Optimal. Leaf size=64 \[ -\frac {a^5}{4 x^4}-\frac {5 a^4 b}{2 x^2}+10 a^3 b^2 \log (x)+5 a^2 b^3 x^2+\frac {5}{4} a b^4 x^4+\frac {b^5 x^6}{6} \]
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Rubi [A] time = 0.04, 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} \begin {gather*} 5 a^2 b^3 x^2+10 a^3 b^2 \log (x)-\frac {5 a^4 b}{2 x^2}-\frac {a^5}{4 x^4}+\frac {5}{4} a b^4 x^4+\frac {b^5 x^6}{6} \end {gather*}
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^5} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^5}{x^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (10 a^2 b^3+\frac {a^5}{x^3}+\frac {5 a^4 b}{x^2}+\frac {10 a^3 b^2}{x}+5 a b^4 x+b^5 x^2\right ) \, dx,x,x^2\right )\\ &=-\frac {a^5}{4 x^4}-\frac {5 a^4 b}{2 x^2}+5 a^2 b^3 x^2+\frac {5}{4} a b^4 x^4+\frac {b^5 x^6}{6}+10 a^3 b^2 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 64, normalized size = 1.00 \begin {gather*} -\frac {a^5}{4 x^4}-\frac {5 a^4 b}{2 x^2}+10 a^3 b^2 \log (x)+5 a^2 b^3 x^2+\frac {5}{4} a b^4 x^4+\frac {b^5 x^6}{6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x^2\right )^5}{x^5} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.32, size = 61, normalized size = 0.95 \begin {gather*} \frac {2 \, b^{5} x^{10} + 15 \, a b^{4} x^{8} + 60 \, a^{2} b^{3} x^{6} + 120 \, a^{3} b^{2} x^{4} \log \relax (x) - 30 \, a^{4} b x^{2} - 3 \, a^{5}}{12 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.88, size = 70, normalized size = 1.09 \begin {gather*} \frac {1}{6} \, b^{5} x^{6} + \frac {5}{4} \, a b^{4} x^{4} + 5 \, a^{2} b^{3} x^{2} + 5 \, a^{3} b^{2} \log \left (x^{2}\right ) - \frac {30 \, a^{3} b^{2} x^{4} + 10 \, a^{4} b x^{2} + a^{5}}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 57, normalized size = 0.89 \begin {gather*} \frac {b^{5} x^{6}}{6}+\frac {5 a \,b^{4} x^{4}}{4}+5 a^{2} b^{3} x^{2}+10 a^{3} b^{2} \ln \relax (x )-\frac {5 a^{4} b}{2 x^{2}}-\frac {a^{5}}{4 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 59, normalized size = 0.92 \begin {gather*} \frac {1}{6} \, b^{5} x^{6} + \frac {5}{4} \, a b^{4} x^{4} + 5 \, a^{2} b^{3} x^{2} + 5 \, a^{3} b^{2} \log \left (x^{2}\right ) - \frac {10 \, a^{4} b x^{2} + a^{5}}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 59, normalized size = 0.92 \begin {gather*} \frac {b^5\,x^6}{6}-\frac {\frac {a^5}{4}+\frac {5\,b\,a^4\,x^2}{2}}{x^4}+\frac {5\,a\,b^4\,x^4}{4}+5\,a^2\,b^3\,x^2+10\,a^3\,b^2\,\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 63, normalized size = 0.98 \begin {gather*} 10 a^{3} b^{2} \log {\relax (x )} + 5 a^{2} b^{3} x^{2} + \frac {5 a b^{4} x^{4}}{4} + \frac {b^{5} x^{6}}{6} + \frac {- a^{5} - 10 a^{4} b x^{2}}{4 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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