Optimal. Leaf size=331 \[ -\frac {49949 x \left (5+\sqrt {13}+2 x^2\right )}{3465 \sqrt {3+5 x^2+x^4}}+\frac {353}{99} x \sqrt {3+5 x^2+x^4}-\frac {x^3 \left (911+890 x^2\right ) \sqrt {3+5 x^2+x^4}}{1155}+\frac {1}{99} x^3 \left (67+27 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac {49949 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) E\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{3465 \sqrt {3+5 x^2+x^4}}-\frac {353 \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{33 \sqrt {6 \left (5+\sqrt {13}\right )} \sqrt {3+5 x^2+x^4}} \]
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Rubi [A]
time = 0.15, antiderivative size = 331, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1287, 1293,
1203, 1113, 1149} \begin {gather*} -\frac {353 \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 (\text {ArcTan}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{33 \sqrt {6 \left (5+\sqrt {13}\right )} \sqrt {x^4+5 x^2+3}}+\frac {49949 \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 (\text {ArcTan}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{3465 \sqrt {x^4+5 x^2+3}}+\frac {353}{99} \sqrt {x^4+5 x^2+3} x-\frac {49949 \left (2 x^2+\sqrt {13}+5\right ) x}{3465 \sqrt {x^4+5 x^2+3}}+\frac {1}{99} \left (27 x^2+67\right ) \left (x^4+5 x^2+3\right )^{3/2} x^3-\frac {\left (890 x^2+911\right ) \sqrt {x^4+5 x^2+3} x^3}{1155} \end {gather*}
Antiderivative was successfully verified.
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Rule 1113
Rule 1149
Rule 1203
Rule 1287
Rule 1293
Rubi steps
\begin {align*} \int x^2 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx &=\frac {1}{99} x^3 \left (67+27 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac {1}{33} \int x^2 \left (-3-178 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx\\ &=-\frac {x^3 \left (911+890 x^2\right ) \sqrt {3+5 x^2+x^4}}{1155}+\frac {1}{99} x^3 \left (67+27 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac {\int \frac {x^2 \left (7884+12355 x^2\right )}{\sqrt {3+5 x^2+x^4}} \, dx}{1155}\\ &=\frac {353}{99} x \sqrt {3+5 x^2+x^4}-\frac {x^3 \left (911+890 x^2\right ) \sqrt {3+5 x^2+x^4}}{1155}+\frac {1}{99} x^3 \left (67+27 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}-\frac {\int \frac {37065+99898 x^2}{\sqrt {3+5 x^2+x^4}} \, dx}{3465}\\ &=\frac {353}{99} x \sqrt {3+5 x^2+x^4}-\frac {x^3 \left (911+890 x^2\right ) \sqrt {3+5 x^2+x^4}}{1155}+\frac {1}{99} x^3 \left (67+27 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}-\frac {353}{33} \int \frac {1}{\sqrt {3+5 x^2+x^4}} \, dx-\frac {99898 \int \frac {x^2}{\sqrt {3+5 x^2+x^4}} \, dx}{3465}\\ &=-\frac {49949 x \left (5+\sqrt {13}+2 x^2\right )}{3465 \sqrt {3+5 x^2+x^4}}+\frac {353}{99} x \sqrt {3+5 x^2+x^4}-\frac {x^3 \left (911+890 x^2\right ) \sqrt {3+5 x^2+x^4}}{1155}+\frac {1}{99} x^3 \left (67+27 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac {49949 \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 )}{3465 \sqrt {3+5 x^2+x^4}}-\frac {353 \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 )}{33 \sqrt {6 \left (5+\sqrt {13}\right )} \sqrt {3+5 x^2+x^4}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 7.46, size = 244, normalized size = 0.74 \begin {gather*} \frac {2 x \left (37065+74681 x^2+69535 x^4+84962 x^6+50075 x^8+11795 x^{10}+945 x^{12}\right )-49949 i \sqrt {2} \left (-5+\sqrt {13}\right ) \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} E\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )+i \sqrt {2} \left (-212680+49949 \sqrt {13}\right ) \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} F\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )}{6930 \sqrt {3+5 x^2+x^4}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.05, size = 277, normalized size = 0.84
method | result | size |
risch | \(\frac {x \left (945 x^{8}+7070 x^{6}+11890 x^{4}+4302 x^{2}+12355\right ) \sqrt {x^{4}+5 x^{2}+3}}{3465}+\frac {399592 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-\EllipticE \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{385 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}-\frac {706 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{11 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}\) | \(231\) |
default | \(\frac {3 x^{9} \sqrt {x^{4}+5 x^{2}+3}}{11}+\frac {202 x^{7} \sqrt {x^{4}+5 x^{2}+3}}{99}+\frac {2378 x^{5} \sqrt {x^{4}+5 x^{2}+3}}{693}+\frac {478 x^{3} \sqrt {x^{4}+5 x^{2}+3}}{385}+\frac {353 x \sqrt {x^{4}+5 x^{2}+3}}{99}-\frac {706 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{11 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}+\frac {399592 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-\EllipticE \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{385 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(277\) |
elliptic | \(\frac {3 x^{9} \sqrt {x^{4}+5 x^{2}+3}}{11}+\frac {202 x^{7} \sqrt {x^{4}+5 x^{2}+3}}{99}+\frac {2378 x^{5} \sqrt {x^{4}+5 x^{2}+3}}{693}+\frac {478 x^{3} \sqrt {x^{4}+5 x^{2}+3}}{385}+\frac {353 x \sqrt {x^{4}+5 x^{2}+3}}{99}-\frac {706 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{11 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}+\frac {399592 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-\EllipticE \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{385 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(277\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int x^{2} \cdot \left (3 x^{2} + 2\right ) \left (x^{4} + 5 x^{2} + 3\right )^{\frac {3}{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^2\,\left (3\,x^2+2\right )\,{\left (x^4+5\,x^2+3\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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