Optimal. Leaf size=263 \[ \frac {41875 \sqrt {x^4+3 x^2+2} x}{84672 \left (5 x^2+7\right )}+\frac {625 \sqrt {x^4+3 x^2+2} x}{1008 \left (5 x^2+7\right )^2}-\frac {5797 \left (x^2+2\right ) x}{28224 \sqrt {x^4+3 x^2+2}}+\frac {\left (23 x^2+50\right ) x}{216 \sqrt {x^4+3 x^2+2}}-\frac {49907 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{56448 \sqrt {2} \sqrt {x^4+3 x^2+2}}+\frac {5797 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{14112 \sqrt {2} \sqrt {x^4+3 x^2+2}}+\frac {192625 \left (x^2+2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{395136 \sqrt {2} \sqrt {\frac {x^2+2}{x^2+1}} \sqrt {x^4+3 x^2+2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.76, antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 11, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {1228, 1178, 1189, 1099, 1135, 1223, 1696, 1716, 1214, 1456, 539} \[ \frac {41875 \sqrt {x^4+3 x^2+2} x}{84672 \left (5 x^2+7\right )}+\frac {625 \sqrt {x^4+3 x^2+2} x}{1008 \left (5 x^2+7\right )^2}-\frac {5797 \left (x^2+2\right ) x}{28224 \sqrt {x^4+3 x^2+2}}+\frac {\left (23 x^2+50\right ) x}{216 \sqrt {x^4+3 x^2+2}}-\frac {49907 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{56448 \sqrt {2} \sqrt {x^4+3 x^2+2}}+\frac {5797 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{14112 \sqrt {2} \sqrt {x^4+3 x^2+2}}+\frac {192625 \left (x^2+2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{395136 \sqrt {2} \sqrt {\frac {x^2+2}{x^2+1}} \sqrt {x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 539
Rule 1099
Rule 1135
Rule 1178
Rule 1189
Rule 1214
Rule 1223
Rule 1228
Rule 1456
Rule 1696
Rule 1716
Rubi steps
\begin {align*} \int \frac {1}{\left (7+5 x^2\right )^3 \left (2+3 x^2+x^4\right )^{3/2}} \, dx &=\int \left (-\frac {-62-35 x^2}{216 \left (2+3 x^2+x^4\right )^{3/2}}-\frac {25}{6 \left (7+5 x^2\right )^3 \sqrt {2+3 x^2+x^4}}-\frac {25}{36 \left (7+5 x^2\right )^2 \sqrt {2+3 x^2+x^4}}-\frac {175}{216 \left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}}\right ) \, dx\\ &=-\left (\frac {1}{216} \int \frac {-62-35 x^2}{\left (2+3 x^2+x^4\right )^{3/2}} \, dx\right )-\frac {25}{36} \int \frac {1}{\left (7+5 x^2\right )^2 \sqrt {2+3 x^2+x^4}} \, dx-\frac {175}{216} \int \frac {1}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx-\frac {25}{6} \int \frac {1}{\left (7+5 x^2\right )^3 \sqrt {2+3 x^2+x^4}} \, dx\\ &=\frac {x \left (50+23 x^2\right )}{216 \sqrt {2+3 x^2+x^4}}+\frac {625 x \sqrt {2+3 x^2+x^4}}{1008 \left (7+5 x^2\right )^2}+\frac {625 x \sqrt {2+3 x^2+x^4}}{3024 \left (7+5 x^2\right )}+\frac {1}{432} \int \frac {-38-46 x^2}{\sqrt {2+3 x^2+x^4}} \, dx-\frac {25 \int \frac {62+70 x^2+25 x^4}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx}{3024}-\frac {25 \int \frac {74-10 x^2-25 x^4}{\left (7+5 x^2\right )^2 \sqrt {2+3 x^2+x^4}} \, dx}{1008}-\frac {175}{432} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {875}{864} \int \frac {2+2 x^2}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx\\ &=\frac {x \left (50+23 x^2\right )}{216 \sqrt {2+3 x^2+x^4}}+\frac {625 x \sqrt {2+3 x^2+x^4}}{1008 \left (7+5 x^2\right )^2}+\frac {41875 x \sqrt {2+3 x^2+x^4}}{84672 \left (7+5 x^2\right )}-\frac {175 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{432 \sqrt {2} \sqrt {2+3 x^2+x^4}}-\frac {25 \int \frac {2838+2310 x^2+975 x^4}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx}{84672}+\frac {\int \frac {-175-125 x^2}{\sqrt {2+3 x^2+x^4}} \, dx}{3024}-\frac {19}{216} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx-\frac {23}{216} \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx-\frac {325 \int \frac {1}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx}{3024}+\frac {\left (875 \sqrt {1+\frac {x^2}{2}} \sqrt {2+2 x^2}\right ) \int \frac {\sqrt {2+2 x^2}}{\sqrt {1+\frac {x^2}{2}} \left (7+5 x^2\right )} \, dx}{864 \sqrt {2+3 x^2+x^4}}\\ &=-\frac {23 x \left (2+x^2\right )}{216 \sqrt {2+3 x^2+x^4}}+\frac {x \left (50+23 x^2\right )}{216 \sqrt {2+3 x^2+x^4}}+\frac {625 x \sqrt {2+3 x^2+x^4}}{1008 \left (7+5 x^2\right )^2}+\frac {41875 x \sqrt {2+3 x^2+x^4}}{84672 \left (7+5 x^2\right )}+\frac {23 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{108 \sqrt {2} \sqrt {2+3 x^2+x^4}}-\frac {71 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{144 \sqrt {2} \sqrt {2+3 x^2+x^4}}+\frac {125 \left (2+x^2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{432 \sqrt {2} \sqrt {\frac {2+x^2}{1+x^2}} \sqrt {2+3 x^2+x^4}}+\frac {\int \frac {-4725-4875 x^2}{\sqrt {2+3 x^2+x^4}} \, dx}{84672}-\frac {125 \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx}{3024}-\frac {325 \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx}{6048}-\frac {25}{432} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {1625 \int \frac {2+2 x^2}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx}{12096}-\frac {12625 \int \frac {1}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx}{28224}\\ &=-\frac {149 x \left (2+x^2\right )}{1008 \sqrt {2+3 x^2+x^4}}+\frac {x \left (50+23 x^2\right )}{216 \sqrt {2+3 x^2+x^4}}+\frac {625 x \sqrt {2+3 x^2+x^4}}{1008 \left (7+5 x^2\right )^2}+\frac {41875 x \sqrt {2+3 x^2+x^4}}{84672 \left (7+5 x^2\right )}+\frac {149 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{504 \sqrt {2} \sqrt {2+3 x^2+x^4}}-\frac {1219 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{2016 \sqrt {2} \sqrt {2+3 x^2+x^4}}+\frac {125 \left (2+x^2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{432 \sqrt {2} \sqrt {\frac {2+x^2}{1+x^2}} \sqrt {2+3 x^2+x^4}}-\frac {25}{448} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx-\frac {1625 \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx}{28224}-\frac {12625 \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx}{56448}+\frac {63125 \int \frac {2+2 x^2}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx}{112896}+\frac {\left (1625 \sqrt {1+\frac {x^2}{2}} \sqrt {2+2 x^2}\right ) \int \frac {\sqrt {2+2 x^2}}{\sqrt {1+\frac {x^2}{2}} \left (7+5 x^2\right )} \, dx}{12096 \sqrt {2+3 x^2+x^4}}\\ &=-\frac {5797 x \left (2+x^2\right )}{28224 \sqrt {2+3 x^2+x^4}}+\frac {x \left (50+23 x^2\right )}{216 \sqrt {2+3 x^2+x^4}}+\frac {625 x \sqrt {2+3 x^2+x^4}}{1008 \left (7+5 x^2\right )^2}+\frac {41875 x \sqrt {2+3 x^2+x^4}}{84672 \left (7+5 x^2\right )}+\frac {5797 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{14112 \sqrt {2} \sqrt {2+3 x^2+x^4}}-\frac {49907 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{56448 \sqrt {2} \sqrt {2+3 x^2+x^4}}+\frac {4625 \left (2+x^2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{14112 \sqrt {2} \sqrt {\frac {2+x^2}{1+x^2}} \sqrt {2+3 x^2+x^4}}+\frac {\left (63125 \sqrt {1+\frac {x^2}{2}} \sqrt {2+2 x^2}\right ) \int \frac {\sqrt {2+2 x^2}}{\sqrt {1+\frac {x^2}{2}} \left (7+5 x^2\right )} \, dx}{112896 \sqrt {2+3 x^2+x^4}}\\ &=-\frac {5797 x \left (2+x^2\right )}{28224 \sqrt {2+3 x^2+x^4}}+\frac {x \left (50+23 x^2\right )}{216 \sqrt {2+3 x^2+x^4}}+\frac {625 x \sqrt {2+3 x^2+x^4}}{1008 \left (7+5 x^2\right )^2}+\frac {41875 x \sqrt {2+3 x^2+x^4}}{84672 \left (7+5 x^2\right )}+\frac {5797 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{14112 \sqrt {2} \sqrt {2+3 x^2+x^4}}-\frac {49907 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{56448 \sqrt {2} \sqrt {2+3 x^2+x^4}}+\frac {192625 \left (2+x^2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{395136 \sqrt {2} \sqrt {\frac {2+x^2}{1+x^2}} \sqrt {2+3 x^2+x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.50, size = 159, normalized size = 0.60 \[ \frac {-742 i \sqrt {x^2+1} \sqrt {x^2+2} F\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )+40579 i \sqrt {x^2+1} \sqrt {x^2+2} E\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )+38525 i \sqrt {x^2+1} \sqrt {x^2+2} \Pi \left (\frac {10}{7};\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )+\frac {7 x \left (144925 x^6+698290 x^4+1089803 x^2+550550\right )}{\left (5 x^2+7\right )^2}}{197568 \sqrt {x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{4} + 3 \, x^{2} + 2}}{125 \, x^{14} + 1275 \, x^{12} + 5510 \, x^{10} + 13078 \, x^{8} + 18413 \, x^{6} + 15379 \, x^{4} + 7056 \, x^{2} + 1372}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.02, size = 209, normalized size = 0.79 \[ \frac {625 \sqrt {x^{4}+3 x^{2}+2}\, x}{1008 \left (5 x^{2}+7\right )^{2}}+\frac {41875 \sqrt {x^{4}+3 x^{2}+2}\, x}{84672 \left (5 x^{2}+7\right )}+\frac {5797 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticE \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{56448 \sqrt {x^{4}+3 x^{2}+2}}-\frac {53 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{28224 \sqrt {x^{4}+3 x^{2}+2}}+\frac {38525 i \sqrt {2}\, \sqrt {\frac {x^{2}}{2}+1}\, \sqrt {x^{2}+1}\, \EllipticPi \left (\frac {i \sqrt {2}\, x}{2}, \frac {10}{7}, \sqrt {2}\right )}{197568 \sqrt {x^{4}+3 x^{2}+2}}-\frac {2 \left (-\frac {23}{432} x^{3}-\frac {25}{216} x \right )}{\sqrt {x^{4}+3 x^{2}+2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (5\,x^2+7\right )}^3\,{\left (x^4+3\,x^2+2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\left (x^{2} + 1\right ) \left (x^{2} + 2\right )\right )^{\frac {3}{2}} \left (5 x^{2} + 7\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________