Optimal. Leaf size=95 \[ \frac {22 \sqrt {1-2 x}}{3125}+\frac {2 (1-2 x)^{3/2}}{1875}-\frac {3897 (1-2 x)^{5/2}}{2500}+\frac {162}{175} (1-2 x)^{7/2}-\frac {3}{20} (1-2 x)^{9/2}-\frac {22 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125} \]
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Rubi [A]
time = 0.02, antiderivative size = 95, 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 {3}{20} (1-2 x)^{9/2}+\frac {162}{175} (1-2 x)^{7/2}-\frac {3897 (1-2 x)^{5/2}}{2500}+\frac {2 (1-2 x)^{3/2}}{1875}+\frac {22 \sqrt {1-2 x}}{3125}-\frac {22 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125} \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)^3}{3+5 x} \, dx &=\int \left (\frac {3897}{500} (1-2 x)^{3/2}-\frac {162}{25} (1-2 x)^{5/2}+\frac {27}{20} (1-2 x)^{7/2}+\frac {(1-2 x)^{3/2}}{125 (3+5 x)}\right ) \, dx\\ &=-\frac {3897 (1-2 x)^{5/2}}{2500}+\frac {162}{175} (1-2 x)^{7/2}-\frac {3}{20} (1-2 x)^{9/2}+\frac {1}{125} \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx\\ &=\frac {2 (1-2 x)^{3/2}}{1875}-\frac {3897 (1-2 x)^{5/2}}{2500}+\frac {162}{175} (1-2 x)^{7/2}-\frac {3}{20} (1-2 x)^{9/2}+\frac {11}{625} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx\\ &=\frac {22 \sqrt {1-2 x}}{3125}+\frac {2 (1-2 x)^{3/2}}{1875}-\frac {3897 (1-2 x)^{5/2}}{2500}+\frac {162}{175} (1-2 x)^{7/2}-\frac {3}{20} (1-2 x)^{9/2}+\frac {121 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{3125}\\ &=\frac {22 \sqrt {1-2 x}}{3125}+\frac {2 (1-2 x)^{3/2}}{1875}-\frac {3897 (1-2 x)^{5/2}}{2500}+\frac {162}{175} (1-2 x)^{7/2}-\frac {3}{20} (1-2 x)^{9/2}-\frac {121 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{3125}\\ &=\frac {22 \sqrt {1-2 x}}{3125}+\frac {2 (1-2 x)^{3/2}}{1875}-\frac {3897 (1-2 x)^{5/2}}{2500}+\frac {162}{175} (1-2 x)^{7/2}-\frac {3}{20} (1-2 x)^{9/2}-\frac {22 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 61, normalized size = 0.64 \begin {gather*} \frac {-5 \sqrt {1-2 x} \left (50858-123295 x-83565 x^2+171000 x^3+157500 x^4\right )-462 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{328125} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 65, normalized size = 0.68
method | result | size |
risch | \(\frac {\left (157500 x^{4}+171000 x^{3}-83565 x^{2}-123295 x +50858\right ) \left (-1+2 x \right )}{65625 \sqrt {1-2 x}}-\frac {22 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15625}\) | \(54\) |
derivativedivides | \(\frac {2 \left (1-2 x \right )^{\frac {3}{2}}}{1875}-\frac {3897 \left (1-2 x \right )^{\frac {5}{2}}}{2500}+\frac {162 \left (1-2 x \right )^{\frac {7}{2}}}{175}-\frac {3 \left (1-2 x \right )^{\frac {9}{2}}}{20}-\frac {22 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15625}+\frac {22 \sqrt {1-2 x}}{3125}\) | \(65\) |
default | \(\frac {2 \left (1-2 x \right )^{\frac {3}{2}}}{1875}-\frac {3897 \left (1-2 x \right )^{\frac {5}{2}}}{2500}+\frac {162 \left (1-2 x \right )^{\frac {7}{2}}}{175}-\frac {3 \left (1-2 x \right )^{\frac {9}{2}}}{20}-\frac {22 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15625}+\frac {22 \sqrt {1-2 x}}{3125}\) | \(65\) |
trager | \(\left (-\frac {12}{5} x^{4}-\frac {456}{175} x^{3}+\frac {5571}{4375} x^{2}+\frac {24659}{13125} x -\frac {50858}{65625}\right ) \sqrt {1-2 x}+\frac {11 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{15625}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 82, normalized size = 0.86 \begin {gather*} -\frac {3}{20} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {162}{175} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {3897}{2500} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {2}{1875} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11}{15625} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {22}{3125} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.16, size = 66, normalized size = 0.69 \begin {gather*} \frac {11}{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}{65625} \, {\left (157500 \, x^{4} + 171000 \, x^{3} - 83565 \, x^{2} - 123295 \, x + 50858\right )} \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 32.90, size = 119, normalized size = 1.25 \begin {gather*} - \frac {3 \left (1 - 2 x\right )^{\frac {9}{2}}}{20} + \frac {162 \left (1 - 2 x\right )^{\frac {7}{2}}}{175} - \frac {3897 \left (1 - 2 x\right )^{\frac {5}{2}}}{2500} + \frac {2 \left (1 - 2 x\right )^{\frac {3}{2}}}{1875} + \frac {22 \sqrt {1 - 2 x}}{3125} + \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 )}{3125} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.96, size = 106, normalized size = 1.12 \begin {gather*} -\frac {3}{20} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {162}{175} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {3897}{2500} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {2}{1875} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11}{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 {22}{3125} \, \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 = 66, normalized size = 0.69 \begin {gather*} \frac {22\,\sqrt {1-2\,x}}{3125}+\frac {2\,{\left (1-2\,x\right )}^{3/2}}{1875}-\frac {3897\,{\left (1-2\,x\right )}^{5/2}}{2500}+\frac {162\,{\left (1-2\,x\right )}^{7/2}}{175}-\frac {3\,{\left (1-2\,x\right )}^{9/2}}{20}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,22{}\mathrm {i}}{15625} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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