Optimal. Leaf size=95 \[ \frac {1}{10} (1-2 x)^{9/2}-\frac {111}{350} (1-2 x)^{7/2}+\frac {2}{625} (1-2 x)^{5/2}+\frac {22 (1-2 x)^{3/2}}{1875}+\frac {242 \sqrt {1-2 x}}{3125}-\frac {242 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125} \]
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Rubi [A] time = 0.03, antiderivative size = 95, 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 = {88, 50, 63, 206} \begin {gather*} \frac {1}{10} (1-2 x)^{9/2}-\frac {111}{350} (1-2 x)^{7/2}+\frac {2}{625} (1-2 x)^{5/2}+\frac {22 (1-2 x)^{3/2}}{1875}+\frac {242 \sqrt {1-2 x}}{3125}-\frac {242 \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 50
Rule 63
Rule 88
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{3+5 x} \, dx &=\int \left (\frac {111}{50} (1-2 x)^{5/2}-\frac {9}{10} (1-2 x)^{7/2}+\frac {(1-2 x)^{5/2}}{25 (3+5 x)}\right ) \, dx\\ &=-\frac {111}{350} (1-2 x)^{7/2}+\frac {1}{10} (1-2 x)^{9/2}+\frac {1}{25} \int \frac {(1-2 x)^{5/2}}{3+5 x} \, dx\\ &=\frac {2}{625} (1-2 x)^{5/2}-\frac {111}{350} (1-2 x)^{7/2}+\frac {1}{10} (1-2 x)^{9/2}+\frac {11}{125} \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx\\ &=\frac {22 (1-2 x)^{3/2}}{1875}+\frac {2}{625} (1-2 x)^{5/2}-\frac {111}{350} (1-2 x)^{7/2}+\frac {1}{10} (1-2 x)^{9/2}+\frac {121}{625} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx\\ &=\frac {242 \sqrt {1-2 x}}{3125}+\frac {22 (1-2 x)^{3/2}}{1875}+\frac {2}{625} (1-2 x)^{5/2}-\frac {111}{350} (1-2 x)^{7/2}+\frac {1}{10} (1-2 x)^{9/2}+\frac {1331 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{3125}\\ &=\frac {242 \sqrt {1-2 x}}{3125}+\frac {22 (1-2 x)^{3/2}}{1875}+\frac {2}{625} (1-2 x)^{5/2}-\frac {111}{350} (1-2 x)^{7/2}+\frac {1}{10} (1-2 x)^{9/2}-\frac {1331 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{3125}\\ &=\frac {242 \sqrt {1-2 x}}{3125}+\frac {22 (1-2 x)^{3/2}}{1875}+\frac {2}{625} (1-2 x)^{5/2}-\frac {111}{350} (1-2 x)^{7/2}+\frac {1}{10} (1-2 x)^{9/2}-\frac {242 \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 = 0.08, size = 61, normalized size = 0.64 \begin {gather*} \frac {5 \sqrt {1-2 x} \left (105000 x^4-43500 x^3-91410 x^2+69995 x-8188\right )-5082 \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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IntegrateAlgebraic [A] time = 0.06, size = 79, normalized size = 0.83 \begin {gather*} \frac {\left (13125 (1-2 x)^4-41625 (1-2 x)^3+420 (1-2 x)^2+1540 (1-2 x)+10164\right ) \sqrt {1-2 x}}{131250}-\frac {242 \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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fricas [A] time = 1.33, size = 66, normalized size = 0.69 \begin {gather*} \frac {121}{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 (105000 \, x^{4} - 43500 \, x^{3} - 91410 \, x^{2} + 69995 \, x - 8188\right )} \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.04, size = 106, normalized size = 1.12 \begin {gather*} \frac {1}{10} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {111}{350} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {2}{625} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {22}{1875} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {121}{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 {242}{3125} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 65, normalized size = 0.68 \begin {gather*} -\frac {242 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{15625}+\frac {22 \left (-2 x +1\right )^{\frac {3}{2}}}{1875}+\frac {2 \left (-2 x +1\right )^{\frac {5}{2}}}{625}-\frac {111 \left (-2 x +1\right )^{\frac {7}{2}}}{350}+\frac {\left (-2 x +1\right )^{\frac {9}{2}}}{10}+\frac {242 \sqrt {-2 x +1}}{3125} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 82, normalized size = 0.86 \begin {gather*} \frac {1}{10} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {111}{350} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {2}{625} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {22}{1875} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {121}{15625} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {242}{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 {242\,\sqrt {1-2\,x}}{3125}+\frac {22\,{\left (1-2\,x\right )}^{3/2}}{1875}+\frac {2\,{\left (1-2\,x\right )}^{5/2}}{625}-\frac {111\,{\left (1-2\,x\right )}^{7/2}}{350}+\frac {{\left (1-2\,x\right )}^{9/2}}{10}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,242{}\mathrm {i}}{15625} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 54.48, size = 124, normalized size = 1.31 \begin {gather*} \frac {\left (1 - 2 x\right )^{\frac {9}{2}}}{10} - \frac {111 \left (1 - 2 x\right )^{\frac {7}{2}}}{350} + \frac {2 \left (1 - 2 x\right )^{\frac {5}{2}}}{625} + \frac {22 \left (1 - 2 x\right )^{\frac {3}{2}}}{1875} + \frac {242 \sqrt {1 - 2 x}}{3125} + \frac {2662 \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.
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