Optimal. Leaf size=242 \[ \frac {3050}{11} \left (x^4+3 x^2+4\right )^{3/2} x+\frac {1}{33} \left (4516 x^2+18727\right ) \sqrt {x^4+3 x^2+4} x+\frac {51665 \sqrt {x^4+3 x^2+4} x}{33 \left (x^2+2\right )}+\frac {33159 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{11 \sqrt {2} \sqrt {x^4+3 x^2+4}}-\frac {51665 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{33 \sqrt {x^4+3 x^2+4}}+\frac {625}{11} \left (x^4+3 x^2+4\right )^{3/2} x^5+\frac {23500}{99} \left (x^4+3 x^2+4\right )^{3/2} x^3 \]
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Rubi [A] time = 0.15, antiderivative size = 242, 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} \[ \frac {625}{11} \left (x^4+3 x^2+4\right )^{3/2} x^5+\frac {23500}{99} \left (x^4+3 x^2+4\right )^{3/2} x^3+\frac {3050}{11} \left (x^4+3 x^2+4\right )^{3/2} x+\frac {1}{33} \left (4516 x^2+18727\right ) \sqrt {x^4+3 x^2+4} x+\frac {51665 \sqrt {x^4+3 x^2+4} x}{33 \left (x^2+2\right )}+\frac {33159 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{11 \sqrt {2} \sqrt {x^4+3 x^2+4}}-\frac {51665 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{33 \sqrt {x^4+3 x^2+4}} \]
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 )^4 \sqrt {4+3 x^2+x^4} \, dx &=\frac {625}{11} x^5 \left (4+3 x^2+x^4\right )^{3/2}+\frac {1}{11} \int \sqrt {4+3 x^2+x^4} \left (26411+75460 x^2+68350 x^4+23500 x^6\right ) \, dx\\ &=\frac {23500}{99} x^3 \left (4+3 x^2+x^4\right )^{3/2}+\frac {625}{11} x^5 \left (4+3 x^2+x^4\right )^{3/2}+\frac {1}{99} \int \sqrt {4+3 x^2+x^4} \left (237699+397140 x^2+192150 x^4\right ) \, dx\\ &=\frac {3050}{11} x \left (4+3 x^2+x^4\right )^{3/2}+\frac {23500}{99} x^3 \left (4+3 x^2+x^4\right )^{3/2}+\frac {625}{11} x^5 \left (4+3 x^2+x^4\right )^{3/2}+\frac {1}{693} \int \left (895293+474180 x^2\right ) \sqrt {4+3 x^2+x^4} \, dx\\ &=\frac {1}{33} x \left (18727+4516 x^2\right ) \sqrt {4+3 x^2+x^4}+\frac {3050}{11} x \left (4+3 x^2+x^4\right )^{3/2}+\frac {23500}{99} x^3 \left (4+3 x^2+x^4\right )^{3/2}+\frac {625}{11} x^5 \left (4+3 x^2+x^4\right )^{3/2}+\frac {\int \frac {30121560+16274475 x^2}{\sqrt {4+3 x^2+x^4}} \, dx}{10395}\\ &=\frac {1}{33} x \left (18727+4516 x^2\right ) \sqrt {4+3 x^2+x^4}+\frac {3050}{11} x \left (4+3 x^2+x^4\right )^{3/2}+\frac {23500}{99} x^3 \left (4+3 x^2+x^4\right )^{3/2}+\frac {625}{11} x^5 \left (4+3 x^2+x^4\right )^{3/2}-\frac {103330}{33} \int \frac {1-\frac {x^2}{2}}{\sqrt {4+3 x^2+x^4}} \, dx+\frac {66318}{11} \int \frac {1}{\sqrt {4+3 x^2+x^4}} \, dx\\ &=\frac {51665 x \sqrt {4+3 x^2+x^4}}{33 \left (2+x^2\right )}+\frac {1}{33} x \left (18727+4516 x^2\right ) \sqrt {4+3 x^2+x^4}+\frac {3050}{11} x \left (4+3 x^2+x^4\right )^{3/2}+\frac {23500}{99} x^3 \left (4+3 x^2+x^4\right )^{3/2}+\frac {625}{11} x^5 \left (4+3 x^2+x^4\right )^{3/2}-\frac {51665 \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 )}{33 \sqrt {4+3 x^2+x^4}}+\frac {33159 \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 )}{11 \sqrt {2} \sqrt {4+3 x^2+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.59, size = 354, normalized size = 1.46 \[ \frac {3 \sqrt {2} \left (51665 \sqrt {7}-36253 i\right ) \sqrt {\frac {-2 i x^2+\sqrt {7}-3 i}{\sqrt {7}-3 i}} \sqrt {\frac {2 i x^2+\sqrt {7}+3 i}{\sqrt {7}+3 i}} F\left (i \sinh ^{-1}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right )|\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )-154995 \sqrt {2} \left (\sqrt {7}+3 i\right ) \sqrt {\frac {-2 i x^2+\sqrt {7}-3 i}{\sqrt {7}-3 i}} \sqrt {\frac {2 i x^2+\sqrt {7}+3 i}{\sqrt {7}+3 i}} E\left (i \sinh ^{-1}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right )|\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )+4 \sqrt {-\frac {i}{\sqrt {7}-3 i}} x \left (5625 x^{12}+57250 x^{10}+264075 x^8+712748 x^6+1217475 x^4+1257535 x^2+663924\right )}{396 \sqrt {-\frac {i}{\sqrt {7}-3 i}} \sqrt {x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (625 \, x^{8} + 3500 \, x^{6} + 7350 \, x^{4} + 6860 \, x^{2} + 2401\right )} \sqrt {x^{4} + 3 \, x^{2} + 4}, x\right ) \]
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 \[ \int \sqrt {x^{4} + 3 \, x^{2} + 4} {\left (5 \, x^{2} + 7\right )}^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.17, size = 292, normalized size = 1.21 \[ \frac {625 \sqrt {x^{4}+3 x^{2}+4}\, x^{9}}{11}+\frac {40375 \sqrt {x^{4}+3 x^{2}+4}\, x^{7}}{99}+\frac {3650 \sqrt {x^{4}+3 x^{2}+4}\, x^{5}}{3}+\frac {189898 \sqrt {x^{4}+3 x^{2}+4}\, x^{3}}{99}+\frac {55327 \sqrt {x^{4}+3 x^{2}+4}\, x}{33}+\frac {382496 \sqrt {-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{33 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {1653280 \sqrt {-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}+1}\, \left (-\EllipticE \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )+\EllipticF \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )\right )}{33 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (i \sqrt {7}+3\right )} \]
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 \[ \int \sqrt {x^{4} + 3 \, x^{2} + 4} {\left (5 \, x^{2} + 7\right )}^{4}\,{d x} \]
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 \[ \int {\left (5\,x^2+7\right )}^4\,\sqrt {x^4+3\,x^2+4} \,d x \]
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 \[ \int \sqrt {\left (x^{2} - x + 2\right ) \left (x^{2} + x + 2\right )} \left (5 x^{2} + 7\right )^{4}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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