Optimal. Leaf size=170 \[ \frac {625}{3} \sqrt {x^4+3 x^2+2} x+\frac {637 \left (x^2+2\right ) x}{2 \sqrt {x^4+3 x^2+2}}+\frac {\left (113 x^2+145\right ) x}{2 \sqrt {x^4+3 x^2+2}}+\frac {1067 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{3 \sqrt {x^4+3 x^2+2}}-\frac {637 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^4+3 x^2+2}} \]
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Rubi [A] time = 0.08, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1205, 1679, 1189, 1099, 1135} \[ \frac {625}{3} \sqrt {x^4+3 x^2+2} x+\frac {637 \left (x^2+2\right ) x}{2 \sqrt {x^4+3 x^2+2}}+\frac {\left (113 x^2+145\right ) x}{2 \sqrt {x^4+3 x^2+2}}+\frac {1067 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{3 \sqrt {x^4+3 x^2+2}}-\frac {637 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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Rule 1099
Rule 1135
Rule 1189
Rule 1205
Rule 1679
Rubi steps
\begin {align*} \int \frac {\left (7+5 x^2\right )^4}{\left (2+3 x^2+x^4\right )^{3/2}} \, dx &=\frac {x \left (145+113 x^2\right )}{2 \sqrt {2+3 x^2+x^4}}-\frac {1}{2} \int \frac {-2256-3137 x^2-1250 x^4}{\sqrt {2+3 x^2+x^4}} \, dx\\ &=\frac {x \left (145+113 x^2\right )}{2 \sqrt {2+3 x^2+x^4}}+\frac {625}{3} x \sqrt {2+3 x^2+x^4}-\frac {1}{6} \int \frac {-4268-1911 x^2}{\sqrt {2+3 x^2+x^4}} \, dx\\ &=\frac {x \left (145+113 x^2\right )}{2 \sqrt {2+3 x^2+x^4}}+\frac {625}{3} x \sqrt {2+3 x^2+x^4}+\frac {637}{2} \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {2134}{3} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx\\ &=\frac {637 x \left (2+x^2\right )}{2 \sqrt {2+3 x^2+x^4}}+\frac {x \left (145+113 x^2\right )}{2 \sqrt {2+3 x^2+x^4}}+\frac {625}{3} x \sqrt {2+3 x^2+x^4}-\frac {637 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {2} \sqrt {2+3 x^2+x^4}}+\frac {1067 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{3 \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 \[ {\rm integral}\left (\frac {{\left (625 \, x^{8} + 3500 \, x^{6} + 7350 \, x^{4} + 6860 \, x^{2} + 2401\right )} \sqrt {x^{4} + 3 \, x^{2} + 2}}{x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4}, x\right ) \]
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 \[ \int \frac {{\left (5 \, x^{2} + 7\right )}^{4}}{{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 234, normalized size = 1.38 \[ \frac {625 \sqrt {x^{4}+3 x^{2}+2}\, x}{3}-\frac {1067 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{3 \sqrt {x^{4}+3 x^{2}+2}}-\frac {1250 \left (-\frac {9}{2} x^{3}-5 x \right )}{\sqrt {x^{4}+3 x^{2}+2}}+\frac {637 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (-\EllipticE \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )+\EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{4 \sqrt {x^{4}+3 x^{2}+2}}-\frac {7000 \left (\frac {5}{2} x^{3}+3 x \right )}{\sqrt {x^{4}+3 x^{2}+2}}-\frac {14700 \left (-\frac {3}{2} x^{3}-2 x \right )}{\sqrt {x^{4}+3 x^{2}+2}}-\frac {13720 \left (x^{3}+\frac {3}{2} x \right )}{\sqrt {x^{4}+3 x^{2}+2}}-\frac {4802 \left (-\frac {3}{4} x^{3}-\frac {5}{4} x \right )}{\sqrt {x^{4}+3 x^{2}+2}} \]
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 \[ \int \frac {{\left (5 \, x^{2} + 7\right )}^{4}}{{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (5\,x^2+7\right )}^4}{{\left (x^4+3\,x^2+2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (5 x^{2} + 7\right )^{4}}{\left (\left (x^{2} + 1\right ) \left (x^{2} + 2\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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