Optimal. Leaf size=121 \[ \frac {242 \sqrt {1-2 x}}{78125}+\frac {22 (1-2 x)^{3/2}}{46875}+\frac {2 (1-2 x)^{5/2}}{15625}-\frac {136419 (1-2 x)^{7/2}}{35000}+\frac {3819 (1-2 x)^{9/2}}{1000}-\frac {2889 (1-2 x)^{11/2}}{2200}+\frac {81}{520} (1-2 x)^{13/2}-\frac {242 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{78125} \]
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Rubi [A]
time = 0.03, antiderivative size = 121, 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 {81}{520} (1-2 x)^{13/2}-\frac {2889 (1-2 x)^{11/2}}{2200}+\frac {3819 (1-2 x)^{9/2}}{1000}-\frac {136419 (1-2 x)^{7/2}}{35000}+\frac {2 (1-2 x)^{5/2}}{15625}+\frac {22 (1-2 x)^{3/2}}{46875}+\frac {242 \sqrt {1-2 x}}{78125}-\frac {242 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{78125} \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)^4}{3+5 x} \, dx &=\int \left (\frac {136419 (1-2 x)^{5/2}}{5000}-\frac {34371 (1-2 x)^{7/2}}{1000}+\frac {2889}{200} (1-2 x)^{9/2}-\frac {81}{40} (1-2 x)^{11/2}+\frac {(1-2 x)^{5/2}}{625 (3+5 x)}\right ) \, dx\\ &=-\frac {136419 (1-2 x)^{7/2}}{35000}+\frac {3819 (1-2 x)^{9/2}}{1000}-\frac {2889 (1-2 x)^{11/2}}{2200}+\frac {81}{520} (1-2 x)^{13/2}+\frac {1}{625} \int \frac {(1-2 x)^{5/2}}{3+5 x} \, dx\\ &=\frac {2 (1-2 x)^{5/2}}{15625}-\frac {136419 (1-2 x)^{7/2}}{35000}+\frac {3819 (1-2 x)^{9/2}}{1000}-\frac {2889 (1-2 x)^{11/2}}{2200}+\frac {81}{520} (1-2 x)^{13/2}+\frac {11 \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx}{3125}\\ &=\frac {22 (1-2 x)^{3/2}}{46875}+\frac {2 (1-2 x)^{5/2}}{15625}-\frac {136419 (1-2 x)^{7/2}}{35000}+\frac {3819 (1-2 x)^{9/2}}{1000}-\frac {2889 (1-2 x)^{11/2}}{2200}+\frac {81}{520} (1-2 x)^{13/2}+\frac {121 \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx}{15625}\\ &=\frac {242 \sqrt {1-2 x}}{78125}+\frac {22 (1-2 x)^{3/2}}{46875}+\frac {2 (1-2 x)^{5/2}}{15625}-\frac {136419 (1-2 x)^{7/2}}{35000}+\frac {3819 (1-2 x)^{9/2}}{1000}-\frac {2889 (1-2 x)^{11/2}}{2200}+\frac {81}{520} (1-2 x)^{13/2}+\frac {1331 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{78125}\\ &=\frac {242 \sqrt {1-2 x}}{78125}+\frac {22 (1-2 x)^{3/2}}{46875}+\frac {2 (1-2 x)^{5/2}}{15625}-\frac {136419 (1-2 x)^{7/2}}{35000}+\frac {3819 (1-2 x)^{9/2}}{1000}-\frac {2889 (1-2 x)^{11/2}}{2200}+\frac {81}{520} (1-2 x)^{13/2}-\frac {1331 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{78125}\\ &=\frac {242 \sqrt {1-2 x}}{78125}+\frac {22 (1-2 x)^{3/2}}{46875}+\frac {2 (1-2 x)^{5/2}}{15625}-\frac {136419 (1-2 x)^{7/2}}{35000}+\frac {3819 (1-2 x)^{9/2}}{1000}-\frac {2889 (1-2 x)^{11/2}}{2200}+\frac {81}{520} (1-2 x)^{13/2}-\frac {242 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{78125}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 71, normalized size = 0.59 \begin {gather*} \frac {5 \sqrt {1-2 x} \left (-289133384+960784285 x+399578370 x^2-2556079875 x^3-1540428750 x^4+2842087500 x^5+2338875000 x^6\right )-726726 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1173046875} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 83, normalized size = 0.69
method | result | size |
risch | \(-\frac {\left (2338875000 x^{6}+2842087500 x^{5}-1540428750 x^{4}-2556079875 x^{3}+399578370 x^{2}+960784285 x -289133384\right ) \left (-1+2 x \right )}{234609375 \sqrt {1-2 x}}-\frac {242 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{390625}\) | \(64\) |
derivativedivides | \(\frac {22 \left (1-2 x \right )^{\frac {3}{2}}}{46875}+\frac {2 \left (1-2 x \right )^{\frac {5}{2}}}{15625}-\frac {136419 \left (1-2 x \right )^{\frac {7}{2}}}{35000}+\frac {3819 \left (1-2 x \right )^{\frac {9}{2}}}{1000}-\frac {2889 \left (1-2 x \right )^{\frac {11}{2}}}{2200}+\frac {81 \left (1-2 x \right )^{\frac {13}{2}}}{520}-\frac {242 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{390625}+\frac {242 \sqrt {1-2 x}}{78125}\) | \(83\) |
default | \(\frac {22 \left (1-2 x \right )^{\frac {3}{2}}}{46875}+\frac {2 \left (1-2 x \right )^{\frac {5}{2}}}{15625}-\frac {136419 \left (1-2 x \right )^{\frac {7}{2}}}{35000}+\frac {3819 \left (1-2 x \right )^{\frac {9}{2}}}{1000}-\frac {2889 \left (1-2 x \right )^{\frac {11}{2}}}{2200}+\frac {81 \left (1-2 x \right )^{\frac {13}{2}}}{520}-\frac {242 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{390625}+\frac {242 \sqrt {1-2 x}}{78125}\) | \(83\) |
trager | \(\left (\frac {648}{65} x^{6}+\frac {43308}{3575} x^{5}-\frac {117366}{17875} x^{4}-\frac {6816213}{625625} x^{3}+\frac {26638558}{15640625} x^{2}+\frac {192156857}{46921875} x -\frac {289133384}{234609375}\right ) \sqrt {1-2 x}-\frac {121 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{390625}\) | \(85\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.56, size = 100, normalized size = 0.83 \begin {gather*} \frac {81}{520} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {2889}{2200} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {3819}{1000} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {136419}{35000} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {2}{15625} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {22}{46875} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {121}{390625} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {242}{78125} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.95, size = 76, normalized size = 0.63 \begin {gather*} \frac {121}{390625} \, \sqrt {11} \sqrt {5} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + \frac {1}{234609375} \, {\left (2338875000 \, x^{6} + 2842087500 \, x^{5} - 1540428750 \, x^{4} - 2556079875 \, x^{3} + 399578370 \, x^{2} + 960784285 \, x - 289133384\right )} \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 61.08, size = 143, normalized size = 1.18 \begin {gather*} \frac {81 \left (1 - 2 x\right )^{\frac {13}{2}}}{520} - \frac {2889 \left (1 - 2 x\right )^{\frac {11}{2}}}{2200} + \frac {3819 \left (1 - 2 x\right )^{\frac {9}{2}}}{1000} - \frac {136419 \left (1 - 2 x\right )^{\frac {7}{2}}}{35000} + \frac {2 \left (1 - 2 x\right )^{\frac {5}{2}}}{15625} + \frac {22 \left (1 - 2 x\right )^{\frac {3}{2}}}{46875} + \frac {242 \sqrt {1 - 2 x}}{78125} + \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 )}{78125} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.23, size = 138, normalized size = 1.14 \begin {gather*} \frac {81}{520} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {2889}{2200} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {3819}{1000} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {136419}{35000} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {2}{15625} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {22}{46875} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {121}{390625} \, \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}{78125} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 84, normalized size = 0.69 \begin {gather*} \frac {242\,\sqrt {1-2\,x}}{78125}+\frac {22\,{\left (1-2\,x\right )}^{3/2}}{46875}+\frac {2\,{\left (1-2\,x\right )}^{5/2}}{15625}-\frac {136419\,{\left (1-2\,x\right )}^{7/2}}{35000}+\frac {3819\,{\left (1-2\,x\right )}^{9/2}}{1000}-\frac {2889\,{\left (1-2\,x\right )}^{11/2}}{2200}+\frac {81\,{\left (1-2\,x\right )}^{13/2}}{520}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,242{}\mathrm {i}}{390625} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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