Optimal. Leaf size=314 \[ -\frac {555 \sqrt {x^4+3 x^2+4} x}{758912 \left (x^2+2\right )}+\frac {2775 \sqrt {x^4+3 x^2+4} x}{758912 \left (5 x^2+7\right )}+\frac {25 \sqrt {x^4+3 x^2+4} x}{1232 \left (5 x^2+7\right )^2}-\frac {3285 \sqrt {\frac {5}{77}} \tan ^{-1}\left (\frac {2 \sqrt {\frac {11}{35}} x}{\sqrt {x^4+3 x^2+4}}\right )}{3035648}-\frac {\left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{8624 \sqrt {2} \sqrt {x^4+3 x^2+4}}+\frac {555 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{379456 \sqrt {2} \sqrt {x^4+3 x^2+4}}-\frac {18615 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \Pi \left (-\frac {9}{280};2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{21249536 \sqrt {2} \sqrt {x^4+3 x^2+4}} \]
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Rubi [A] time = 0.29, antiderivative size = 314, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1223, 1696, 1714, 1195, 1708, 1103, 1706} \[ -\frac {555 \sqrt {x^4+3 x^2+4} x}{758912 \left (x^2+2\right )}+\frac {2775 \sqrt {x^4+3 x^2+4} x}{758912 \left (5 x^2+7\right )}+\frac {25 \sqrt {x^4+3 x^2+4} x}{1232 \left (5 x^2+7\right )^2}-\frac {3285 \sqrt {\frac {5}{77}} \tan ^{-1}\left (\frac {2 \sqrt {\frac {11}{35}} x}{\sqrt {x^4+3 x^2+4}}\right )}{3035648}-\frac {\left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{8624 \sqrt {2} \sqrt {x^4+3 x^2+4}}+\frac {555 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{379456 \sqrt {2} \sqrt {x^4+3 x^2+4}}-\frac {18615 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \Pi \left (-\frac {9}{280};2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{21249536 \sqrt {2} \sqrt {x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
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Rule 1103
Rule 1195
Rule 1223
Rule 1696
Rule 1706
Rule 1708
Rule 1714
Rubi steps
\begin {align*} \int \frac {1}{\left (7+5 x^2\right )^3 \sqrt {4+3 x^2+x^4}} \, dx &=\frac {25 x \sqrt {4+3 x^2+x^4}}{1232 \left (7+5 x^2\right )^2}-\frac {\int \frac {-76-10 x^2-25 x^4}{\left (7+5 x^2\right )^2 \sqrt {4+3 x^2+x^4}} \, dx}{1232}\\ &=\frac {25 x \sqrt {4+3 x^2+x^4}}{1232 \left (7+5 x^2\right )^2}+\frac {2775 x \sqrt {4+3 x^2+x^4}}{758912 \left (7+5 x^2\right )}+\frac {\int \frac {-4412-4690 x^2-2775 x^4}{\left (7+5 x^2\right ) \sqrt {4+3 x^2+x^4}} \, dx}{758912}\\ &=\frac {25 x \sqrt {4+3 x^2+x^4}}{1232 \left (7+5 x^2\right )^2}+\frac {2775 x \sqrt {4+3 x^2+x^4}}{758912 \left (7+5 x^2\right )}+\frac {\int \frac {-60910-31775 x^2}{\left (7+5 x^2\right ) \sqrt {4+3 x^2+x^4}} \, dx}{3794560}+\frac {555 \int \frac {1-\frac {x^2}{2}}{\sqrt {4+3 x^2+x^4}} \, dx}{379456}\\ &=-\frac {555 x \sqrt {4+3 x^2+x^4}}{758912 \left (2+x^2\right )}+\frac {25 x \sqrt {4+3 x^2+x^4}}{1232 \left (7+5 x^2\right )^2}+\frac {2775 x \sqrt {4+3 x^2+x^4}}{758912 \left (7+5 x^2\right )}+\frac {555 \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 )}{379456 \sqrt {2} \sqrt {4+3 x^2+x^4}}-\frac {\int \frac {1}{\sqrt {4+3 x^2+x^4}} \, dx}{4312}-\frac {5475 \int \frac {1+\frac {x^2}{2}}{\left (7+5 x^2\right ) \sqrt {4+3 x^2+x^4}} \, dx}{379456}\\ &=-\frac {555 x \sqrt {4+3 x^2+x^4}}{758912 \left (2+x^2\right )}+\frac {25 x \sqrt {4+3 x^2+x^4}}{1232 \left (7+5 x^2\right )^2}+\frac {2775 x \sqrt {4+3 x^2+x^4}}{758912 \left (7+5 x^2\right )}-\frac {3285 \sqrt {\frac {5}{77}} \tan ^{-1}\left (\frac {2 \sqrt {\frac {11}{35}} x}{\sqrt {4+3 x^2+x^4}}\right )}{3035648}+\frac {555 \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 )}{379456 \sqrt {2} \sqrt {4+3 x^2+x^4}}-\frac {\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 )}{8624 \sqrt {2} \sqrt {4+3 x^2+x^4}}-\frac {18615 \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} \Pi \left (-\frac {9}{280};2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{21249536 \sqrt {2} \sqrt {4+3 x^2+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.89, size = 308, normalized size = 0.98 \[ \frac {\frac {700 x \left (555 x^2+1393\right ) \left (x^4+3 x^2+4\right )}{\left (5 x^2+7\right )^2}+i \sqrt {6+2 i \sqrt {7}} \sqrt {1-\frac {2 i x^2}{\sqrt {7}-3 i}} \sqrt {1+\frac {2 i x^2}{\sqrt {7}+3 i}} \left (\left (-9401+3885 i \sqrt {7}\right ) F\left (i \sinh ^{-1}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right )|\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )+3885 \left (3-i \sqrt {7}\right ) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right )|\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )+6570 \Pi \left (\frac {5}{14} \left (3+i \sqrt {7}\right );i \sinh ^{-1}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right )|\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )\right )}{21249536 \sqrt {x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{4} + 3 \, x^{2} + 4}}{125 \, x^{10} + 900 \, x^{8} + 2810 \, x^{6} + 4648 \, x^{4} + 3969 \, x^{2} + 1372}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {x^{4} + 3 \, x^{2} + 4} {\left (5 \, x^{2} + 7\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 434, normalized size = 1.38 \[ \frac {25 \sqrt {x^{4}+3 x^{2}+4}\, x}{1232 \left (5 x^{2}+7\right )^{2}}+\frac {2775 \sqrt {x^{4}+3 x^{2}+4}\, x}{758912 \left (5 x^{2}+7\right )}-\frac {555 \sqrt {\frac {3 x^{2}}{8}-\frac {i \sqrt {7}\, x^{2}}{8}+1}\, \sqrt {\frac {3 x^{2}}{8}+\frac {i \sqrt {7}\, x^{2}}{8}+1}\, \EllipticE \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{23716 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (i \sqrt {7}+3\right )}-\frac {23 \sqrt {\frac {3 x^{2}}{8}-\frac {i \sqrt {7}\, x^{2}}{8}+1}\, \sqrt {\frac {3 x^{2}}{8}+\frac {i \sqrt {7}\, x^{2}}{8}+1}\, \EllipticF \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{27104 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}+\frac {555 \sqrt {\frac {3 x^{2}}{8}-\frac {i \sqrt {7}\, x^{2}}{8}+1}\, \sqrt {\frac {3 x^{2}}{8}+\frac {i \sqrt {7}\, x^{2}}{8}+1}\, \EllipticF \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{23716 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (i \sqrt {7}+3\right )}-\frac {3285 \sqrt {\frac {3 x^{2}}{8}-\frac {i \sqrt {7}\, x^{2}}{8}+1}\, \sqrt {\frac {3 x^{2}}{8}+\frac {i \sqrt {7}\, x^{2}}{8}+1}\, \EllipticPi \left (\sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, x , -\frac {5}{7 \left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right )}, \frac {\sqrt {-\frac {3}{8}-\frac {i \sqrt {7}}{8}}}{\sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}}\right )}{5312384 \sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, \sqrt {x^{4}+3 x^{2}+4}} \]
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 \[ \int \frac {1}{\sqrt {x^{4} + 3 \, x^{2} + 4} {\left (5 \, x^{2} + 7\right )}^{3}}\,{d x} \]
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 \[ \int \frac {1}{{\left (5\,x^2+7\right )}^3\,\sqrt {x^4+3\,x^2+4}} \,d x \]
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 \[ \int \frac {1}{\sqrt {\left (x^{2} - x + 2\right ) \left (x^{2} + x + 2\right )} \left (5 x^{2} + 7\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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