Optimal. Leaf size=330 \[ \frac {7}{16} x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}} \sqrt [3]{b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}+\frac {\left (-a x^2-10\right ) \sqrt [3]{b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}{16 b}-\frac {5 \log \left (\sqrt [3]{2} \sqrt [3]{b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}-1\right )}{24 \sqrt [3]{2} b}+\frac {5 \log \left (2^{2/3} \left (b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2\right )^{2/3}+\sqrt [3]{2} \sqrt [3]{b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}+1\right )}{48 \sqrt [3]{2} b}+\frac {5 \tan ^{-1}\left (\frac {2 \sqrt [3]{2} \sqrt [3]{b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{8 \sqrt [3]{2} \sqrt {3} b} \]
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Rubi [F] time = 0.47, 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 \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt [3]{a x^2+b x \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 \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx &=\int \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx\\ \end {align*}
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Mathematica [C] time = 20.84, size = 10907, normalized size = 33.05 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 3.72, size = 333, normalized size = 1.01 \begin {gather*} \frac {7}{16} x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}} \sqrt [3]{b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}+\frac {\left (-a x^2-10\right ) \sqrt [3]{b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}{16 b}-\frac {5 \log \left (\sqrt [3]{2} b \sqrt [3]{b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}-b\right )}{24 \sqrt [3]{2} b}+\frac {5 \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 )}{48 \sqrt [3]{2} b}+\frac {5 \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 )}{8 \sqrt [3]{2} \sqrt {3} b} \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(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.49, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}\, \left (a \,x^{2}+b x \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}\right )^{\frac {1}{3}}\, 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 {\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 {\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 \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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