Optimal. Leaf size=232 \[ -\frac{b \sqrt{c} \sqrt{a+\frac{b}{x^2}} \text{EllipticF}\left (\cot ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right ),1-\frac{b c}{a d}\right )}{a \sqrt{d} \sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left (a+\frac{b}{x^2}\right )}{a \left (c+\frac{d}{x^2}\right )}}}-\frac{d \sqrt{a+\frac{b}{x^2}}}{c x \sqrt{c+\frac{d}{x^2}}}+\frac{x \sqrt{a+\frac{b}{x^2}} \sqrt{c+\frac{d}{x^2}}}{c}+\frac{\sqrt{d} \sqrt{a+\frac{b}{x^2}} E\left (\cot ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{c} \sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left (a+\frac{b}{x^2}\right )}{a \left (c+\frac{d}{x^2}\right )}}} \]
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Rubi [A] time = 0.204726, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {375, 475, 21, 422, 418, 492, 411} \[ -\frac{d \sqrt{a+\frac{b}{x^2}}}{c x \sqrt{c+\frac{d}{x^2}}}+\frac{x \sqrt{a+\frac{b}{x^2}} \sqrt{c+\frac{d}{x^2}}}{c}-\frac{b \sqrt{c} \sqrt{a+\frac{b}{x^2}} F\left (\cot ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )|1-\frac{b c}{a d}\right )}{a \sqrt{d} \sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left (a+\frac{b}{x^2}\right )}{a \left (c+\frac{d}{x^2}\right )}}}+\frac{\sqrt{d} \sqrt{a+\frac{b}{x^2}} E\left (\cot ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{c} \sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left (a+\frac{b}{x^2}\right )}{a \left (c+\frac{d}{x^2}\right )}}} \]
Antiderivative was successfully verified.
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Rule 375
Rule 475
Rule 21
Rule 422
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{c+\frac{d}{x^2}}} \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt{a+b x^2}}{x^2 \sqrt{c+d x^2}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{\sqrt{a+\frac{b}{x^2}} \sqrt{c+\frac{d}{x^2}} x}{c}-\frac{\operatorname{Subst}\left (\int \frac{b c+b d x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=\frac{\sqrt{a+\frac{b}{x^2}} \sqrt{c+\frac{d}{x^2}} x}{c}-\frac{b \operatorname{Subst}\left (\int \frac{\sqrt{c+d x^2}}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=\frac{\sqrt{a+\frac{b}{x^2}} \sqrt{c+\frac{d}{x^2}} x}{c}-b \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx,x,\frac{1}{x}\right )-\frac{(b d) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=-\frac{d \sqrt{a+\frac{b}{x^2}}}{c \sqrt{c+\frac{d}{x^2}} x}+\frac{\sqrt{a+\frac{b}{x^2}} \sqrt{c+\frac{d}{x^2}} x}{c}-\frac{b \sqrt{c} \sqrt{a+\frac{b}{x^2}} F\left (\cot ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )|1-\frac{b c}{a d}\right )}{a \sqrt{d} \sqrt{\frac{c \left (a+\frac{b}{x^2}\right )}{a \left (c+\frac{d}{x^2}\right )}} \sqrt{c+\frac{d}{x^2}}}+d \operatorname{Subst}\left (\int \frac{\sqrt{a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{d \sqrt{a+\frac{b}{x^2}}}{c \sqrt{c+\frac{d}{x^2}} x}+\frac{\sqrt{a+\frac{b}{x^2}} \sqrt{c+\frac{d}{x^2}} x}{c}+\frac{\sqrt{d} \sqrt{a+\frac{b}{x^2}} E\left (\cot ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{c} \sqrt{\frac{c \left (a+\frac{b}{x^2}\right )}{a \left (c+\frac{d}{x^2}\right )}} \sqrt{c+\frac{d}{x^2}}}-\frac{b \sqrt{c} \sqrt{a+\frac{b}{x^2}} F\left (\cot ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )|1-\frac{b c}{a d}\right )}{a \sqrt{d} \sqrt{\frac{c \left (a+\frac{b}{x^2}\right )}{a \left (c+\frac{d}{x^2}\right )}} \sqrt{c+\frac{d}{x^2}}}\\ \end{align*}
Mathematica [A] time = 0.0609942, size = 86, normalized size = 0.37 \[ \frac{\sqrt{a+\frac{b}{x^2}} \sqrt{\frac{c x^2+d}{d}} E\left (\sin ^{-1}\left (\sqrt{-\frac{c}{d}} x\right )|\frac{a d}{b c}\right )}{\sqrt{-\frac{c}{d}} \sqrt{\frac{a x^2+b}{b}} \sqrt{c+\frac{d}{x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 94, normalized size = 0.4 \begin{align*}{\frac{b}{a{x}^{2}+b}\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}}{\it EllipticE} \left ( x\sqrt{-{\frac{c}{d}}},\sqrt{{\frac{ad}{bc}}} \right ) \sqrt{{\frac{a{x}^{2}+b}{b}}}\sqrt{{\frac{c{x}^{2}+d}{d}}}{\frac{1}{\sqrt{-{\frac{c}{d}}}}}{\frac{1}{\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + \frac{b}{x^{2}}}}{\sqrt{c + \frac{d}{x^{2}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{c x^{2} + d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + \frac{b}{x^{2}}}}{\sqrt{c + \frac{d}{x^{2}}}}\, dx \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{a + \frac{b}{x^{2}}}}{\sqrt{c + \frac{d}{x^{2}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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