Optimal. Leaf size=61 \[ \frac {5 x^6}{6}-\frac {27 x^4}{4}+49 x^2-\frac {5}{2} \log \left (x^2+1\right )-144 \log \left (x^2+2\right )-\frac {207 x^2+206}{2 \left (x^4+3 x^2+2\right )} \]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {1663, 1660, 1657, 632, 31} \begin {gather*} \frac {5 x^6}{6}-\frac {27 x^4}{4}+49 x^2-\frac {207 x^2+206}{2 \left (x^4+3 x^2+2\right )}-\frac {5}{2} \log \left (x^2+1\right )-144 \log \left (x^2+2\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 632
Rule 1657
Rule 1660
Rule 1663
Rubi steps
\begin {align*} \int \frac {x^7 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^3 \left (4+x+3 x^2+5 x^3\right )}{\left (2+3 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=-\frac {206+207 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {102+53 x-27 x^2+12 x^3-5 x^4}{2+3 x+x^2} \, dx,x,x^2\right )\\ &=-\frac {206+207 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac {1}{2} \operatorname {Subst}\left (\int \left (-98+27 x-5 x^2+\frac {298+293 x}{2+3 x+x^2}\right ) \, dx,x,x^2\right )\\ &=49 x^2-\frac {27 x^4}{4}+\frac {5 x^6}{6}-\frac {206+207 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {298+293 x}{2+3 x+x^2} \, dx,x,x^2\right )\\ &=49 x^2-\frac {27 x^4}{4}+\frac {5 x^6}{6}-\frac {206+207 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac {5}{2} \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,x^2\right )-144 \operatorname {Subst}\left (\int \frac {1}{2+x} \, dx,x,x^2\right )\\ &=49 x^2-\frac {27 x^4}{4}+\frac {5 x^6}{6}-\frac {206+207 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac {5}{2} \log \left (1+x^2\right )-144 \log \left (2+x^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 61, normalized size = 1.00 \begin {gather*} \frac {5 x^6}{6}-\frac {27 x^4}{4}+49 x^2-\frac {5}{2} \log \left (x^2+1\right )-144 \log \left (x^2+2\right )+\frac {-207 x^2-206}{2 \left (x^4+3 x^2+2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^7 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.64, size = 77, normalized size = 1.26 \begin {gather*} \frac {10 \, x^{10} - 51 \, x^{8} + 365 \, x^{6} + 1602 \, x^{4} - 66 \, x^{2} - 1728 \, {\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 2\right ) - 30 \, {\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 1\right ) - 1236}{12 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.32, size = 58, normalized size = 0.95 \begin {gather*} \frac {5}{6} \, x^{6} - \frac {27}{4} \, x^{4} + 49 \, x^{2} + \frac {293 \, x^{4} + 465 \, x^{2} + 174}{4 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} - 144 \, \log \left (x^{2} + 2\right ) - \frac {5}{2} \, \log \left (x^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 51, normalized size = 0.84 \begin {gather*} \frac {5 x^{6}}{6}-\frac {27 x^{4}}{4}+49 x^{2}-\frac {5 \ln \left (x^{2}+1\right )}{2}-144 \ln \left (x^{2}+2\right )+\frac {1}{2 x^{2}+2}-\frac {104}{x^{2}+2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.72, size = 53, normalized size = 0.87 \begin {gather*} \frac {5}{6} \, x^{6} - \frac {27}{4} \, x^{4} + 49 \, x^{2} - \frac {207 \, x^{2} + 206}{2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} - 144 \, \log \left (x^{2} + 2\right ) - \frac {5}{2} \, \log \left (x^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 53, normalized size = 0.87 \begin {gather*} 49\,x^2-144\,\ln \left (x^2+2\right )-\frac {\frac {207\,x^2}{2}+103}{x^4+3\,x^2+2}-\frac {5\,\ln \left (x^2+1\right )}{2}-\frac {27\,x^4}{4}+\frac {5\,x^6}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.17, size = 56, normalized size = 0.92 \begin {gather*} \frac {5 x^{6}}{6} - \frac {27 x^{4}}{4} + 49 x^{2} + \frac {- 207 x^{2} - 206}{2 x^{4} + 6 x^{2} + 4} - \frac {5 \log {\left (x^{2} + 1 \right )}}{2} - 144 \log {\left (x^{2} + 2 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________