Optimal. Leaf size=241 \[ -\frac{\sqrt{2} \sqrt{b x^2+2} \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{3}}\right ),1-\frac{3 b}{2 d}\right )}{d^{3/2} \sqrt{d x^2+3} \sqrt{\frac{b x^2+2}{d x^2+3}}}+\frac{2 \sqrt{2} (3 b-d) \sqrt{b x^2+2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{3}}\right )|1-\frac{3 b}{2 d}\right )}{3 b d^{3/2} \sqrt{d x^2+3} \sqrt{\frac{b x^2+2}{d x^2+3}}}+\frac{x \sqrt{b x^2+2} \sqrt{d x^2+3}}{3 d}-\frac{2 x (3 b-d) \sqrt{b x^2+2}}{3 b d \sqrt{d x^2+3}} \]
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Rubi [A] time = 0.149874, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {478, 531, 418, 492, 411} \[ -\frac{\sqrt{2} \sqrt{b x^2+2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{3}}\right )|1-\frac{3 b}{2 d}\right )}{d^{3/2} \sqrt{d x^2+3} \sqrt{\frac{b x^2+2}{d x^2+3}}}+\frac{2 \sqrt{2} (3 b-d) \sqrt{b x^2+2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{3}}\right )|1-\frac{3 b}{2 d}\right )}{3 b d^{3/2} \sqrt{d x^2+3} \sqrt{\frac{b x^2+2}{d x^2+3}}}+\frac{x \sqrt{b x^2+2} \sqrt{d x^2+3}}{3 d}-\frac{2 x (3 b-d) \sqrt{b x^2+2}}{3 b d \sqrt{d x^2+3}} \]
Antiderivative was successfully verified.
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Rule 478
Rule 531
Rule 418
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} \sqrt{3+d x^2}}{3 d}-\frac{\int \frac{6+2 (3 b-d) x^2}{\sqrt{2+b x^2} \sqrt{3+d x^2}} \, dx}{3 d}\\ &=\frac{x \sqrt{2+b x^2} \sqrt{3+d x^2}}{3 d}-\frac{2 \int \frac{1}{\sqrt{2+b x^2} \sqrt{3+d x^2}} \, dx}{d}-\frac{(2 (3 b-d)) \int \frac{x^2}{\sqrt{2+b x^2} \sqrt{3+d x^2}} \, dx}{3 d}\\ &=-\frac{2 (3 b-d) x \sqrt{2+b x^2}}{3 b d \sqrt{3+d x^2}}+\frac{x \sqrt{2+b x^2} \sqrt{3+d x^2}}{3 d}-\frac{\sqrt{2} \sqrt{2+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{3}}\right )|1-\frac{3 b}{2 d}\right )}{d^{3/2} \sqrt{\frac{2+b x^2}{3+d x^2}} \sqrt{3+d x^2}}+\frac{(2 (3 b-d)) \int \frac{\sqrt{2+b x^2}}{\left (3+d x^2\right )^{3/2}} \, dx}{b d}\\ &=-\frac{2 (3 b-d) x \sqrt{2+b x^2}}{3 b d \sqrt{3+d x^2}}+\frac{x \sqrt{2+b x^2} \sqrt{3+d x^2}}{3 d}+\frac{2 \sqrt{2} (3 b-d) \sqrt{2+b x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{3}}\right )|1-\frac{3 b}{2 d}\right )}{3 b d^{3/2} \sqrt{\frac{2+b x^2}{3+d x^2}} \sqrt{3+d x^2}}-\frac{\sqrt{2} \sqrt{2+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{3}}\right )|1-\frac{3 b}{2 d}\right )}{d^{3/2} \sqrt{\frac{2+b x^2}{3+d x^2}} \sqrt{3+d x^2}}\\ \end{align*}
Mathematica [C] time = 0.112274, size = 127, normalized size = 0.53 \[ \frac{-2 i \sqrt{3} (3 b-2 d) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2}}\right ),\frac{2 d}{3 b}\right )+\sqrt{b} d x \sqrt{b x^2+2} \sqrt{d x^2+3}+2 i \sqrt{3} (3 b-d) E\left (i \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2}}\right )|\frac{2 d}{3 b}\right )}{3 \sqrt{b} d^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 306, normalized size = 1.3 \begin{align*}{\frac{1}{ \left ( 3\,bd{x}^{4}+9\,b{x}^{2}+6\,d{x}^{2}+18 \right ) db}\sqrt{b{x}^{2}+2}\sqrt{d{x}^{2}+3} \left ({x}^{5}{b}^{2}d\sqrt{-d}+3\,{x}^{3}{b}^{2}\sqrt{-d}+2\,{x}^{3}bd\sqrt{-d}+3\,\sqrt{2}{\it EllipticF} \left ( 1/3\,x\sqrt{3}\sqrt{-d},1/2\,\sqrt{2}\sqrt{3}\sqrt{{\frac{b}{d}}} \right ) b\sqrt{b{x}^{2}+2}\sqrt{d{x}^{2}+3}-2\,\sqrt{2}{\it EllipticF} \left ( 1/3\,x\sqrt{3}\sqrt{-d},1/2\,\sqrt{2}\sqrt{3}\sqrt{{\frac{b}{d}}} \right ) d\sqrt{b{x}^{2}+2}\sqrt{d{x}^{2}+3}-6\,\sqrt{2}{\it EllipticE} \left ( 1/3\,x\sqrt{3}\sqrt{-d},1/2\,\sqrt{2}\sqrt{3}\sqrt{{\frac{b}{d}}} \right ) b\sqrt{b{x}^{2}+2}\sqrt{d{x}^{2}+3}+2\,\sqrt{2}{\it EllipticE} \left ( 1/3\,x\sqrt{3}\sqrt{-d},1/2\,\sqrt{2}\sqrt{3}\sqrt{{\frac{b}{d}}} \right ) d\sqrt{b{x}^{2}+2}\sqrt{d{x}^{2}+3}+6\,xb\sqrt{-d} \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{\sqrt{b x^{2} + 2} x^{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} x^{2}}{\sqrt{d x^{2} + 3}}, 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{\sqrt{b x^{2} + 2} x^{2}}{\sqrt{d x^{2} + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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