Optimal. Leaf size=150 \[ \frac{4 \sqrt{c+d x^2} \text{EllipticF}\left (\tan ^{-1}\left (\frac{x}{2}\right ),1-\frac{4 d}{c}\right )}{c \sqrt{x^2+4} \sqrt{\frac{c+d x^2}{c \left (x^2+4\right )}}}+\frac{x \sqrt{c+d x^2}}{d \sqrt{x^2+4}}-\frac{\sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{x}{2}\right )|1-\frac{4 d}{c}\right )}{d \sqrt{x^2+4} \sqrt{\frac{c+d x^2}{c \left (x^2+4\right )}}} \]
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Rubi [A] time = 0.0591805, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {422, 418, 492, 411} \[ \frac{x \sqrt{c+d x^2}}{d \sqrt{x^2+4}}+\frac{4 \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{x}{2}\right )|1-\frac{4 d}{c}\right )}{c \sqrt{x^2+4} \sqrt{\frac{c+d x^2}{c \left (x^2+4\right )}}}-\frac{\sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{x}{2}\right )|1-\frac{4 d}{c}\right )}{d \sqrt{x^2+4} \sqrt{\frac{c+d x^2}{c \left (x^2+4\right )}}} \]
Antiderivative was successfully verified.
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Rule 422
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{\sqrt{4+x^2}}{\sqrt{c+d x^2}} \, dx &=4 \int \frac{1}{\sqrt{4+x^2} \sqrt{c+d x^2}} \, dx+\int \frac{x^2}{\sqrt{4+x^2} \sqrt{c+d x^2}} \, dx\\ &=\frac{x \sqrt{c+d x^2}}{d \sqrt{4+x^2}}+\frac{4 \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{x}{2}\right )|1-\frac{4 d}{c}\right )}{c \sqrt{4+x^2} \sqrt{\frac{c+d x^2}{c \left (4+x^2\right )}}}-\frac{4 \int \frac{\sqrt{c+d x^2}}{\left (4+x^2\right )^{3/2}} \, dx}{d}\\ &=\frac{x \sqrt{c+d x^2}}{d \sqrt{4+x^2}}-\frac{\sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{x}{2}\right )|1-\frac{4 d}{c}\right )}{d \sqrt{4+x^2} \sqrt{\frac{c+d x^2}{c \left (4+x^2\right )}}}+\frac{4 \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{x}{2}\right )|1-\frac{4 d}{c}\right )}{c \sqrt{4+x^2} \sqrt{\frac{c+d x^2}{c \left (4+x^2\right )}}}\\ \end{align*}
Mathematica [A] time = 0.0311412, size = 60, normalized size = 0.4 \[ \frac{2 \sqrt{\frac{c+d x^2}{c}} E\left (\sin ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|\frac{c}{4 d}\right )}{\sqrt{-\frac{d}{c}} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 53, normalized size = 0.4 \begin{align*} 2\,{\frac{1}{\sqrt{d{x}^{2}+c}}{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},1/2\,\sqrt{{\frac{c}{d}}} \right ) \sqrt{{\frac{d{x}^{2}+c}{c}}}{\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \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{x^{2} + 4}}{\sqrt{d x^{2} + c}}\,{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{\sqrt{x^{2} + 4}}{\sqrt{d x^{2} + c}}, 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{x^{2} + 4}}{\sqrt{c + 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{x^{2} + 4}}{\sqrt{d x^{2} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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