Optimal. Leaf size=179 \[ \frac{197 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} \text{EllipticF}\left (\tan ^{-1}(x),\frac{1}{2}\right )}{35 \sqrt{x^4+3 x^2+2}}+\frac{1}{63} x \left (35 x^2+108\right ) \left (x^4+3 x^2+2\right )^{3/2}+\frac{1}{105} x \left (149 x^2+519\right ) \sqrt{x^4+3 x^2+2}+\frac{116 x \left (x^2+2\right )}{15 \sqrt{x^4+3 x^2+2}}-\frac{116 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{15 \sqrt{x^4+3 x^2+2}} \]
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Rubi [A] time = 0.0639407, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {1176, 1189, 1099, 1135} \[ \frac{1}{63} x \left (35 x^2+108\right ) \left (x^4+3 x^2+2\right )^{3/2}+\frac{1}{105} x \left (149 x^2+519\right ) \sqrt{x^4+3 x^2+2}+\frac{116 x \left (x^2+2\right )}{15 \sqrt{x^4+3 x^2+2}}+\frac{197 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{35 \sqrt{x^4+3 x^2+2}}-\frac{116 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{15 \sqrt{x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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Rule 1176
Rule 1189
Rule 1099
Rule 1135
Rubi steps
\begin{align*} \int \left (7+5 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2} \, dx &=\frac{1}{63} x \left (108+35 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}+\frac{1}{21} \int \left (222+149 x^2\right ) \sqrt{2+3 x^2+x^4} \, dx\\ &=\frac{1}{105} x \left (519+149 x^2\right ) \sqrt{2+3 x^2+x^4}+\frac{1}{63} x \left (108+35 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}+\frac{1}{315} \int \frac{3546+2436 x^2}{\sqrt{2+3 x^2+x^4}} \, dx\\ &=\frac{1}{105} x \left (519+149 x^2\right ) \sqrt{2+3 x^2+x^4}+\frac{1}{63} x \left (108+35 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}+\frac{116}{15} \int \frac{x^2}{\sqrt{2+3 x^2+x^4}} \, dx+\frac{394}{35} \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx\\ &=\frac{116 x \left (2+x^2\right )}{15 \sqrt{2+3 x^2+x^4}}+\frac{1}{105} x \left (519+149 x^2\right ) \sqrt{2+3 x^2+x^4}+\frac{1}{63} x \left (108+35 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}-\frac{116 \sqrt{2} \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{15 \sqrt{2+3 x^2+x^4}}+\frac{197 \sqrt{2} \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{35 \sqrt{2+3 x^2+x^4}}\\ \end{align*}
Mathematica [F] time = 0, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
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Maple [C] time = 0.005, size = 172, normalized size = 1. \begin{align*}{\frac{5\,{x}^{7}}{9}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{71\,{x}^{5}}{21}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{2417\,{x}^{3}}{315}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{293\,x}{35}\sqrt{{x}^{4}+3\,{x}^{2}+2}}-{{\frac{197\,i}{35}}\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{58\,i}{15}}\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}}}} \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{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac{3}{2}}{\left (5 \, x^{2} + 7\right )}\,{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 ({\left (5 \, x^{6} + 22 \, x^{4} + 31 \, x^{2} + 14\right )} \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 \left (\left (x^{2} + 1\right ) \left (x^{2} + 2\right )\right )^{\frac{3}{2}} \left (5 x^{2} + 7\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{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac{3}{2}}{\left (5 \, x^{2} + 7\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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