Optimal. Leaf size=64 \[ -\frac {a^5}{2 x^2}+5 a^4 b \log (x)+5 a^3 b^2 x^2+\frac {5}{2} a^2 b^3 x^4+\frac {5}{6} a b^4 x^6+\frac {b^5 x^8}{8} \]
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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*} \frac {5}{2} a^2 b^3 x^4+5 a^3 b^2 x^2+5 a^4 b \log (x)-\frac {a^5}{2 x^2}+\frac {5}{6} a b^4 x^6+\frac {b^5 x^8}{8} \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^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^5}{x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (10 a^3 b^2+\frac {a^5}{x^2}+\frac {5 a^4 b}{x}+10 a^2 b^3 x+5 a b^4 x^2+b^5 x^3\right ) \, dx,x,x^2\right )\\ &=-\frac {a^5}{2 x^2}+5 a^3 b^2 x^2+\frac {5}{2} a^2 b^3 x^4+\frac {5}{6} a b^4 x^6+\frac {b^5 x^8}{8}+5 a^4 b \log (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 64, normalized size = 1.00 \begin {gather*} -\frac {a^5}{2 x^2}+5 a^4 b \log (x)+5 a^3 b^2 x^2+\frac {5}{2} a^2 b^3 x^4+\frac {5}{6} a b^4 x^6+\frac {b^5 x^8}{8} \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^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.93, size = 61, normalized size = 0.95 \begin {gather*} \frac {3 \, b^{5} x^{10} + 20 \, a b^{4} x^{8} + 60 \, a^{2} b^{3} x^{6} + 120 \, a^{3} b^{2} x^{4} + 120 \, a^{4} b x^{2} \log \relax (x) - 12 \, a^{5}}{24 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.06, size = 68, normalized size = 1.06 \begin {gather*} \frac {1}{8} \, b^{5} x^{8} + \frac {5}{6} \, a b^{4} x^{6} + \frac {5}{2} \, a^{2} b^{3} x^{4} + 5 \, a^{3} b^{2} x^{2} + \frac {5}{2} \, a^{4} b \log \left (x^{2}\right ) - \frac {5 \, a^{4} b x^{2} + a^{5}}{2 \, x^{2}} \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^{8}}{8}+\frac {5 a \,b^{4} x^{6}}{6}+\frac {5 a^{2} b^{3} x^{4}}{2}+5 a^{3} b^{2} x^{2}+5 a^{4} b \ln \relax (x )-\frac {a^{5}}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 58, normalized size = 0.91 \begin {gather*} \frac {1}{8} \, b^{5} x^{8} + \frac {5}{6} \, a b^{4} x^{6} + \frac {5}{2} \, a^{2} b^{3} x^{4} + 5 \, a^{3} b^{2} x^{2} + \frac {5}{2} \, a^{4} b \log \left (x^{2}\right ) - \frac {a^{5}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 56, normalized size = 0.88 \begin {gather*} \frac {b^5\,x^8}{8}-\frac {a^5}{2\,x^2}+\frac {5\,a\,b^4\,x^6}{6}+5\,a^4\,b\,\ln \relax (x)+5\,a^3\,b^2\,x^2+\frac {5\,a^2\,b^3\,x^4}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 63, normalized size = 0.98 \begin {gather*} - \frac {a^{5}}{2 x^{2}} + 5 a^{4} b \log {\relax (x )} + 5 a^{3} b^{2} x^{2} + \frac {5 a^{2} b^{3} x^{4}}{2} + \frac {5 a b^{4} x^{6}}{6} + \frac {b^{5} x^{8}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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