Optimal. Leaf size=68 \[ -\frac {2}{9} (2 x+3)^{3/2}+\frac {62}{9} \sqrt {2 x+3}+12 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {170}{9} \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {824, 826, 1166, 207} \begin {gather*} -\frac {2}{9} (2 x+3)^{3/2}+\frac {62}{9} \sqrt {2 x+3}+12 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {170}{9} \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 207
Rule 824
Rule 826
Rule 1166
Rubi steps
\begin {align*} \int \frac {(5-x) (3+2 x)^{3/2}}{2+5 x+3 x^2} \, dx &=-\frac {2}{9} (3+2 x)^{3/2}+\frac {1}{3} \int \frac {\sqrt {3+2 x} (49+31 x)}{2+5 x+3 x^2} \, dx\\ &=\frac {62}{9} \sqrt {3+2 x}-\frac {2}{9} (3+2 x)^{3/2}+\frac {1}{9} \int \frac {317+263 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx\\ &=\frac {62}{9} \sqrt {3+2 x}-\frac {2}{9} (3+2 x)^{3/2}+\frac {2}{9} \operatorname {Subst}\left (\int \frac {-155+263 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt {3+2 x}\right )\\ &=\frac {62}{9} \sqrt {3+2 x}-\frac {2}{9} (3+2 x)^{3/2}-36 \operatorname {Subst}\left (\int \frac {1}{-3+3 x^2} \, dx,x,\sqrt {3+2 x}\right )+\frac {850}{9} \operatorname {Subst}\left (\int \frac {1}{-5+3 x^2} \, dx,x,\sqrt {3+2 x}\right )\\ &=\frac {62}{9} \sqrt {3+2 x}-\frac {2}{9} (3+2 x)^{3/2}+12 \tanh ^{-1}\left (\sqrt {3+2 x}\right )-\frac {170}{9} \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 56, normalized size = 0.82 \begin {gather*} -\frac {2}{27} \left (6 \sqrt {2 x+3} (x-14)-162 \tanh ^{-1}\left (\sqrt {2 x+3}\right )+85 \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 = 0.12, size = 60, normalized size = 0.88 \begin {gather*} -\frac {2}{9} \sqrt {2 x+3} (2 x-28)+12 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {170}{9} \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 74, normalized size = 1.09 \begin {gather*} \frac {85}{27} \, \sqrt {5} \sqrt {3} \log \left (-\frac {\sqrt {5} \sqrt {3} \sqrt {2 \, x + 3} - 3 \, x - 7}{3 \, x + 2}\right ) - \frac {4}{9} \, \sqrt {2 \, x + 3} {\left (x - 14\right )} + 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]
________________________________________________________________________________________
giac [A] time = 0.18, size = 83, normalized size = 1.22 \begin {gather*} -\frac {2}{9} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + \frac {85}{27} \, \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 {62}{9} \, \sqrt {2 \, x + 3} + 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 = 0.01, size = 62, normalized size = 0.91 \begin {gather*} -\frac {170 \sqrt {15}\, \arctanh \left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{27}-6 \ln \left (-1+\sqrt {2 x +3}\right )+6 \ln \left (\sqrt {2 x +3}+1\right )-\frac {2 \left (2 x +3\right )^{\frac {3}{2}}}{9}+\frac {62 \sqrt {2 x +3}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.20, size = 79, normalized size = 1.16 \begin {gather*} -\frac {2}{9} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + \frac {85}{27} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) + \frac {62}{9} \, \sqrt {2 \, x + 3} + 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 = 53, normalized size = 0.78 \begin {gather*} \frac {62\,\sqrt {2\,x+3}}{9}-\frac {2\,{\left (2\,x+3\right )}^{3/2}}{9}-\mathrm {atan}\left (\sqrt {2\,x+3}\,1{}\mathrm {i}\right )\,12{}\mathrm {i}+\frac {\sqrt {15}\,\mathrm {atan}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}\,1{}\mathrm {i}}{5}\right )\,170{}\mathrm {i}}{27} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 59.21, size = 114, normalized size = 1.68 \begin {gather*} - \frac {2 \left (2 x + 3\right )^{\frac {3}{2}}}{9} + \frac {62 \sqrt {2 x + 3}}{9} + \frac {850 \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 )}{9} - 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]
________________________________________________________________________________________