Optimal. Leaf size=64 \[ -\frac {a^5}{6 x^6}-\frac {5 a^4 b}{4 x^4}-\frac {5 a^3 b^2}{x^2}+10 a^2 b^3 \log (x)+\frac {5}{2} a b^4 x^2+\frac {b^5 x^4}{4} \]
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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} \begin {gather*} -\frac {5 a^3 b^2}{x^2}+10 a^2 b^3 \log (x)-\frac {5 a^4 b}{4 x^4}-\frac {a^5}{6 x^6}+\frac {5}{2} a b^4 x^2+\frac {b^5 x^4}{4} \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^7} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^5}{x^4} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (5 a b^4+\frac {a^5}{x^4}+\frac {5 a^4 b}{x^3}+\frac {10 a^3 b^2}{x^2}+\frac {10 a^2 b^3}{x}+b^5 x\right ) \, dx,x,x^2\right )\\ &=-\frac {a^5}{6 x^6}-\frac {5 a^4 b}{4 x^4}-\frac {5 a^3 b^2}{x^2}+\frac {5}{2} a b^4 x^2+\frac {b^5 x^4}{4}+10 a^2 b^3 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 64, normalized size = 1.00 \begin {gather*} -\frac {a^5}{6 x^6}-\frac {5 a^4 b}{4 x^4}-\frac {5 a^3 b^2}{x^2}+10 a^2 b^3 \log (x)+\frac {5}{2} a b^4 x^2+\frac {b^5 x^4}{4} \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^7} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.15, size = 61, normalized size = 0.95 \begin {gather*} \frac {3 \, b^{5} x^{10} + 30 \, a b^{4} x^{8} + 120 \, a^{2} b^{3} x^{6} \log \relax (x) - 60 \, a^{3} b^{2} x^{4} - 15 \, a^{4} b x^{2} - 2 \, a^{5}}{12 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.19, size = 72, normalized size = 1.12 \begin {gather*} \frac {1}{4} \, b^{5} x^{4} + \frac {5}{2} \, a b^{4} x^{2} + 5 \, a^{2} b^{3} \log \left (x^{2}\right ) - \frac {110 \, a^{2} b^{3} x^{6} + 60 \, a^{3} b^{2} x^{4} + 15 \, a^{4} b x^{2} + 2 \, a^{5}}{12 \, x^{6}} \end {gather*}
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 \begin {gather*} \frac {b^{5} x^{4}}{4}+\frac {5 a \,b^{4} x^{2}}{2}+10 a^{2} b^{3} \ln \relax (x )-\frac {5 a^{3} b^{2}}{x^{2}}-\frac {5 a^{4} b}{4 x^{4}}-\frac {a^{5}}{6 x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 61, normalized size = 0.95 \begin {gather*} \frac {1}{4} \, b^{5} x^{4} + \frac {5}{2} \, a b^{4} x^{2} + 5 \, a^{2} b^{3} \log \left (x^{2}\right ) - \frac {60 \, a^{3} b^{2} x^{4} + 15 \, a^{4} b x^{2} + 2 \, a^{5}}{12 \, x^{6}} \end {gather*}
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 \begin {gather*} \frac {b^5\,x^4}{4}-\frac {\frac {a^5}{6}+\frac {5\,a^4\,b\,x^2}{4}+5\,a^3\,b^2\,x^4}{x^6}+\frac {5\,a\,b^4\,x^2}{2}+10\,a^2\,b^3\,\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.28, size = 65, normalized size = 1.02 \begin {gather*} 10 a^{2} b^{3} \log {\relax (x )} + \frac {5 a b^{4} x^{2}}{2} + \frac {b^{5} x^{4}}{4} + \frac {- 2 a^{5} - 15 a^{4} b x^{2} - 60 a^{3} b^{2} x^{4}}{12 x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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