Optimal. Leaf size=179 \[ \frac {1}{63} x \left (35 x^2+108\right ) \left (x^4+3 x^2+2\right )^{3/2}+\frac {1}{105} x \left (149 x^2+519\right ) \sqrt {x^4+3 x^2+2}+\frac {116 x \left (x^2+2\right )}{15 \sqrt {x^4+3 x^2+2}}+\frac {197 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{35 \sqrt {x^4+3 x^2+2}}-\frac {116 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{15 \sqrt {x^4+3 x^2+2}} \]
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Rubi [A] time = 0.06, antiderivative size = 179, normalized size of antiderivative = 1.00, 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, 1189, 1099, 1135} \[ \frac {1}{63} x \left (35 x^2+108\right ) \left (x^4+3 x^2+2\right )^{3/2}+\frac {1}{105} x \left (149 x^2+519\right ) \sqrt {x^4+3 x^2+2}+\frac {116 x \left (x^2+2\right )}{15 \sqrt {x^4+3 x^2+2}}+\frac {197 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{35 \sqrt {x^4+3 x^2+2}}-\frac {116 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{15 \sqrt {x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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Rule 1099
Rule 1135
Rule 1176
Rule 1189
Rubi steps
\begin {align*} \int \left (7+5 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2} \, dx &=\frac {1}{63} x \left (108+35 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}+\frac {1}{21} \int \left (222+149 x^2\right ) \sqrt {2+3 x^2+x^4} \, dx\\ &=\frac {1}{105} x \left (519+149 x^2\right ) \sqrt {2+3 x^2+x^4}+\frac {1}{63} x \left (108+35 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}+\frac {1}{315} \int \frac {3546+2436 x^2}{\sqrt {2+3 x^2+x^4}} \, dx\\ &=\frac {1}{105} x \left (519+149 x^2\right ) \sqrt {2+3 x^2+x^4}+\frac {1}{63} x \left (108+35 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}+\frac {116}{15} \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {394}{35} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx\\ &=\frac {116 x \left (2+x^2\right )}{15 \sqrt {2+3 x^2+x^4}}+\frac {1}{105} x \left (519+149 x^2\right ) \sqrt {2+3 x^2+x^4}+\frac {1}{63} x \left (108+35 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}-\frac {116 \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 {197 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{35 \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 = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (5 \, x^{6} + 22 \, x^{4} + 31 \, x^{2} + 14\right )} \sqrt {x^{4} + 3 \, x^{2} + 2}, x\right ) \]
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 \[ \int {\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.00, size = 172, normalized size = 0.96 \[ \frac {5 \sqrt {x^{4}+3 x^{2}+2}\, x^{7}}{9}+\frac {71 \sqrt {x^{4}+3 x^{2}+2}\, x^{5}}{21}+\frac {2417 \sqrt {x^{4}+3 x^{2}+2}\, x^{3}}{315}+\frac {293 \sqrt {x^{4}+3 x^{2}+2}\, x}{35}-\frac {197 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{35 \sqrt {x^{4}+3 x^{2}+2}}+\frac {58 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (-\EllipticE \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )+\EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{15 \sqrt {x^{4}+3 x^{2}+2}} \]
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 \[ \int {\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}\,{d x} \]
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 \[ \int \left (5\,x^2+7\right )\,{\left (x^4+3\,x^2+2\right )}^{3/2} \,d x \]
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 \[ \int \left (\left (x^{2} + 1\right ) \left (x^{2} + 2\right )\right )^{\frac {3}{2}} \left (5 x^{2} + 7\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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