Optimal. Leaf size=88 \[ \frac {(1-2 x)^{5/2}}{63 (2+3 x)^3}-\frac {52 (1-2 x)^{3/2}}{189 (2+3 x)^2}+\frac {52 \sqrt {1-2 x}}{189 (2+3 x)}-\frac {104 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{189 \sqrt {21}} \]
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Rubi [A]
time = 0.01, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {79, 43, 65, 212}
\begin {gather*} \frac {(1-2 x)^{5/2}}{63 (3 x+2)^3}-\frac {52 (1-2 x)^{3/2}}{189 (3 x+2)^2}+\frac {52 \sqrt {1-2 x}}{189 (3 x+2)}-\frac {104 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{189 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 65
Rule 79
Rule 212
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^4} \, dx &=\frac {(1-2 x)^{5/2}}{63 (2+3 x)^3}+\frac {104}{63} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3} \, dx\\ &=\frac {(1-2 x)^{5/2}}{63 (2+3 x)^3}-\frac {52 (1-2 x)^{3/2}}{189 (2+3 x)^2}-\frac {52}{63} \int \frac {\sqrt {1-2 x}}{(2+3 x)^2} \, dx\\ &=\frac {(1-2 x)^{5/2}}{63 (2+3 x)^3}-\frac {52 (1-2 x)^{3/2}}{189 (2+3 x)^2}+\frac {52 \sqrt {1-2 x}}{189 (2+3 x)}+\frac {52}{189} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {(1-2 x)^{5/2}}{63 (2+3 x)^3}-\frac {52 (1-2 x)^{3/2}}{189 (2+3 x)^2}+\frac {52 \sqrt {1-2 x}}{189 (2+3 x)}-\frac {52}{189} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {(1-2 x)^{5/2}}{63 (2+3 x)^3}-\frac {52 (1-2 x)^{3/2}}{189 (2+3 x)^2}+\frac {52 \sqrt {1-2 x}}{189 (2+3 x)}-\frac {104 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{189 \sqrt {21}}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 60, normalized size = 0.68 \begin {gather*} \frac {8 \left (\frac {21 \sqrt {1-2 x} \left (107+664 x+792 x^2\right )}{8 (2+3 x)^3}-13 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )}{3969} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 57, normalized size = 0.65
method | result | size |
risch | \(-\frac {1584 x^{3}+536 x^{2}-450 x -107}{189 \left (2+3 x \right )^{3} \sqrt {1-2 x}}-\frac {104 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3969}\) | \(51\) |
derivativedivides | \(\frac {-\frac {176 \left (1-2 x \right )^{\frac {5}{2}}}{21}+\frac {832 \left (1-2 x \right )^{\frac {3}{2}}}{27}-\frac {728 \sqrt {1-2 x}}{27}}{\left (-4-6 x \right )^{3}}-\frac {104 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3969}\) | \(57\) |
default | \(\frac {-\frac {176 \left (1-2 x \right )^{\frac {5}{2}}}{21}+\frac {832 \left (1-2 x \right )^{\frac {3}{2}}}{27}-\frac {728 \sqrt {1-2 x}}{27}}{\left (-4-6 x \right )^{3}}-\frac {104 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3969}\) | \(57\) |
trager | \(\frac {\left (792 x^{2}+664 x +107\right ) \sqrt {1-2 x}}{189 \left (2+3 x \right )^{3}}+\frac {52 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{3969}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 92, normalized size = 1.05 \begin {gather*} \frac {52}{3969} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {8 \, {\left (198 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 728 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 637 \, \sqrt {-2 \, x + 1}\right )}}{189 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.68, size = 84, normalized size = 0.95 \begin {gather*} \frac {52 \, \sqrt {21} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (792 \, x^{2} + 664 \, x + 107\right )} \sqrt {-2 \, x + 1}}{3969 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, 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 = 3.43, size = 84, normalized size = 0.95 \begin {gather*} \frac {52}{3969} \, \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 {198 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 728 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 637 \, \sqrt {-2 \, x + 1}}{189 \, {\left (3 \, x + 2\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 71, normalized size = 0.81 \begin {gather*} \frac {\frac {728\,\sqrt {1-2\,x}}{729}-\frac {832\,{\left (1-2\,x\right )}^{3/2}}{729}+\frac {176\,{\left (1-2\,x\right )}^{5/2}}{567}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}}-\frac {104\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{3969} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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