Optimal. Leaf size=81 \[ -\frac {1194}{125 \sqrt {2 x+3}}-\frac {66}{25 (2 x+3)^{3/2}}-\frac {26}{25 (2 x+3)^{5/2}}+12 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {306}{125} \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {828, 826, 1166, 207} \begin {gather*} -\frac {1194}{125 \sqrt {2 x+3}}-\frac {66}{25 (2 x+3)^{3/2}}-\frac {26}{25 (2 x+3)^{5/2}}+12 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {306}{125} \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 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 )} \, dx &=-\frac {26}{25 (3+2 x)^{5/2}}+\frac {1}{5} \int \frac {-9-39 x}{(3+2 x)^{5/2} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac {26}{25 (3+2 x)^{5/2}}-\frac {66}{25 (3+2 x)^{3/2}}+\frac {1}{25} \int \frac {-147-297 x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac {26}{25 (3+2 x)^{5/2}}-\frac {66}{25 (3+2 x)^{3/2}}-\frac {1194}{125 \sqrt {3+2 x}}+\frac {1}{125} \int \frac {-1041-1791 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac {26}{25 (3+2 x)^{5/2}}-\frac {66}{25 (3+2 x)^{3/2}}-\frac {1194}{125 \sqrt {3+2 x}}+\frac {2}{125} \operatorname {Subst}\left (\int \frac {3291-1791 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt {3+2 x}\right )\\ &=-\frac {26}{25 (3+2 x)^{5/2}}-\frac {66}{25 (3+2 x)^{3/2}}-\frac {1194}{125 \sqrt {3+2 x}}+\frac {918}{125} \operatorname {Subst}\left (\int \frac {1}{-5+3 x^2} \, dx,x,\sqrt {3+2 x}\right )-36 \operatorname {Subst}\left (\int \frac {1}{-3+3 x^2} \, dx,x,\sqrt {3+2 x}\right )\\ &=-\frac {26}{25 (3+2 x)^{5/2}}-\frac {66}{25 (3+2 x)^{3/2}}-\frac {1194}{125 \sqrt {3+2 x}}+12 \tanh ^{-1}\left (\sqrt {3+2 x}\right )-\frac {306}{125} \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 63, normalized size = 0.78 \begin {gather*} \frac {2}{625} \left (-\frac {5 \left (2388 x^2+7494 x+5933\right )}{(2 x+3)^{5/2}}+3750 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-153 \sqrt {15} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.14, size = 73, normalized size = 0.90 \begin {gather*} -\frac {2 \left (597 (2 x+3)^2+165 (2 x+3)+65\right )}{125 (2 x+3)^{5/2}}+12 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {306}{125} \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 = 145, normalized size = 1.79 \begin {gather*} \frac {153 \, \sqrt {5} \sqrt {3} {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (-\frac {\sqrt {5} \sqrt {3} \sqrt {2 \, x + 3} - 3 \, x - 7}{3 \, x + 2}\right ) + 3750 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (\sqrt {2 \, x + 3} + 1\right ) - 3750 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (\sqrt {2 \, x + 3} - 1\right ) - 10 \, {\left (2388 \, x^{2} + 7494 \, x + 5933\right )} \sqrt {2 \, x + 3}}{625 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 88, normalized size = 1.09 \begin {gather*} \frac {153}{625} \, \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 {2 \, {\left (597 \, {\left (2 \, x + 3\right )}^{2} + 330 \, x + 560\right )}}{125 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}}} + 6 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 6 \, \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 = 71, normalized size = 0.88 \begin {gather*} -\frac {306 \sqrt {15}\, \arctanh \left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{625}-6 \ln \left (-1+\sqrt {2 x +3}\right )+6 \ln \left (\sqrt {2 x +3}+1\right )-\frac {26}{25 \left (2 x +3\right )^{\frac {5}{2}}}-\frac {66}{25 \left (2 x +3\right )^{\frac {3}{2}}}-\frac {1194}{125 \sqrt {2 x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.04, size = 84, normalized size = 1.04 \begin {gather*} \frac {153}{625} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) - \frac {2 \, {\left (597 \, {\left (2 \, x + 3\right )}^{2} + 330 \, x + 560\right )}}{125 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}}} + 6 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 6 \, \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 = 52, normalized size = 0.64 \begin {gather*} 12\,\mathrm {atanh}\left (\sqrt {2\,x+3}\right )-\frac {\frac {132\,x}{25}+\frac {1194\,{\left (2\,x+3\right )}^2}{125}+\frac {224}{25}}{{\left (2\,x+3\right )}^{5/2}}-\frac {306\,\sqrt {15}\,\mathrm {atanh}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}}{5}\right )}{625} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 94.03, size = 126, normalized size = 1.56 \begin {gather*} \frac {918 \left (\begin {cases} - \frac {\sqrt {15} \operatorname {acoth}{\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} \right )}}{15} & \text {for}\: 2 x + 3 > \frac {5}{3} \\- \frac {\sqrt {15} \operatorname {atanh}{\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} \right )}}{15} & \text {for}\: 2 x + 3 < \frac {5}{3} \end {cases}\right )}{125} - 6 \log {\left (\sqrt {2 x + 3} - 1 \right )} + 6 \log {\left (\sqrt {2 x + 3} + 1 \right )} - \frac {1194}{125 \sqrt {2 x + 3}} - \frac {66}{25 \left (2 x + 3\right )^{\frac {3}{2}}} - \frac {26}{25 \left (2 x + 3\right )^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________