Optimal. Leaf size=108 \[ \frac {242 \sqrt {1-2 x}}{15625}+\frac {22 (1-2 x)^{3/2}}{9375}+\frac {2 (1-2 x)^{5/2}}{3125}-\frac {3897 (1-2 x)^{7/2}}{3500}+\frac {18}{25} (1-2 x)^{9/2}-\frac {27}{220} (1-2 x)^{11/2}-\frac {242 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625} \]
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Rubi [A]
time = 0.03, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {90, 52, 65, 212}
\begin {gather*} -\frac {27}{220} (1-2 x)^{11/2}+\frac {18}{25} (1-2 x)^{9/2}-\frac {3897 (1-2 x)^{7/2}}{3500}+\frac {2 (1-2 x)^{5/2}}{3125}+\frac {22 (1-2 x)^{3/2}}{9375}+\frac {242 \sqrt {1-2 x}}{15625}-\frac {242 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 90
Rule 212
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (2+3 x)^3}{3+5 x} \, dx &=\int \left (\frac {3897}{500} (1-2 x)^{5/2}-\frac {162}{25} (1-2 x)^{7/2}+\frac {27}{20} (1-2 x)^{9/2}+\frac {(1-2 x)^{5/2}}{125 (3+5 x)}\right ) \, dx\\ &=-\frac {3897 (1-2 x)^{7/2}}{3500}+\frac {18}{25} (1-2 x)^{9/2}-\frac {27}{220} (1-2 x)^{11/2}+\frac {1}{125} \int \frac {(1-2 x)^{5/2}}{3+5 x} \, dx\\ &=\frac {2 (1-2 x)^{5/2}}{3125}-\frac {3897 (1-2 x)^{7/2}}{3500}+\frac {18}{25} (1-2 x)^{9/2}-\frac {27}{220} (1-2 x)^{11/2}+\frac {11}{625} \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx\\ &=\frac {22 (1-2 x)^{3/2}}{9375}+\frac {2 (1-2 x)^{5/2}}{3125}-\frac {3897 (1-2 x)^{7/2}}{3500}+\frac {18}{25} (1-2 x)^{9/2}-\frac {27}{220} (1-2 x)^{11/2}+\frac {121 \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx}{3125}\\ &=\frac {242 \sqrt {1-2 x}}{15625}+\frac {22 (1-2 x)^{3/2}}{9375}+\frac {2 (1-2 x)^{5/2}}{3125}-\frac {3897 (1-2 x)^{7/2}}{3500}+\frac {18}{25} (1-2 x)^{9/2}-\frac {27}{220} (1-2 x)^{11/2}+\frac {1331 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{15625}\\ &=\frac {242 \sqrt {1-2 x}}{15625}+\frac {22 (1-2 x)^{3/2}}{9375}+\frac {2 (1-2 x)^{5/2}}{3125}-\frac {3897 (1-2 x)^{7/2}}{3500}+\frac {18}{25} (1-2 x)^{9/2}-\frac {27}{220} (1-2 x)^{11/2}-\frac {1331 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{15625}\\ &=\frac {242 \sqrt {1-2 x}}{15625}+\frac {22 (1-2 x)^{3/2}}{9375}+\frac {2 (1-2 x)^{5/2}}{3125}-\frac {3897 (1-2 x)^{7/2}}{3500}+\frac {18}{25} (1-2 x)^{9/2}-\frac {27}{220} (1-2 x)^{11/2}-\frac {242 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 66, normalized size = 0.61 \begin {gather*} \frac {5 \sqrt {1-2 x} \left (-1796318+7726195 x-3564885 x^2-15572250 x^3+6142500 x^4+14175000 x^5\right )-55902 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{18046875} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 74, normalized size = 0.69
method | result | size |
risch | \(-\frac {\left (14175000 x^{5}+6142500 x^{4}-15572250 x^{3}-3564885 x^{2}+7726195 x -1796318\right ) \left (-1+2 x \right )}{3609375 \sqrt {1-2 x}}-\frac {242 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{78125}\) | \(59\) |
derivativedivides | \(\frac {22 \left (1-2 x \right )^{\frac {3}{2}}}{9375}+\frac {2 \left (1-2 x \right )^{\frac {5}{2}}}{3125}-\frac {3897 \left (1-2 x \right )^{\frac {7}{2}}}{3500}+\frac {18 \left (1-2 x \right )^{\frac {9}{2}}}{25}-\frac {27 \left (1-2 x \right )^{\frac {11}{2}}}{220}-\frac {242 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{78125}+\frac {242 \sqrt {1-2 x}}{15625}\) | \(74\) |
default | \(\frac {22 \left (1-2 x \right )^{\frac {3}{2}}}{9375}+\frac {2 \left (1-2 x \right )^{\frac {5}{2}}}{3125}-\frac {3897 \left (1-2 x \right )^{\frac {7}{2}}}{3500}+\frac {18 \left (1-2 x \right )^{\frac {9}{2}}}{25}-\frac {27 \left (1-2 x \right )^{\frac {11}{2}}}{220}-\frac {242 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{78125}+\frac {242 \sqrt {1-2 x}}{15625}\) | \(74\) |
trager | \(\left (\frac {216}{55} x^{5}+\frac {468}{275} x^{4}-\frac {41526}{9625} x^{3}-\frac {237659}{240625} x^{2}+\frac {1545239}{721875} x -\frac {1796318}{3609375}\right ) \sqrt {1-2 x}+\frac {121 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{78125}\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 91, normalized size = 0.84 \begin {gather*} -\frac {27}{220} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {18}{25} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {3897}{3500} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {2}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {22}{9375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {121}{78125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {242}{15625} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.77, size = 71, normalized size = 0.66 \begin {gather*} \frac {121}{78125} \, \sqrt {11} \sqrt {5} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + \frac {1}{3609375} \, {\left (14175000 \, x^{5} + 6142500 \, x^{4} - 15572250 \, x^{3} - 3564885 \, x^{2} + 7726195 \, x - 1796318\right )} \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 40.66, size = 131, normalized size = 1.21 \begin {gather*} - \frac {27 \left (1 - 2 x\right )^{\frac {11}{2}}}{220} + \frac {18 \left (1 - 2 x\right )^{\frac {9}{2}}}{25} - \frac {3897 \left (1 - 2 x\right )^{\frac {7}{2}}}{3500} + \frac {2 \left (1 - 2 x\right )^{\frac {5}{2}}}{3125} + \frac {22 \left (1 - 2 x\right )^{\frac {3}{2}}}{9375} + \frac {242 \sqrt {1 - 2 x}}{15625} + \frac {2662 \left (\begin {cases} - \frac {\sqrt {55} \operatorname {acoth}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: x < - \frac {3}{5} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: x > - \frac {3}{5} \end {cases}\right )}{15625} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.99, size = 122, normalized size = 1.13 \begin {gather*} \frac {27}{220} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {18}{25} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {3897}{3500} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {2}{3125} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {22}{9375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {121}{78125} \, \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 {242}{15625} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 75, normalized size = 0.69 \begin {gather*} \frac {242\,\sqrt {1-2\,x}}{15625}+\frac {22\,{\left (1-2\,x\right )}^{3/2}}{9375}+\frac {2\,{\left (1-2\,x\right )}^{5/2}}{3125}-\frac {3897\,{\left (1-2\,x\right )}^{7/2}}{3500}+\frac {18\,{\left (1-2\,x\right )}^{9/2}}{25}-\frac {27\,{\left (1-2\,x\right )}^{11/2}}{220}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,242{}\mathrm {i}}{78125} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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