Optimal. Leaf size=181 \[ -\frac {19 \sqrt {x^4+3 x^2+4} x}{28 \left (x^2+2\right )}+\frac {\left (19 x^2+53\right ) x}{28 \sqrt {x^4+3 x^2+4}}-\frac {3 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{4 \sqrt {2} \sqrt {x^4+3 x^2+4}}+\frac {19 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{14 \sqrt {2} \sqrt {x^4+3 x^2+4}} \]
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Rubi [A] time = 0.05, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1178, 1197, 1103, 1195} \[ -\frac {19 \sqrt {x^4+3 x^2+4} x}{28 \left (x^2+2\right )}+\frac {\left (19 x^2+53\right ) x}{28 \sqrt {x^4+3 x^2+4}}-\frac {3 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{4 \sqrt {2} \sqrt {x^4+3 x^2+4}}+\frac {19 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{14 \sqrt {2} \sqrt {x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
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Rule 1103
Rule 1178
Rule 1195
Rule 1197
Rubi steps
\begin {align*} \int \frac {7+5 x^2}{\left (4+3 x^2+x^4\right )^{3/2}} \, dx &=\frac {x \left (53+19 x^2\right )}{28 \sqrt {4+3 x^2+x^4}}+\frac {1}{28} \int \frac {-4-19 x^2}{\sqrt {4+3 x^2+x^4}} \, dx\\ &=\frac {x \left (53+19 x^2\right )}{28 \sqrt {4+3 x^2+x^4}}+\frac {19}{14} \int \frac {1-\frac {x^2}{2}}{\sqrt {4+3 x^2+x^4}} \, dx-\frac {3}{2} \int \frac {1}{\sqrt {4+3 x^2+x^4}} \, dx\\ &=\frac {x \left (53+19 x^2\right )}{28 \sqrt {4+3 x^2+x^4}}-\frac {19 x \sqrt {4+3 x^2+x^4}}{28 \left (2+x^2\right )}+\frac {19 \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 )}{14 \sqrt {2} \sqrt {4+3 x^2+x^4}}-\frac {3 \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 )}{4 \sqrt {2} \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 \[ {\rm integral}\left (\frac {\sqrt {x^{4} + 3 \, x^{2} + 4} {\left (5 \, x^{2} + 7\right )}}{x^{8} + 6 \, x^{6} + 17 \, x^{4} + 24 \, x^{2} + 16}, x\right ) \]
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 \[ \int \frac {5 \, x^{2} + 7}{{\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 255, normalized size = 1.41 \[ -\frac {4 \sqrt {-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{7 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {10 \left (-\frac {1}{7} x^{3}-\frac {3}{14} x \right )}{\sqrt {x^{4}+3 x^{2}+4}}+\frac {152 \sqrt {-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}+1}\, \left (-\EllipticE \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )+\EllipticF \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )\right )}{7 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (i \sqrt {7}+3\right )}-\frac {14 \left (\frac {3}{56} x^{3}+\frac {1}{56} x \right )}{\sqrt {x^{4}+3 x^{2}+4}} \]
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 \[ \int \frac {5 \, x^{2} + 7}{{\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {5\,x^2+7}{{\left (x^4+3\,x^2+4\right )}^{3/2}} \,d x \]
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 \[ \int \frac {5 x^{2} + 7}{\left (\left (x^{2} - x + 2\right ) \left (x^{2} + x + 2\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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