Optimal. Leaf size=127 \[ \frac{(b c-a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{2 c \left (a-\frac{c \left (a+b x^2\right )}{c+d x^2}\right )}-\frac{\sqrt{e} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt{a} \sqrt{e}}\right )}{2 \sqrt{a} c^{3/2}} \]
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Rubi [A] time = 0.086561, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1960, 288, 208} \[ \frac{(b c-a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{2 c \left (a-\frac{c \left (a+b x^2\right )}{c+d x^2}\right )}-\frac{\sqrt{e} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt{a} \sqrt{e}}\right )}{2 \sqrt{a} c^{3/2}} \]
Antiderivative was successfully verified.
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Rule 1960
Rule 288
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{x^3} \, dx &=((b c-a d) e) \operatorname{Subst}\left (\int \frac{x^2}{\left (-a e+c x^2\right )^2} \, dx,x,\sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}\right )\\ &=\frac{(b c-a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{2 c \left (a-\frac{c \left (a+b x^2\right )}{c+d x^2}\right )}+\frac{((b c-a d) e) \operatorname{Subst}\left (\int \frac{1}{-a e+c x^2} \, dx,x,\sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}\right )}{2 c}\\ &=\frac{(b c-a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{2 c \left (a-\frac{c \left (a+b x^2\right )}{c+d x^2}\right )}-\frac{(b c-a d) \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt{a} \sqrt{e}}\right )}{2 \sqrt{a} c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.100646, size = 133, normalized size = 1.05 \[ \frac{\sqrt{c+d x^2} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (-\frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{\sqrt{a} c^{3/2}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{c x^2}\right )}{2 \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.02, size = 326, normalized size = 2.6 \begin{align*} -{\frac{d{x}^{2}+c}{4\,a{c}^{2}{x}^{2}}\sqrt{{\frac{e \left ( b{x}^{2}+a \right ) }{d{x}^{2}+c}}} \left ( -2\,bd\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}{x}^{4}\sqrt{ac}-{a}^{2}\ln \left ({\frac{1}{{x}^{2}} \left ( ad{x}^{2}+bc{x}^{2}+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}+2\,ac \right ) } \right ) dc{x}^{2}+{c}^{2}\ln \left ({\frac{1}{{x}^{2}} \left ( ad{x}^{2}+bc{x}^{2}+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}+2\,ac \right ) } \right ) ba{x}^{2}-2\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}da{x}^{2}\sqrt{ac}-2\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}bc{x}^{2}\sqrt{ac}+2\, \left ( bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac \right ) ^{3/2}\sqrt{ac} \right ){\frac{1}{\sqrt{ \left ( d{x}^{2}+c \right ) \left ( b{x}^{2}+a \right ) }}}{\frac{1}{\sqrt{ac}}}} \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 = 3.48498, size = 697, normalized size = 5.49 \begin{align*} \left [-\frac{{\left (b c - a d\right )} x^{2} \sqrt{\frac{e}{a c}} \log \left (\frac{{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} e x^{4} + 8 \, a^{2} c^{2} e + 8 \,{\left (a b c^{2} + a^{2} c d\right )} e x^{2} + 4 \,{\left (2 \, a^{2} c^{3} +{\left (a b c^{2} d + a^{2} c d^{2}\right )} x^{4} +{\left (a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt{\frac{b e x^{2} + a e}{d x^{2} + c}} \sqrt{\frac{e}{a c}}}{x^{4}}\right ) + 4 \,{\left (d x^{2} + c\right )} \sqrt{\frac{b e x^{2} + a e}{d x^{2} + c}}}{8 \, c x^{2}}, \frac{{\left (b c - a d\right )} x^{2} \sqrt{-\frac{e}{a c}} \arctan \left (\frac{{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt{\frac{b e x^{2} + a e}{d x^{2} + c}} \sqrt{-\frac{e}{a c}}}{2 \,{\left (b e x^{2} + a e\right )}}\right ) - 2 \,{\left (d x^{2} + c\right )} \sqrt{\frac{b e x^{2} + a e}{d x^{2} + c}}}{4 \, c x^{2}}\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{{\left (b x^{2} + a\right )} e}{d x^{2} + c}}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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