Optimal. Leaf size=67 \[ -\frac {9}{20} (1-2 x)^{3/2}+\frac {162}{25} \sqrt {1-2 x}+\frac {343}{44 \sqrt {1-2 x}}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{275 \sqrt {55}} \]
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Rubi [A] time = 0.03, antiderivative size = 67, 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 = {87, 43, 63, 206} \[ -\frac {9}{20} (1-2 x)^{3/2}+\frac {162}{25} \sqrt {1-2 x}+\frac {343}{44 \sqrt {1-2 x}}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{275 \sqrt {55}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 63
Rule 87
Rule 206
Rubi steps
\begin {align*} \int \frac {(2+3 x)^3}{(1-2 x)^{3/2} (3+5 x)} \, dx &=\int \left (\frac {343}{44 (1-2 x)^{3/2}}-\frac {513}{100 \sqrt {1-2 x}}-\frac {27 x}{10 \sqrt {1-2 x}}+\frac {1}{275 \sqrt {1-2 x} (3+5 x)}\right ) \, dx\\ &=\frac {343}{44 \sqrt {1-2 x}}+\frac {513}{100} \sqrt {1-2 x}+\frac {1}{275} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx-\frac {27}{10} \int \frac {x}{\sqrt {1-2 x}} \, dx\\ &=\frac {343}{44 \sqrt {1-2 x}}+\frac {513}{100} \sqrt {1-2 x}-\frac {1}{275} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {27}{10} \int \left (\frac {1}{2 \sqrt {1-2 x}}-\frac {1}{2} \sqrt {1-2 x}\right ) \, dx\\ &=\frac {343}{44 \sqrt {1-2 x}}+\frac {162}{25} \sqrt {1-2 x}-\frac {9}{20} (1-2 x)^{3/2}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{275 \sqrt {55}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 45, normalized size = 0.67 \[ \frac {2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {5}{11} (1-2 x)\right )-99 \left (25 x^2+155 x-192\right )}{1375 \sqrt {1-2 x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 63, normalized size = 0.94 \[ \frac {\sqrt {55} {\left (2 \, x - 1\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \, {\left (495 \, x^{2} + 3069 \, x - 3802\right )} \sqrt {-2 \, x + 1}}{15125 \, {\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.23, size = 67, normalized size = 1.00 \[ -\frac {9}{20} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{15125} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {162}{25} \, \sqrt {-2 \, x + 1} + \frac {343}{44 \, \sqrt {-2 \, x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 47, normalized size = 0.70 \[ -\frac {2 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{15125}-\frac {9 \left (-2 x +1\right )^{\frac {3}{2}}}{20}+\frac {343}{44 \sqrt {-2 x +1}}+\frac {162 \sqrt {-2 x +1}}{25} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 64, normalized size = 0.96 \[ -\frac {9}{20} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{15125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {162}{25} \, \sqrt {-2 \, x + 1} + \frac {343}{44 \, \sqrt {-2 \, x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 48, normalized size = 0.72 \[ \frac {343}{44\,\sqrt {1-2\,x}}+\frac {162\,\sqrt {1-2\,x}}{25}-\frac {9\,{\left (1-2\,x\right )}^{3/2}}{20}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,2{}\mathrm {i}}{15125} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 62.00, size = 102, normalized size = 1.52 \[ - \frac {9 \left (1 - 2 x\right )^{\frac {3}{2}}}{20} + \frac {162 \sqrt {1 - 2 x}}{25} + \frac {2 \left (\begin {cases} - \frac {\sqrt {55} \operatorname {acoth}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: 2 x - 1 < - \frac {11}{5} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: 2 x - 1 > - \frac {11}{5} \end {cases}\right )}{275} + \frac {343}{44 \sqrt {1 - 2 x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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