Optimal. Leaf size=128 \[ -\frac {(1-2 x)^{3/2}}{315 (2+3 x)^5}+\frac {7 (1-2 x)^{3/2}}{180 (2+3 x)^4}-\frac {31 \sqrt {1-2 x}}{108 (2+3 x)^3}+\frac {31 \sqrt {1-2 x}}{1512 (2+3 x)^2}+\frac {31 \sqrt {1-2 x}}{3528 (2+3 x)}+\frac {31 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1764 \sqrt {21}} \]
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Rubi [A]
time = 0.03, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {91, 79, 43, 44,
65, 212} \begin {gather*} \frac {7 (1-2 x)^{3/2}}{180 (3 x+2)^4}-\frac {(1-2 x)^{3/2}}{315 (3 x+2)^5}+\frac {31 \sqrt {1-2 x}}{3528 (3 x+2)}+\frac {31 \sqrt {1-2 x}}{1512 (3 x+2)^2}-\frac {31 \sqrt {1-2 x}}{108 (3 x+2)^3}+\frac {31 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1764 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 44
Rule 65
Rule 79
Rule 91
Rule 212
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^6} \, dx &=-\frac {(1-2 x)^{3/2}}{315 (2+3 x)^5}+\frac {1}{315} \int \frac {\sqrt {1-2 x} (1407+2625 x)}{(2+3 x)^5} \, dx\\ &=-\frac {(1-2 x)^{3/2}}{315 (2+3 x)^5}+\frac {7 (1-2 x)^{3/2}}{180 (2+3 x)^4}+\frac {31}{12} \int \frac {\sqrt {1-2 x}}{(2+3 x)^4} \, dx\\ &=-\frac {(1-2 x)^{3/2}}{315 (2+3 x)^5}+\frac {7 (1-2 x)^{3/2}}{180 (2+3 x)^4}-\frac {31 \sqrt {1-2 x}}{108 (2+3 x)^3}-\frac {31}{108} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=-\frac {(1-2 x)^{3/2}}{315 (2+3 x)^5}+\frac {7 (1-2 x)^{3/2}}{180 (2+3 x)^4}-\frac {31 \sqrt {1-2 x}}{108 (2+3 x)^3}+\frac {31 \sqrt {1-2 x}}{1512 (2+3 x)^2}-\frac {31}{504} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {(1-2 x)^{3/2}}{315 (2+3 x)^5}+\frac {7 (1-2 x)^{3/2}}{180 (2+3 x)^4}-\frac {31 \sqrt {1-2 x}}{108 (2+3 x)^3}+\frac {31 \sqrt {1-2 x}}{1512 (2+3 x)^2}+\frac {31 \sqrt {1-2 x}}{3528 (2+3 x)}-\frac {31 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{3528}\\ &=-\frac {(1-2 x)^{3/2}}{315 (2+3 x)^5}+\frac {7 (1-2 x)^{3/2}}{180 (2+3 x)^4}-\frac {31 \sqrt {1-2 x}}{108 (2+3 x)^3}+\frac {31 \sqrt {1-2 x}}{1512 (2+3 x)^2}+\frac {31 \sqrt {1-2 x}}{3528 (2+3 x)}+\frac {31 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{3528}\\ &=-\frac {(1-2 x)^{3/2}}{315 (2+3 x)^5}+\frac {7 (1-2 x)^{3/2}}{180 (2+3 x)^4}-\frac {31 \sqrt {1-2 x}}{108 (2+3 x)^3}+\frac {31 \sqrt {1-2 x}}{1512 (2+3 x)^2}+\frac {31 \sqrt {1-2 x}}{3528 (2+3 x)}+\frac {31 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1764 \sqrt {21}}\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 68, normalized size = 0.53 \begin {gather*} \frac {\frac {21 \sqrt {1-2 x} \left (-13564-33434 x+3324 x^2+43245 x^3+12555 x^4\right )}{(2+3 x)^5}+310 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{370440} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 75, normalized size = 0.59
method | result | size |
risch | \(-\frac {25110 x^{5}+73935 x^{4}-36597 x^{3}-70192 x^{2}+6306 x +13564}{17640 \left (2+3 x \right )^{5} \sqrt {1-2 x}}+\frac {31 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{37044}\) | \(61\) |
derivativedivides | \(-\frac {3888 \left (\frac {31 \left (1-2 x \right )^{\frac {9}{2}}}{84672}-\frac {31 \left (1-2 x \right )^{\frac {7}{2}}}{7776}+\frac {37 \left (1-2 x \right )^{\frac {5}{2}}}{3645}-\frac {983 \left (1-2 x \right )^{\frac {3}{2}}}{489888}-\frac {1519 \sqrt {1-2 x}}{139968}\right )}{\left (-4-6 x \right )^{5}}+\frac {31 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{37044}\) | \(75\) |
default | \(-\frac {3888 \left (\frac {31 \left (1-2 x \right )^{\frac {9}{2}}}{84672}-\frac {31 \left (1-2 x \right )^{\frac {7}{2}}}{7776}+\frac {37 \left (1-2 x \right )^{\frac {5}{2}}}{3645}-\frac {983 \left (1-2 x \right )^{\frac {3}{2}}}{489888}-\frac {1519 \sqrt {1-2 x}}{139968}\right )}{\left (-4-6 x \right )^{5}}+\frac {31 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{37044}\) | \(75\) |
trager | \(\frac {\left (12555 x^{4}+43245 x^{3}+3324 x^{2}-33434 x -13564\right ) \sqrt {1-2 x}}{17640 \left (2+3 x \right )^{5}}+\frac {31 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{74088}\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 128, normalized size = 1.00 \begin {gather*} -\frac {31}{74088} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {12555 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 136710 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 348096 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 68810 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 372155 \, \sqrt {-2 \, x + 1}}{8820 \, {\left (243 \, {\left (2 \, x - 1\right )}^{5} + 2835 \, {\left (2 \, x - 1\right )}^{4} + 13230 \, {\left (2 \, x - 1\right )}^{3} + 30870 \, {\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.82, size = 115, normalized size = 0.90 \begin {gather*} \frac {155 \, \sqrt {21} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (12555 \, x^{4} + 43245 \, x^{3} + 3324 \, x^{2} - 33434 \, x - 13564\right )} \sqrt {-2 \, x + 1}}{370440 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\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 = 0.54, size = 116, normalized size = 0.91 \begin {gather*} -\frac {31}{74088} \, \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 {12555 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 136710 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 348096 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 68810 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 372155 \, \sqrt {-2 \, x + 1}}{282240 \, {\left (3 \, x + 2\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 108, normalized size = 0.84 \begin {gather*} \frac {31\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{37044}-\frac {\frac {1519\,\sqrt {1-2\,x}}{8748}+\frac {983\,{\left (1-2\,x\right )}^{3/2}}{30618}-\frac {592\,{\left (1-2\,x\right )}^{5/2}}{3645}+\frac {31\,{\left (1-2\,x\right )}^{7/2}}{486}-\frac {31\,{\left (1-2\,x\right )}^{9/2}}{5292}}{\frac {24010\,x}{81}+\frac {3430\,{\left (2\,x-1\right )}^2}{27}+\frac {490\,{\left (2\,x-1\right )}^3}{9}+\frac {35\,{\left (2\,x-1\right )}^4}{3}+{\left (2\,x-1\right )}^5-\frac {19208}{243}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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