Optimal. Leaf size=49 \[ \frac{\tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{\sqrt{c} \sqrt{b c-a d}} \]
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Rubi [A] time = 0.0215087, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {377, 208} \[ \frac{\tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{\sqrt{c} \sqrt{b c-a d}} \]
Antiderivative was successfully verified.
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Rule 377
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+b x^2} \left (c+d x^2\right )} \, dx &=\operatorname{Subst}\left (\int \frac{1}{c-(b c-a d) x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{c} \sqrt{a+b x^2}}\right )}{\sqrt{c} \sqrt{b c-a d}}\\ \end{align*}
Mathematica [A] time = 0.0183035, size = 49, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{\sqrt{c} \sqrt{b c-a d}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 300, normalized size = 6.1 \begin{align*} -{\frac{1}{2}\ln \left ({ \left ( 2\,{\frac{ad-bc}{d}}+2\,{\frac{b\sqrt{-cd}}{d} \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) }+2\,\sqrt{{\frac{ad-bc}{d}}}\sqrt{ \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) ^{2}b+2\,{\frac{b\sqrt{-cd}}{d} \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) }+{\frac{ad-bc}{d}}} \right ) \left ( x-{\frac{1}{d}\sqrt{-cd}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-cd}}}{\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}}+{\frac{1}{2}\ln \left ({ \left ( 2\,{\frac{ad-bc}{d}}-2\,{\frac{b\sqrt{-cd}}{d} \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) }+2\,\sqrt{{\frac{ad-bc}{d}}}\sqrt{ \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) ^{2}b-2\,{\frac{b\sqrt{-cd}}{d} \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) }+{\frac{ad-bc}{d}}} \right ) \left ( x+{\frac{1}{d}\sqrt{-cd}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-cd}}}{\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}} \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 [B] time = 1.9915, size = 513, normalized size = 10.47 \begin{align*} \left [\frac{\log \left (\frac{{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \,{\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} + 4 \,{\left ({\left (2 \, b c - a d\right )} x^{3} + a c x\right )} \sqrt{b c^{2} - a c d} \sqrt{b x^{2} + a}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{4 \, \sqrt{b c^{2} - a c d}}, -\frac{\sqrt{-b c^{2} + a c d} \arctan \left (\frac{\sqrt{-b c^{2} + a c d}{\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt{b x^{2} + a}}{2 \,{\left ({\left (b^{2} c^{2} - a b c d\right )} x^{3} +{\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right )}{2 \,{\left (b c^{2} - a c d\right )}}\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{1}{\sqrt{a + b x^{2}} \left (c + d x^{2}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14349, size = 95, normalized size = 1.94 \begin{align*} -\frac{\sqrt{b} \arctan \left (\frac{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt{-b^{2} c^{2} + a b c d}}\right )}{\sqrt{-b^{2} c^{2} + a b c d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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