Optimal. Leaf size=222 \[ -\frac {3 \sqrt {x^4+3 x^2+2} x}{175 \left (5 x^2+7\right )}+\frac {1}{75} \sqrt {x^4+3 x^2+2} x+\frac {9 \left (x^2+2\right ) x}{175 \sqrt {x^4+3 x^2+2}}+\frac {59 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{2 x^2+2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{1050 \sqrt {x^4+3 x^2+2}}-\frac {9 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{175 \sqrt {x^4+3 x^2+2}}+\frac {9 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{2 x^2+2}} \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{2450 \sqrt {x^4+3 x^2+2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.44, antiderivative size = 333, normalized size of antiderivative = 1.50, number of steps used = 21, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {1228, 1099, 1135, 1122, 1189, 1223, 1716, 1214, 1456, 539} \[ -\frac {3 \sqrt {x^4+3 x^2+2} x}{175 \left (5 x^2+7\right )}+\frac {1}{75} \sqrt {x^4+3 x^2+2} x+\frac {9 \left (x^2+2\right ) x}{175 \sqrt {x^4+3 x^2+2}}+\frac {44 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{1875 \sqrt {x^4+3 x^2+2}}+\frac {81 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{8750 \sqrt {2} \sqrt {x^4+3 x^2+2}}-\frac {9 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{175 \sqrt {x^4+3 x^2+2}}+\frac {3 \sqrt {2} \left (x^2+2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{875 \sqrt {\frac {x^2+2}{x^2+1}} \sqrt {x^4+3 x^2+2}}-\frac {39 \left (x^2+2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{12250 \sqrt {2} \sqrt {\frac {x^2+2}{x^2+1}} \sqrt {x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 539
Rule 1099
Rule 1122
Rule 1135
Rule 1189
Rule 1214
Rule 1223
Rule 1228
Rule 1456
Rule 1716
Rubi steps
\begin {align*} \int \frac {\left (2+3 x^2+x^4\right )^{3/2}}{\left (7+5 x^2\right )^2} \, dx &=\int \left (\frac {52}{625 \sqrt {2+3 x^2+x^4}}+\frac {16 x^2}{125 \sqrt {2+3 x^2+x^4}}+\frac {x^4}{25 \sqrt {2+3 x^2+x^4}}+\frac {36}{625 \left (7+5 x^2\right )^2 \sqrt {2+3 x^2+x^4}}-\frac {12}{625 \left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}}\right ) \, dx\\ &=-\left (\frac {12}{625} \int \frac {1}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx\right )+\frac {1}{25} \int \frac {x^4}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {36}{625} \int \frac {1}{\left (7+5 x^2\right )^2 \sqrt {2+3 x^2+x^4}} \, dx+\frac {52}{625} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {16}{125} \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx\\ &=\frac {16 x \left (2+x^2\right )}{125 \sqrt {2+3 x^2+x^4}}+\frac {1}{75} x \sqrt {2+3 x^2+x^4}-\frac {3 x \sqrt {2+3 x^2+x^4}}{175 \left (7+5 x^2\right )}-\frac {16 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{125 \sqrt {2+3 x^2+x^4}}+\frac {26 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{625 \sqrt {2+3 x^2+x^4}}+\frac {3 \int \frac {62+70 x^2+25 x^4}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx}{4375}-\frac {6}{625} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx-\frac {1}{75} \int \frac {2+6 x^2}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {3}{125} \int \frac {2+2 x^2}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx\\ &=\frac {16 x \left (2+x^2\right )}{125 \sqrt {2+3 x^2+x^4}}+\frac {1}{75} x \sqrt {2+3 x^2+x^4}-\frac {3 x \sqrt {2+3 x^2+x^4}}{175 \left (7+5 x^2\right )}-\frac {16 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{125 \sqrt {2+3 x^2+x^4}}+\frac {23 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{625 \sqrt {2+3 x^2+x^4}}-\frac {3 \int \frac {-175-125 x^2}{\sqrt {2+3 x^2+x^4}} \, dx}{109375}+\frac {39 \int \frac {1}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx}{4375}-\frac {2}{75} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx-\frac {2}{25} \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {\left (3 \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}{125 \sqrt {2+3 x^2+x^4}}\\ &=\frac {6 x \left (2+x^2\right )}{125 \sqrt {2+3 x^2+x^4}}+\frac {1}{75} x \sqrt {2+3 x^2+x^4}-\frac {3 x \sqrt {2+3 x^2+x^4}}{175 \left (7+5 x^2\right )}-\frac {6 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{125 \sqrt {2+3 x^2+x^4}}+\frac {44 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{1875 \sqrt {2+3 x^2+x^4}}+\frac {3 \sqrt {2} \left (2+x^2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{875 \sqrt {\frac {2+x^2}{1+x^2}} \sqrt {2+3 x^2+x^4}}+\frac {3}{875} \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {39 \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx}{8750}+\frac {3}{625} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx-\frac {39 \int \frac {2+2 x^2}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx}{3500}\\ &=\frac {9 x \left (2+x^2\right )}{175 \sqrt {2+3 x^2+x^4}}+\frac {1}{75} x \sqrt {2+3 x^2+x^4}-\frac {3 x \sqrt {2+3 x^2+x^4}}{175 \left (7+5 x^2\right )}-\frac {9 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{175 \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 )}{8750 \sqrt {2} \sqrt {2+3 x^2+x^4}}+\frac {44 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{1875 \sqrt {2+3 x^2+x^4}}+\frac {3 \sqrt {2} \left (2+x^2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{875 \sqrt {\frac {2+x^2}{1+x^2}} \sqrt {2+3 x^2+x^4}}-\frac {\left (39 \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}{3500 \sqrt {2+3 x^2+x^4}}\\ &=\frac {9 x \left (2+x^2\right )}{175 \sqrt {2+3 x^2+x^4}}+\frac {1}{75} x \sqrt {2+3 x^2+x^4}-\frac {3 x \sqrt {2+3 x^2+x^4}}{175 \left (7+5 x^2\right )}-\frac {9 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{175 \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 )}{8750 \sqrt {2} \sqrt {2+3 x^2+x^4}}+\frac {44 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{1875 \sqrt {2+3 x^2+x^4}}-\frac {39 \left (2+x^2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{12250 \sqrt {2} \sqrt {\frac {2+x^2}{1+x^2}} \sqrt {2+3 x^2+x^4}}+\frac {3 \sqrt {2} \left (2+x^2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{875 \sqrt {\frac {2+x^2}{1+x^2}} \sqrt {2+3 x^2+x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.29, size = 213, normalized size = 0.96 \[ \frac {1225 x^7+5075 x^5+6650 x^3-182 i \sqrt {x^2+1} \sqrt {x^2+2} \left (5 x^2+7\right ) F\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-945 i \sqrt {x^2+1} \sqrt {x^2+2} \left (5 x^2+7\right ) E\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )+135 i \sqrt {x^2+1} \sqrt {x^2+2} x^2 \Pi \left (\frac {10}{7};\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )+189 i \sqrt {x^2+1} \sqrt {x^2+2} \Pi \left (\frac {10}{7};\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )+2800 x}{18375 \left (5 x^2+7\right ) \sqrt {x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{25 \, x^{4} + 70 \, x^{2} + 49}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{{\left (5 \, x^{2} + 7\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.02, size = 177, normalized size = 0.80 \[ -\frac {3 \sqrt {x^{4}+3 x^{2}+2}\, x}{175 \left (5 x^{2}+7\right )}+\frac {\sqrt {x^{4}+3 x^{2}+2}\, x}{75}-\frac {9 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticE \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{350 \sqrt {x^{4}+3 x^{2}+2}}-\frac {13 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{2625 \sqrt {x^{4}+3 x^{2}+2}}+\frac {9 i \sqrt {2}\, \sqrt {\frac {x^{2}}{2}+1}\, \sqrt {x^{2}+1}\, \EllipticPi \left (\frac {i \sqrt {2}\, x}{2}, \frac {10}{7}, \sqrt {2}\right )}{6125 \sqrt {x^{4}+3 x^{2}+2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{{\left (5 \, x^{2} + 7\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (x^4+3\,x^2+2\right )}^{3/2}}{{\left (5\,x^2+7\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\left (x^{2} + 1\right ) \left (x^{2} + 2\right )\right )^{\frac {3}{2}}}{\left (5 x^{2} + 7\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________