Optimal. Leaf size=49 \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} \sqrt{d e-c f}} \]
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Rubi [A] time = 0.0200121, 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, 205} \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} \sqrt{d e-c f}} \]
Antiderivative was successfully verified.
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Rule 377
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\left (c+d x^2\right ) \sqrt{e+f x^2}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{c-(-d e+c f) x^2} \, dx,x,\frac{x}{\sqrt{e+f x^2}}\right )\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} \sqrt{d e-c f}}\\ \end{align*}
Mathematica [A] time = 0.0107023, size = 49, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} \sqrt{d e-c f}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 306, normalized size = 6.2 \begin{align*} -{\frac{1}{2}\ln \left ({ \left ( -2\,{\frac{cf-de}{d}}+2\,{\frac{f\sqrt{-cd}}{d} \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) }+2\,\sqrt{-{\frac{cf-de}{d}}}\sqrt{ \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) ^{2}f+2\,{\frac{f\sqrt{-cd}}{d} \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) }-{\frac{cf-de}{d}}} \right ) \left ( x-{\frac{1}{d}\sqrt{-cd}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-cd}}}{\frac{1}{\sqrt{-{\frac{cf-de}{d}}}}}}+{\frac{1}{2}\ln \left ({ \left ( -2\,{\frac{cf-de}{d}}-2\,{\frac{f\sqrt{-cd}}{d} \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) }+2\,\sqrt{-{\frac{cf-de}{d}}}\sqrt{ \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) ^{2}f-2\,{\frac{f\sqrt{-cd}}{d} \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) }-{\frac{cf-de}{d}}} \right ) \left ( x+{\frac{1}{d}\sqrt{-cd}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-cd}}}{\frac{1}{\sqrt{-{\frac{cf-de}{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 = 1.70522, size = 513, normalized size = 10.47 \begin{align*} \left [-\frac{\sqrt{-c d e + c^{2} f} \log \left (\frac{{\left (d^{2} e^{2} - 8 \, c d e f + 8 \, c^{2} f^{2}\right )} x^{4} + c^{2} e^{2} - 2 \,{\left (3 \, c d e^{2} - 4 \, c^{2} e f\right )} x^{2} - 4 \,{\left ({\left (d e - 2 \, c f\right )} x^{3} - c e x\right )} \sqrt{-c d e + c^{2} f} \sqrt{f x^{2} + e}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{4 \,{\left (c d e - c^{2} f\right )}}, \frac{\arctan \left (\frac{\sqrt{c d e - c^{2} f}{\left ({\left (d e - 2 \, c f\right )} x^{2} - c e\right )} \sqrt{f x^{2} + e}}{2 \,{\left ({\left (c d e f - c^{2} f^{2}\right )} x^{3} +{\left (c d e^{2} - c^{2} e f\right )} x\right )}}\right )}{2 \, \sqrt{c d e - c^{2} f}}\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}{\left (c + d x^{2}\right ) \sqrt{e + f x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.54156, size = 100, normalized size = 2.04 \begin{align*} -\frac{\sqrt{f} \arctan \left (\frac{{\left (\sqrt{f} x - \sqrt{f x^{2} + e}\right )}^{2} d + 2 \, c f - d e}{2 \, \sqrt{-c^{2} f^{2} + c d f e}}\right )}{\sqrt{-c^{2} f^{2} + c d f e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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