Optimal. Leaf size=148 \[ -\frac {(1-2 x)^{3/2} (3 x+2)^5}{5 (5 x+3)}+\frac {39}{275} (1-2 x)^{3/2} (3 x+2)^4+\frac {38 (1-2 x)^{3/2} (3 x+2)^3}{4125}-\frac {4016 (1-2 x)^{3/2} (3 x+2)^2}{48125}-\frac {2 (1-2 x)^{3/2} (204777 x+298462)}{515625}+\frac {324 \sqrt {1-2 x}}{78125}-\frac {324 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{78125} \]
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Rubi [A] time = 0.05, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {97, 153, 147, 50, 63, 206} \[ -\frac {(1-2 x)^{3/2} (3 x+2)^5}{5 (5 x+3)}+\frac {39}{275} (1-2 x)^{3/2} (3 x+2)^4+\frac {38 (1-2 x)^{3/2} (3 x+2)^3}{4125}-\frac {4016 (1-2 x)^{3/2} (3 x+2)^2}{48125}-\frac {2 (1-2 x)^{3/2} (204777 x+298462)}{515625}+\frac {324 \sqrt {1-2 x}}{78125}-\frac {324 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{78125} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 97
Rule 147
Rule 153
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (2+3 x)^5}{(3+5 x)^2} \, dx &=-\frac {(1-2 x)^{3/2} (2+3 x)^5}{5 (3+5 x)}+\frac {1}{5} \int \frac {(9-39 x) \sqrt {1-2 x} (2+3 x)^4}{3+5 x} \, dx\\ &=\frac {39}{275} (1-2 x)^{3/2} (2+3 x)^4-\frac {(1-2 x)^{3/2} (2+3 x)^5}{5 (3+5 x)}-\frac {1}{275} \int \frac {\sqrt {1-2 x} (2+3 x)^3 (-288+114 x)}{3+5 x} \, dx\\ &=\frac {38 (1-2 x)^{3/2} (2+3 x)^3}{4125}+\frac {39}{275} (1-2 x)^{3/2} (2+3 x)^4-\frac {(1-2 x)^{3/2} (2+3 x)^5}{5 (3+5 x)}+\frac {\int \frac {\sqrt {1-2 x} (2+3 x)^2 (24894+36144 x)}{3+5 x} \, dx}{12375}\\ &=-\frac {4016 (1-2 x)^{3/2} (2+3 x)^2}{48125}+\frac {38 (1-2 x)^{3/2} (2+3 x)^3}{4125}+\frac {39}{275} (1-2 x)^{3/2} (2+3 x)^4-\frac {(1-2 x)^{3/2} (2+3 x)^5}{5 (3+5 x)}-\frac {\int \frac {(-1742580-2866878 x) \sqrt {1-2 x} (2+3 x)}{3+5 x} \, dx}{433125}\\ &=-\frac {4016 (1-2 x)^{3/2} (2+3 x)^2}{48125}+\frac {38 (1-2 x)^{3/2} (2+3 x)^3}{4125}+\frac {39}{275} (1-2 x)^{3/2} (2+3 x)^4-\frac {(1-2 x)^{3/2} (2+3 x)^5}{5 (3+5 x)}-\frac {2 (1-2 x)^{3/2} (298462+204777 x)}{515625}+\frac {162 \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx}{15625}\\ &=\frac {324 \sqrt {1-2 x}}{78125}-\frac {4016 (1-2 x)^{3/2} (2+3 x)^2}{48125}+\frac {38 (1-2 x)^{3/2} (2+3 x)^3}{4125}+\frac {39}{275} (1-2 x)^{3/2} (2+3 x)^4-\frac {(1-2 x)^{3/2} (2+3 x)^5}{5 (3+5 x)}-\frac {2 (1-2 x)^{3/2} (298462+204777 x)}{515625}+\frac {1782 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{78125}\\ &=\frac {324 \sqrt {1-2 x}}{78125}-\frac {4016 (1-2 x)^{3/2} (2+3 x)^2}{48125}+\frac {38 (1-2 x)^{3/2} (2+3 x)^3}{4125}+\frac {39}{275} (1-2 x)^{3/2} (2+3 x)^4-\frac {(1-2 x)^{3/2} (2+3 x)^5}{5 (3+5 x)}-\frac {2 (1-2 x)^{3/2} (298462+204777 x)}{515625}-\frac {1782 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{78125}\\ &=\frac {324 \sqrt {1-2 x}}{78125}-\frac {4016 (1-2 x)^{3/2} (2+3 x)^2}{48125}+\frac {38 (1-2 x)^{3/2} (2+3 x)^3}{4125}+\frac {39}{275} (1-2 x)^{3/2} (2+3 x)^4-\frac {(1-2 x)^{3/2} (2+3 x)^5}{5 (3+5 x)}-\frac {2 (1-2 x)^{3/2} (298462+204777 x)}{515625}-\frac {324 \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 = 0.11, size = 78, normalized size = 0.53 \[ \frac {-\frac {5 \sqrt {1-2 x} \left (106312500 x^6+270112500 x^5+181738125 x^4-76760550 x^3-135193430 x^2-2532130 x+23061496\right )}{5 x+3}-24948 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{30078125} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.24, size = 90, normalized size = 0.61 \[ \frac {12474 \, \sqrt {11} \sqrt {5} {\left (5 \, x + 3\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 5 \, {\left (106312500 \, x^{6} + 270112500 \, x^{5} + 181738125 \, x^{4} - 76760550 \, x^{3} - 135193430 \, x^{2} - 2532130 \, x + 23061496\right )} \sqrt {-2 \, x + 1}}{30078125 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.09, size = 138, normalized size = 0.93 \[ -\frac {243}{2200} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {981}{1000} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {107109}{35000} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {434043}{125000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {2}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {162}{390625} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {326}{78125} \, \sqrt {-2 \, x + 1} - \frac {11 \, \sqrt {-2 \, x + 1}}{78125 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 90, normalized size = 0.61 \[ -\frac {324 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{390625}+\frac {243 \left (-2 x +1\right )^{\frac {11}{2}}}{2200}-\frac {981 \left (-2 x +1\right )^{\frac {9}{2}}}{1000}+\frac {107109 \left (-2 x +1\right )^{\frac {7}{2}}}{35000}-\frac {434043 \left (-2 x +1\right )^{\frac {5}{2}}}{125000}+\frac {2 \left (-2 x +1\right )^{\frac {3}{2}}}{3125}+\frac {326 \sqrt {-2 x +1}}{78125}+\frac {22 \sqrt {-2 x +1}}{390625 \left (-2 x -\frac {6}{5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.12, size = 107, normalized size = 0.72 \[ \frac {243}{2200} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {981}{1000} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {107109}{35000} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {434043}{125000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {2}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {162}{390625} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {326}{78125} \, \sqrt {-2 \, x + 1} - \frac {11 \, \sqrt {-2 \, x + 1}}{78125 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 91, normalized size = 0.61 \[ \frac {326\,\sqrt {1-2\,x}}{78125}-\frac {22\,\sqrt {1-2\,x}}{390625\,\left (2\,x+\frac {6}{5}\right )}+\frac {2\,{\left (1-2\,x\right )}^{3/2}}{3125}-\frac {434043\,{\left (1-2\,x\right )}^{5/2}}{125000}+\frac {107109\,{\left (1-2\,x\right )}^{7/2}}{35000}-\frac {981\,{\left (1-2\,x\right )}^{9/2}}{1000}+\frac {243\,{\left (1-2\,x\right )}^{11/2}}{2200}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,324{}\mathrm {i}}{390625} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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