Optimal. Leaf size=44 \[ \frac {\left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 b \sqrt {a^2+2 a b x^3+b^2 x^6}} \]
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Rubi [A] time = 0.03, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1352, 608, 31} \begin {gather*} \frac {\left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 b \sqrt {a^2+2 a b x^3+b^2 x^6}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 608
Rule 1352
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx,x,x^3\right )\\ &=\frac {\left (a b+b^2 x^3\right ) \operatorname {Subst}\left (\int \frac {1}{a b+b^2 x} \, dx,x,x^3\right )}{3 \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ &=\frac {\left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 b \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 35, normalized size = 0.80 \begin {gather*} \frac {\left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 b \sqrt {\left (a+b x^3\right )^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.27, size = 149, normalized size = 3.39 \begin {gather*} -\frac {\log \left (\sqrt {a^2+2 a b x^3+b^2 x^6}-a-\sqrt {b^2} x^3\right )}{6 \sqrt {b^2}}-\frac {\log \left (\sqrt {a^2+2 a b x^3+b^2 x^6}+a-\sqrt {b^2} x^3\right )}{6 \sqrt {b^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b^2} x^3}{a}-\frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{a}\right )}{3 b} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.07, size = 13, normalized size = 0.30 \begin {gather*} \frac {\log \left (b x^{3} + a\right )}{3 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 22, normalized size = 0.50 \begin {gather*} \frac {\log \left ({\left | b x^{3} + a \right |}\right ) \mathrm {sgn}\left (b x^{3} + a\right )}{3 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 32, normalized size = 0.73 \begin {gather*} \frac {\left (b \,x^{3}+a \right ) \ln \left (b \,x^{3}+a \right )}{3 \sqrt {\left (b \,x^{3}+a \right )^{2}}\, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.93, size = 15, normalized size = 0.34 \begin {gather*} \frac {\log \left (x^{3} + \frac {a}{b}\right )}{3 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.39, size = 33, normalized size = 0.75 \begin {gather*} \frac {\ln \left (b^2\,x^3+a\,b\right )\,\mathrm {sign}\left (2\,b^2\,x^3+2\,a\,b\right )}{3\,\sqrt {b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 10, normalized size = 0.23 \begin {gather*} \frac {\log {\left (a + b x^{3} \right )}}{3 b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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