Optimal. Leaf size=146 \[ \frac{3 \sqrt{d} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt{b} \sqrt{e}}\right )}{2 b^{5/2} e^{3/2}}-\frac{3 (b c-a d)}{2 b^2 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{c+d x^2}{2 b e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}} \]
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Rubi [A] time = 0.0983798, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1960, 290, 325, 208} \[ \frac{3 \sqrt{d} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt{b} \sqrt{e}}\right )}{2 b^{5/2} e^{3/2}}-\frac{3 (b c-a d)}{2 b^2 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{c+d x^2}{2 b e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}} \]
Antiderivative was successfully verified.
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Rule 1960
Rule 290
Rule 325
Rule 208
Rubi steps
\begin{align*} \int \frac{x}{\left (\frac{e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx &=((b c-a d) e) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (b e-d x^2\right )^2} \, dx,x,\sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}\right )\\ &=\frac{c+d x^2}{2 b e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{(3 (b c-a d)) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (b e-d x^2\right )} \, dx,x,\sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}\right )}{2 b}\\ &=-\frac{3 (b c-a d)}{2 b^2 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{c+d x^2}{2 b e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{(3 d (b c-a d)) \operatorname{Subst}\left (\int \frac{1}{b e-d x^2} \, dx,x,\sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}\right )}{2 b^2 e}\\ &=-\frac{3 (b c-a d)}{2 b^2 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{c+d x^2}{2 b e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{3 \sqrt{d} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt{b} \sqrt{e}}\right )}{2 b^{5/2} e^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0702331, size = 86, normalized size = 0.59 \[ -\frac{\left (a+b x^2\right ) \, _2F_1\left (-\frac{3}{2},-\frac{1}{2};\frac{1}{2};\frac{d \left (b x^2+a\right )}{a d-b c}\right )}{b \left (\frac{b \left (c+d x^2\right )}{b c-a d}\right )^{3/2} \left (\frac{e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 432, normalized size = 3. \begin{align*}{\frac{b{x}^{2}+a}{4\,{b}^{2} \left ( d{x}^{2}+c \right ) } \left ( -3\,\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}ab{d}^{2}+3\,\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{b}^{2}cd+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}\sqrt{bd}{x}^{2}bd-3\,{d}^{2}\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}+3\,\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) acbd+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}\sqrt{bd}ad+4\,\sqrt{bd}\sqrt{ \left ( d{x}^{2}+c \right ) \left ( b{x}^{2}+a \right ) }ad-4\,\sqrt{bd}\sqrt{ \left ( d{x}^{2}+c \right ) \left ( b{x}^{2}+a \right ) }bc \right ){\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ \left ( d{x}^{2}+c \right ) \left ( b{x}^{2}+a \right ) }}} \left ({\frac{e \left ( b{x}^{2}+a \right ) }{d{x}^{2}+c}} \right ) ^{-{\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 = 3.93855, size = 914, normalized size = 6.26 \begin{align*} \left [-\frac{3 \,{\left ({\left (b^{2} c - a b d\right )} e x^{2} +{\left (a b c - a^{2} d\right )} e\right )} \sqrt{\frac{d}{b e}} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x^{2} - 4 \,{\left (2 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + a b c d +{\left (3 \, b^{2} c d + a b d^{2}\right )} x^{2}\right )} \sqrt{\frac{b e x^{2} + a e}{d x^{2} + c}} \sqrt{\frac{d}{b e}}\right ) - 4 \,{\left (b d^{2} x^{4} - 2 \, b c^{2} + 3 \, a c d -{\left (b c d - 3 \, a d^{2}\right )} x^{2}\right )} \sqrt{\frac{b e x^{2} + a e}{d x^{2} + c}}}{8 \,{\left (b^{3} e^{2} x^{2} + a b^{2} e^{2}\right )}}, -\frac{3 \,{\left ({\left (b^{2} c - a b d\right )} e x^{2} +{\left (a b c - a^{2} d\right )} e\right )} \sqrt{-\frac{d}{b e}} \arctan \left (\frac{{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt{\frac{b e x^{2} + a e}{d x^{2} + c}} \sqrt{-\frac{d}{b e}}}{2 \,{\left (b d x^{2} + a d\right )}}\right ) - 2 \,{\left (b d^{2} x^{4} - 2 \, b c^{2} + 3 \, a c d -{\left (b c d - 3 \, a d^{2}\right )} x^{2}\right )} \sqrt{\frac{b e x^{2} + a e}{d x^{2} + c}}}{4 \,{\left (b^{3} e^{2} x^{2} + a b^{2} e^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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