Optimal. Leaf size=69 \[ -\frac {3}{25} (1-2 x)^{5/2}+\frac {2}{75} (1-2 x)^{3/2}+\frac {22}{125} \sqrt {1-2 x}-\frac {22}{125} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {80, 50, 63, 206} \begin {gather*} -\frac {3}{25} (1-2 x)^{5/2}+\frac {2}{75} (1-2 x)^{3/2}+\frac {22}{125} \sqrt {1-2 x}-\frac {22}{125} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 80
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (2+3 x)}{3+5 x} \, dx &=-\frac {3}{25} (1-2 x)^{5/2}+\frac {1}{5} \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx\\ &=\frac {2}{75} (1-2 x)^{3/2}-\frac {3}{25} (1-2 x)^{5/2}+\frac {11}{25} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx\\ &=\frac {22}{125} \sqrt {1-2 x}+\frac {2}{75} (1-2 x)^{3/2}-\frac {3}{25} (1-2 x)^{5/2}+\frac {121}{125} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {22}{125} \sqrt {1-2 x}+\frac {2}{75} (1-2 x)^{3/2}-\frac {3}{25} (1-2 x)^{5/2}-\frac {121}{125} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {22}{125} \sqrt {1-2 x}+\frac {2}{75} (1-2 x)^{3/2}-\frac {3}{25} (1-2 x)^{5/2}-\frac {22}{125} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 51, normalized size = 0.74 \begin {gather*} \frac {5 \sqrt {1-2 x} \left (-180 x^2+160 x+31\right )-66 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1875} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.06, size = 68, normalized size = 0.99 \begin {gather*} \frac {1}{375} \left (-45 (1-2 x)^{5/2}+10 (1-2 x)^{3/2}+66 \sqrt {1-2 x}\right )-\frac {22}{125} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.61, size = 56, normalized size = 0.81 \begin {gather*} \frac {11}{625} \, \sqrt {11} \sqrt {5} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - \frac {1}{375} \, {\left (180 \, x^{2} - 160 \, x - 31\right )} \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.01, size = 74, normalized size = 1.07 \begin {gather*} -\frac {3}{25} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {2}{75} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11}{625} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {22}{125} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 47, normalized size = 0.68 \begin {gather*} -\frac {22 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{625}+\frac {2 \left (-2 x +1\right )^{\frac {3}{2}}}{75}-\frac {3 \left (-2 x +1\right )^{\frac {5}{2}}}{25}+\frac {22 \sqrt {-2 x +1}}{125} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.07, size = 64, normalized size = 0.93 \begin {gather*} -\frac {3}{25} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {2}{75} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11}{625} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {22}{125} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.07, size = 48, normalized size = 0.70 \begin {gather*} \frac {22\,\sqrt {1-2\,x}}{125}+\frac {2\,{\left (1-2\,x\right )}^{3/2}}{75}-\frac {3\,{\left (1-2\,x\right )}^{5/2}}{25}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,22{}\mathrm {i}}{625} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 22.08, size = 102, normalized size = 1.48 \begin {gather*} - \frac {3 \left (1 - 2 x\right )^{\frac {5}{2}}}{25} + \frac {2 \left (1 - 2 x\right )^{\frac {3}{2}}}{75} + \frac {22 \sqrt {1 - 2 x}}{125} + \frac {242 \left (\begin {cases} - \frac {\sqrt {55} \operatorname {acoth}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: 2 x - 1 < - \frac {11}{5} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: 2 x - 1 > - \frac {11}{5} \end {cases}\right )}{125} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________