Optimal. Leaf size=57 \[ \frac {421 (6 x+5)}{6 \left (3 x^2+5 x+2\right )}-\frac {139 x+121}{6 \left (3 x^2+5 x+2\right )^2}-421 \log (x+1)+421 \log (3 x+2) \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {777, 614, 616, 31} \begin {gather*} \frac {421 (6 x+5)}{6 \left (3 x^2+5 x+2\right )}-\frac {139 x+121}{6 \left (3 x^2+5 x+2\right )^2}-421 \log (x+1)+421 \log (3 x+2) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 614
Rule 616
Rule 777
Rubi steps
\begin {align*} \int \frac {(5-x) (3+2 x)}{\left (2+5 x+3 x^2\right )^3} \, dx &=-\frac {121+139 x}{6 \left (2+5 x+3 x^2\right )^2}-\frac {421}{6} \int \frac {1}{\left (2+5 x+3 x^2\right )^2} \, dx\\ &=-\frac {121+139 x}{6 \left (2+5 x+3 x^2\right )^2}+\frac {421 (5+6 x)}{6 \left (2+5 x+3 x^2\right )}+421 \int \frac {1}{2+5 x+3 x^2} \, dx\\ &=-\frac {121+139 x}{6 \left (2+5 x+3 x^2\right )^2}+\frac {421 (5+6 x)}{6 \left (2+5 x+3 x^2\right )}+1263 \int \frac {1}{2+3 x} \, dx-1263 \int \frac {1}{3+3 x} \, dx\\ &=-\frac {121+139 x}{6 \left (2+5 x+3 x^2\right )^2}+\frac {421 (5+6 x)}{6 \left (2+5 x+3 x^2\right )}-421 \log (1+x)+421 \log (2+3 x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 57, normalized size = 1.00 \begin {gather*} \frac {421 (6 x+5)}{6 \left (3 x^2+5 x+2\right )}-\frac {139 x+121}{6 \left (3 x^2+5 x+2\right )^2}-421 \log (x+1)+421 \log (3 x+2) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(5-x) (3+2 x)}{\left (2+5 x+3 x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.37, size = 93, normalized size = 1.63 \begin {gather*} \frac {2526 \, x^{3} + 6315 \, x^{2} + 842 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (3 \, x + 2\right ) - 842 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (x + 1\right ) + 5146 \, x + 1363}{2 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.20, size = 46, normalized size = 0.81 \begin {gather*} \frac {2526 \, x^{3} + 6315 \, x^{2} + 5146 \, x + 1363}{2 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{2}} + 421 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - 421 \, \log \left ({\left | x + 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 48, normalized size = 0.84 \begin {gather*} 421 \ln \left (3 x +2\right )-421 \ln \left (x +1\right )-\frac {85}{2 \left (3 x +2\right )^{2}}+\frac {226}{3 x +2}+\frac {3}{\left (x +1\right )^{2}}+\frac {65}{x +1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.48, size = 54, normalized size = 0.95 \begin {gather*} \frac {2526 \, x^{3} + 6315 \, x^{2} + 5146 \, x + 1363}{2 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} + 421 \, \log \left (3 \, x + 2\right ) - 421 \, \log \left (x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.30, size = 45, normalized size = 0.79 \begin {gather*} \frac {\frac {421\,x^3}{3}+\frac {2105\,x^2}{6}+\frac {2573\,x}{9}+\frac {1363}{18}}{x^4+\frac {10\,x^3}{3}+\frac {37\,x^2}{9}+\frac {20\,x}{9}+\frac {4}{9}}-842\,\mathrm {atanh}\left (6\,x+5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.16, size = 51, normalized size = 0.89 \begin {gather*} - \frac {- 2526 x^{3} - 6315 x^{2} - 5146 x - 1363}{18 x^{4} + 60 x^{3} + 74 x^{2} + 40 x + 8} + 421 \log {\left (x + \frac {2}{3} \right )} - 421 \log {\left (x + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________