Optimal. Leaf size=247 \[ \frac {3825}{143} \left (x^4+3 x^2+4\right )^{5/2} x+\frac {\left (15365 x^2+53504\right ) \left (x^4+3 x^2+4\right )^{3/2} x}{1001}+\frac {\left (435441 x^2+1653701\right ) \sqrt {x^4+3 x^2+4} x}{5005}+\frac {4525662 \sqrt {x^4+3 x^2+4} x}{5005 \left (x^2+2\right )}+\frac {121826 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{143 \sqrt {x^4+3 x^2+4}}-\frac {4525662 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{5005 \sqrt {x^4+3 x^2+4}}+\frac {125}{13} \left (x^4+3 x^2+4\right )^{5/2} x^3 \]
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Rubi [A] time = 0.13, antiderivative size = 247, 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, 1197, 1103, 1195} \[ \frac {125}{13} \left (x^4+3 x^2+4\right )^{5/2} x^3+\frac {3825}{143} \left (x^4+3 x^2+4\right )^{5/2} x+\frac {\left (15365 x^2+53504\right ) \left (x^4+3 x^2+4\right )^{3/2} x}{1001}+\frac {\left (435441 x^2+1653701\right ) \sqrt {x^4+3 x^2+4} x}{5005}+\frac {4525662 \sqrt {x^4+3 x^2+4} x}{5005 \left (x^2+2\right )}+\frac {121826 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{143 \sqrt {x^4+3 x^2+4}}-\frac {4525662 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{5005 \sqrt {x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
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Rule 1103
Rule 1176
Rule 1195
Rule 1197
Rule 1206
Rule 1679
Rubi steps
\begin {align*} \int \left (7+5 x^2\right )^3 \left (4+3 x^2+x^4\right )^{3/2} \, dx &=\frac {125}{13} x^3 \left (4+3 x^2+x^4\right )^{5/2}+\frac {1}{13} \int \left (4+3 x^2+x^4\right )^{3/2} \left (4459+8055 x^2+3825 x^4\right ) \, dx\\ &=\frac {3825}{143} x \left (4+3 x^2+x^4\right )^{5/2}+\frac {125}{13} x^3 \left (4+3 x^2+x^4\right )^{5/2}+\frac {1}{143} \int \left (33749+19755 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2} \, dx\\ &=\frac {x \left (53504+15365 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2}}{1001}+\frac {3825}{143} x \left (4+3 x^2+x^4\right )^{5/2}+\frac {125}{13} x^3 \left (4+3 x^2+x^4\right )^{5/2}+\frac {\int \left (2192868+1306323 x^2\right ) \sqrt {4+3 x^2+x^4} \, dx}{3003}\\ &=\frac {x \left (1653701+435441 x^2\right ) \sqrt {4+3 x^2+x^4}}{5005}+\frac {x \left (53504+15365 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2}}{1001}+\frac {3825}{143} x \left (4+3 x^2+x^4\right )^{5/2}+\frac {125}{13} x^3 \left (4+3 x^2+x^4\right )^{5/2}+\frac {\int \frac {72038844+40730958 x^2}{\sqrt {4+3 x^2+x^4}} \, dx}{45045}\\ &=\frac {x \left (1653701+435441 x^2\right ) \sqrt {4+3 x^2+x^4}}{5005}+\frac {x \left (53504+15365 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2}}{1001}+\frac {3825}{143} x \left (4+3 x^2+x^4\right )^{5/2}+\frac {125}{13} x^3 \left (4+3 x^2+x^4\right )^{5/2}-\frac {9051324 \int \frac {1-\frac {x^2}{2}}{\sqrt {4+3 x^2+x^4}} \, dx}{5005}+\frac {487304}{143} \int \frac {1}{\sqrt {4+3 x^2+x^4}} \, dx\\ &=\frac {4525662 x \sqrt {4+3 x^2+x^4}}{5005 \left (2+x^2\right )}+\frac {x \left (1653701+435441 x^2\right ) \sqrt {4+3 x^2+x^4}}{5005}+\frac {x \left (53504+15365 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2}}{1001}+\frac {3825}{143} x \left (4+3 x^2+x^4\right )^{5/2}+\frac {125}{13} x^3 \left (4+3 x^2+x^4\right )^{5/2}-\frac {4525662 \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 )}{5005 \sqrt {4+3 x^2+x^4}}+\frac {121826 \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 )}{143 \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 \[ {\rm integral}\left ({\left (125 \, x^{10} + 900 \, x^{8} + 2810 \, x^{6} + 4648 \, x^{4} + 3969 \, x^{2} + 1372\right )} \sqrt {x^{4} + 3 \, x^{2} + 4}, x\right ) \]
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 \[ \int {\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 309, normalized size = 1.25 \[ \frac {125 \sqrt {x^{4}+3 x^{2}+4}\, x^{11}}{13}+\frac {12075 \sqrt {x^{4}+3 x^{2}+4}\, x^{9}}{143}+\frac {48520 \sqrt {x^{4}+3 x^{2}+4}\, x^{7}}{143}+\frac {71434 \sqrt {x^{4}+3 x^{2}+4}\, x^{5}}{91}+\frac {5528301 \sqrt {x^{4}+3 x^{2}+4}\, x^{3}}{5005}+\frac {4865781 \sqrt {x^{4}+3 x^{2}+4}\, x}{5005}+\frac {32017264 \sqrt {-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{5005 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {144821184 \sqrt {-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}+1}\, \left (-\EllipticE \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )+\EllipticF \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )\right )}{5005 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (i \sqrt {7}+3\right )} \]
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 \[ \int {\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}^{3}\,{d x} \]
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 \[ \int {\left (5\,x^2+7\right )}^3\,{\left (x^4+3\,x^2+4\right )}^{3/2} \,d x \]
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 \[ \int \left (\left (x^{2} - x + 2\right ) \left (x^{2} + x + 2\right )\right )^{\frac {3}{2}} \left (5 x^{2} + 7\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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