Optimal. Leaf size=219 \[ \frac {3825}{143} \left (x^4+3 x^2+2\right )^{5/2} x+\frac {\left (65345 x^2+208212\right ) \left (x^4+3 x^2+2\right )^{3/2} x}{3003}+\frac {\left (297911 x^2+1032541\right ) \sqrt {x^4+3 x^2+2} x}{5005}+\frac {20884 \left (x^2+2\right ) x}{65 \sqrt {x^4+3 x^2+2}}+\frac {1171349 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{5005 \sqrt {x^4+3 x^2+2}}-\frac {20884 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{65 \sqrt {x^4+3 x^2+2}}+\frac {125}{13} \left (x^4+3 x^2+2\right )^{5/2} x^3 \]
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Rubi [A] time = 0.12, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1206, 1679, 1176, 1189, 1099, 1135} \[ \frac {125}{13} \left (x^4+3 x^2+2\right )^{5/2} x^3+\frac {3825}{143} \left (x^4+3 x^2+2\right )^{5/2} x+\frac {\left (65345 x^2+208212\right ) \left (x^4+3 x^2+2\right )^{3/2} x}{3003}+\frac {\left (297911 x^2+1032541\right ) \sqrt {x^4+3 x^2+2} x}{5005}+\frac {20884 \left (x^2+2\right ) x}{65 \sqrt {x^4+3 x^2+2}}+\frac {1171349 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{5005 \sqrt {x^4+3 x^2+2}}-\frac {20884 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{65 \sqrt {x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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Rule 1099
Rule 1135
Rule 1176
Rule 1189
Rule 1206
Rule 1679
Rubi steps
\begin {align*} \int \left (7+5 x^2\right )^3 \left (2+3 x^2+x^4\right )^{3/2} \, dx &=\frac {125}{13} x^3 \left (2+3 x^2+x^4\right )^{5/2}+\frac {1}{13} \int \left (2+3 x^2+x^4\right )^{3/2} \left (4459+8805 x^2+3825 x^4\right ) \, dx\\ &=\frac {3825}{143} x \left (2+3 x^2+x^4\right )^{5/2}+\frac {125}{13} x^3 \left (2+3 x^2+x^4\right )^{5/2}+\frac {1}{143} \int \left (41399+28005 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2} \, dx\\ &=\frac {x \left (208212+65345 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}}{3003}+\frac {3825}{143} x \left (2+3 x^2+x^4\right )^{5/2}+\frac {125}{13} x^3 \left (2+3 x^2+x^4\right )^{5/2}+\frac {\int \left (1322334+893733 x^2\right ) \sqrt {2+3 x^2+x^4} \, dx}{3003}\\ &=\frac {x \left (1032541+297911 x^2\right ) \sqrt {2+3 x^2+x^4}}{5005}+\frac {x \left (208212+65345 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}}{3003}+\frac {3825}{143} x \left (2+3 x^2+x^4\right )^{5/2}+\frac {125}{13} x^3 \left (2+3 x^2+x^4\right )^{5/2}+\frac {\int \frac {21084282+14472612 x^2}{\sqrt {2+3 x^2+x^4}} \, dx}{45045}\\ &=\frac {x \left (1032541+297911 x^2\right ) \sqrt {2+3 x^2+x^4}}{5005}+\frac {x \left (208212+65345 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}}{3003}+\frac {3825}{143} x \left (2+3 x^2+x^4\right )^{5/2}+\frac {125}{13} x^3 \left (2+3 x^2+x^4\right )^{5/2}+\frac {20884}{65} \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {2342698 \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx}{5005}\\ &=\frac {20884 x \left (2+x^2\right )}{65 \sqrt {2+3 x^2+x^4}}+\frac {x \left (1032541+297911 x^2\right ) \sqrt {2+3 x^2+x^4}}{5005}+\frac {x \left (208212+65345 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}}{3003}+\frac {3825}{143} x \left (2+3 x^2+x^4\right )^{5/2}+\frac {125}{13} x^3 \left (2+3 x^2+x^4\right )^{5/2}-\frac {20884 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{65 \sqrt {2+3 x^2+x^4}}+\frac {1171349 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{5005 \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 = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (125 \, x^{10} + 900 \, x^{8} + 2560 \, x^{6} + 3598 \, x^{4} + 2499 \, x^{2} + 686\right )} \sqrt {x^{4} + 3 \, x^{2} + 2}, x\right ) \]
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 \[ \int {\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 206, normalized size = 0.94 \[ \frac {125 \sqrt {x^{4}+3 x^{2}+2}\, x^{11}}{13}+\frac {12075 \sqrt {x^{4}+3 x^{2}+2}\, x^{9}}{143}+\frac {131810 \sqrt {x^{4}+3 x^{2}+2}\, x^{7}}{429}+\frac {598324 \sqrt {x^{4}+3 x^{2}+2}\, x^{5}}{1001}+\frac {10067363 \sqrt {x^{4}+3 x^{2}+2}\, x^{3}}{15015}+\frac {2262081 \sqrt {x^{4}+3 x^{2}+2}\, x}{5005}-\frac {1171349 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{5005 \sqrt {x^{4}+3 x^{2}+2}}+\frac {10442 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (-\EllipticE \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )+\EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{65 \sqrt {x^{4}+3 x^{2}+2}} \]
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 \[ \int {\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}^{3}\,{d x} \]
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 \[ \int {\left (5\,x^2+7\right )}^3\,{\left (x^4+3\,x^2+2\right )}^{3/2} \,d x \]
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 \[ \int \left (\left (x^{2} + 1\right ) \left (x^{2} + 2\right )\right )^{\frac {3}{2}} \left (5 x^{2} + 7\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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