Optimal. Leaf size=120 \[ -\frac {3755 \sqrt {1-2 x}}{3087 (2+3 x)^2}-\frac {3755 \sqrt {1-2 x}}{7203 (2+3 x)}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^3}+\frac {2 (1346+2027 x)}{441 \sqrt {1-2 x} (2+3 x)^3}-\frac {7510 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7203 \sqrt {21}} \]
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Rubi [A]
time = 0.02, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {100, 149, 44,
65, 212} \begin {gather*} \frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^3}+\frac {2 (2027 x+1346)}{441 \sqrt {1-2 x} (3 x+2)^3}-\frac {3755 \sqrt {1-2 x}}{7203 (3 x+2)}-\frac {3755 \sqrt {1-2 x}}{3087 (3 x+2)^2}-\frac {7510 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7203 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 65
Rule 100
Rule 149
Rule 212
Rubi steps
\begin {align*} \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^4} \, dx &=\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {1}{21} \int \frac {(-68-150 x) (3+5 x)}{(1-2 x)^{3/2} (2+3 x)^4} \, dx\\ &=\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^3}+\frac {2 (1346+2027 x)}{441 \sqrt {1-2 x} (2+3 x)^3}+\frac {7510}{441} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=-\frac {3755 \sqrt {1-2 x}}{3087 (2+3 x)^2}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^3}+\frac {2 (1346+2027 x)}{441 \sqrt {1-2 x} (2+3 x)^3}+\frac {3755 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{1029}\\ &=-\frac {3755 \sqrt {1-2 x}}{3087 (2+3 x)^2}-\frac {3755 \sqrt {1-2 x}}{7203 (2+3 x)}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^3}+\frac {2 (1346+2027 x)}{441 \sqrt {1-2 x} (2+3 x)^3}+\frac {3755 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{7203}\\ &=-\frac {3755 \sqrt {1-2 x}}{3087 (2+3 x)^2}-\frac {3755 \sqrt {1-2 x}}{7203 (2+3 x)}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^3}+\frac {2 (1346+2027 x)}{441 \sqrt {1-2 x} (2+3 x)^3}-\frac {3755 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{7203}\\ &=-\frac {3755 \sqrt {1-2 x}}{3087 (2+3 x)^2}-\frac {3755 \sqrt {1-2 x}}{7203 (2+3 x)}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^3}+\frac {2 (1346+2027 x)}{441 \sqrt {1-2 x} (2+3 x)^3}-\frac {7510 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7203 \sqrt {21}}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 68, normalized size = 0.57 \begin {gather*} \frac {-\frac {21 \left (-45383-150295 x-83306 x^2+150200 x^3+135180 x^4\right )}{(1-2 x)^{3/2} (2+3 x)^3}-7510 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{151263} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 75, normalized size = 0.62
method | result | size |
risch | \(\frac {135180 x^{4}+150200 x^{3}-83306 x^{2}-150295 x -45383}{7203 \left (-1+2 x \right ) \sqrt {1-2 x}\, \left (2+3 x \right )^{3}}-\frac {7510 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{151263}\) | \(63\) |
derivativedivides | \(\frac {2662}{7203 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {6534}{16807 \sqrt {1-2 x}}+\frac {-\frac {18708 \left (1-2 x \right )^{\frac {5}{2}}}{16807}+\frac {5260 \left (1-2 x \right )^{\frac {3}{2}}}{1029}-\frac {6040 \sqrt {1-2 x}}{1029}}{\left (-4-6 x \right )^{3}}-\frac {7510 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{151263}\) | \(75\) |
default | \(\frac {2662}{7203 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {6534}{16807 \sqrt {1-2 x}}+\frac {-\frac {18708 \left (1-2 x \right )^{\frac {5}{2}}}{16807}+\frac {5260 \left (1-2 x \right )^{\frac {3}{2}}}{1029}-\frac {6040 \sqrt {1-2 x}}{1029}}{\left (-4-6 x \right )^{3}}-\frac {7510 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{151263}\) | \(75\) |
trager | \(-\frac {\left (135180 x^{4}+150200 x^{3}-83306 x^{2}-150295 x -45383\right ) \sqrt {1-2 x}}{7203 \left (2+3 x \right )^{3} \left (-1+2 x \right )^{2}}+\frac {3755 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{151263}\) | \(89\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 110, normalized size = 0.92 \begin {gather*} \frac {3755}{151263} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (33795 \, {\left (2 \, x - 1\right )}^{4} + 210280 \, {\left (2 \, x - 1\right )}^{3} + 344764 \, {\left (2 \, x - 1\right )}^{2} - 213444 \, x - 349811\right )}}{7203 \, {\left (27 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 189 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 441 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 343 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.08, size = 114, normalized size = 0.95 \begin {gather*} \frac {3755 \, \sqrt {21} {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (135180 \, x^{4} + 150200 \, x^{3} - 83306 \, x^{2} - 150295 \, x - 45383\right )} \sqrt {-2 \, x + 1}}{151263 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\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 = 1.92, size = 95, normalized size = 0.79 \begin {gather*} \frac {3755}{151263} \, \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 {2 \, {\left (33795 \, {\left (2 \, x - 1\right )}^{4} + 210280 \, {\left (2 \, x - 1\right )}^{3} + 344764 \, {\left (2 \, x - 1\right )}^{2} - 213444 \, x - 349811\right )}}{7203 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 7 \, \sqrt {-2 \, x + 1}\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 92, normalized size = 0.77 \begin {gather*} -\frac {\frac {14072\,{\left (2\,x-1\right )}^2}{3969}-\frac {968\,x}{441}+\frac {60080\,{\left (2\,x-1\right )}^3}{27783}+\frac {7510\,{\left (2\,x-1\right )}^4}{21609}-\frac {14278}{3969}}{\frac {343\,{\left (1-2\,x\right )}^{3/2}}{27}-\frac {49\,{\left (1-2\,x\right )}^{5/2}}{3}+7\,{\left (1-2\,x\right )}^{7/2}-{\left (1-2\,x\right )}^{9/2}}-\frac {7510\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{151263} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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