Optimal. Leaf size=103 \[ -\frac {1}{x \sqrt {\sqrt {a^2 x^2+b^2}+a x}}+\frac {a \tan ^{-1}\left (\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {b}}\right )}{b^{3/2}}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {b}}\right )}{b^{3/2}} \]
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Rubi [A] time = 0.21, antiderivative size = 130, normalized size of antiderivative = 1.26, number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2119, 457, 329, 212, 206, 203} \begin {gather*} \frac {2 a \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{b^2-\left (\sqrt {a^2 x^2+b^2}+a x\right )^2}+\frac {a \tan ^{-1}\left (\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {b}}\right )}{b^{3/2}}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {b}}\right )}{b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 329
Rule 457
Rule 2119
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx &=(2 a) \operatorname {Subst}\left (\int \frac {b^2+x^2}{\sqrt {x} \left (-b^2+x^2\right )^2} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )\\ &=\frac {2 a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-\left (a x+\sqrt {b^2+a^2 x^2}\right )^2}-a \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (-b^2+x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )\\ &=\frac {2 a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-\left (a x+\sqrt {b^2+a^2 x^2}\right )^2}-(2 a) \operatorname {Subst}\left (\int \frac {1}{-b^2+x^4} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )\\ &=\frac {2 a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-\left (a x+\sqrt {b^2+a^2 x^2}\right )^2}+\frac {a \operatorname {Subst}\left (\int \frac {1}{b-x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{b}+\frac {a \operatorname {Subst}\left (\int \frac {1}{b+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{b}\\ &=\frac {2 a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-\left (a x+\sqrt {b^2+a^2 x^2}\right )^2}+\frac {a \tan ^{-1}\left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {b}}\right )}{b^{3/2}}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {b}}\right )}{b^{3/2}}\\ \end {align*}
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Mathematica [C] time = 23.02, size = 9150, normalized size = 88.83 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.18, size = 103, normalized size = 1.00 \begin {gather*} -\frac {1}{x \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {a \tan ^{-1}\left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {b}}\right )}{b^{3/2}}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {b}}\right )}{b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 335, normalized size = 3.25 \begin {gather*} \left [\frac {2 \, a \sqrt {b} x \arctan \left (\frac {\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{\sqrt {b}}\right ) + a \sqrt {b} x \log \left (\frac {b^{2} - \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} {\left ({\left (a x - b\right )} \sqrt {b} - \sqrt {a^{2} x^{2} + b^{2}} \sqrt {b}\right )} + \sqrt {a^{2} x^{2} + b^{2}} b}{x}\right ) + 2 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} {\left (a x - \sqrt {a^{2} x^{2} + b^{2}}\right )}}{2 \, b^{2} x}, -\frac {2 \, a \sqrt {-b} x \arctan \left (\frac {\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} \sqrt {-b}}{b}\right ) + a \sqrt {-b} x \log \left (-\frac {b^{2} + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} {\left ({\left (a x + b\right )} \sqrt {-b} - \sqrt {a^{2} x^{2} + b^{2}} \sqrt {-b}\right )} - \sqrt {a^{2} x^{2} + b^{2}} b}{x}\right ) - 2 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} {\left (a x - \sqrt {a^{2} x^{2} + b^{2}}\right )}}{2 \, b^{2} x}\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 {1}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{2} \sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}\, dx\]
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 {1}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} 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 {1}{x^2\,\sqrt {a\,x+\sqrt {a^2\,x^2+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 {1}{x^{2} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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