Optimal. Leaf size=75 \[ -\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{c^{3/2}}+\frac{(b c-a d)^2}{c d^2 \sqrt{c+d x^2}}+\frac{b^2 \sqrt{c+d x^2}}{d^2} \]
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Rubi [A] time = 0.0750599, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {446, 87, 63, 208} \[ -\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{c^{3/2}}+\frac{(b c-a d)^2}{c d^2 \sqrt{c+d x^2}}+\frac{b^2 \sqrt{c+d x^2}}{d^2} \]
Antiderivative was successfully verified.
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Rule 446
Rule 87
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2}{x \left (c+d x^2\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^2}{x (c+d x)^{3/2}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{(b c-a d)^2}{c d (c+d x)^{3/2}}+\frac{b^2}{d \sqrt{c+d x}}+\frac{a^2}{c x \sqrt{c+d x}}\right ) \, dx,x,x^2\right )\\ &=\frac{(b c-a d)^2}{c d^2 \sqrt{c+d x^2}}+\frac{b^2 \sqrt{c+d x^2}}{d^2}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+d x}} \, dx,x,x^2\right )}{2 c}\\ &=\frac{(b c-a d)^2}{c d^2 \sqrt{c+d x^2}}+\frac{b^2 \sqrt{c+d x^2}}{d^2}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x^2}\right )}{c d}\\ &=\frac{(b c-a d)^2}{c d^2 \sqrt{c+d x^2}}+\frac{b^2 \sqrt{c+d x^2}}{d^2}-\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{c^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.040323, size = 62, normalized size = 0.83 \[ \frac{a^2 d^2 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{d x^2}{c}+1\right )+b c \left (-2 a d+2 b c+b d x^2\right )}{c d^2 \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 102, normalized size = 1.4 \begin{align*}{\frac{{b}^{2}{x}^{2}}{d}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+2\,{\frac{{b}^{2}c}{{d}^{2}\sqrt{d{x}^{2}+c}}}-2\,{\frac{ab}{d\sqrt{d{x}^{2}+c}}}+{\frac{{a}^{2}}{c}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-{{a}^{2}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.34391, size = 489, normalized size = 6.52 \begin{align*} \left [\frac{{\left (a^{2} d^{3} x^{2} + a^{2} c d^{2}\right )} \sqrt{c} \log \left (-\frac{d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{c} + 2 \, c}{x^{2}}\right ) + 2 \,{\left (b^{2} c^{2} d x^{2} + 2 \, b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt{d x^{2} + c}}{2 \,{\left (c^{2} d^{3} x^{2} + c^{3} d^{2}\right )}}, \frac{{\left (a^{2} d^{3} x^{2} + a^{2} c d^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right ) +{\left (b^{2} c^{2} d x^{2} + 2 \, b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt{d x^{2} + c}}{c^{2} d^{3} x^{2} + c^{3} d^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 17.9242, size = 70, normalized size = 0.93 \begin{align*} \frac{a^{2} \operatorname{atan}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{- c}} \right )}}{c \sqrt{- c}} + \frac{b^{2} \sqrt{c + d x^{2}}}{d^{2}} + \frac{\left (a d - b c\right )^{2}}{c d^{2} \sqrt{c + d x^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13746, size = 111, normalized size = 1.48 \begin{align*} \frac{a^{2} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c} + \frac{\sqrt{d x^{2} + c} b^{2}}{d^{2}} + \frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{\sqrt{d x^{2} + c} c d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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