Optimal. Leaf size=140 \[ -a \log \left (-a x+\sqrt {b+a^2 x^2}\right )+3 a \text {RootSum}\left [b^2-2 b \text {$\#$1}^6-4 a^2 \text {$\#$1}^7+\text {$\#$1}^{12}\& ,\frac {b \log \left (\sqrt [3]{a x-\sqrt {b+a^2 x^2}}-\text {$\#$1}\right )+a^2 \log \left (\sqrt [3]{a x-\sqrt {b+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}}{3 b+7 a^2 \text {$\#$1}-3 \text {$\#$1}^6}\& \right ] \]
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Rubi [F]
time = 6.15, 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 {b+a^2 x^2}}{x^2-\sqrt [3]{a 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 {\sqrt {b+a^2 x^2}}{x^2-\sqrt [3]{a x-\sqrt {b+a^2 x^2}}} \, dx &=\int \left (\frac {x^4 \left (b+a^2 x^2\right )}{b+2 a x^7-x^{12}}+\frac {x^5 \sqrt {b+a^2 x^2} \left (-a+x^5\right )}{-b-2 a x^7+x^{12}}+\frac {a x^3 \sqrt {b+a^2 x^2} \sqrt [3]{a x-\sqrt {b+a^2 x^2}}}{b+2 a x^7-x^{12}}-\frac {x^8 \sqrt {b+a^2 x^2} \sqrt [3]{a x-\sqrt {b+a^2 x^2}}}{b+2 a x^7-x^{12}}+\frac {x^2 \left (b+a^2 x^2\right ) \sqrt [3]{a x-\sqrt {b+a^2 x^2}}}{b+2 a x^7-x^{12}}+\frac {a x \sqrt {b+a^2 x^2} \left (a x-\sqrt {b+a^2 x^2}\right )^{2/3}}{b+2 a x^7-x^{12}}-\frac {x^6 \sqrt {b+a^2 x^2} \left (a x-\sqrt {b+a^2 x^2}\right )^{2/3}}{b+2 a x^7-x^{12}}+\frac {\left (b+a^2 x^2\right ) \left (a x-\sqrt {b+a^2 x^2}\right )^{2/3}}{b+2 a x^7-x^{12}}\right ) \, dx\\ &=a \int \frac {x^3 \sqrt {b+a^2 x^2} \sqrt [3]{a x-\sqrt {b+a^2 x^2}}}{b+2 a x^7-x^{12}} \, dx+a \int \frac {x \sqrt {b+a^2 x^2} \left (a x-\sqrt {b+a^2 x^2}\right )^{2/3}}{b+2 a x^7-x^{12}} \, dx+\int \frac {x^4 \left (b+a^2 x^2\right )}{b+2 a x^7-x^{12}} \, dx+\int \frac {x^5 \sqrt {b+a^2 x^2} \left (-a+x^5\right )}{-b-2 a x^7+x^{12}} \, dx-\int \frac {x^8 \sqrt {b+a^2 x^2} \sqrt [3]{a x-\sqrt {b+a^2 x^2}}}{b+2 a x^7-x^{12}} \, dx+\int \frac {x^2 \left (b+a^2 x^2\right ) \sqrt [3]{a x-\sqrt {b+a^2 x^2}}}{b+2 a x^7-x^{12}} \, dx-\int \frac {x^6 \sqrt {b+a^2 x^2} \left (a x-\sqrt {b+a^2 x^2}\right )^{2/3}}{b+2 a x^7-x^{12}} \, dx+\int \frac {\left (b+a^2 x^2\right ) \left (a x-\sqrt {b+a^2 x^2}\right )^{2/3}}{b+2 a x^7-x^{12}} \, dx\\ &=a \int \frac {x^3 \sqrt {b+a^2 x^2} \sqrt [3]{a x-\sqrt {b+a^2 x^2}}}{b+2 a x^7-x^{12}} \, dx+a \int \frac {x \sqrt {b+a^2 x^2} \left (a x-\sqrt {b+a^2 x^2}\right )^{2/3}}{b+2 a x^7-x^{12}} \, dx-\int \frac {x^8 \sqrt {b+a^2 x^2} \sqrt [3]{a x-\sqrt {b+a^2 x^2}}}{b+2 a x^7-x^{12}} \, dx-\int \frac {x^6 \sqrt {b+a^2 x^2} \left (a x-\sqrt {b+a^2 x^2}\right )^{2/3}}{b+2 a x^7-x^{12}} \, dx+\int \left (\frac {b x^4}{b+2 a x^7-x^{12}}+\frac {a^2 x^6}{b+2 a x^7-x^{12}}\right ) \, dx+\int \left (\frac {a x^5 \sqrt {b+a^2 x^2}}{b+2 a x^7-x^{12}}+\frac {x^{10} \sqrt {b+a^2 x^2}}{-b-2 a x^7+x^{12}}\right ) \, dx+\int \left (\frac {b x^2 \sqrt [3]{a x-\sqrt {b+a^2 x^2}}}{b+2 a x^7-x^{12}}+\frac {a^2 x^4 \sqrt [3]{a x-\sqrt {b+a^2 x^2}}}{b+2 a x^7-x^{12}}\right ) \, dx+\int \left (\frac {b \left (a x-\sqrt {b+a^2 x^2}\right )^{2/3}}{b+2 a x^7-x^{12}}+\frac {a^2 x^2 \left (a x-\sqrt {b+a^2 x^2}\right )^{2/3}}{b+2 a x^7-x^{12}}\right ) \, dx\\ &=a \int \frac {x^5 \sqrt {b+a^2 x^2}}{b+2 a x^7-x^{12}} \, dx+a \int \frac {x^3 \sqrt {b+a^2 x^2} \sqrt [3]{a x-\sqrt {b+a^2 x^2}}}{b+2 a x^7-x^{12}} \, dx+a \int \frac {x \sqrt {b+a^2 x^2} \left (a x-\sqrt {b+a^2 x^2}\right )^{2/3}}{b+2 a x^7-x^{12}} \, dx+a^2 \int \frac {x^6}{b+2 a x^7-x^{12}} \, dx+a^2 \int \frac {x^4 \sqrt [3]{a x-\sqrt {b+a^2 x^2}}}{b+2 a x^7-x^{12}} \, dx+a^2 \int \frac {x^2 \left (a x-\sqrt {b+a^2 x^2}\right )^{2/3}}{b+2 a x^7-x^{12}} \, dx+b \int \frac {x^4}{b+2 a x^7-x^{12}} \, dx+b \int \frac {x^2 \sqrt [3]{a x-\sqrt {b+a^2 x^2}}}{b+2 a x^7-x^{12}} \, dx+b \int \frac {\left (a x-\sqrt {b+a^2 x^2}\right )^{2/3}}{b+2 a x^7-x^{12}} \, dx+\int \frac {x^{10} \sqrt {b+a^2 x^2}}{-b-2 a x^7+x^{12}} \, dx-\int \frac {x^8 \sqrt {b+a^2 x^2} \sqrt [3]{a x-\sqrt {b+a^2 x^2}}}{b+2 a x^7-x^{12}} \, dx-\int \frac {x^6 \sqrt {b+a^2 x^2} \left (a x-\sqrt {b+a^2 x^2}\right )^{2/3}}{b+2 a x^7-x^{12}} \, dx\\ \end {align*}
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Mathematica [A]
time = 2.90, size = 140, normalized size = 1.00 \begin {gather*} -a \log \left (-a x+\sqrt {b+a^2 x^2}\right )+3 a \text {RootSum}\left [b^2-2 b \text {$\#$1}^6-4 a^2 \text {$\#$1}^7+\text {$\#$1}^{12}\&,\frac {b \log \left (\sqrt [3]{a x-\sqrt {b+a^2 x^2}}-\text {$\#$1}\right )+a^2 \log \left (\sqrt [3]{a x-\sqrt {b+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}}{3 b+7 a^2 \text {$\#$1}-3 \text {$\#$1}^6}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a^{2} x^{2}+b}}{x^{2}-\left (a x -\sqrt {a^{2} x^{2}+b}\right )^{\frac {1}{3}}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a^{2} x^{2} + b}}{x^{2} - \sqrt [3]{a x - \sqrt {a^{2} x^{2} + b}}}\, dx \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*} \text {could not integrate} \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 = -0.01 \begin {gather*} -\int \frac {\sqrt {a^2\,x^2+b}}{{\left (a\,x-\sqrt {a^2\,x^2+b}\right )}^{1/3}-x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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