Optimal. Leaf size=207 \[ \frac{74 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right ),\frac{1}{8}\right )}{3 \sqrt{x^4+3 x^2+4}}+\frac{1}{63} x \left (35 x^2+108\right ) \left (x^4+3 x^2+4\right )^{3/2}+\frac{1}{105} x \left (289 x^2+1029\right ) \sqrt{x^4+3 x^2+4}+\frac{2798 x \sqrt{x^4+3 x^2+4}}{105 \left (x^2+2\right )}-\frac{2798 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{105 \sqrt{x^4+3 x^2+4}} \]
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Rubi [A] time = 0.0712722, antiderivative size = 207, 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, 1197, 1103, 1195} \[ \frac{1}{63} x \left (35 x^2+108\right ) \left (x^4+3 x^2+4\right )^{3/2}+\frac{1}{105} x \left (289 x^2+1029\right ) \sqrt{x^4+3 x^2+4}+\frac{2798 x \sqrt{x^4+3 x^2+4}}{105 \left (x^2+2\right )}+\frac{74 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{3 \sqrt{x^4+3 x^2+4}}-\frac{2798 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{105 \sqrt{x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
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Rule 1176
Rule 1197
Rule 1103
Rule 1195
Rubi steps
\begin{align*} \int \left (7+5 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2} \, dx &=\frac{1}{63} x \left (108+35 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2}+\frac{1}{21} \int \left (444+289 x^2\right ) \sqrt{4+3 x^2+x^4} \, dx\\ &=\frac{1}{105} x \left (1029+289 x^2\right ) \sqrt{4+3 x^2+x^4}+\frac{1}{63} x \left (108+35 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2}+\frac{1}{315} \int \frac{14292+8394 x^2}{\sqrt{4+3 x^2+x^4}} \, dx\\ &=\frac{1}{105} x \left (1029+289 x^2\right ) \sqrt{4+3 x^2+x^4}+\frac{1}{63} x \left (108+35 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2}-\frac{5596}{105} \int \frac{1-\frac{x^2}{2}}{\sqrt{4+3 x^2+x^4}} \, dx+\frac{296}{3} \int \frac{1}{\sqrt{4+3 x^2+x^4}} \, dx\\ &=\frac{2798 x \sqrt{4+3 x^2+x^4}}{105 \left (2+x^2\right )}+\frac{1}{105} x \left (1029+289 x^2\right ) \sqrt{4+3 x^2+x^4}+\frac{1}{63} x \left (108+35 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2}-\frac{2798 \sqrt{2} \left (2+x^2\right ) \sqrt{\frac{4+3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{105 \sqrt{4+3 x^2+x^4}}+\frac{74 \sqrt{2} \left (2+x^2\right ) \sqrt{\frac{4+3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{3 \sqrt{4+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 = 0.007, size = 275, normalized size = 1.3 \begin{align*}{\frac{5\,{x}^{7}}{9}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{71\,{x}^{5}}{21}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{3187\,{x}^{3}}{315}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{583\,x}{35}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{6352}{35\,\sqrt{-6+2\,i\sqrt{7}}}\sqrt{1- \left ( -{\frac{3}{8}}+{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{3}{8}}-{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}-{\frac{89536}{105\,\sqrt{-6+2\,i\sqrt{7}} \left ( i\sqrt{7}+3 \right ) }\sqrt{1- \left ( -{\frac{3}{8}}+{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{3}{8}}-{\frac{i}{8}}\sqrt{7} \right ){x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ) \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}} \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} + 4\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} + 41 \, x^{2} + 28\right )} \sqrt{x^{4} + 3 \, x^{2} + 4}, 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} - x + 2\right ) \left (x^{2} + x + 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} + 4\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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