Optimal. Leaf size=64 \[ -\frac {a^5}{8 x^8}-\frac {5 a^4 b}{6 x^6}-\frac {5 a^3 b^2}{2 x^4}-\frac {5 a^2 b^3}{x^2}+5 a b^4 \log (x)+\frac {b^5 x^2}{2} \]
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Rubi [A] time = 0.03, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \[ -\frac {5 a^3 b^2}{2 x^4}-\frac {5 a^2 b^3}{x^2}-\frac {5 a^4 b}{6 x^6}-\frac {a^5}{8 x^8}+5 a b^4 \log (x)+\frac {b^5 x^2}{2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^5}{x^9} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^5}{x^5} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (b^5+\frac {a^5}{x^5}+\frac {5 a^4 b}{x^4}+\frac {10 a^3 b^2}{x^3}+\frac {10 a^2 b^3}{x^2}+\frac {5 a b^4}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^5}{8 x^8}-\frac {5 a^4 b}{6 x^6}-\frac {5 a^3 b^2}{2 x^4}-\frac {5 a^2 b^3}{x^2}+\frac {b^5 x^2}{2}+5 a b^4 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 64, normalized size = 1.00 \[ -\frac {a^5}{8 x^8}-\frac {5 a^4 b}{6 x^6}-\frac {5 a^3 b^2}{2 x^4}-\frac {5 a^2 b^3}{x^2}+5 a b^4 \log (x)+\frac {b^5 x^2}{2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 61, normalized size = 0.95 \[ \frac {12 \, b^{5} x^{10} + 120 \, a b^{4} x^{8} \log \relax (x) - 120 \, a^{2} b^{3} x^{6} - 60 \, a^{3} b^{2} x^{4} - 20 \, a^{4} b x^{2} - 3 \, a^{5}}{24 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.05, size = 70, normalized size = 1.09 \[ \frac {1}{2} \, b^{5} x^{2} + \frac {5}{2} \, a b^{4} \log \left (x^{2}\right ) - \frac {125 \, a b^{4} x^{8} + 120 \, a^{2} b^{3} x^{6} + 60 \, a^{3} b^{2} x^{4} + 20 \, a^{4} b x^{2} + 3 \, a^{5}}{24 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 57, normalized size = 0.89 \[ \frac {b^{5} x^{2}}{2}+5 a \,b^{4} \ln \relax (x )-\frac {5 a^{2} b^{3}}{x^{2}}-\frac {5 a^{3} b^{2}}{2 x^{4}}-\frac {5 a^{4} b}{6 x^{6}}-\frac {a^{5}}{8 x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.41, size = 61, normalized size = 0.95 \[ \frac {1}{2} \, b^{5} x^{2} + \frac {5}{2} \, a b^{4} \log \left (x^{2}\right ) - \frac {120 \, a^{2} b^{3} x^{6} + 60 \, a^{3} b^{2} x^{4} + 20 \, a^{4} b x^{2} + 3 \, a^{5}}{24 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 59, normalized size = 0.92 \[ \frac {b^5\,x^2}{2}-\frac {\frac {a^5}{8}+\frac {5\,a^4\,b\,x^2}{6}+\frac {5\,a^3\,b^2\,x^4}{2}+5\,a^2\,b^3\,x^6}{x^8}+5\,a\,b^4\,\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.36, size = 63, normalized size = 0.98 \[ 5 a b^{4} \log {\relax (x )} + \frac {b^{5} x^{2}}{2} + \frac {- 3 a^{5} - 20 a^{4} b x^{2} - 60 a^{3} b^{2} x^{4} - 120 a^{2} b^{3} x^{6}}{24 x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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