Optimal. Leaf size=248 \[ -\frac {\log \left (\frac {\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{x \sqrt {a^2 x^2-b}+a x^2}}{\sqrt [3]{b}}+\frac {2^{2/3} a^{2/3} \left (x \sqrt {a^2 x^2-b}+a x^2\right )^{2/3}}{b^{2/3}}+1\right )}{2\ 2^{2/3} a^{2/3} \sqrt [3]{b}}+\frac {\log \left (\frac {\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{x \sqrt {a^2 x^2-b}+a x^2}}{\sqrt [3]{b}}-1\right )}{2^{2/3} a^{2/3} \sqrt [3]{b}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{x \sqrt {a^2 x^2-b}+a x^2}}{\sqrt {3} \sqrt [3]{b}}+\frac {1}{\sqrt {3}}\right )}{2^{2/3} a^{2/3} \sqrt [3]{b}} \]
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Rubi [F] time = 0.33, 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 {1}{\sqrt {-b+a^2 x^2} \sqrt [3]{a x^2+x \sqrt {-b+a^2 x^2}}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [3]{a x^2+x \sqrt {-b+a^2 x^2}}} \, dx &=\int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [3]{a x^2+x \sqrt {-b+a^2 x^2}}} \, dx\\ \end {align*}
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Mathematica [A] time = 13.65, size = 225, normalized size = 0.91 \begin {gather*} \frac {\sqrt {a^2 x^2-b} \left (a x \left (\sqrt {a^2 x^2-b}+a x\right )\right )^{4/3} \left (-2 \log \left (\sqrt {a^2 x^2-b}+a x\right )+3 \log \left (\sqrt [3]{b}-\sqrt [3]{\left (\sqrt {a^2 x^2-b}+a x\right )^2+b}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\left (\sqrt {a^2 x^2-b}+a x\right )^2+b}}{\sqrt [3]{b}}+1}{\sqrt {3}}\right )\right )}{2\ 2^{2/3} a^2 \sqrt [3]{b} x \sqrt [3]{x \left (\sqrt {a^2 x^2-b}+a x\right )} \left (a x \left (\sqrt {a^2 x^2-b}+a x\right )-b\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.45, size = 248, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a x^2+x \sqrt {-b+a^2 x^2}}}{\sqrt {3} \sqrt [3]{b}}\right )}{2^{2/3} a^{2/3} \sqrt [3]{b}}+\frac {\log \left (-1+\frac {\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a x^2+x \sqrt {-b+a^2 x^2}}}{\sqrt [3]{b}}\right )}{2^{2/3} a^{2/3} \sqrt [3]{b}}-\frac {\log \left (1+\frac {\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a x^2+x \sqrt {-b+a^2 x^2}}}{\sqrt [3]{b}}+\frac {2^{2/3} a^{2/3} \left (a x^2+x \sqrt {-b+a^2 x^2}\right )^{2/3}}{b^{2/3}}\right )}{2\ 2^{2/3} a^{2/3} \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] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a^{2} x^{2} - b} {\left (a x^{2} + \sqrt {a^{2} x^{2} - b} x\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {a^{2} x^{2}-b}\, \left (a \,x^{2}+x \sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}}\, dx\]
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 {1}{\sqrt {a^{2} x^{2} - b} {\left (a x^{2} + \sqrt {a^{2} x^{2} - b} x\right )}^{\frac {1}{3}}}\,{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 {1}{{\left (x\,\sqrt {a^2\,x^2-b}+a\,x^2\right )}^{1/3}\,\sqrt {a^2\,x^2-b}} \,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 {1}{\sqrt [3]{x \left (a x + \sqrt {a^{2} x^{2} - b}\right )} \sqrt {a^{2} x^{2} - b}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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