Optimal. Leaf size=80 \[ \frac {2401}{88 \sqrt {1-2 x}}+\frac {34371 \sqrt {1-2 x}}{1000}-\frac {963}{200} (1-2 x)^{3/2}+\frac {81}{200} (1-2 x)^{5/2}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1375 \sqrt {55}} \]
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Rubi [A]
time = 0.03, antiderivative size = 80, 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 {81}{200} (1-2 x)^{5/2}-\frac {963}{200} (1-2 x)^{3/2}+\frac {34371 \sqrt {1-2 x}}{1000}+\frac {2401}{88 \sqrt {1-2 x}}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1375 \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)^4}{(1-2 x)^{3/2} (3+5 x)} \, dx &=\int \left (\frac {2401}{88 (1-2 x)^{3/2}}-\frac {21951}{1000 \sqrt {1-2 x}}-\frac {2079 x}{100 \sqrt {1-2 x}}-\frac {81 x^2}{10 \sqrt {1-2 x}}+\frac {1}{1375 \sqrt {1-2 x} (3+5 x)}\right ) \, dx\\ &=\frac {2401}{88 \sqrt {1-2 x}}+\frac {21951 \sqrt {1-2 x}}{1000}+\frac {\int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{1375}-\frac {81}{10} \int \frac {x^2}{\sqrt {1-2 x}} \, dx-\frac {2079}{100} \int \frac {x}{\sqrt {1-2 x}} \, dx\\ &=\frac {2401}{88 \sqrt {1-2 x}}+\frac {21951 \sqrt {1-2 x}}{1000}-\frac {\text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{1375}-\frac {81}{10} \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 {2079}{100} \int \left (\frac {1}{2 \sqrt {1-2 x}}-\frac {1}{2} \sqrt {1-2 x}\right ) \, dx\\ &=\frac {2401}{88 \sqrt {1-2 x}}+\frac {34371 \sqrt {1-2 x}}{1000}-\frac {963}{200} (1-2 x)^{3/2}+\frac {81}{200} (1-2 x)^{5/2}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1375 \sqrt {55}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 56, normalized size = 0.70 \begin {gather*} \frac {-\frac {55 \left (-78712+71379 x+19800 x^2+4455 x^3\right )}{\sqrt {1-2 x}}-2 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{75625} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 56, normalized size = 0.70
method | result | size |
risch | \(-\frac {4455 x^{3}+19800 x^{2}+71379 x -78712}{1375 \sqrt {1-2 x}}-\frac {2 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{75625}\) | \(44\) |
derivativedivides | \(-\frac {963 \left (1-2 x \right )^{\frac {3}{2}}}{200}+\frac {81 \left (1-2 x \right )^{\frac {5}{2}}}{200}-\frac {2 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{75625}+\frac {2401}{88 \sqrt {1-2 x}}+\frac {34371 \sqrt {1-2 x}}{1000}\) | \(56\) |
default | \(-\frac {963 \left (1-2 x \right )^{\frac {3}{2}}}{200}+\frac {81 \left (1-2 x \right )^{\frac {5}{2}}}{200}-\frac {2 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{75625}+\frac {2401}{88 \sqrt {1-2 x}}+\frac {34371 \sqrt {1-2 x}}{1000}\) | \(56\) |
trager | \(\frac {\left (4455 x^{3}+19800 x^{2}+71379 x -78712\right ) \sqrt {1-2 x}}{-1375+2750 x}+\frac {\RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{75625}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 73, normalized size = 0.91 \begin {gather*} \frac {81}{200} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {963}{200} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{75625} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {34371}{1000} \, \sqrt {-2 \, x + 1} + \frac {2401}{88 \, \sqrt {-2 \, x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.70, size = 68, normalized size = 0.85 \begin {gather*} \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 (4455 \, x^{3} + 19800 \, x^{2} + 71379 \, x - 78712\right )} \sqrt {-2 \, x + 1}}{75625 \, {\left (2 \, x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 40.83, size = 107, normalized size = 1.34 \begin {gather*} \frac {81 \left (1 - 2 x\right )^{\frac {5}{2}}}{200} - \frac {963 \left (1 - 2 x\right )^{\frac {3}{2}}}{200} + \frac {34371 \sqrt {1 - 2 x}}{1000} + \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 )}{1375} + \frac {2401}{88 \sqrt {1 - 2 x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.60, size = 83, normalized size = 1.04 \begin {gather*} \frac {81}{200} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {963}{200} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{75625} \, \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 {34371}{1000} \, \sqrt {-2 \, x + 1} + \frac {2401}{88 \, \sqrt {-2 \, x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 57, normalized size = 0.71 \begin {gather*} \frac {2401}{88\,\sqrt {1-2\,x}}+\frac {34371\,\sqrt {1-2\,x}}{1000}-\frac {963\,{\left (1-2\,x\right )}^{3/2}}{200}+\frac {81\,{\left (1-2\,x\right )}^{5/2}}{200}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,2{}\mathrm {i}}{75625} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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