Optimal. Leaf size=237 \[ \frac {11 \sqrt {x^4+3 x^2+2} x}{2352 \left (5 x^2+7\right )}+\frac {\sqrt {x^4+3 x^2+2} x}{28 \left (5 x^2+7\right )^2}-\frac {11 \left (x^2+2\right ) x}{11760 \sqrt {x^4+3 x^2+2}}+\frac {81 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{7840 \sqrt {2} \sqrt {x^4+3 x^2+2}}+\frac {11 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{5880 \sqrt {2} \sqrt {x^4+3 x^2+2}}-\frac {1201 \left (x^2+2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{164640 \sqrt {2} \sqrt {\frac {x^2+2}{x^2+1}} \sqrt {x^4+3 x^2+2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.60, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {1228, 1223, 1696, 1716, 1189, 1099, 1135, 1214, 1456, 539} \[ \frac {11 \sqrt {x^4+3 x^2+2} x}{2352 \left (5 x^2+7\right )}+\frac {\sqrt {x^4+3 x^2+2} x}{28 \left (5 x^2+7\right )^2}-\frac {11 \left (x^2+2\right ) x}{11760 \sqrt {x^4+3 x^2+2}}+\frac {81 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{7840 \sqrt {2} \sqrt {x^4+3 x^2+2}}+\frac {11 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{5880 \sqrt {2} \sqrt {x^4+3 x^2+2}}-\frac {1201 \left (x^2+2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{164640 \sqrt {2} \sqrt {\frac {x^2+2}{x^2+1}} \sqrt {x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 539
Rule 1099
Rule 1135
Rule 1189
Rule 1214
Rule 1223
Rule 1228
Rule 1456
Rule 1696
Rule 1716
Rubi steps
\begin {align*} \int \frac {\sqrt {2+3 x^2+x^4}}{\left (7+5 x^2\right )^3} \, dx &=\int \left (-\frac {6}{25 \left (7+5 x^2\right )^3 \sqrt {2+3 x^2+x^4}}+\frac {1}{25 \left (7+5 x^2\right )^2 \sqrt {2+3 x^2+x^4}}+\frac {1}{25 \left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}}\right ) \, dx\\ &=\frac {1}{25} \int \frac {1}{\left (7+5 x^2\right )^2 \sqrt {2+3 x^2+x^4}} \, dx+\frac {1}{25} \int \frac {1}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx-\frac {6}{25} \int \frac {1}{\left (7+5 x^2\right )^3 \sqrt {2+3 x^2+x^4}} \, dx\\ &=\frac {x \sqrt {2+3 x^2+x^4}}{28 \left (7+5 x^2\right )^2}-\frac {x \sqrt {2+3 x^2+x^4}}{84 \left (7+5 x^2\right )}+\frac {\int \frac {62+70 x^2+25 x^4}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx}{2100}-\frac {1}{700} \int \frac {74-10 x^2-25 x^4}{\left (7+5 x^2\right )^2 \sqrt {2+3 x^2+x^4}} \, dx+\frac {1}{50} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx-\frac {1}{20} \int \frac {2+2 x^2}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx\\ &=\frac {x \sqrt {2+3 x^2+x^4}}{28 \left (7+5 x^2\right )^2}+\frac {11 x \sqrt {2+3 x^2+x^4}}{2352 \left (7+5 x^2\right )}+\frac {\left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{50 \sqrt {2} \sqrt {2+3 x^2+x^4}}-\frac {\int \frac {2838+2310 x^2+975 x^4}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx}{58800}-\frac {\int \frac {-175-125 x^2}{\sqrt {2+3 x^2+x^4}} \, dx}{52500}+\frac {13 \int \frac {1}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx}{2100}-\frac {\left (\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}{20 \sqrt {2+3 x^2+x^4}}\\ &=\frac {x \sqrt {2+3 x^2+x^4}}{28 \left (7+5 x^2\right )^2}+\frac {11 x \sqrt {2+3 x^2+x^4}}{2352 \left (7+5 x^2\right )}+\frac {\left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{50 \sqrt {2} \sqrt {2+3 x^2+x^4}}-\frac {\left (2+x^2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{70 \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}{1470000}+\frac {1}{420} \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {13 \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx}{4200}+\frac {1}{300} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx-\frac {13 \int \frac {2+2 x^2}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx}{1680}-\frac {101 \int \frac {1}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx}{3920}\\ &=\frac {x \left (2+x^2\right )}{420 \sqrt {2+3 x^2+x^4}}+\frac {x \sqrt {2+3 x^2+x^4}}{28 \left (7+5 x^2\right )^2}+\frac {11 x \sqrt {2+3 x^2+x^4}}{2352 \left (7+5 x^2\right )}-\frac {\left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{210 \sqrt {2} \sqrt {2+3 x^2+x^4}}+\frac {37 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{1400 \sqrt {2} \sqrt {2+3 x^2+x^4}}-\frac {\left (2+x^2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{70 \sqrt {2} \sqrt {\frac {2+x^2}{1+x^2}} \sqrt {2+3 x^2+x^4}}-\frac {9 \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx}{2800}-\frac {13 \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx}{3920}-\frac {101 \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx}{7840}+\frac {101 \int \frac {2+2 x^2}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx}{3136}-\frac {\left (13 \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}{1680 \sqrt {2+3 x^2+x^4}}\\ &=-\frac {11 x \left (2+x^2\right )}{11760 \sqrt {2+3 x^2+x^4}}+\frac {x \sqrt {2+3 x^2+x^4}}{28 \left (7+5 x^2\right )^2}+\frac {11 x \sqrt {2+3 x^2+x^4}}{2352 \left (7+5 x^2\right )}+\frac {11 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{5880 \sqrt {2} \sqrt {2+3 x^2+x^4}}+\frac {81 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{7840 \sqrt {2} \sqrt {2+3 x^2+x^4}}-\frac {97 \left (2+x^2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{5880 \sqrt {2} \sqrt {\frac {2+x^2}{1+x^2}} \sqrt {2+3 x^2+x^4}}+\frac {\left (101 \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}{3136 \sqrt {2+3 x^2+x^4}}\\ &=-\frac {11 x \left (2+x^2\right )}{11760 \sqrt {2+3 x^2+x^4}}+\frac {x \sqrt {2+3 x^2+x^4}}{28 \left (7+5 x^2\right )^2}+\frac {11 x \sqrt {2+3 x^2+x^4}}{2352 \left (7+5 x^2\right )}+\frac {11 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{5880 \sqrt {2} \sqrt {2+3 x^2+x^4}}+\frac {81 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{7840 \sqrt {2} \sqrt {2+3 x^2+x^4}}-\frac {1201 \left (2+x^2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{164640 \sqrt {2} \sqrt {\frac {2+x^2}{1+x^2}} \sqrt {2+3 x^2+x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.35, size = 174, normalized size = 0.73 \[ \frac {-434 i \sqrt {x^2+1} \sqrt {x^2+2} F\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )+385 i \sqrt {x^2+1} \sqrt {x^2+2} E\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-1201 i \sqrt {x^2+1} \sqrt {x^2+2} \Pi \left (\frac {10}{7};\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )+\frac {1925 x \left (x^4+3 x^2+2\right )}{5 x^2+7}+\frac {14700 x \left (x^4+3 x^2+2\right )}{\left (5 x^2+7\right )^2}}{411600 \sqrt {x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{4} + 3 \, x^{2} + 2}}{125 \, x^{6} + 525 \, x^{4} + 735 \, x^{2} + 343}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x^{4} + 3 \, x^{2} + 2}}{{\left (5 \, x^{2} + 7\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.02, size = 186, normalized size = 0.78 \[ \frac {\sqrt {x^{4}+3 x^{2}+2}\, x}{28 \left (5 x^{2}+7\right )^{2}}+\frac {11 \sqrt {x^{4}+3 x^{2}+2}\, x}{2352 \left (5 x^{2}+7\right )}+\frac {11 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticE \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{23520 \sqrt {x^{4}+3 x^{2}+2}}-\frac {31 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{58800 \sqrt {x^{4}+3 x^{2}+2}}-\frac {1201 i \sqrt {2}\, \sqrt {\frac {x^{2}}{2}+1}\, \sqrt {x^{2}+1}\, \EllipticPi \left (\frac {i \sqrt {2}\, x}{2}, \frac {10}{7}, \sqrt {2}\right )}{411600 \sqrt {x^{4}+3 x^{2}+2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x^{4} + 3 \, x^{2} + 2}}{{\left (5 \, x^{2} + 7\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {x^4+3\,x^2+2}}{{\left (5\,x^2+7\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\left (x^{2} + 1\right ) \left (x^{2} + 2\right )}}{\left (5 x^{2} + 7\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________