Optimal. Leaf size=226 \[ \frac {25}{11} x \left (x^4+3 x^2+4\right )^{5/2}+\frac {1}{693} x \left (2240 x^2+6831\right ) \left (x^4+3 x^2+4\right )^{3/2}+\frac {x \left (18253 x^2+64533\right ) \sqrt {x^4+3 x^2+4}}{1155}+\frac {175346 x \sqrt {x^4+3 x^2+4}}{1155 \left (x^2+2\right )}+\frac {4628 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{33 \sqrt {x^4+3 x^2+4}}-\frac {175346 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{1155 \sqrt {x^4+3 x^2+4}} \]
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Rubi [A] time = 0.10, antiderivative size = 226, 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, 1197, 1103, 1195} \[ \frac {25}{11} x \left (x^4+3 x^2+4\right )^{5/2}+\frac {1}{693} x \left (2240 x^2+6831\right ) \left (x^4+3 x^2+4\right )^{3/2}+\frac {x \left (18253 x^2+64533\right ) \sqrt {x^4+3 x^2+4}}{1155}+\frac {175346 x \sqrt {x^4+3 x^2+4}}{1155 \left (x^2+2\right )}+\frac {4628 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{33 \sqrt {x^4+3 x^2+4}}-\frac {175346 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{1155 \sqrt {x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
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Rule 1103
Rule 1176
Rule 1195
Rule 1197
Rule 1206
Rubi steps
\begin {align*} \int \left (7+5 x^2\right )^2 \left (4+3 x^2+x^4\right )^{3/2} \, dx &=\frac {25}{11} x \left (4+3 x^2+x^4\right )^{5/2}+\frac {1}{11} \int \left (439+320 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2} \, dx\\ &=\frac {1}{693} x \left (6831+2240 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2}+\frac {25}{11} x \left (4+3 x^2+x^4\right )^{5/2}+\frac {1}{231} \int \left (27768+18253 x^2\right ) \sqrt {4+3 x^2+x^4} \, dx\\ &=\frac {x \left (64533+18253 x^2\right ) \sqrt {4+3 x^2+x^4}}{1155}+\frac {1}{693} x \left (6831+2240 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2}+\frac {25}{11} x \left (4+3 x^2+x^4\right )^{5/2}+\frac {\int \frac {891684+526038 x^2}{\sqrt {4+3 x^2+x^4}} \, dx}{3465}\\ &=\frac {x \left (64533+18253 x^2\right ) \sqrt {4+3 x^2+x^4}}{1155}+\frac {1}{693} x \left (6831+2240 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2}+\frac {25}{11} x \left (4+3 x^2+x^4\right )^{5/2}-\frac {350692 \int \frac {1-\frac {x^2}{2}}{\sqrt {4+3 x^2+x^4}} \, dx}{1155}+\frac {18512}{33} \int \frac {1}{\sqrt {4+3 x^2+x^4}} \, dx\\ &=\frac {175346 x \sqrt {4+3 x^2+x^4}}{1155 \left (2+x^2\right )}+\frac {x \left (64533+18253 x^2\right ) \sqrt {4+3 x^2+x^4}}{1155}+\frac {1}{693} x \left (6831+2240 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2}+\frac {25}{11} x \left (4+3 x^2+x^4\right )^{5/2}-\frac {175346 \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 )}{1155 \sqrt {4+3 x^2+x^4}}+\frac {4628 \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 )}{33 \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 = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (25 \, x^{8} + 145 \, x^{6} + 359 \, x^{4} + 427 \, x^{2} + 196\right )} \sqrt {x^{4} + 3 \, x^{2} + 4}, x\right ) \]
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 \[ \int {\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 292, normalized size = 1.29 \[ \frac {25 \sqrt {x^{4}+3 x^{2}+4}\, x^{9}}{11}+\frac {1670 \sqrt {x^{4}+3 x^{2}+4}\, x^{7}}{99}+\frac {1222 \sqrt {x^{4}+3 x^{2}+4}\, x^{5}}{21}+\frac {391024 \sqrt {x^{4}+3 x^{2}+4}\, x^{3}}{3465}+\frac {50691 \sqrt {x^{4}+3 x^{2}+4}\, x}{385}+\frac {396304 \sqrt {-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{385 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {5611072 \sqrt {-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}+1}\, \left (-\EllipticE \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )+\EllipticF \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )\right )}{1155 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (i \sqrt {7}+3\right )} \]
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 \[ \int {\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (5\,x^2+7\right )}^2\,{\left (x^4+3\,x^2+4\right )}^{3/2} \,d x \]
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 \[ \int \left (\left (x^{2} - x + 2\right ) \left (x^{2} + x + 2\right )\right )^{\frac {3}{2}} \left (5 x^{2} + 7\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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