Optimal. Leaf size=198 \[ \frac {25}{11} x \left (x^4+3 x^2+2\right )^{5/2}+\frac {1}{693} x \left (2240 x^2+7281\right ) \left (x^4+3 x^2+2\right )^{3/2}+\frac {x \left (10643 x^2+36783\right ) \sqrt {x^4+3 x^2+2}}{1155}+\frac {742 x \left (x^2+2\right )}{15 \sqrt {x^4+3 x^2+2}}+\frac {13879 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{385 \sqrt {x^4+3 x^2+2}}-\frac {742 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{15 \sqrt {x^4+3 x^2+2}} \]
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Rubi [A] time = 0.09, antiderivative size = 198, normalized size of antiderivative = 1.00, 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, 1189, 1099, 1135} \[ \frac {25}{11} x \left (x^4+3 x^2+2\right )^{5/2}+\frac {1}{693} x \left (2240 x^2+7281\right ) \left (x^4+3 x^2+2\right )^{3/2}+\frac {x \left (10643 x^2+36783\right ) \sqrt {x^4+3 x^2+2}}{1155}+\frac {742 x \left (x^2+2\right )}{15 \sqrt {x^4+3 x^2+2}}+\frac {13879 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{385 \sqrt {x^4+3 x^2+2}}-\frac {742 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{15 \sqrt {x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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Rule 1099
Rule 1135
Rule 1176
Rule 1189
Rule 1206
Rubi steps
\begin {align*} \int \left (7+5 x^2\right )^2 \left (2+3 x^2+x^4\right )^{3/2} \, dx &=\frac {25}{11} x \left (2+3 x^2+x^4\right )^{5/2}+\frac {1}{11} \int \left (489+320 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2} \, dx\\ &=\frac {1}{693} x \left (7281+2240 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}+\frac {25}{11} x \left (2+3 x^2+x^4\right )^{5/2}+\frac {1}{231} \int \left (15684+10643 x^2\right ) \sqrt {2+3 x^2+x^4} \, dx\\ &=\frac {x \left (36783+10643 x^2\right ) \sqrt {2+3 x^2+x^4}}{1155}+\frac {1}{693} x \left (7281+2240 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}+\frac {25}{11} x \left (2+3 x^2+x^4\right )^{5/2}+\frac {\int \frac {249822+171402 x^2}{\sqrt {2+3 x^2+x^4}} \, dx}{3465}\\ &=\frac {x \left (36783+10643 x^2\right ) \sqrt {2+3 x^2+x^4}}{1155}+\frac {1}{693} x \left (7281+2240 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}+\frac {25}{11} x \left (2+3 x^2+x^4\right )^{5/2}+\frac {742}{15} \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {27758}{385} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx\\ &=\frac {742 x \left (2+x^2\right )}{15 \sqrt {2+3 x^2+x^4}}+\frac {x \left (36783+10643 x^2\right ) \sqrt {2+3 x^2+x^4}}{1155}+\frac {1}{693} x \left (7281+2240 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}+\frac {25}{11} x \left (2+3 x^2+x^4\right )^{5/2}-\frac {742 \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 {13879 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{385 \sqrt {2+3 x^2+x^4}}\\ \end {align*}
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Mathematica [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]
Verification is Not applicable to the result.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (25 \, x^{8} + 145 \, x^{6} + 309 \, x^{4} + 287 \, x^{2} + 98\right )} \sqrt {x^{4} + 3 \, x^{2} + 2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 189, normalized size = 0.95 \[ \frac {25 \sqrt {x^{4}+3 x^{2}+2}\, x^{9}}{11}+\frac {1670 \sqrt {x^{4}+3 x^{2}+2}\, x^{7}}{99}+\frac {11492 \sqrt {x^{4}+3 x^{2}+2}\, x^{5}}{231}+\frac {258044 \sqrt {x^{4}+3 x^{2}+2}\, x^{3}}{3465}+\frac {23851 \sqrt {x^{4}+3 x^{2}+2}\, x}{385}-\frac {13879 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{385 \sqrt {x^{4}+3 x^{2}+2}}+\frac {371 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (-\EllipticE \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )+\EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{15 \sqrt {x^{4}+3 x^{2}+2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (5\,x^2+7\right )}^2\,{\left (x^4+3\,x^2+2\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\left (x^{2} + 1\right ) \left (x^{2} + 2\right )\right )^{\frac {3}{2}} \left (5 x^{2} + 7\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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