Optimal. Leaf size=95 \[ \frac {9}{40} (1-2 x)^{9/2}-\frac {2889 (1-2 x)^{7/2}}{1400}+\frac {34371 (1-2 x)^{5/2}}{5000}-\frac {45473 (1-2 x)^{3/2}}{5000}+\frac {2 \sqrt {1-2 x}}{3125}-\frac {2 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125} \]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {88, 50, 63, 206} \begin {gather*} \frac {9}{40} (1-2 x)^{9/2}-\frac {2889 (1-2 x)^{7/2}}{1400}+\frac {34371 (1-2 x)^{5/2}}{5000}-\frac {45473 (1-2 x)^{3/2}}{5000}+\frac {2 \sqrt {1-2 x}}{3125}-\frac {2 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 88
Rule 206
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (2+3 x)^4}{3+5 x} \, dx &=\int \left (\frac {136419 \sqrt {1-2 x}}{5000}-\frac {34371 (1-2 x)^{3/2}}{1000}+\frac {2889}{200} (1-2 x)^{5/2}-\frac {81}{40} (1-2 x)^{7/2}+\frac {\sqrt {1-2 x}}{625 (3+5 x)}\right ) \, dx\\ &=-\frac {45473 (1-2 x)^{3/2}}{5000}+\frac {34371 (1-2 x)^{5/2}}{5000}-\frac {2889 (1-2 x)^{7/2}}{1400}+\frac {9}{40} (1-2 x)^{9/2}+\frac {1}{625} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx\\ &=\frac {2 \sqrt {1-2 x}}{3125}-\frac {45473 (1-2 x)^{3/2}}{5000}+\frac {34371 (1-2 x)^{5/2}}{5000}-\frac {2889 (1-2 x)^{7/2}}{1400}+\frac {9}{40} (1-2 x)^{9/2}+\frac {11 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{3125}\\ &=\frac {2 \sqrt {1-2 x}}{3125}-\frac {45473 (1-2 x)^{3/2}}{5000}+\frac {34371 (1-2 x)^{5/2}}{5000}-\frac {2889 (1-2 x)^{7/2}}{1400}+\frac {9}{40} (1-2 x)^{9/2}-\frac {11 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{3125}\\ &=\frac {2 \sqrt {1-2 x}}{3125}-\frac {45473 (1-2 x)^{3/2}}{5000}+\frac {34371 (1-2 x)^{5/2}}{5000}-\frac {2889 (1-2 x)^{7/2}}{1400}+\frac {9}{40} (1-2 x)^{9/2}-\frac {2 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 61, normalized size = 0.64 \begin {gather*} \frac {5 \sqrt {1-2 x} \left (78750 x^4+203625 x^3+177930 x^2+27865 x-88776\right )-14 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{109375} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.07, size = 79, normalized size = 0.83 \begin {gather*} \frac {\left (39375 (1-2 x)^4-361125 (1-2 x)^3+1202985 (1-2 x)^2-1591555 (1-2 x)+112\right ) \sqrt {1-2 x}}{175000}-\frac {2 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.52, size = 66, normalized size = 0.69 \begin {gather*} \frac {1}{15625} \, \sqrt {11} \sqrt {5} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + \frac {1}{21875} \, {\left (78750 \, x^{4} + 203625 \, x^{3} + 177930 \, x^{2} + 27865 \, x - 88776\right )} \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.27, size = 106, normalized size = 1.12 \begin {gather*} \frac {9}{40} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {2889}{1400} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {34371}{5000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {45473}{5000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{15625} \, \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 {2}{3125} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 65, normalized size = 0.68 \begin {gather*} -\frac {2 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{15625}-\frac {45473 \left (-2 x +1\right )^{\frac {3}{2}}}{5000}+\frac {34371 \left (-2 x +1\right )^{\frac {5}{2}}}{5000}-\frac {2889 \left (-2 x +1\right )^{\frac {7}{2}}}{1400}+\frac {9 \left (-2 x +1\right )^{\frac {9}{2}}}{40}+\frac {2 \sqrt {-2 x +1}}{3125} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.18, size = 82, normalized size = 0.86 \begin {gather*} \frac {9}{40} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {2889}{1400} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {34371}{5000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {45473}{5000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{15625} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2}{3125} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.18, size = 66, normalized size = 0.69 \begin {gather*} \frac {2\,\sqrt {1-2\,x}}{3125}-\frac {45473\,{\left (1-2\,x\right )}^{3/2}}{5000}+\frac {34371\,{\left (1-2\,x\right )}^{5/2}}{5000}-\frac {2889\,{\left (1-2\,x\right )}^{7/2}}{1400}+\frac {9\,{\left (1-2\,x\right )}^{9/2}}{40}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,2{}\mathrm {i}}{15625} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 9.67, size = 126, normalized size = 1.33 \begin {gather*} \frac {9 \left (1 - 2 x\right )^{\frac {9}{2}}}{40} - \frac {2889 \left (1 - 2 x\right )^{\frac {7}{2}}}{1400} + \frac {34371 \left (1 - 2 x\right )^{\frac {5}{2}}}{5000} - \frac {45473 \left (1 - 2 x\right )^{\frac {3}{2}}}{5000} + \frac {2 \sqrt {1 - 2 x}}{3125} + \frac {22 \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 )}{3125} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________