Optimal. Leaf size=314 \[ -\frac {13 \left (24-5 x^2\right ) \sqrt {x^4+5 x^2+3}}{15 x}+\frac {949 x \left (2 x^2+\sqrt {13}+5\right )}{30 \sqrt {x^4+5 x^2+3}}+\frac {65 \sqrt {\frac {2}{3 \left (5+\sqrt {13}\right )}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}}-\frac {949 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{30 \sqrt {x^4+5 x^2+3}}-\frac {\left (10-9 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{15 x^3} \]
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Rubi [A] time = 0.16, antiderivative size = 314, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1271, 1189, 1099, 1135} \[ -\frac {\left (10-9 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{15 x^3}-\frac {13 \left (24-5 x^2\right ) \sqrt {x^4+5 x^2+3}}{15 x}+\frac {949 x \left (2 x^2+\sqrt {13}+5\right )}{30 \sqrt {x^4+5 x^2+3}}+\frac {65 \sqrt {\frac {2}{3 \left (5+\sqrt {13}\right )}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}}-\frac {949 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{30 \sqrt {x^4+5 x^2+3}} \]
Antiderivative was successfully verified.
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Rule 1099
Rule 1135
Rule 1189
Rule 1271
Rubi steps
\begin {align*} \int \frac {\left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{x^4} \, dx &=-\frac {\left (10-9 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{15 x^3}-\frac {1}{5} \int \frac {\left (-104-65 x^2\right ) \sqrt {3+5 x^2+x^4}}{x^2} \, dx\\ &=-\frac {13 \left (24-5 x^2\right ) \sqrt {3+5 x^2+x^4}}{15 x}-\frac {\left (10-9 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{15 x^3}+\frac {1}{15} \int \frac {1950+949 x^2}{\sqrt {3+5 x^2+x^4}} \, dx\\ &=-\frac {13 \left (24-5 x^2\right ) \sqrt {3+5 x^2+x^4}}{15 x}-\frac {\left (10-9 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{15 x^3}+\frac {949}{15} \int \frac {x^2}{\sqrt {3+5 x^2+x^4}} \, dx+130 \int \frac {1}{\sqrt {3+5 x^2+x^4}} \, dx\\ &=\frac {949 x \left (5+\sqrt {13}+2 x^2\right )}{30 \sqrt {3+5 x^2+x^4}}-\frac {13 \left (24-5 x^2\right ) \sqrt {3+5 x^2+x^4}}{15 x}-\frac {\left (10-9 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{15 x^3}-\frac {949 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) E\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{30 \sqrt {3+5 x^2+x^4}}+\frac {65 \sqrt {\frac {2}{3 \left (5+\sqrt {13}\right )}} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {3+5 x^2+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.29, size = 247, normalized size = 0.79 \[ \frac {-13 i \sqrt {2} \left (73 \sqrt {13}-65\right ) \sqrt {\frac {-2 x^2+\sqrt {13}-5}{\sqrt {13}-5}} \sqrt {2 x^2+\sqrt {13}+5} x^3 F\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )+949 i \sqrt {2} \left (\sqrt {13}-5\right ) \sqrt {\frac {-2 x^2+\sqrt {13}-5}{\sqrt {13}-5}} \sqrt {2 x^2+\sqrt {13}+5} x^3 E\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )+4 \left (9 x^{10}+145 x^8+192 x^6-1405 x^4-1155 x^2-90\right )}{60 x^3 \sqrt {x^4+5 x^2+3}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (3 \, x^{6} + 17 \, x^{4} + 19 \, x^{2} + 6\right )} \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{4}}, x\right ) \]
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 \[ \int \frac {{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} {\left (3 \, x^{2} + 2\right )}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 260, normalized size = 0.83 \[ \frac {3 \sqrt {x^{4}+5 x^{2}+3}\, x^{3}}{5}+\frac {20 \sqrt {x^{4}+5 x^{2}+3}\, x}{3}+\frac {780 \sqrt {-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {-30+6 \sqrt {13}}\, x}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {67 \sqrt {x^{4}+5 x^{2}+3}}{3 x}-\frac {2 \sqrt {x^{4}+5 x^{2}+3}}{x^{3}}-\frac {11388 \sqrt {-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}+1}\, \left (-\EllipticE \left (\frac {\sqrt {-30+6 \sqrt {13}}\, x}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )+\EllipticF \left (\frac {\sqrt {-30+6 \sqrt {13}}\, x}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{5 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (\sqrt {13}+5\right )} \]
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 \[ \int \frac {{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} {\left (3 \, x^{2} + 2\right )}}{x^{4}}\,{d x} \]
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 \[ \int \frac {\left (3\,x^2+2\right )\,{\left (x^4+5\,x^2+3\right )}^{3/2}}{x^4} \,d x \]
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 \[ \int \frac {\left (3 x^{2} + 2\right ) \left (x^{4} + 5 x^{2} + 3\right )^{\frac {3}{2}}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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