Optimal. Leaf size=121 \[ \frac {655}{28812 \sqrt {1-2 x}}+\frac {1}{84 \sqrt {1-2 x} (2+3 x)^4}-\frac {131}{1764 \sqrt {1-2 x} (2+3 x)^3}-\frac {131}{3528 \sqrt {1-2 x} (2+3 x)^2}-\frac {655}{24696 \sqrt {1-2 x} (2+3 x)}-\frac {655 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9604 \sqrt {21}} \]
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Rubi [A]
time = 0.02, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {79, 44, 53, 65,
212} \begin {gather*} \frac {655}{28812 \sqrt {1-2 x}}-\frac {655}{24696 \sqrt {1-2 x} (3 x+2)}-\frac {131}{3528 \sqrt {1-2 x} (3 x+2)^2}-\frac {131}{1764 \sqrt {1-2 x} (3 x+2)^3}+\frac {1}{84 \sqrt {1-2 x} (3 x+2)^4}-\frac {655 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9604 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 53
Rule 65
Rule 79
Rule 212
Rubi steps
\begin {align*} \int \frac {3+5 x}{(1-2 x)^{3/2} (2+3 x)^5} \, dx &=\frac {1}{84 \sqrt {1-2 x} (2+3 x)^4}+\frac {131}{84} \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^4} \, dx\\ &=\frac {1}{84 \sqrt {1-2 x} (2+3 x)^4}+\frac {131}{294 \sqrt {1-2 x} (2+3 x)^3}+\frac {131}{28} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=\frac {1}{84 \sqrt {1-2 x} (2+3 x)^4}+\frac {131}{294 \sqrt {1-2 x} (2+3 x)^3}-\frac {131 \sqrt {1-2 x}}{588 (2+3 x)^3}+\frac {655}{588} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=\frac {1}{84 \sqrt {1-2 x} (2+3 x)^4}+\frac {131}{294 \sqrt {1-2 x} (2+3 x)^3}-\frac {131 \sqrt {1-2 x}}{588 (2+3 x)^3}-\frac {655 \sqrt {1-2 x}}{8232 (2+3 x)^2}+\frac {655 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{2744}\\ &=\frac {1}{84 \sqrt {1-2 x} (2+3 x)^4}+\frac {131}{294 \sqrt {1-2 x} (2+3 x)^3}-\frac {131 \sqrt {1-2 x}}{588 (2+3 x)^3}-\frac {655 \sqrt {1-2 x}}{8232 (2+3 x)^2}-\frac {655 \sqrt {1-2 x}}{19208 (2+3 x)}+\frac {655 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{19208}\\ &=\frac {1}{84 \sqrt {1-2 x} (2+3 x)^4}+\frac {131}{294 \sqrt {1-2 x} (2+3 x)^3}-\frac {131 \sqrt {1-2 x}}{588 (2+3 x)^3}-\frac {655 \sqrt {1-2 x}}{8232 (2+3 x)^2}-\frac {655 \sqrt {1-2 x}}{19208 (2+3 x)}-\frac {655 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{19208}\\ &=\frac {1}{84 \sqrt {1-2 x} (2+3 x)^4}+\frac {131}{294 \sqrt {1-2 x} (2+3 x)^3}-\frac {131 \sqrt {1-2 x}}{588 (2+3 x)^3}-\frac {655 \sqrt {1-2 x}}{8232 (2+3 x)^2}-\frac {655 \sqrt {1-2 x}}{19208 (2+3 x)}-\frac {655 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9604 \sqrt {21}}\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 70, normalized size = 0.58 \begin {gather*} \frac {\frac {21 \left (-2566+10742 x+60391 x^2+80565 x^3+35370 x^4\right )}{2 \sqrt {1-2 x} (2+3 x)^4}-655 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{201684} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 75, normalized size = 0.62
method | result | size |
risch | \(\frac {35370 x^{4}+80565 x^{3}+60391 x^{2}+10742 x -2566}{19208 \left (2+3 x \right )^{4} \sqrt {1-2 x}}-\frac {655 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{201684}\) | \(56\) |
derivativedivides | \(\frac {176}{16807 \sqrt {1-2 x}}+\frac {\frac {66771 \left (1-2 x \right )^{\frac {7}{2}}}{67228}-\frac {75273 \left (1-2 x \right )^{\frac {5}{2}}}{9604}+\frac {28925 \left (1-2 x \right )^{\frac {3}{2}}}{1372}-\frac {3735 \sqrt {1-2 x}}{196}}{\left (-4-6 x \right )^{4}}-\frac {655 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{201684}\) | \(75\) |
default | \(\frac {176}{16807 \sqrt {1-2 x}}+\frac {\frac {66771 \left (1-2 x \right )^{\frac {7}{2}}}{67228}-\frac {75273 \left (1-2 x \right )^{\frac {5}{2}}}{9604}+\frac {28925 \left (1-2 x \right )^{\frac {3}{2}}}{1372}-\frac {3735 \sqrt {1-2 x}}{196}}{\left (-4-6 x \right )^{4}}-\frac {655 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{201684}\) | \(75\) |
trager | \(-\frac {\left (35370 x^{4}+80565 x^{3}+60391 x^{2}+10742 x -2566\right ) \sqrt {1-2 x}}{19208 \left (2+3 x \right )^{4} \left (-1+2 x \right )}+\frac {655 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{403368}\) | \(89\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 119, normalized size = 0.98 \begin {gather*} \frac {655}{403368} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {17685 \, {\left (2 \, x - 1\right )}^{4} + 151305 \, {\left (2 \, x - 1\right )}^{3} + 468587 \, {\left (2 \, x - 1\right )}^{2} + 1193934 \, x - 355495}{9604 \, {\left (81 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 756 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 2646 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 4116 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 2401 \, \sqrt {-2 \, x + 1}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.84, size = 114, normalized size = 0.94 \begin {gather*} \frac {655 \, \sqrt {21} {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (35370 \, x^{4} + 80565 \, x^{3} + 60391 \, x^{2} + 10742 \, x - 2566\right )} \sqrt {-2 \, x + 1}}{403368 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, 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 = 1.24, size = 109, normalized size = 0.90 \begin {gather*} \frac {655}{403368} \, \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 {176}{16807 \, \sqrt {-2 \, x + 1}} - \frac {66771 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 526911 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 1417325 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 1281105 \, \sqrt {-2 \, x + 1}}{1075648 \, {\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 = 1.26, size = 98, normalized size = 0.81 \begin {gather*} \frac {\frac {4061\,x}{2646}+\frac {9563\,{\left (2\,x-1\right )}^2}{15876}+\frac {7205\,{\left (2\,x-1\right )}^3}{37044}+\frac {655\,{\left (2\,x-1\right )}^4}{28812}-\frac {7255}{15876}}{\frac {2401\,\sqrt {1-2\,x}}{81}-\frac {1372\,{\left (1-2\,x\right )}^{3/2}}{27}+\frac {98\,{\left (1-2\,x\right )}^{5/2}}{3}-\frac {28\,{\left (1-2\,x\right )}^{7/2}}{3}+{\left (1-2\,x\right )}^{9/2}}-\frac {655\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{201684} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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