Optimal. Leaf size=82 \[ \frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{d}-\frac{\sqrt{b c-a d} \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{\sqrt{c} d} \]
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Rubi [A] time = 0.0542935, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {402, 217, 206, 377, 208} \[ \frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{d}-\frac{\sqrt{b c-a d} \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{\sqrt{c} d} \]
Antiderivative was successfully verified.
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Rule 402
Rule 217
Rule 206
Rule 377
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x^2}}{c+d x^2} \, dx &=\frac{b \int \frac{1}{\sqrt{a+b x^2}} \, dx}{d}-\frac{(b c-a d) \int \frac{1}{\sqrt{a+b x^2} \left (c+d x^2\right )} \, dx}{d}\\ &=\frac{b \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{d}-\frac{(b c-a d) \operatorname{Subst}\left (\int \frac{1}{c-(b c-a d) x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{d}\\ &=\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{d}-\frac{\sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{c} \sqrt{a+b x^2}}\right )}{\sqrt{c} d}\\ \end{align*}
Mathematica [A] time = 0.0446294, size = 84, normalized size = 1.02 \[ \frac{\sqrt{a d-b c} \tan ^{-1}\left (\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{\sqrt{c} d}+\frac{\sqrt{b} \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.033, size = 932, normalized size = 11.4 \begin{align*} \text{result too large to display} \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 = 1.91823, size = 1288, normalized size = 15.71 \begin{align*} \left [\frac{2 \, \sqrt{b} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + \sqrt{\frac{b c - a d}{c}} \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 (a c^{2} x +{\left (2 \, b c^{2} - a c d\right )} x^{3}\right )} \sqrt{b x^{2} + a} \sqrt{\frac{b c - a d}{c}}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{4 \, d}, -\frac{4 \, \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) - \sqrt{\frac{b c - a d}{c}} \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 (a c^{2} x +{\left (2 \, b c^{2} - a c d\right )} x^{3}\right )} \sqrt{b x^{2} + a} \sqrt{\frac{b c - a d}{c}}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{4 \, d}, \frac{\sqrt{-\frac{b c - a d}{c}} \arctan \left (\frac{{\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt{b x^{2} + a} \sqrt{-\frac{b c - a d}{c}}}{2 \,{\left ({\left (b^{2} c - a b d\right )} x^{3} +{\left (a b c - a^{2} d\right )} x\right )}}\right ) + \sqrt{b} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right )}{2 \, d}, -\frac{2 \, \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) - \sqrt{-\frac{b c - a d}{c}} \arctan \left (\frac{{\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt{b x^{2} + a} \sqrt{-\frac{b c - a d}{c}}}{2 \,{\left ({\left (b^{2} c - a b d\right )} x^{3} +{\left (a b c - a^{2} d\right )} x\right )}}\right )}{2 \, d}\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 + b x^{2}}}{c + d x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1362, size = 150, normalized size = 1.83 \begin{align*} \frac{{\left (b^{\frac{3}{2}} c - a \sqrt{b} d\right )} \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} d} - \frac{\sqrt{b} \log \left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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