Optimal. Leaf size=207 \[ \frac {1}{63} x \left (35 x^2+108\right ) \left (x^4+3 x^2+4\right )^{3/2}+\frac {1}{105} x \left (289 x^2+1029\right ) \sqrt {x^4+3 x^2+4}+\frac {2798 x \sqrt {x^4+3 x^2+4}}{105 \left (x^2+2\right )}+\frac {74 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{3 \sqrt {x^4+3 x^2+4}}-\frac {2798 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{105 \sqrt {x^4+3 x^2+4}} \]
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Rubi [A] time = 0.07, antiderivative size = 207, 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, 1197, 1103, 1195} \[ \frac {1}{63} x \left (35 x^2+108\right ) \left (x^4+3 x^2+4\right )^{3/2}+\frac {1}{105} x \left (289 x^2+1029\right ) \sqrt {x^4+3 x^2+4}+\frac {2798 x \sqrt {x^4+3 x^2+4}}{105 \left (x^2+2\right )}+\frac {74 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{3 \sqrt {x^4+3 x^2+4}}-\frac {2798 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{105 \sqrt {x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
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Rule 1103
Rule 1176
Rule 1195
Rule 1197
Rubi steps
\begin {align*} \int \left (7+5 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2} \, dx &=\frac {1}{63} x \left (108+35 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2}+\frac {1}{21} \int \left (444+289 x^2\right ) \sqrt {4+3 x^2+x^4} \, dx\\ &=\frac {1}{105} x \left (1029+289 x^2\right ) \sqrt {4+3 x^2+x^4}+\frac {1}{63} x \left (108+35 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2}+\frac {1}{315} \int \frac {14292+8394 x^2}{\sqrt {4+3 x^2+x^4}} \, dx\\ &=\frac {1}{105} x \left (1029+289 x^2\right ) \sqrt {4+3 x^2+x^4}+\frac {1}{63} x \left (108+35 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2}-\frac {5596}{105} \int \frac {1-\frac {x^2}{2}}{\sqrt {4+3 x^2+x^4}} \, dx+\frac {296}{3} \int \frac {1}{\sqrt {4+3 x^2+x^4}} \, dx\\ &=\frac {2798 x \sqrt {4+3 x^2+x^4}}{105 \left (2+x^2\right )}+\frac {1}{105} x \left (1029+289 x^2\right ) \sqrt {4+3 x^2+x^4}+\frac {1}{63} x \left (108+35 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2}-\frac {2798 \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 )}{105 \sqrt {4+3 x^2+x^4}}+\frac {74 \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 )}{3 \sqrt {4+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 = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (5 \, x^{6} + 22 \, x^{4} + 41 \, x^{2} + 28\right )} \sqrt {x^{4} + 3 \, x^{2} + 4}, x\right ) \]
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 \[ \int {\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 275, normalized size = 1.33 \[ \frac {5 \sqrt {x^{4}+3 x^{2}+4}\, x^{7}}{9}+\frac {71 \sqrt {x^{4}+3 x^{2}+4}\, x^{5}}{21}+\frac {3187 \sqrt {x^{4}+3 x^{2}+4}\, x^{3}}{315}+\frac {583 \sqrt {x^{4}+3 x^{2}+4}\, x}{35}+\frac {6352 \sqrt {-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{35 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {89536 \sqrt {-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}+1}\, \left (-\EllipticE \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )+\EllipticF \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )\right )}{105 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (i \sqrt {7}+3\right )} \]
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 \[ \int {\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (5\,x^2+7\right )\,{\left (x^4+3\,x^2+4\right )}^{3/2} \,d x \]
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 \[ \int \left (\left (x^{2} - x + 2\right ) \left (x^{2} + x + 2\right )\right )^{\frac {3}{2}} \left (5 x^{2} + 7\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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