Optimal. Leaf size=110 \[ \frac{x \sqrt{b x^2+2}}{b \sqrt{d x^2+3}}-\frac{\sqrt{2} \sqrt{b x^2+2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{3}}\right )|1-\frac{3 b}{2 d}\right )}{b \sqrt{d} \sqrt{d x^2+3} \sqrt{\frac{b x^2+2}{d x^2+3}}} \]
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Rubi [A] time = 0.0451851, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {492, 411} \[ \frac{x \sqrt{b x^2+2}}{b \sqrt{d x^2+3}}-\frac{\sqrt{2} \sqrt{b x^2+2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{3}}\right )|1-\frac{3 b}{2 d}\right )}{b \sqrt{d} \sqrt{d x^2+3} \sqrt{\frac{b x^2+2}{d x^2+3}}} \]
Antiderivative was successfully verified.
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Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt{2+b x^2} \sqrt{3+d x^2}} \, dx &=\frac{x \sqrt{2+b x^2}}{b \sqrt{3+d x^2}}-\frac{3 \int \frac{\sqrt{2+b x^2}}{\left (3+d x^2\right )^{3/2}} \, dx}{b}\\ &=\frac{x \sqrt{2+b x^2}}{b \sqrt{3+d x^2}}-\frac{\sqrt{2} \sqrt{2+b x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{3}}\right )|1-\frac{3 b}{2 d}\right )}{b \sqrt{d} \sqrt{\frac{2+b x^2}{3+d x^2}} \sqrt{3+d x^2}}\\ \end{align*}
Mathematica [C] time = 0.0477002, size = 72, normalized size = 0.65 \[ -\frac{i \sqrt{3} \left (E\left (i \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2}}\right )|\frac{2 d}{3 b}\right )-\text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2}}\right ),\frac{2 d}{3 b}\right )\right )}{\sqrt{b} d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 70, normalized size = 0.6 \begin{align*}{\frac{\sqrt{2}}{b} \left ( -{\it EllipticF} \left ({\frac{x\sqrt{3}}{3}\sqrt{-d}},{\frac{\sqrt{2}\sqrt{3}}{2}\sqrt{{\frac{b}{d}}}} \right ) +{\it EllipticE} \left ({\frac{x\sqrt{3}}{3}\sqrt{-d}},{\frac{\sqrt{2}\sqrt{3}}{2}\sqrt{{\frac{b}{d}}}} \right ) \right ){\frac{1}{\sqrt{-d}}}} \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{b x^{2} + 2} \sqrt{d x^{2} + 3}}\,{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{b x^{2} + 2} \sqrt{d x^{2} + 3} x^{2}}{b d x^{4} +{\left (3 \, b + 2 \, d\right )} x^{2} + 6}, 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{b x^{2} + 2} \sqrt{d x^{2} + 3}}\, 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{b x^{2} + 2} \sqrt{d x^{2} + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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