Optimal. Leaf size=229 \[ \frac{\left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} \left (5 d^3-10 d e^2+8 e^3\right ) \text{EllipticF}\left (\tan ^{-1}(x),\frac{1}{2}\right )}{5 \sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{3 e \left (x^2+2\right ) x \left (5 d^2-10 d e+6 e^2\right )}{5 \sqrt{x^4+3 x^2+2}}-\frac{3 \sqrt{2} e \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} \left (5 d^2-10 d e+6 e^2\right ) E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{5 \sqrt{x^4+3 x^2+2}}+\frac{1}{5} e^2 \sqrt{x^4+3 x^2+2} x (5 d-4 e)+\frac{1}{5} e^3 \sqrt{x^4+3 x^2+2} x^3 \]
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Rubi [A] time = 0.153778, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {1206, 1679, 1189, 1099, 1135} \[ \frac{3 e \left (x^2+2\right ) x \left (5 d^2-10 d e+6 e^2\right )}{5 \sqrt{x^4+3 x^2+2}}+\frac{\left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} \left (5 d^3-10 d e^2+8 e^3\right ) F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{5 \sqrt{2} \sqrt{x^4+3 x^2+2}}-\frac{3 \sqrt{2} e \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} \left (5 d^2-10 d e+6 e^2\right ) E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{5 \sqrt{x^4+3 x^2+2}}+\frac{1}{5} e^2 \sqrt{x^4+3 x^2+2} x (5 d-4 e)+\frac{1}{5} e^3 \sqrt{x^4+3 x^2+2} x^3 \]
Antiderivative was successfully verified.
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Rule 1206
Rule 1679
Rule 1189
Rule 1099
Rule 1135
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right )^3}{\sqrt{2+3 x^2+x^4}} \, dx &=\frac{1}{5} e^3 x^3 \sqrt{2+3 x^2+x^4}+\frac{1}{5} \int \frac{5 d^3+3 e \left (5 d^2-2 e^2\right ) x^2+3 (5 d-4 e) e^2 x^4}{\sqrt{2+3 x^2+x^4}} \, dx\\ &=\frac{1}{5} (5 d-4 e) e^2 x \sqrt{2+3 x^2+x^4}+\frac{1}{5} e^3 x^3 \sqrt{2+3 x^2+x^4}+\frac{1}{15} \int \frac{3 \left (5 d^3-10 d e^2+8 e^3\right )+9 e \left (5 d^2-10 d e+6 e^2\right ) x^2}{\sqrt{2+3 x^2+x^4}} \, dx\\ &=\frac{1}{5} (5 d-4 e) e^2 x \sqrt{2+3 x^2+x^4}+\frac{1}{5} e^3 x^3 \sqrt{2+3 x^2+x^4}+\frac{1}{5} \left (3 e \left (5 d^2-10 d e+6 e^2\right )\right ) \int \frac{x^2}{\sqrt{2+3 x^2+x^4}} \, dx+\frac{1}{5} \left (5 d^3-10 d e^2+8 e^3\right ) \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx\\ &=\frac{3 e \left (5 d^2-10 d e+6 e^2\right ) x \left (2+x^2\right )}{5 \sqrt{2+3 x^2+x^4}}+\frac{1}{5} (5 d-4 e) e^2 x \sqrt{2+3 x^2+x^4}+\frac{1}{5} e^3 x^3 \sqrt{2+3 x^2+x^4}-\frac{3 \sqrt{2} e \left (5 d^2-10 d e+6 e^2\right ) \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{5 \sqrt{2+3 x^2+x^4}}+\frac{\left (5 d^3-10 d e^2+8 e^3\right ) \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{5 \sqrt{2} \sqrt{2+3 x^2+x^4}}\\ \end{align*}
Mathematica [C] time = 0.229417, size = 154, normalized size = 0.67 \[ \frac{-5 i \sqrt{x^2+1} \sqrt{x^2+2} \left (-3 d^2 e+d^3+4 d e^2-2 e^3\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right ),2\right )-3 i e \sqrt{x^2+1} \sqrt{x^2+2} \left (5 d^2-10 d e+6 e^2\right ) E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+e^2 x \left (x^4+3 x^2+2\right ) \left (5 d+e \left (x^2-4\right )\right )}{5 \sqrt{x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.01, size = 380, normalized size = 1.7 \begin{align*}{e}^{3} \left ({\frac{{x}^{3}}{5}\sqrt{{x}^{4}+3\,{x}^{2}+2}}-{\frac{4\,x}{5}\sqrt{{x}^{4}+3\,{x}^{2}+2}}-{{\frac{4\,i}{5}}\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+{{\frac{9\,i}{5}}\sqrt{2} \left ({\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}} \right ) +3\,d{e}^{2} \left ( 1/3\,x\sqrt{{x}^{4}+3\,{x}^{2}+2}+{\frac{i/3\sqrt{2}\sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ( i/2x\sqrt{2},\sqrt{2} \right ) }{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}-{\frac{i\sqrt{2}\sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1} \left ({\it EllipticF} \left ( i/2x\sqrt{2},\sqrt{2} \right ) -{\it EllipticE} \left ( i/2x\sqrt{2},\sqrt{2} \right ) \right ) }{\sqrt{{x}^{4}+3\,{x}^{2}+2}}} \right ) +{{\frac{3\,i}{2}}{d}^{2}e\sqrt{2} \left ({\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}-{{\frac{i}{2}}{d}^{3}\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}} \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{{\left (e x^{2} + d\right )}^{3}}{\sqrt{x^{4} + 3 \, x^{2} + 2}}\,{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{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}{\sqrt{x^{4} + 3 \, x^{2} + 2}}, 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{\left (d + e x^{2}\right )^{3}}{\sqrt{\left (x^{2} + 1\right ) \left (x^{2} + 2\right )}}\, 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{{\left (e x^{2} + d\right )}^{3}}{\sqrt{x^{4} + 3 \, x^{2} + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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