Optimal. Leaf size=43 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2}}-\frac {x}{b \sqrt {a+b x^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {288, 217, 206} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2}}-\frac {x}{b \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 288
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a+b x^2\right )^{3/2}} \, dx &=-\frac {x}{b \sqrt {a+b x^2}}+\frac {\int \frac {1}{\sqrt {a+b x^2}} \, dx}{b}\\ &=-\frac {x}{b \sqrt {a+b x^2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{b}\\ &=-\frac {x}{b \sqrt {a+b x^2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 59, normalized size = 1.37 \[ \frac {\sqrt {a} \sqrt {\frac {b x^2}{a}+1} \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )-\sqrt {b} x}{b^{3/2} \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.24, size = 130, normalized size = 3.02 \[ \left [-\frac {2 \, \sqrt {b x^{2} + a} b x - {\left (b x^{2} + a\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right )}{2 \, {\left (b^{3} x^{2} + a b^{2}\right )}}, -\frac {\sqrt {b x^{2} + a} b x + {\left (b x^{2} + a\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right )}{b^{3} x^{2} + a b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.23, size = 39, normalized size = 0.91 \[ -\frac {x}{\sqrt {b x^{2} + a} b} - \frac {\log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{b^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 37, normalized size = 0.86 \[ -\frac {x}{\sqrt {b \,x^{2}+a}\, b}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 29, normalized size = 0.67 \[ -\frac {x}{\sqrt {b x^{2} + a} b} + \frac {\operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 36, normalized size = 0.84 \[ \frac {\ln \left (\sqrt {b}\,x+\sqrt {b\,x^2+a}\right )}{b^{3/2}}-\frac {x}{b\,\sqrt {b\,x^2+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.71, size = 37, normalized size = 0.86 \[ \frac {\operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} - \frac {x}{\sqrt {a} b \sqrt {1 + \frac {b x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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