Optimal. Leaf size=88 \[ -\frac {\sqrt {2} b \log \left (b \left (-\sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}\right )+\sqrt {2} \sqrt {a} \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}-a x\right )}{\sqrt {a}} \]
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Rubi [A] time = 0.64, antiderivative size = 46, normalized size of antiderivative = 0.52, number of steps used = 2, number of rules used = 2, integrand size = 59, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {2130, 215} \begin {gather*} \frac {\sqrt {2} b \sinh ^{-1}\left (\frac {b \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 215
Rule 2130
Rubi steps
\begin {align*} \int \frac {\sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx &=\frac {\left (\sqrt {2} b\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{a}}} \, dx,x,a x+b \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )}{a}\\ &=\frac {\sqrt {2} b \sinh ^{-1}\left (\frac {a x+b \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a}}\right )}{\sqrt {a}}\\ \end {align*}
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Mathematica [A] time = 1.08, size = 148, normalized size = 1.68 \begin {gather*} \frac {\sqrt {2} x \sqrt {a x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )} \left (b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x^2-1\right ) \tanh ^{-1}\left (\frac {\sqrt {a x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )}}{\sqrt {2} a x}\right )}{\sqrt {\frac {a \left (a x^2-1\right )}{b^2}} \left (x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 3.47, size = 88, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {2} b \log \left (-a x-b \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}+\sqrt {2} \sqrt {a} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 10.21, size = 161, normalized size = 1.83 \begin {gather*} \left [\frac {\sqrt {2} b \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 (\sqrt {2} \sqrt {a} x + \frac {\sqrt {2} b \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}}{\sqrt {a}}\right )} + 1\right )}{2 \, \sqrt {a}}, -\sqrt {2} b \sqrt {-\frac {1}{a}} \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} \sqrt {-\frac {1}{a}}}{2 \, x}\right )\right ] \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 {a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} b x}}{\sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} x}\,{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 \[\int \frac {\sqrt {a \,x^{2}+b x \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}}{x \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}\, dx\]
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 {a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} b x}}{\sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} x}\,{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 {a\,x^2+b\,x\,\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}}}{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 {x \left (a x + b \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}\right )}}{x \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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