Optimal. Leaf size=111 \[ -\frac {3 (47 x+37)}{5 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )}-\frac {24626}{625 \sqrt {2 x+3}}-\frac {7042}{375 (2 x+3)^{3/2}}-\frac {2114}{125 (2 x+3)^{5/2}}+14 \tanh ^{-1}\left (\sqrt {2 x+3}\right )+\frac {15876}{625} \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {822, 828, 826, 1166, 207} \begin {gather*} -\frac {3 (47 x+37)}{5 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )}-\frac {24626}{625 \sqrt {2 x+3}}-\frac {7042}{375 (2 x+3)^{3/2}}-\frac {2114}{125 (2 x+3)^{5/2}}+14 \tanh ^{-1}\left (\sqrt {2 x+3}\right )+\frac {15876}{625} \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 822
Rule 826
Rule 828
Rule 1166
Rubi steps
\begin {align*} \int \frac {5-x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^2} \, dx &=-\frac {3 (37+47 x)}{5 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}-\frac {1}{5} \int \frac {952+987 x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac {2114}{125 (3+2 x)^{5/2}}-\frac {3 (37+47 x)}{5 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}-\frac {1}{25} \int \frac {2996+3171 x}{(3+2 x)^{5/2} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac {2114}{125 (3+2 x)^{5/2}}-\frac {7042}{375 (3+2 x)^{3/2}}-\frac {3 (37+47 x)}{5 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}-\frac {1}{125} \int \frac {9688+10563 x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac {2114}{125 (3+2 x)^{5/2}}-\frac {7042}{375 (3+2 x)^{3/2}}-\frac {24626}{625 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{5 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}-\frac {1}{625} \int \frac {32564+36939 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac {2114}{125 (3+2 x)^{5/2}}-\frac {7042}{375 (3+2 x)^{3/2}}-\frac {24626}{625 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{5 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}-\frac {2}{625} \operatorname {Subst}\left (\int \frac {-45689+36939 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt {3+2 x}\right )\\ &=-\frac {2114}{125 (3+2 x)^{5/2}}-\frac {7042}{375 (3+2 x)^{3/2}}-\frac {24626}{625 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{5 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}-42 \operatorname {Subst}\left (\int \frac {1}{-3+3 x^2} \, dx,x,\sqrt {3+2 x}\right )-\frac {47628}{625} \operatorname {Subst}\left (\int \frac {1}{-5+3 x^2} \, dx,x,\sqrt {3+2 x}\right )\\ &=-\frac {2114}{125 (3+2 x)^{5/2}}-\frac {7042}{375 (3+2 x)^{3/2}}-\frac {24626}{625 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{5 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}+14 \tanh ^{-1}\left (\sqrt {3+2 x}\right )+\frac {15876}{625} \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.20, size = 86, normalized size = 0.77 \begin {gather*} \frac {47628 \sqrt {15} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )-\frac {5 \left (886536 x^4+4348428 x^3+7782530 x^2+5977997 x+1646109\right )}{(2 x+3)^{5/2} \left (3 x^2+5 x+2\right )}}{9375}+14 \tanh ^{-1}\left (\sqrt {2 x+3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.20, size = 111, normalized size = 1.00 \begin {gather*} -\frac {2 \left (110817 (2 x+3)^4-242697 (2 x+3)^3+91420 (2 x+3)^2+14060 (2 x+3)+3900\right )}{1875 (2 x+3)^{5/2} \left (3 (2 x+3)^2-8 (2 x+3)+5\right )}+14 \tanh ^{-1}\left (\sqrt {2 x+3}\right )+\frac {15876}{625} \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 = 0.41, size = 194, normalized size = 1.75 \begin {gather*} \frac {23814 \, \sqrt {5} \sqrt {3} {\left (24 \, x^{5} + 148 \, x^{4} + 358 \, x^{3} + 423 \, x^{2} + 243 \, x + 54\right )} \log \left (\frac {\sqrt {5} \sqrt {3} \sqrt {2 \, x + 3} + 3 \, x + 7}{3 \, x + 2}\right ) + 65625 \, {\left (24 \, x^{5} + 148 \, x^{4} + 358 \, x^{3} + 423 \, x^{2} + 243 \, x + 54\right )} \log \left (\sqrt {2 \, x + 3} + 1\right ) - 65625 \, {\left (24 \, x^{5} + 148 \, x^{4} + 358 \, x^{3} + 423 \, x^{2} + 243 \, x + 54\right )} \log \left (\sqrt {2 \, x + 3} - 1\right ) - 5 \, {\left (886536 \, x^{4} + 4348428 \, x^{3} + 7782530 \, x^{2} + 5977997 \, x + 1646109\right )} \sqrt {2 \, x + 3}}{9375 \, {\left (24 \, x^{5} + 148 \, x^{4} + 358 \, x^{3} + 423 \, x^{2} + 243 \, x + 54\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.20, size = 125, normalized size = 1.13 \begin {gather*} -\frac {7938}{3125} \, \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 {6 \, {\left (4209 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 6709 \, \sqrt {2 \, x + 3}\right )}}{625 \, {\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}} - \frac {16 \, {\left (3039 \, {\left (2 \, x + 3\right )}^{2} + 1015 \, x + 1620\right )}}{1875 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}}} + 7 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 7 \, \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 = 113, normalized size = 1.02 \begin {gather*} \frac {15876 \sqrt {15}\, \arctanh \left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{3125}-7 \ln \left (-1+\sqrt {2 x +3}\right )+7 \ln \left (\sqrt {2 x +3}+1\right )-\frac {918 \sqrt {2 x +3}}{625 \left (2 x +\frac {4}{3}\right )}-\frac {6}{\sqrt {2 x +3}+1}-\frac {104}{125 \left (2 x +3\right )^{\frac {5}{2}}}-\frac {1624}{375 \left (2 x +3\right )^{\frac {3}{2}}}-\frac {16208}{625 \sqrt {2 x +3}}-\frac {6}{-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 = 125, normalized size = 1.13 \begin {gather*} -\frac {7938}{3125} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) - \frac {2 \, {\left (110817 \, {\left (2 \, x + 3\right )}^{4} - 242697 \, {\left (2 \, x + 3\right )}^{3} + 91420 \, {\left (2 \, x + 3\right )}^{2} + 28120 \, x + 46080\right )}}{1875 \, {\left (3 \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} - 8 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + 5 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}}\right )}} + 7 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 7 \, \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 = 91, normalized size = 0.82 \begin {gather*} 14\,\mathrm {atanh}\left (\sqrt {2\,x+3}\right )+\frac {15876\,\sqrt {15}\,\mathrm {atanh}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}}{5}\right )}{3125}-\frac {\frac {11248\,x}{1125}+\frac {36568\,{\left (2\,x+3\right )}^2}{1125}-\frac {161798\,{\left (2\,x+3\right )}^3}{1875}+\frac {24626\,{\left (2\,x+3\right )}^4}{625}+\frac {2048}{125}}{\frac {5\,{\left (2\,x+3\right )}^{5/2}}{3}-\frac {8\,{\left (2\,x+3\right )}^{7/2}}{3}+{\left (2\,x+3\right )}^{9/2}} \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]
________________________________________________________________________________________