Optimal. Leaf size=40 \[ \frac {b^2 \log \left (a x^2+b\right )}{2 a^3}-\frac {b x^2}{2 a^2}+\frac {x^4}{4 a} \]
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Rubi [A] time = 0.03, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {263, 266, 43} \[ \frac {b^2 \log \left (a x^2+b\right )}{2 a^3}-\frac {b x^2}{2 a^2}+\frac {x^4}{4 a} \]
Antiderivative was successfully verified.
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Rule 43
Rule 263
Rule 266
Rubi steps
\begin {align*} \int \frac {x^3}{a+\frac {b}{x^2}} \, dx &=\int \frac {x^5}{b+a x^2} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{b+a x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {b}{a^2}+\frac {x}{a}+\frac {b^2}{a^2 (b+a x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {b x^2}{2 a^2}+\frac {x^4}{4 a}+\frac {b^2 \log \left (b+a x^2\right )}{2 a^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 40, normalized size = 1.00 \[ \frac {b^2 \log \left (a x^2+b\right )}{2 a^3}-\frac {b x^2}{2 a^2}+\frac {x^4}{4 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 33, normalized size = 0.82 \[ \frac {a^{2} x^{4} - 2 \, a b x^{2} + 2 \, b^{2} \log \left (a x^{2} + b\right )}{4 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 35, normalized size = 0.88 \[ \frac {b^{2} \log \left ({\left | a x^{2} + b \right |}\right )}{2 \, a^{3}} + \frac {a x^{4} - 2 \, b x^{2}}{4 \, a^{2}} \]
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 \[ \frac {x^{4}}{4 a}-\frac {b \,x^{2}}{2 a^{2}}+\frac {b^{2} \ln \left (a \,x^{2}+b \right )}{2 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.83, size = 34, normalized size = 0.85 \[ \frac {b^{2} \log \left (a x^{2} + b\right )}{2 \, a^{3}} + \frac {a x^{4} - 2 \, b x^{2}}{4 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.09, size = 33, normalized size = 0.82 \[ \frac {2\,b^2\,\ln \left (a\,x^2+b\right )+a^2\,x^4-2\,a\,b\,x^2}{4\,a^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 32, normalized size = 0.80 \[ \frac {x^{4}}{4 a} - \frac {b x^{2}}{2 a^{2}} + \frac {b^{2} \log {\left (a x^{2} + b \right )}}{2 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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