Optimal. Leaf size=100 \[ -\frac {\sqrt {2 x+3} (139 x+121)}{6 \left (3 x^2+5 x+2\right )^2}+\frac {\sqrt {2 x+3} (2529 x+2090)}{6 \left (3 x^2+5 x+2\right )}+966 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-1247 \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {818, 822, 826, 1166, 207} \begin {gather*} -\frac {\sqrt {2 x+3} (139 x+121)}{6 \left (3 x^2+5 x+2\right )^2}+\frac {\sqrt {2 x+3} (2529 x+2090)}{6 \left (3 x^2+5 x+2\right )}+966 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-1247 \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 207
Rule 818
Rule 822
Rule 826
Rule 1166
Rubi steps
\begin {align*} \int \frac {(5-x) (3+2 x)^{3/2}}{\left (2+5 x+3 x^2\right )^3} \, dx &=-\frac {\sqrt {3+2 x} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {1}{6} \int \frac {-1142-703 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )^2} \, dx\\ &=-\frac {\sqrt {3+2 x} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {\sqrt {3+2 x} (2090+2529 x)}{6 \left (2+5 x+3 x^2\right )}-\frac {1}{30} \int \frac {-27135-12645 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac {\sqrt {3+2 x} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {\sqrt {3+2 x} (2090+2529 x)}{6 \left (2+5 x+3 x^2\right )}-\frac {1}{15} \operatorname {Subst}\left (\int \frac {-16335-12645 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt {3+2 x}\right )\\ &=-\frac {\sqrt {3+2 x} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {\sqrt {3+2 x} (2090+2529 x)}{6 \left (2+5 x+3 x^2\right )}-2898 \operatorname {Subst}\left (\int \frac {1}{-3+3 x^2} \, dx,x,\sqrt {3+2 x}\right )+3741 \operatorname {Subst}\left (\int \frac {1}{-5+3 x^2} \, dx,x,\sqrt {3+2 x}\right )\\ &=-\frac {\sqrt {3+2 x} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {\sqrt {3+2 x} (2090+2529 x)}{6 \left (2+5 x+3 x^2\right )}+966 \tanh ^{-1}\left (\sqrt {3+2 x}\right )-1247 \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 80, normalized size = 0.80 \begin {gather*} \frac {\sqrt {2 x+3} \left (2529 x^3+6305 x^2+5123 x+1353\right )}{2 \left (3 x^2+5 x+2\right )^2}+966 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-1247 \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.30, size = 97, normalized size = 0.97 \begin {gather*} \frac {\sqrt {2 x+3} \left (2529 (2 x+3)^3-10151 (2 x+3)^2+13115 (2 x+3)-5445\right )}{\left (3 (2 x+3)^2-8 (2 x+3)+5\right )^2}+966 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-1247 \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.40, size = 170, normalized size = 1.70 \begin {gather*} \frac {1247 \, \sqrt {5} \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (-\frac {\sqrt {5} \sqrt {3} \sqrt {2 \, x + 3} - 3 \, x - 7}{3 \, x + 2}\right ) + 4830 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt {2 \, x + 3} + 1\right ) - 4830 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt {2 \, x + 3} - 1\right ) + 5 \, {\left (2529 \, x^{3} + 6305 \, x^{2} + 5123 \, x + 1353\right )} \sqrt {2 \, x + 3}}{10 \, {\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 = 0.18, size = 119, normalized size = 1.19 \begin {gather*} \frac {1247}{10} \, \sqrt {15} \log \left (\frac {{\left | -2 \, \sqrt {15} + 6 \, \sqrt {2 \, x + 3} \right |}}{2 \, {\left (\sqrt {15} + 3 \, \sqrt {2 \, x + 3}\right )}}\right ) + \frac {2529 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - 10151 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + 13115 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 5445 \, \sqrt {2 \, x + 3}}{{\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}^{2}} + 483 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 483 \, \log \left ({\left | \sqrt {2 \, x + 3} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 124, normalized size = 1.24 \begin {gather*} -\frac {1247 \sqrt {15}\, \arctanh \left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{5}-483 \ln \left (-1+\sqrt {2 x +3}\right )+483 \ln \left (\sqrt {2 x +3}+1\right )+\frac {1305 \left (2 x +3\right )^{\frac {3}{2}}-2345 \sqrt {2 x +3}}{\left (6 x +4\right )^{2}}-\frac {3}{\left (\sqrt {2 x +3}+1\right )^{2}}+\frac {68}{\sqrt {2 x +3}+1}+\frac {3}{\left (-1+\sqrt {2 x +3}\right )^{2}}+\frac {68}{-1+\sqrt {2 x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.31, size = 133, normalized size = 1.33 \begin {gather*} \frac {1247}{10} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) + \frac {2529 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - 10151 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + 13115 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 5445 \, \sqrt {2 \, x + 3}}{9 \, {\left (2 \, x + 3\right )}^{4} - 48 \, {\left (2 \, x + 3\right )}^{3} + 94 \, {\left (2 \, x + 3\right )}^{2} - 160 \, x - 215} + 483 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 483 \, \log \left (\sqrt {2 \, x + 3} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.07, size = 101, normalized size = 1.01 \begin {gather*} 966\,\mathrm {atanh}\left (\sqrt {2\,x+3}\right )+\frac {605\,\sqrt {2\,x+3}-\frac {13115\,{\left (2\,x+3\right )}^{3/2}}{9}+\frac {10151\,{\left (2\,x+3\right )}^{5/2}}{9}-281\,{\left (2\,x+3\right )}^{7/2}}{\frac {160\,x}{9}-\frac {94\,{\left (2\,x+3\right )}^2}{9}+\frac {16\,{\left (2\,x+3\right )}^3}{3}-{\left (2\,x+3\right )}^4+\frac {215}{9}}-\frac {1247\,\sqrt {15}\,\mathrm {atanh}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}}{5}\right )}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________