Optimal. Leaf size=131 \[ \frac {13 \sqrt {1-2 x} (3+5 x)^2}{56 (2+3 x)^2}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{12 (2+3 x)^4}+\frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^3}-\frac {\sqrt {1-2 x} (18187+26775 x)}{1176 (2+3 x)}+\frac {13243 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{588 \sqrt {21}} \]
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Rubi [A]
time = 0.03, antiderivative size = 131, 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 = {99, 154, 151,
65, 212} \begin {gather*} \frac {\sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^3}-\frac {(1-2 x)^{3/2} (5 x+3)^3}{12 (3 x+2)^4}+\frac {13 \sqrt {1-2 x} (5 x+3)^2}{56 (3 x+2)^2}-\frac {\sqrt {1-2 x} (26775 x+18187)}{1176 (3 x+2)}+\frac {13243 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{588 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 99
Rule 151
Rule 154
Rule 212
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^5} \, dx &=-\frac {(1-2 x)^{3/2} (3+5 x)^3}{12 (2+3 x)^4}+\frac {1}{12} \int \frac {(6-45 x) \sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^4} \, dx\\ &=-\frac {(1-2 x)^{3/2} (3+5 x)^3}{12 (2+3 x)^4}+\frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^3}-\frac {1}{108} \int \frac {(-189-810 x) (3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=\frac {13 \sqrt {1-2 x} (3+5 x)^2}{56 (2+3 x)^2}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{12 (2+3 x)^4}+\frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^3}-\frac {\int \frac {(-14013-61965 x) (3+5 x)}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{4536}\\ &=\frac {13 \sqrt {1-2 x} (3+5 x)^2}{56 (2+3 x)^2}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{12 (2+3 x)^4}+\frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^3}-\frac {\sqrt {1-2 x} (18187+26775 x)}{1176 (2+3 x)}-\frac {13243 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{1176}\\ &=\frac {13 \sqrt {1-2 x} (3+5 x)^2}{56 (2+3 x)^2}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{12 (2+3 x)^4}+\frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^3}-\frac {\sqrt {1-2 x} (18187+26775 x)}{1176 (2+3 x)}+\frac {13243 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{1176}\\ &=\frac {13 \sqrt {1-2 x} (3+5 x)^2}{56 (2+3 x)^2}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{12 (2+3 x)^4}+\frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^3}-\frac {\sqrt {1-2 x} (18187+26775 x)}{1176 (2+3 x)}+\frac {13243 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{588 \sqrt {21}}\\ \end {align*}
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Mathematica [A]
time = 0.26, size = 68, normalized size = 0.52 \begin {gather*} \frac {-\frac {21 \sqrt {1-2 x} \left (74810+401850 x+788415 x^2+661639 x^3+196000 x^4\right )}{(2+3 x)^4}+26486 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{24696} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 75, normalized size = 0.57
method | result | size |
risch | \(\frac {392000 x^{5}+1127278 x^{4}+915191 x^{3}+15285 x^{2}-252230 x -74810}{1176 \left (2+3 x \right )^{4} \sqrt {1-2 x}}+\frac {13243 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{12348}\) | \(61\) |
derivativedivides | \(-\frac {500 \sqrt {1-2 x}}{243}-\frac {4 \left (-\frac {416917 \left (1-2 x \right )^{\frac {7}{2}}}{2352}+\frac {406463 \left (1-2 x \right )^{\frac {5}{2}}}{336}-\frac {1189171 \left (1-2 x \right )^{\frac {3}{2}}}{432}+\frac {2706781 \sqrt {1-2 x}}{1296}\right )}{3 \left (-4-6 x \right )^{4}}+\frac {13243 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{12348}\) | \(75\) |
default | \(-\frac {500 \sqrt {1-2 x}}{243}-\frac {4 \left (-\frac {416917 \left (1-2 x \right )^{\frac {7}{2}}}{2352}+\frac {406463 \left (1-2 x \right )^{\frac {5}{2}}}{336}-\frac {1189171 \left (1-2 x \right )^{\frac {3}{2}}}{432}+\frac {2706781 \sqrt {1-2 x}}{1296}\right )}{3 \left (-4-6 x \right )^{4}}+\frac {13243 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{12348}\) | \(75\) |
trager | \(-\frac {\left (196000 x^{4}+661639 x^{3}+788415 x^{2}+401850 x +74810\right ) \sqrt {1-2 x}}{1176 \left (2+3 x \right )^{4}}+\frac {13243 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{24696}\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 119, normalized size = 0.91 \begin {gather*} -\frac {13243}{24696} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {500}{243} \, \sqrt {-2 \, x + 1} + \frac {11256759 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 76821507 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 174808137 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 132632269 \, \sqrt {-2 \, x + 1}}{47628 \, {\left (81 \, {\left (2 \, x - 1\right )}^{4} + 756 \, {\left (2 \, x - 1\right )}^{3} + 2646 \, {\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.35, size = 105, normalized size = 0.80 \begin {gather*} \frac {13243 \, \sqrt {21} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (196000 \, x^{4} + 661639 \, x^{3} + 788415 \, x^{2} + 401850 \, x + 74810\right )} \sqrt {-2 \, x + 1}}{24696 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\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 = 1.19, size = 109, normalized size = 0.83 \begin {gather*} -\frac {13243}{24696} \, \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 {500}{243} \, \sqrt {-2 \, x + 1} - \frac {11256759 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 76821507 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 174808137 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 132632269 \, \sqrt {-2 \, x + 1}}{762048 \, {\left (3 \, x + 2\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.18, size = 99, normalized size = 0.76 \begin {gather*} \frac {13243\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{12348}-\frac {500\,\sqrt {1-2\,x}}{243}-\frac {\frac {2706781\,\sqrt {1-2\,x}}{78732}-\frac {1189171\,{\left (1-2\,x\right )}^{3/2}}{26244}+\frac {406463\,{\left (1-2\,x\right )}^{5/2}}{20412}-\frac {416917\,{\left (1-2\,x\right )}^{7/2}}{142884}}{\frac {2744\,x}{27}+\frac {98\,{\left (2\,x-1\right )}^2}{3}+\frac {28\,{\left (2\,x-1\right )}^3}{3}+{\left (2\,x-1\right )}^4-\frac {1715}{81}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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