Optimal. Leaf size=219 \[ -\frac {220779 \sqrt {x^4+3 x^2+4} x}{28 \left (x^2+2\right )}+\frac {5000}{3} \sqrt {x^4+3 x^2+4} x+\frac {\left (45779 x^2+99493\right ) x}{28 \sqrt {x^4+3 x^2+4}}-\frac {130729 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{12 \sqrt {2} \sqrt {x^4+3 x^2+4}}+\frac {220779 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{14 \sqrt {2} \sqrt {x^4+3 x^2+4}}+625 \sqrt {x^4+3 x^2+4} x^3 \]
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Rubi [A] time = 0.12, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1205, 1679, 1197, 1103, 1195} \[ 625 \sqrt {x^4+3 x^2+4} x^3-\frac {220779 \sqrt {x^4+3 x^2+4} x}{28 \left (x^2+2\right )}+\frac {5000}{3} \sqrt {x^4+3 x^2+4} x+\frac {\left (45779 x^2+99493\right ) x}{28 \sqrt {x^4+3 x^2+4}}-\frac {130729 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{12 \sqrt {2} \sqrt {x^4+3 x^2+4}}+\frac {220779 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{14 \sqrt {2} \sqrt {x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
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Rule 1103
Rule 1195
Rule 1197
Rule 1205
Rule 1679
Rubi steps
\begin {align*} \int \frac {\left (7+5 x^2\right )^5}{\left (4+3 x^2+x^4\right )^{3/2}} \, dx &=\frac {x \left (99493+45779 x^2\right )}{28 \sqrt {4+3 x^2+x^4}}+\frac {1}{28} \int \frac {18156+269221 x^2+350000 x^4+87500 x^6}{\sqrt {4+3 x^2+x^4}} \, dx\\ &=\frac {x \left (99493+45779 x^2\right )}{28 \sqrt {4+3 x^2+x^4}}+625 x^3 \sqrt {4+3 x^2+x^4}+\frac {1}{140} \int \frac {90780+296105 x^2+700000 x^4}{\sqrt {4+3 x^2+x^4}} \, dx\\ &=\frac {x \left (99493+45779 x^2\right )}{28 \sqrt {4+3 x^2+x^4}}+\frac {5000}{3} x \sqrt {4+3 x^2+x^4}+625 x^3 \sqrt {4+3 x^2+x^4}+\frac {1}{420} \int \frac {-2527660-3311685 x^2}{\sqrt {4+3 x^2+x^4}} \, dx\\ &=\frac {x \left (99493+45779 x^2\right )}{28 \sqrt {4+3 x^2+x^4}}+\frac {5000}{3} x \sqrt {4+3 x^2+x^4}+625 x^3 \sqrt {4+3 x^2+x^4}+\frac {220779}{14} \int \frac {1-\frac {x^2}{2}}{\sqrt {4+3 x^2+x^4}} \, dx-\frac {130729}{6} \int \frac {1}{\sqrt {4+3 x^2+x^4}} \, dx\\ &=\frac {x \left (99493+45779 x^2\right )}{28 \sqrt {4+3 x^2+x^4}}+\frac {5000}{3} x \sqrt {4+3 x^2+x^4}+625 x^3 \sqrt {4+3 x^2+x^4}-\frac {220779 x \sqrt {4+3 x^2+x^4}}{28 \left (2+x^2\right )}+\frac {220779 \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 {130729 \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 )}{12 \sqrt {2} \sqrt {4+3 x^2+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.52, size = 339, normalized size = 1.55 \[ \frac {-\sqrt {2} \left (662337 \sqrt {7}+975947 i\right ) \sqrt {\frac {-2 i x^2+\sqrt {7}-3 i}{\sqrt {7}-3 i}} \sqrt {\frac {2 i x^2+\sqrt {7}+3 i}{\sqrt {7}+3 i}} F\left (i \sinh ^{-1}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right )|\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )+662337 \sqrt {2} \left (\sqrt {7}+3 i\right ) \sqrt {\frac {-2 i x^2+\sqrt {7}-3 i}{\sqrt {7}-3 i}} \sqrt {\frac {2 i x^2+\sqrt {7}+3 i}{\sqrt {7}+3 i}} E\left (i \sinh ^{-1}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right )|\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )+4 \sqrt {-\frac {i}{\sqrt {7}-3 i}} x \left (52500 x^6+297500 x^4+767337 x^2+858479\right )}{336 \sqrt {-\frac {i}{\sqrt {7}-3 i}} \sqrt {x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (3125 \, x^{10} + 21875 \, x^{8} + 61250 \, x^{6} + 85750 \, x^{4} + 60025 \, x^{2} + 16807\right )} \sqrt {x^{4} + 3 \, x^{2} + 4}}{x^{8} + 6 \, x^{6} + 17 \, x^{4} + 24 \, x^{2} + 16}, x\right ) \]
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 \[ \int \frac {{\left (5 \, x^{2} + 7\right )}^{5}}{{\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.05, size = 379, normalized size = 1.73 \[ 625 \sqrt {x^{4}+3 x^{2}+4}\, x^{3}+\frac {5000 \sqrt {x^{4}+3 x^{2}+4}\, x}{3}-\frac {505532 \sqrt {-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{21 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {6250 \left (\frac {31}{14} x^{3}+\frac {18}{7} x \right )}{\sqrt {x^{4}+3 x^{2}+4}}+\frac {1766232 \sqrt {-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}+1}\, \left (-\EllipticE \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )+\EllipticF \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )\right )}{7 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (i \sqrt {7}+3\right )}-\frac {43750 \left (-\frac {9}{14} x^{3}+\frac {2}{7} x \right )}{\sqrt {x^{4}+3 x^{2}+4}}-\frac {122500 \left (-\frac {1}{14} x^{3}-\frac {6}{7} x \right )}{\sqrt {x^{4}+3 x^{2}+4}}-\frac {171500 \left (\frac {3}{14} x^{3}+\frac {4}{7} x \right )}{\sqrt {x^{4}+3 x^{2}+4}}-\frac {120050 \left (-\frac {1}{7} x^{3}-\frac {3}{14} x \right )}{\sqrt {x^{4}+3 x^{2}+4}}-\frac {33614 \left (\frac {3}{56} x^{3}+\frac {1}{56} x \right )}{\sqrt {x^{4}+3 x^{2}+4}} \]
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 \[ \int \frac {{\left (5 \, x^{2} + 7\right )}^{5}}{{\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (5\,x^2+7\right )}^5}{{\left (x^4+3\,x^2+4\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (5 x^{2} + 7\right )^{5}}{\left (\left (x^{2} - x + 2\right ) \left (x^{2} + x + 2\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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