Optimal. Leaf size=80 \[ \frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)^2}+\frac {5 \sqrt {1-2 x} (541+857 x)}{2058 (2+3 x)^2}+\frac {2245 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}} \]
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Rubi [A]
time = 0.01, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {100, 150, 65,
212} \begin {gather*} \frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)^2}+\frac {5 \sqrt {1-2 x} (857 x+541)}{2058 (3 x+2)^2}+\frac {2245 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 100
Rule 150
Rule 212
Rubi steps
\begin {align*} \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^3} \, dx &=\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)^2}-\frac {1}{7} \int \frac {(3+5 x) (25+5 x)}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)^2}+\frac {5 \sqrt {1-2 x} (541+857 x)}{2058 (2+3 x)^2}-\frac {2245 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{2058}\\ &=\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)^2}+\frac {5 \sqrt {1-2 x} (541+857 x)}{2058 (2+3 x)^2}+\frac {2245 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{2058}\\ &=\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)^2}+\frac {5 \sqrt {1-2 x} (541+857 x)}{2058 (2+3 x)^2}+\frac {2245 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 70, normalized size = 0.88 \begin {gather*} \frac {195657-168175 (1-2 x)+36140 (1-2 x)^2}{1029 (-7+3 (1-2 x))^2 \sqrt {1-2 x}}+\frac {2245 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 57, normalized size = 0.71
method | result | size |
risch | \(\frac {72280 x^{2}+95895 x +31811}{2058 \left (2+3 x \right )^{2} \sqrt {1-2 x}}+\frac {2245 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21609}\) | \(46\) |
derivativedivides | \(\frac {1331}{343 \sqrt {1-2 x}}-\frac {18 \left (-\frac {203 \left (1-2 x \right )^{\frac {3}{2}}}{54}+\frac {469 \sqrt {1-2 x}}{54}\right )}{343 \left (-4-6 x \right )^{2}}+\frac {2245 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21609}\) | \(57\) |
default | \(\frac {1331}{343 \sqrt {1-2 x}}-\frac {18 \left (-\frac {203 \left (1-2 x \right )^{\frac {3}{2}}}{54}+\frac {469 \sqrt {1-2 x}}{54}\right )}{343 \left (-4-6 x \right )^{2}}+\frac {2245 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21609}\) | \(57\) |
trager | \(-\frac {\left (72280 x^{2}+95895 x +31811\right ) \sqrt {1-2 x}}{2058 \left (2+3 x \right )^{2} \left (-1+2 x \right )}-\frac {2245 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{43218}\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 83, normalized size = 1.04 \begin {gather*} -\frac {2245}{43218} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (18070 \, {\left (2 \, x - 1\right )}^{2} + 168175 \, x + 13741\right )}}{1029 \, {\left (9 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 42 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 49 \, \sqrt {-2 \, x + 1}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.76, size = 85, normalized size = 1.06 \begin {gather*} \frac {2245 \, \sqrt {21} {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (72280 \, x^{2} + 95895 \, x + 31811\right )} \sqrt {-2 \, x + 1}}{43218 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.36, size = 77, normalized size = 0.96 \begin {gather*} -\frac {2245}{43218} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {1331}{343 \, \sqrt {-2 \, x + 1}} + \frac {29 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 67 \, \sqrt {-2 \, x + 1}}{588 \, {\left (3 \, x + 2\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 62, normalized size = 0.78 \begin {gather*} \frac {2245\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{21609}+\frac {\frac {48050\,x}{1323}+\frac {36140\,{\left (2\,x-1\right )}^2}{9261}+\frac {3926}{1323}}{\frac {49\,\sqrt {1-2\,x}}{9}-\frac {14\,{\left (1-2\,x\right )}^{3/2}}{3}+{\left (1-2\,x\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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