B a^{4} b x^{4} + 150 \, A a^{3} b^{2} x^{4} + 12 \, B a^{5} x^{2} + 60 \, A a^{4} b x^{2} + 10 \, A a^{5}}{120 \, x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 124, normalized size = 1.36 \[ B \,b^{5} \ln \relax (x )-\frac {A \,b^{5}}{2 x^{2}}-\frac {5 B a \,b^{4}}{2 x^{2}}-\frac {5 A a \,b^{4}}{4 x^{4}}-\frac {5 B \,a^{2} b^{3}}{2 x^{4}}-\frac {5 A \,a^{2} b^{3}}{3 x^{6}}-\frac {5 B \,a^{3} b^{2}}{3 x^{6}}-\frac {5 A \,a^{3} b^{2}}{4 x^{8}}-\frac {5 B \,a^{4} b}{8 x^{8}}-\frac {A \,a^{4} b}{2 x^{10}}-\frac {B \,a^{5}}{10 x^{10}}-\frac {A \,a^{5}}{12 x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.09, size = 123, normalized size = 1.35 \[ \frac {1}{2} \, B b^{5} \log \left (x^{2}\right ) - \frac {60 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 150 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 200 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 10 \, A a^{5} + 75 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 12 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{120 \, x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 121, normalized size = 1.33 \[ B\,b^5\,\ln \relax (x)-\frac {\frac {A\,a^5}{12}+x^8\,\left (\frac {5\,B\,a^2\,b^3}{2}+\frac {5\,A\,a\,b^4}{4}\right )+x^4\,\left (\frac {5\,B\,a^4\,b}{8}+\frac {5\,A\,a^3\,b^2}{4}\right )+x^2\,\left (\frac {B\,a^5}{10}+\frac {A\,b\,a^4}{2}\right )+x^{10}\,\left (\frac {A\,b^5}{2}+\frac {5\,B\,a\,b^4}{2}\right )+x^6\,\left (\frac {5\,B\,a^3\,b^2}{3}+\frac {5\,A\,a^2\,b^3}{3}\right )}{x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.68, size = 133, normalized size = 1.46 \[ B b^{5} \log {\relax (x )} + \frac {- 10 A a^{5} + x^{10} \left (- 60 A b^{5} - 300 B a b^{4}\right ) + x^{8} \left (- 150 A a b^{4} - 300 B a^{2} b^{3}\right ) + x^{6} \left (- 200 A a^{2} b^{3} - 200 B a^{3} b^{2}\right ) + x^{4} \left (- 150 A a^{3} b^{2} - 75 B a^{4} b\right ) + x^{2} \left (- 60 A a^{4} b - 12 B a^{5}\right )}{120 x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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