Optimal. Leaf size=140 \[ 3 a \text {RootSum}\left [\text {$\#$1}^{12}-4 \text {$\#$1}^7 a^2-2 \text {$\#$1}^6 b+b^2\& ,\frac {\text {$\#$1} a^2 \log \left (\sqrt [3]{a x-\sqrt {a^2 x^2+b}}-\text {$\#$1}\right )+b \log \left (\sqrt [3]{a x-\sqrt {a^2 x^2+b}}-\text {$\#$1}\right )}{-3 \text {$\#$1}^6+7 \text {$\#$1} a^2+3 b}\& \right ]-a \log \left (\sqrt {a^2 x^2+b}-a x\right ) \]
________________________________________________________________________________________
Rubi [F] time = 9.56, 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.
[In]
[Out]
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*}
________________________________________________________________________________________
Mathematica [F] time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 3.92, 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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a^{2} x^{2} + b}}{x^{2} - {\left (a x - \sqrt {a^{2} x^{2} + b}\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.09, 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.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a^{2} x^{2} + b}}{x^{2} - {\left (a x - \sqrt {a^{2} x^{2} + b}\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________