Optimal. Leaf size=93 \[ \frac {16807}{528 (1-2 x)^{3/2}}-\frac {156065}{968 \sqrt {1-2 x}}-\frac {51057}{500} \sqrt {1-2 x}+\frac {1917}{200} (1-2 x)^{3/2}-\frac {243}{400} (1-2 x)^{5/2}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15125 \sqrt {55}} \]
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Rubi [A]
time = 0.04, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {89, 45, 65, 212}
\begin {gather*} -\frac {243}{400} (1-2 x)^{5/2}+\frac {1917}{200} (1-2 x)^{3/2}-\frac {51057}{500} \sqrt {1-2 x}-\frac {156065}{968 \sqrt {1-2 x}}+\frac {16807}{528 (1-2 x)^{3/2}}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15125 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 65
Rule 89
Rule 212
Rubi steps
\begin {align*} \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)} \, dx &=\int \left (\frac {16807}{176 (1-2 x)^{5/2}}-\frac {156065}{968 (1-2 x)^{3/2}}+\frac {152793}{2000 \sqrt {1-2 x}}+\frac {1134 x}{25 \sqrt {1-2 x}}+\frac {243 x^2}{20 \sqrt {1-2 x}}+\frac {1}{15125 \sqrt {1-2 x} (3+5 x)}\right ) \, dx\\ &=\frac {16807}{528 (1-2 x)^{3/2}}-\frac {156065}{968 \sqrt {1-2 x}}-\frac {152793 \sqrt {1-2 x}}{2000}+\frac {\int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{15125}+\frac {243}{20} \int \frac {x^2}{\sqrt {1-2 x}} \, dx+\frac {1134}{25} \int \frac {x}{\sqrt {1-2 x}} \, dx\\ &=\frac {16807}{528 (1-2 x)^{3/2}}-\frac {156065}{968 \sqrt {1-2 x}}-\frac {152793 \sqrt {1-2 x}}{2000}-\frac {\text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{15125}+\frac {243}{20} \int \left (\frac {1}{4 \sqrt {1-2 x}}-\frac {1}{2} \sqrt {1-2 x}+\frac {1}{4} (1-2 x)^{3/2}\right ) \, dx+\frac {1134}{25} \int \left (\frac {1}{2 \sqrt {1-2 x}}-\frac {1}{2} \sqrt {1-2 x}\right ) \, dx\\ &=\frac {16807}{528 (1-2 x)^{3/2}}-\frac {156065}{968 \sqrt {1-2 x}}-\frac {51057}{500} \sqrt {1-2 x}+\frac {1917}{200} (1-2 x)^{3/2}-\frac {243}{400} (1-2 x)^{5/2}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15125 \sqrt {55}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 61, normalized size = 0.66 \begin {gather*} \frac {-\frac {55 \left (10097264-30775791 x+13976226 x^2+2597265 x^3+441045 x^4\right )}{(1-2 x)^{3/2}}-6 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{2495625} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 65, normalized size = 0.70
method | result | size |
derivativedivides | \(\frac {16807}{528 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {1917 \left (1-2 x \right )^{\frac {3}{2}}}{200}-\frac {243 \left (1-2 x \right )^{\frac {5}{2}}}{400}-\frac {2 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{831875}-\frac {156065}{968 \sqrt {1-2 x}}-\frac {51057 \sqrt {1-2 x}}{500}\) | \(65\) |
default | \(\frac {16807}{528 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {1917 \left (1-2 x \right )^{\frac {3}{2}}}{200}-\frac {243 \left (1-2 x \right )^{\frac {5}{2}}}{400}-\frac {2 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{831875}-\frac {156065}{968 \sqrt {1-2 x}}-\frac {51057 \sqrt {1-2 x}}{500}\) | \(65\) |
trager | \(-\frac {\left (441045 x^{4}+2597265 x^{3}+13976226 x^{2}-30775791 x +10097264\right ) \sqrt {1-2 x}}{45375 \left (-1+2 x \right )^{2}}-\frac {\RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{831875}\) | \(83\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 78, normalized size = 0.84 \begin {gather*} -\frac {243}{400} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {1917}{200} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{831875} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {51057}{500} \, \sqrt {-2 \, x + 1} + \frac {2401 \, {\left (780 \, x - 313\right )}}{5808 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.11, size = 84, normalized size = 0.90 \begin {gather*} \frac {3 \, \sqrt {55} {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (441045 \, x^{4} + 2597265 \, x^{3} + 13976226 \, x^{2} - 30775791 \, x + 10097264\right )} \sqrt {-2 \, x + 1}}{2495625 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 54.06, size = 119, normalized size = 1.28 \begin {gather*} - \frac {243 \left (1 - 2 x\right )^{\frac {5}{2}}}{400} + \frac {1917 \left (1 - 2 x\right )^{\frac {3}{2}}}{200} - \frac {51057 \sqrt {1 - 2 x}}{500} + \frac {2 \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 )}{15125} - \frac {156065}{968 \sqrt {1 - 2 x}} + \frac {16807}{528 \left (1 - 2 x\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.79, size = 95, normalized size = 1.02 \begin {gather*} -\frac {243}{400} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {1917}{200} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{831875} \, \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 {51057}{500} \, \sqrt {-2 \, x + 1} - \frac {2401 \, {\left (780 \, x - 313\right )}}{5808 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.21, size = 61, normalized size = 0.66 \begin {gather*} \frac {\frac {156065\,x}{484}-\frac {751513}{5808}}{{\left (1-2\,x\right )}^{3/2}}-\frac {51057\,\sqrt {1-2\,x}}{500}+\frac {1917\,{\left (1-2\,x\right )}^{3/2}}{200}-\frac {243\,{\left (1-2\,x\right )}^{5/2}}{400}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,2{}\mathrm {i}}{831875} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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