Optimal. Leaf size=108 \[ \frac {22 \sqrt {1-2 x}}{15625}+\frac {2 (1-2 x)^{3/2}}{9375}-\frac {136419 (1-2 x)^{5/2}}{25000}+\frac {34371 (1-2 x)^{7/2}}{7000}-\frac {321}{200} (1-2 x)^{9/2}+\frac {81}{440} (1-2 x)^{11/2}-\frac {22 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625} \]
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Rubi [A]
time = 0.02, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps
used = 6, 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}{440} (1-2 x)^{11/2}-\frac {321}{200} (1-2 x)^{9/2}+\frac {34371 (1-2 x)^{7/2}}{7000}-\frac {136419 (1-2 x)^{5/2}}{25000}+\frac {2 (1-2 x)^{3/2}}{9375}+\frac {22 \sqrt {1-2 x}}{15625}-\frac {22 \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)^{3/2} (2+3 x)^4}{3+5 x} \, dx &=\int \left (\frac {136419 (1-2 x)^{3/2}}{5000}-\frac {34371 (1-2 x)^{5/2}}{1000}+\frac {2889}{200} (1-2 x)^{7/2}-\frac {81}{40} (1-2 x)^{9/2}+\frac {(1-2 x)^{3/2}}{625 (3+5 x)}\right ) \, dx\\ &=-\frac {136419 (1-2 x)^{5/2}}{25000}+\frac {34371 (1-2 x)^{7/2}}{7000}-\frac {321}{200} (1-2 x)^{9/2}+\frac {81}{440} (1-2 x)^{11/2}+\frac {1}{625} \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx\\ &=\frac {2 (1-2 x)^{3/2}}{9375}-\frac {136419 (1-2 x)^{5/2}}{25000}+\frac {34371 (1-2 x)^{7/2}}{7000}-\frac {321}{200} (1-2 x)^{9/2}+\frac {81}{440} (1-2 x)^{11/2}+\frac {11 \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx}{3125}\\ &=\frac {22 \sqrt {1-2 x}}{15625}+\frac {2 (1-2 x)^{3/2}}{9375}-\frac {136419 (1-2 x)^{5/2}}{25000}+\frac {34371 (1-2 x)^{7/2}}{7000}-\frac {321}{200} (1-2 x)^{9/2}+\frac {81}{440} (1-2 x)^{11/2}+\frac {121 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{15625}\\ &=\frac {22 \sqrt {1-2 x}}{15625}+\frac {2 (1-2 x)^{3/2}}{9375}-\frac {136419 (1-2 x)^{5/2}}{25000}+\frac {34371 (1-2 x)^{7/2}}{7000}-\frac {321}{200} (1-2 x)^{9/2}+\frac {81}{440} (1-2 x)^{11/2}-\frac {121 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{15625}\\ &=\frac {22 \sqrt {1-2 x}}{15625}+\frac {2 (1-2 x)^{3/2}}{9375}-\frac {136419 (1-2 x)^{5/2}}{25000}+\frac {34371 (1-2 x)^{7/2}}{7000}-\frac {321}{200} (1-2 x)^{9/2}+\frac {81}{440} (1-2 x)^{11/2}-\frac {22 \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 = 0.07, size = 66, normalized size = 0.61 \begin {gather*} \frac {-5 \sqrt {1-2 x} \left (7095688-12144995 x-21433590 x^2+9559125 x^3+39532500 x^4+21262500 x^5\right )-5082 \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 | \(\frac {\left (21262500 x^{5}+39532500 x^{4}+9559125 x^{3}-21433590 x^{2}-12144995 x +7095688\right ) \left (-1+2 x \right )}{3609375 \sqrt {1-2 x}}-\frac {22 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{78125}\) | \(59\) |
derivativedivides | \(\frac {2 \left (1-2 x \right )^{\frac {3}{2}}}{9375}-\frac {136419 \left (1-2 x \right )^{\frac {5}{2}}}{25000}+\frac {34371 \left (1-2 x \right )^{\frac {7}{2}}}{7000}-\frac {321 \left (1-2 x \right )^{\frac {9}{2}}}{200}+\frac {81 \left (1-2 x \right )^{\frac {11}{2}}}{440}-\frac {22 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{78125}+\frac {22 \sqrt {1-2 x}}{15625}\) | \(74\) |
default | \(\frac {2 \left (1-2 x \right )^{\frac {3}{2}}}{9375}-\frac {136419 \left (1-2 x \right )^{\frac {5}{2}}}{25000}+\frac {34371 \left (1-2 x \right )^{\frac {7}{2}}}{7000}-\frac {321 \left (1-2 x \right )^{\frac {9}{2}}}{200}+\frac {81 \left (1-2 x \right )^{\frac {11}{2}}}{440}-\frac {22 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{78125}+\frac {22 \sqrt {1-2 x}}{15625}\) | \(74\) |
trager | \(\left (-\frac {324}{55} x^{5}-\frac {3012}{275} x^{4}-\frac {25491}{9625} x^{3}+\frac {1428906}{240625} x^{2}+\frac {2428999}{721875} x -\frac {7095688}{3609375}\right ) \sqrt {1-2 x}-\frac {11 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{78125}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 91, normalized size = 0.84 \begin {gather*} \frac {81}{440} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {321}{200} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {34371}{7000} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {136419}{25000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {2}{9375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11}{78125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {22}{15625} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.28, size = 71, normalized size = 0.66 \begin {gather*} \frac {11}{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 (21262500 \, x^{5} + 39532500 \, x^{4} + 9559125 \, x^{3} - 21433590 \, x^{2} - 12144995 \, x + 7095688\right )} \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 46.95, size = 131, normalized size = 1.21 \begin {gather*} \frac {81 \left (1 - 2 x\right )^{\frac {11}{2}}}{440} - \frac {321 \left (1 - 2 x\right )^{\frac {9}{2}}}{200} + \frac {34371 \left (1 - 2 x\right )^{\frac {7}{2}}}{7000} - \frac {136419 \left (1 - 2 x\right )^{\frac {5}{2}}}{25000} + \frac {2 \left (1 - 2 x\right )^{\frac {3}{2}}}{9375} + \frac {22 \sqrt {1 - 2 x}}{15625} + \frac {242 \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 = 1.13, size = 122, normalized size = 1.13 \begin {gather*} -\frac {81}{440} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {321}{200} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {34371}{7000} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {136419}{25000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {2}{9375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11}{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 {22}{15625} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.17, size = 75, normalized size = 0.69 \begin {gather*} \frac {22\,\sqrt {1-2\,x}}{15625}+\frac {2\,{\left (1-2\,x\right )}^{3/2}}{9375}-\frac {136419\,{\left (1-2\,x\right )}^{5/2}}{25000}+\frac {34371\,{\left (1-2\,x\right )}^{7/2}}{7000}-\frac {321\,{\left (1-2\,x\right )}^{9/2}}{200}+\frac {81\,{\left (1-2\,x\right )}^{11/2}}{440}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,22{}\mathrm {i}}{78125} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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