Optimal. Leaf size=88 \[ \frac{x \sqrt{c+d x^2}}{d \sqrt{x^2+4}}-\frac{\sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{x}{2}\right )|1-\frac{4 d}{c}\right )}{d \sqrt{x^2+4} \sqrt{\frac{c+d x^2}{c \left (x^2+4\right )}}} \]
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Rubi [A] time = 0.0361522, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {492, 411} \[ \frac{x \sqrt{c+d x^2}}{d \sqrt{x^2+4}}-\frac{\sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{x}{2}\right )|1-\frac{4 d}{c}\right )}{d \sqrt{x^2+4} \sqrt{\frac{c+d x^2}{c \left (x^2+4\right )}}} \]
Antiderivative was successfully verified.
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Rule 492
Rule 411
Rubi steps
\begin{align*} \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 \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 )}}}\\ \end{align*}
Mathematica [C] time = 0.0489789, size = 70, normalized size = 0.8 \[ -\frac{i c \sqrt{\frac{d x^2}{c}+1} \left (E\left (i \sinh ^{-1}\left (\frac{x}{2}\right )|\frac{4 d}{c}\right )-\text{EllipticF}\left (i \sinh ^{-1}\left (\frac{x}{2}\right ),\frac{4 d}{c}\right )\right )}{d \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 76, normalized size = 0.9 \begin{align*} -2\,{\frac{1}{\sqrt{d{x}^{2}+c}}\sqrt{{\frac{d{x}^{2}+c}{c}}} \left ({\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},1/2\,\sqrt{{\frac{c}{d}}} \right ) -{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},1/2\,\sqrt{{\frac{c}{d}}} \right ) \right ){\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{x^{2}}{\sqrt{d x^{2} + c} \sqrt{x^{2} + 4}}\,{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{d x^{2} + c} \sqrt{x^{2} + 4} x^{2}}{d x^{4} +{\left (c + 4 \, d\right )} x^{2} + 4 \, 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{x^{2}}{\sqrt{c + d x^{2}} \sqrt{x^{2} + 4}}\, 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{x^{2}}{\sqrt{d x^{2} + c} \sqrt{x^{2} + 4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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