Optimal. Leaf size=121 \[ -\frac {243 (1-2 x)^{13/2}}{1040}+\frac {5751 (1-2 x)^{11/2}}{2200}-\frac {5673}{500} (1-2 x)^{9/2}+\frac {806121 (1-2 x)^{7/2}}{35000}-\frac {4774713 (1-2 x)^{5/2}}{250000}+\frac {2 (1-2 x)^{3/2}}{46875}+\frac {22 \sqrt {1-2 x}}{78125}-\frac {22 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{78125} \]
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Rubi [A] time = 0.04, antiderivative size = 121, 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 = {88, 50, 63, 206} \[ -\frac {243 (1-2 x)^{13/2}}{1040}+\frac {5751 (1-2 x)^{11/2}}{2200}-\frac {5673}{500} (1-2 x)^{9/2}+\frac {806121 (1-2 x)^{7/2}}{35000}-\frac {4774713 (1-2 x)^{5/2}}{250000}+\frac {2 (1-2 x)^{3/2}}{46875}+\frac {22 \sqrt {1-2 x}}{78125}-\frac {22 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{78125} \]
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)^{3/2} (2+3 x)^5}{3+5 x} \, dx &=\int \left (\frac {4774713 (1-2 x)^{3/2}}{50000}-\frac {806121 (1-2 x)^{5/2}}{5000}+\frac {51057}{500} (1-2 x)^{7/2}-\frac {5751}{200} (1-2 x)^{9/2}+\frac {243}{80} (1-2 x)^{11/2}+\frac {(1-2 x)^{3/2}}{3125 (3+5 x)}\right ) \, dx\\ &=-\frac {4774713 (1-2 x)^{5/2}}{250000}+\frac {806121 (1-2 x)^{7/2}}{35000}-\frac {5673}{500} (1-2 x)^{9/2}+\frac {5751 (1-2 x)^{11/2}}{2200}-\frac {243 (1-2 x)^{13/2}}{1040}+\frac {\int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx}{3125}\\ &=\frac {2 (1-2 x)^{3/2}}{46875}-\frac {4774713 (1-2 x)^{5/2}}{250000}+\frac {806121 (1-2 x)^{7/2}}{35000}-\frac {5673}{500} (1-2 x)^{9/2}+\frac {5751 (1-2 x)^{11/2}}{2200}-\frac {243 (1-2 x)^{13/2}}{1040}+\frac {11 \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx}{15625}\\ &=\frac {22 \sqrt {1-2 x}}{78125}+\frac {2 (1-2 x)^{3/2}}{46875}-\frac {4774713 (1-2 x)^{5/2}}{250000}+\frac {806121 (1-2 x)^{7/2}}{35000}-\frac {5673}{500} (1-2 x)^{9/2}+\frac {5751 (1-2 x)^{11/2}}{2200}-\frac {243 (1-2 x)^{13/2}}{1040}+\frac {121 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{78125}\\ &=\frac {22 \sqrt {1-2 x}}{78125}+\frac {2 (1-2 x)^{3/2}}{46875}-\frac {4774713 (1-2 x)^{5/2}}{250000}+\frac {806121 (1-2 x)^{7/2}}{35000}-\frac {5673}{500} (1-2 x)^{9/2}+\frac {5751 (1-2 x)^{11/2}}{2200}-\frac {243 (1-2 x)^{13/2}}{1040}-\frac {121 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{78125}\\ &=\frac {22 \sqrt {1-2 x}}{78125}+\frac {2 (1-2 x)^{3/2}}{46875}-\frac {4774713 (1-2 x)^{5/2}}{250000}+\frac {806121 (1-2 x)^{7/2}}{35000}-\frac {5673}{500} (1-2 x)^{9/2}+\frac {5751 (1-2 x)^{11/2}}{2200}-\frac {243 (1-2 x)^{13/2}}{1040}-\frac {22 \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 = 0.08, size = 71, normalized size = 0.59 \[ \frac {-5 \sqrt {1-2 x} \left (3508312500 x^6+9100350000 x^5+6683000625 x^4-1659418875 x^3-4276774170 x^2-1321809935 x+1180568944\right )-66066 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1173046875} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 76, normalized size = 0.63 \[ \frac {11}{390625} \, \sqrt {11} \sqrt {5} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - \frac {1}{234609375} \, {\left (3508312500 \, x^{6} + 9100350000 \, x^{5} + 6683000625 \, x^{4} - 1659418875 \, x^{3} - 4276774170 \, x^{2} - 1321809935 \, x + 1180568944\right )} \sqrt {-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.12, size = 138, normalized size = 1.14 \[ -\frac {243}{1040} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} - \frac {5751}{2200} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {5673}{500} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {806121}{35000} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {4774713}{250000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {2}{46875} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11}{390625} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {22}{78125} \, \sqrt {-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 83, normalized size = 0.69 \[ -\frac {22 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{390625}+\frac {2 \left (-2 x +1\right )^{\frac {3}{2}}}{46875}-\frac {4774713 \left (-2 x +1\right )^{\frac {5}{2}}}{250000}+\frac {806121 \left (-2 x +1\right )^{\frac {7}{2}}}{35000}-\frac {5673 \left (-2 x +1\right )^{\frac {9}{2}}}{500}+\frac {5751 \left (-2 x +1\right )^{\frac {11}{2}}}{2200}-\frac {243 \left (-2 x +1\right )^{\frac {13}{2}}}{1040}+\frac {22 \sqrt {-2 x +1}}{78125} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.16, size = 100, normalized size = 0.83 \[ -\frac {243}{1040} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} + \frac {5751}{2200} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {5673}{500} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {806121}{35000} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {4774713}{250000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {2}{46875} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11}{390625} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {22}{78125} \, \sqrt {-2 \, x + 1} \]
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 \[ \frac {22\,\sqrt {1-2\,x}}{78125}+\frac {2\,{\left (1-2\,x\right )}^{3/2}}{46875}-\frac {4774713\,{\left (1-2\,x\right )}^{5/2}}{250000}+\frac {806121\,{\left (1-2\,x\right )}^{7/2}}{35000}-\frac {5673\,{\left (1-2\,x\right )}^{9/2}}{500}+\frac {5751\,{\left (1-2\,x\right )}^{11/2}}{2200}-\frac {243\,{\left (1-2\,x\right )}^{13/2}}{1040}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,22{}\mathrm {i}}{390625} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 107.20, size = 150, normalized size = 1.24 \[ - \frac {243 \left (1 - 2 x\right )^{\frac {13}{2}}}{1040} + \frac {5751 \left (1 - 2 x\right )^{\frac {11}{2}}}{2200} - \frac {5673 \left (1 - 2 x\right )^{\frac {9}{2}}}{500} + \frac {806121 \left (1 - 2 x\right )^{\frac {7}{2}}}{35000} - \frac {4774713 \left (1 - 2 x\right )^{\frac {5}{2}}}{250000} + \frac {2 \left (1 - 2 x\right )^{\frac {3}{2}}}{46875} + \frac {22 \sqrt {1 - 2 x}}{78125} + \frac {242 \left (\begin {cases} - \frac {\sqrt {55} \operatorname {acoth}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: 2 x - 1 < - \frac {11}{5} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: 2 x - 1 > - \frac {11}{5} \end {cases}\right )}{78125} \]
Verification of antiderivative is not currently implemented for this CAS.
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