Optimal. Leaf size=226 \[ \frac{4628 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right ),\frac{1}{8}\right )}{33 \sqrt{x^4+3 x^2+4}}+\frac{25}{11} x \left (x^4+3 x^2+4\right )^{5/2}+\frac{1}{693} x \left (2240 x^2+6831\right ) \left (x^4+3 x^2+4\right )^{3/2}+\frac{x \left (18253 x^2+64533\right ) \sqrt{x^4+3 x^2+4}}{1155}+\frac{175346 x \sqrt{x^4+3 x^2+4}}{1155 \left (x^2+2\right )}-\frac{175346 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{1155 \sqrt{x^4+3 x^2+4}} \]
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Rubi [A] time = 0.0979694, antiderivative size = 226, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {1206, 1176, 1197, 1103, 1195} \[ \frac{25}{11} x \left (x^4+3 x^2+4\right )^{5/2}+\frac{1}{693} x \left (2240 x^2+6831\right ) \left (x^4+3 x^2+4\right )^{3/2}+\frac{x \left (18253 x^2+64533\right ) \sqrt{x^4+3 x^2+4}}{1155}+\frac{175346 x \sqrt{x^4+3 x^2+4}}{1155 \left (x^2+2\right )}+\frac{4628 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{33 \sqrt{x^4+3 x^2+4}}-\frac{175346 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{1155 \sqrt{x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
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Rule 1206
Rule 1176
Rule 1197
Rule 1103
Rule 1195
Rubi steps
\begin{align*} \int \left (7+5 x^2\right )^2 \left (4+3 x^2+x^4\right )^{3/2} \, dx &=\frac{25}{11} x \left (4+3 x^2+x^4\right )^{5/2}+\frac{1}{11} \int \left (439+320 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2} \, dx\\ &=\frac{1}{693} x \left (6831+2240 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2}+\frac{25}{11} x \left (4+3 x^2+x^4\right )^{5/2}+\frac{1}{231} \int \left (27768+18253 x^2\right ) \sqrt{4+3 x^2+x^4} \, dx\\ &=\frac{x \left (64533+18253 x^2\right ) \sqrt{4+3 x^2+x^4}}{1155}+\frac{1}{693} x \left (6831+2240 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2}+\frac{25}{11} x \left (4+3 x^2+x^4\right )^{5/2}+\frac{\int \frac{891684+526038 x^2}{\sqrt{4+3 x^2+x^4}} \, dx}{3465}\\ &=\frac{x \left (64533+18253 x^2\right ) \sqrt{4+3 x^2+x^4}}{1155}+\frac{1}{693} x \left (6831+2240 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2}+\frac{25}{11} x \left (4+3 x^2+x^4\right )^{5/2}-\frac{350692 \int \frac{1-\frac{x^2}{2}}{\sqrt{4+3 x^2+x^4}} \, dx}{1155}+\frac{18512}{33} \int \frac{1}{\sqrt{4+3 x^2+x^4}} \, dx\\ &=\frac{175346 x \sqrt{4+3 x^2+x^4}}{1155 \left (2+x^2\right )}+\frac{x \left (64533+18253 x^2\right ) \sqrt{4+3 x^2+x^4}}{1155}+\frac{1}{693} x \left (6831+2240 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2}+\frac{25}{11} x \left (4+3 x^2+x^4\right )^{5/2}-\frac{175346 \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 )}{1155 \sqrt{4+3 x^2+x^4}}+\frac{4628 \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 )}{33 \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 = 292, normalized size = 1.3 \begin{align*}{\frac{25\,{x}^{9}}{11}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{1670\,{x}^{7}}{99}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{1222\,{x}^{5}}{21}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{391024\,{x}^{3}}{3465}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{50691\,x}{385}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{396304}{385\,\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{5611072}{1155\,\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 )}^{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 ({\left (25 \, x^{8} + 145 \, x^{6} + 359 \, x^{4} + 427 \, x^{2} + 196\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 )^{2}\, 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 )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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