Optimal. Leaf size=80 \[ \frac {2401}{264 (1-2 x)^{3/2}}-\frac {33271}{968 \sqrt {1-2 x}}-\frac {2889}{200} \sqrt {1-2 x}+\frac {27}{40} (1-2 x)^{3/2}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3025 \sqrt {55}} \]
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Rubi [A]
time = 0.03, antiderivative size = 80, 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 = {89, 45, 65, 212}
\begin {gather*} \frac {27}{40} (1-2 x)^{3/2}-\frac {2889}{200} \sqrt {1-2 x}-\frac {33271}{968 \sqrt {1-2 x}}+\frac {2401}{264 (1-2 x)^{3/2}}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3025 \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)^{5/2} (3+5 x)} \, dx &=\int \left (\frac {2401}{88 (1-2 x)^{5/2}}-\frac {33271}{968 (1-2 x)^{3/2}}+\frac {621}{50 \sqrt {1-2 x}}+\frac {81 x}{20 \sqrt {1-2 x}}+\frac {1}{3025 \sqrt {1-2 x} (3+5 x)}\right ) \, dx\\ &=\frac {2401}{264 (1-2 x)^{3/2}}-\frac {33271}{968 \sqrt {1-2 x}}-\frac {621}{50} \sqrt {1-2 x}+\frac {\int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{3025}+\frac {81}{20} \int \frac {x}{\sqrt {1-2 x}} \, dx\\ &=\frac {2401}{264 (1-2 x)^{3/2}}-\frac {33271}{968 \sqrt {1-2 x}}-\frac {621}{50} \sqrt {1-2 x}-\frac {\text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{3025}+\frac {81}{20} \int \left (\frac {1}{2 \sqrt {1-2 x}}-\frac {1}{2} \sqrt {1-2 x}\right ) \, dx\\ &=\frac {2401}{264 (1-2 x)^{3/2}}-\frac {33271}{968 \sqrt {1-2 x}}-\frac {2889}{200} \sqrt {1-2 x}+\frac {27}{40} (1-2 x)^{3/2}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3025 \sqrt {55}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 56, normalized size = 0.70 \begin {gather*} \frac {-\frac {55 \left (354344-1111431 x+450846 x^2+49005 x^3\right )}{(1-2 x)^{3/2}}-6 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{499125} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 56, normalized size = 0.70
method | result | size |
derivativedivides | \(\frac {2401}{264 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {27 \left (1-2 x \right )^{\frac {3}{2}}}{40}-\frac {2 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{166375}-\frac {33271}{968 \sqrt {1-2 x}}-\frac {2889 \sqrt {1-2 x}}{200}\) | \(56\) |
default | \(\frac {2401}{264 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {27 \left (1-2 x \right )^{\frac {3}{2}}}{40}-\frac {2 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{166375}-\frac {33271}{968 \sqrt {1-2 x}}-\frac {2889 \sqrt {1-2 x}}{200}\) | \(56\) |
trager | \(-\frac {\left (49005 x^{3}+450846 x^{2}-1111431 x +354344\right ) \sqrt {1-2 x}}{9075 \left (-1+2 x \right )^{2}}-\frac {\RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{166375}\) | \(78\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 69, normalized size = 0.86 \begin {gather*} \frac {27}{40} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{166375} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2889}{200} \, \sqrt {-2 \, x + 1} + \frac {343 \, {\left (291 \, x - 107\right )}}{1452 \, {\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 = 1.34, size = 79, normalized size = 0.99 \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 (49005 \, x^{3} + 450846 \, x^{2} - 1111431 \, x + 354344\right )} \sqrt {-2 \, x + 1}}{499125 \, {\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 = 41.63, size = 107, normalized size = 1.34 \begin {gather*} \frac {27 \left (1 - 2 x\right )^{\frac {3}{2}}}{40} - \frac {2889 \sqrt {1 - 2 x}}{200} + \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 )}{3025} - \frac {33271}{968 \sqrt {1 - 2 x}} + \frac {2401}{264 \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 = 1.25, size = 79, normalized size = 0.99 \begin {gather*} \frac {27}{40} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{166375} \, \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 {2889}{200} \, \sqrt {-2 \, x + 1} - \frac {343 \, {\left (291 \, x - 107\right )}}{1452 \, {\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 = 0.07, size = 52, normalized size = 0.65 \begin {gather*} \frac {\frac {33271\,x}{484}-\frac {36701}{1452}}{{\left (1-2\,x\right )}^{3/2}}-\frac {2889\,\sqrt {1-2\,x}}{200}+\frac {27\,{\left (1-2\,x\right )}^{3/2}}{40}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,2{}\mathrm {i}}{166375} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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