Optimal. Leaf size=167 \[ \frac {2 \left (2 a x^2-1\right ) \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}{3 b x^2}-\frac {4 \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}} \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}{3 x}-\frac {\sqrt {2} a \tanh ^{-1}\left (\sqrt {2} \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}\right )}{b} \]
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Rubi [F] time = 1.78, 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 {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x^2 \sqrt {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 \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x^2 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx &=\int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x^2 \sqrt {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 = 5.41, size = 341, normalized size = 2.04 \begin {gather*} \frac {\frac {3 \sqrt {\frac {a \left (a x^2-1\right )}{b^2}} \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right ) \left (\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};2 x \left (\sqrt {\frac {a \left (a x^2-1\right )}{b^2}} b+a x\right )\right )+2 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+2 a x^2+3\right )}{b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x^2-1}-\frac {4 a \left (3 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+3 a x^2-1\right )}{b x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )}-\frac {3 \sqrt {2} \sqrt {a x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )} \left (\sqrt {2} \sqrt {a x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )}+\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {\left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )^2+a}}{\sqrt {a}}\right )\right )}{b}}{6 \sqrt {x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 3.30, size = 223, normalized size = 1.34 \begin {gather*} \frac {2 \left (-1+2 a x^2\right ) \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{3 b x^2}-\frac {4 \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{3 x}+\frac {a \log \left (-1+\sqrt {2} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}\right )}{\sqrt {2} b}-\frac {a \log \left (b+\sqrt {2} b \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}\right )}{\sqrt {2} b} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 37.49, size = 175, normalized size = 1.05 \begin {gather*} \frac {3 \, \sqrt {2} a x^{2} \log \left (4 \, a x^{2} - 4 \, b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} + 2 \, \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} {\left (2 \, \sqrt {2} b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} - \sqrt {2} {\left (2 \, a x^{2} - 1\right )}\right )} - 1\right ) + 4 \, {\left (2 \, a x^{2} - 2 \, b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} - 1\right )} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}}}{6 \, b x^{2}} \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 {\sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}}{\sqrt {a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} b x} x^{2}}\,{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 \[\int \frac {\sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}{x^{2} \sqrt {a \,x^{2}+b x \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}}\, dx\]
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 {\sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}}{\sqrt {a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} b x} x^{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 = -0.01 \begin {gather*} \int \frac {\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}}{x^2\,\sqrt {a\,x^2+b\,x\,\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 {\frac {a \left (a x^{2} - 1\right )}{b^{2}}}}{x^{2} \sqrt {x \left (a x + b \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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