Optimal. Leaf size=219 \[ \frac{1171349 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} \text{EllipticF}\left (\tan ^{-1}(x),\frac{1}{2}\right )}{5005 \sqrt{x^4+3 x^2+2}}+\frac{125}{13} \left (x^4+3 x^2+2\right )^{5/2} x^3+\frac{3825}{143} \left (x^4+3 x^2+2\right )^{5/2} x+\frac{\left (65345 x^2+208212\right ) \left (x^4+3 x^2+2\right )^{3/2} x}{3003}+\frac{\left (297911 x^2+1032541\right ) \sqrt{x^4+3 x^2+2} x}{5005}+\frac{20884 \left (x^2+2\right ) x}{65 \sqrt{x^4+3 x^2+2}}-\frac{20884 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{65 \sqrt{x^4+3 x^2+2}} \]
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Rubi [A] time = 0.12395, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1206, 1679, 1176, 1189, 1099, 1135} \[ \frac{125}{13} \left (x^4+3 x^2+2\right )^{5/2} x^3+\frac{3825}{143} \left (x^4+3 x^2+2\right )^{5/2} x+\frac{\left (65345 x^2+208212\right ) \left (x^4+3 x^2+2\right )^{3/2} x}{3003}+\frac{\left (297911 x^2+1032541\right ) \sqrt{x^4+3 x^2+2} x}{5005}+\frac{20884 \left (x^2+2\right ) x}{65 \sqrt{x^4+3 x^2+2}}+\frac{1171349 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{5005 \sqrt{x^4+3 x^2+2}}-\frac{20884 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{65 \sqrt{x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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Rule 1206
Rule 1679
Rule 1176
Rule 1189
Rule 1099
Rule 1135
Rubi steps
\begin{align*} \int \left (7+5 x^2\right )^3 \left (2+3 x^2+x^4\right )^{3/2} \, dx &=\frac{125}{13} x^3 \left (2+3 x^2+x^4\right )^{5/2}+\frac{1}{13} \int \left (2+3 x^2+x^4\right )^{3/2} \left (4459+8805 x^2+3825 x^4\right ) \, dx\\ &=\frac{3825}{143} x \left (2+3 x^2+x^4\right )^{5/2}+\frac{125}{13} x^3 \left (2+3 x^2+x^4\right )^{5/2}+\frac{1}{143} \int \left (41399+28005 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2} \, dx\\ &=\frac{x \left (208212+65345 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}}{3003}+\frac{3825}{143} x \left (2+3 x^2+x^4\right )^{5/2}+\frac{125}{13} x^3 \left (2+3 x^2+x^4\right )^{5/2}+\frac{\int \left (1322334+893733 x^2\right ) \sqrt{2+3 x^2+x^4} \, dx}{3003}\\ &=\frac{x \left (1032541+297911 x^2\right ) \sqrt{2+3 x^2+x^4}}{5005}+\frac{x \left (208212+65345 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}}{3003}+\frac{3825}{143} x \left (2+3 x^2+x^4\right )^{5/2}+\frac{125}{13} x^3 \left (2+3 x^2+x^4\right )^{5/2}+\frac{\int \frac{21084282+14472612 x^2}{\sqrt{2+3 x^2+x^4}} \, dx}{45045}\\ &=\frac{x \left (1032541+297911 x^2\right ) \sqrt{2+3 x^2+x^4}}{5005}+\frac{x \left (208212+65345 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}}{3003}+\frac{3825}{143} x \left (2+3 x^2+x^4\right )^{5/2}+\frac{125}{13} x^3 \left (2+3 x^2+x^4\right )^{5/2}+\frac{20884}{65} \int \frac{x^2}{\sqrt{2+3 x^2+x^4}} \, dx+\frac{2342698 \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx}{5005}\\ &=\frac{20884 x \left (2+x^2\right )}{65 \sqrt{2+3 x^2+x^4}}+\frac{x \left (1032541+297911 x^2\right ) \sqrt{2+3 x^2+x^4}}{5005}+\frac{x \left (208212+65345 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}}{3003}+\frac{3825}{143} x \left (2+3 x^2+x^4\right )^{5/2}+\frac{125}{13} x^3 \left (2+3 x^2+x^4\right )^{5/2}-\frac{20884 \sqrt{2} \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{65 \sqrt{2+3 x^2+x^4}}+\frac{1171349 \sqrt{2} \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{5005 \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 = 0.017, size = 206, normalized size = 0.9 \begin{align*}{\frac{10067363\,{x}^{3}}{15015}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{2262081\,x}{5005}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{131810\,{x}^{7}}{429}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{598324\,{x}^{5}}{1001}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{12075\,{x}^{9}}{143}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{125\,{x}^{11}}{13}\sqrt{{x}^{4}+3\,{x}^{2}+2}}-{{\frac{1171349\,i}{5005}}\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{10442\,i}{65}}\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 )}^{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 ({\left (125 \, x^{10} + 900 \, x^{8} + 2560 \, x^{6} + 3598 \, x^{4} + 2499 \, x^{2} + 686\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 )^{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{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac{3}{2}}{\left (5 \, x^{2} + 7\right )}^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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