Optimal. Leaf size=108 \[ \frac {\sqrt {1-2 x}}{84 (2+3 x)^4}-\frac {19 \sqrt {1-2 x}}{252 (2+3 x)^3}-\frac {95 \sqrt {1-2 x}}{3528 (2+3 x)^2}-\frac {95 \sqrt {1-2 x}}{8232 (2+3 x)}-\frac {95 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{4116 \sqrt {21}} \]
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Rubi [A]
time = 0.02, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {79, 44, 65, 212}
\begin {gather*} -\frac {95 \sqrt {1-2 x}}{8232 (3 x+2)}-\frac {95 \sqrt {1-2 x}}{3528 (3 x+2)^2}-\frac {19 \sqrt {1-2 x}}{252 (3 x+2)^3}+\frac {\sqrt {1-2 x}}{84 (3 x+2)^4}-\frac {95 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{4116 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 65
Rule 79
Rule 212
Rubi steps
\begin {align*} \int \frac {3+5 x}{\sqrt {1-2 x} (2+3 x)^5} \, dx &=\frac {\sqrt {1-2 x}}{84 (2+3 x)^4}+\frac {19}{12} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=\frac {\sqrt {1-2 x}}{84 (2+3 x)^4}-\frac {19 \sqrt {1-2 x}}{252 (2+3 x)^3}+\frac {95}{252} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=\frac {\sqrt {1-2 x}}{84 (2+3 x)^4}-\frac {19 \sqrt {1-2 x}}{252 (2+3 x)^3}-\frac {95 \sqrt {1-2 x}}{3528 (2+3 x)^2}+\frac {95 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{1176}\\ &=\frac {\sqrt {1-2 x}}{84 (2+3 x)^4}-\frac {19 \sqrt {1-2 x}}{252 (2+3 x)^3}-\frac {95 \sqrt {1-2 x}}{3528 (2+3 x)^2}-\frac {95 \sqrt {1-2 x}}{8232 (2+3 x)}+\frac {95 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{8232}\\ &=\frac {\sqrt {1-2 x}}{84 (2+3 x)^4}-\frac {19 \sqrt {1-2 x}}{252 (2+3 x)^3}-\frac {95 \sqrt {1-2 x}}{3528 (2+3 x)^2}-\frac {95 \sqrt {1-2 x}}{8232 (2+3 x)}-\frac {95 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{8232}\\ &=\frac {\sqrt {1-2 x}}{84 (2+3 x)^4}-\frac {19 \sqrt {1-2 x}}{252 (2+3 x)^3}-\frac {95 \sqrt {1-2 x}}{3528 (2+3 x)^2}-\frac {95 \sqrt {1-2 x}}{8232 (2+3 x)}-\frac {95 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{4116 \sqrt {21}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 65, normalized size = 0.60 \begin {gather*} \frac {-\frac {21 \sqrt {1-2 x} \left (2790+7942 x+7125 x^2+2565 x^3\right )}{2 (2+3 x)^4}-95 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{86436} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 66, normalized size = 0.61
method | result | size |
risch | \(\frac {5130 x^{4}+11685 x^{3}+8759 x^{2}-2362 x -2790}{8232 \left (2+3 x \right )^{4} \sqrt {1-2 x}}-\frac {95 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{86436}\) | \(56\) |
derivativedivides | \(-\frac {1296 \left (-\frac {95 \left (1-2 x \right )^{\frac {7}{2}}}{197568}+\frac {1045 \left (1-2 x \right )^{\frac {5}{2}}}{254016}-\frac {1387 \left (1-2 x \right )^{\frac {3}{2}}}{108864}+\frac {1447 \sqrt {1-2 x}}{108864}\right )}{\left (-4-6 x \right )^{4}}-\frac {95 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{86436}\) | \(66\) |
default | \(-\frac {1296 \left (-\frac {95 \left (1-2 x \right )^{\frac {7}{2}}}{197568}+\frac {1045 \left (1-2 x \right )^{\frac {5}{2}}}{254016}-\frac {1387 \left (1-2 x \right )^{\frac {3}{2}}}{108864}+\frac {1447 \sqrt {1-2 x}}{108864}\right )}{\left (-4-6 x \right )^{4}}-\frac {95 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{86436}\) | \(66\) |
trager | \(-\frac {\left (2565 x^{3}+7125 x^{2}+7942 x +2790\right ) \sqrt {1-2 x}}{8232 \left (2+3 x \right )^{4}}-\frac {95 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{172872}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 110, normalized size = 1.02 \begin {gather*} \frac {95}{172872} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2565 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 21945 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 67963 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 70903 \, \sqrt {-2 \, x + 1}}{4116 \, {\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 = 1.10, size = 99, normalized size = 0.92 \begin {gather*} \frac {95 \, \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 (2565 \, x^{3} + 7125 \, x^{2} + 7942 \, x + 2790\right )} \sqrt {-2 \, x + 1}}{172872 \, {\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 = 0.96, size = 100, normalized size = 0.93 \begin {gather*} \frac {95}{172872} \, \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 {2565 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 21945 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 67963 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 70903 \, \sqrt {-2 \, x + 1}}{65856 \, {\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 = 1.20, size = 90, normalized size = 0.83 \begin {gather*} -\frac {95\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{86436}-\frac {\frac {1447\,\sqrt {1-2\,x}}{6804}-\frac {1387\,{\left (1-2\,x\right )}^{3/2}}{6804}+\frac {1045\,{\left (1-2\,x\right )}^{5/2}}{15876}-\frac {95\,{\left (1-2\,x\right )}^{7/2}}{12348}}{\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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