Optimal. Leaf size=266 \[ \frac {3 b \sqrt [3]{b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}{2 a}+\frac {b \log \left (\sqrt [3]{2} \sqrt [3]{b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}-1\right )}{2 \sqrt [3]{2} a}-\frac {b \log \left (2^{2/3} \left (b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2\right )^{2/3}+\sqrt [3]{2} \sqrt [3]{b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}+1\right )}{4 \sqrt [3]{2} a}-\frac {\sqrt {3} b \tan ^{-1}\left (\frac {2 \sqrt [3]{2} \sqrt [3]{b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt [3]{2} a} \]
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Rubi [F] time = 0.50, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx &=\int \frac {\sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx\\ \end {align*}
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Mathematica [A] time = 1.44, size = 331, normalized size = 1.24 \begin {gather*} \frac {a x \sqrt [3]{x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )} \left (b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x^2-1\right ) \left (-\sqrt [3]{a} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{\left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )^2+a}+\left (\left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )^2+a\right )^{2/3}\right )+6 \sqrt [3]{2} \sqrt [3]{a x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )}+2 \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{\left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )^2+a}\right )-2 \sqrt {3} \sqrt [3]{a} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )^2+a}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )\right )}{4 \sqrt [3]{2} \sqrt {\frac {a \left (a x^2-1\right )}{b^2}} \left (a x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )\right )^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 3.85, size = 266, normalized size = 1.00 \begin {gather*} \frac {3 b \sqrt [3]{b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}{2 a}+\frac {b \log \left (\sqrt [3]{2} \sqrt [3]{b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}-1\right )}{2 \sqrt [3]{2} a}-\frac {b \log \left (2^{2/3} \left (b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2\right )^{2/3}+\sqrt [3]{2} \sqrt [3]{b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}+1\right )}{4 \sqrt [3]{2} a}-\frac {\sqrt {3} b \tan ^{-1}\left (\frac {2 \sqrt [3]{2} \sqrt [3]{b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt [3]{2} a} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} b x\right )}^{\frac {1}{3}}}{\sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.48, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a \,x^{2}+b x \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}\right )^{\frac {1}{3}}}{\sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} b x\right )}^{\frac {1}{3}}}{\sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a\,x^2+b\,x\,\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}\right )}^{1/3}}{\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{x \left (a x + b \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}\right )}}{\sqrt {\frac {a \left (a x^{2} - 1\right )}{b^{2}}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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