Optimal. Leaf size=40 \[ \frac {a^2 \log \left (a+b x^2\right )}{2 b^3}-\frac {a x^2}{2 b^2}+\frac {x^4}{4 b} \]
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Rubi [A] time = 0.03, antiderivative size = 40, 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} \[ \frac {a^2 \log \left (a+b x^2\right )}{2 b^3}-\frac {a x^2}{2 b^2}+\frac {x^4}{4 b} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^5}{a+b x^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{a+b x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {a}{b^2}+\frac {x}{b}+\frac {a^2}{b^2 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {a x^2}{2 b^2}+\frac {x^4}{4 b}+\frac {a^2 \log \left (a+b x^2\right )}{2 b^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 40, normalized size = 1.00 \[ \frac {a^2 \log \left (a+b x^2\right )}{2 b^3}-\frac {a x^2}{2 b^2}+\frac {x^4}{4 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.15, size = 33, normalized size = 0.82 \[ \frac {b^{2} x^{4} - 2 \, a b x^{2} + 2 \, a^{2} \log \left (b x^{2} + a\right )}{4 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.12, size = 35, normalized size = 0.88 \[ \frac {a^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{3}} + \frac {b x^{4} - 2 \, a x^{2}}{4 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 35, normalized size = 0.88 \[ \frac {x^{4}}{4 b}-\frac {a \,x^{2}}{2 b^{2}}+\frac {a^{2} \ln \left (b \,x^{2}+a \right )}{2 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 34, normalized size = 0.85 \[ \frac {a^{2} \log \left (b x^{2} + a\right )}{2 \, b^{3}} + \frac {b x^{4} - 2 \, a x^{2}}{4 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.65, size = 33, normalized size = 0.82 \[ \frac {2\,a^2\,\ln \left (b\,x^2+a\right )+b^2\,x^4-2\,a\,b\,x^2}{4\,b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 32, normalized size = 0.80 \[ \frac {a^{2} \log {\left (a + b x^{2} \right )}}{2 b^{3}} - \frac {a x^{2}}{2 b^{2}} + \frac {x^{4}}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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