Optimal. Leaf size=128 \[ -\frac {(1-2 x)^{5/2}}{315 (2+3 x)^5}+\frac {23 (1-2 x)^{5/2}}{588 (2+3 x)^4}-\frac {4693 (1-2 x)^{3/2}}{15876 (2+3 x)^3}+\frac {4693 \sqrt {1-2 x}}{31752 (2+3 x)^2}-\frac {4693 \sqrt {1-2 x}}{222264 (2+3 x)}-\frac {4693 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{111132 \sqrt {21}} \]
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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 {23 (1-2 x)^{5/2}}{588 (3 x+2)^4}-\frac {(1-2 x)^{5/2}}{315 (3 x+2)^5}-\frac {4693 (1-2 x)^{3/2}}{15876 (3 x+2)^3}-\frac {4693 \sqrt {1-2 x}}{222264 (3 x+2)}+\frac {4693 \sqrt {1-2 x}}{31752 (3 x+2)^2}-\frac {4693 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{111132 \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 {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^6} \, dx &=-\frac {(1-2 x)^{5/2}}{315 (2+3 x)^5}+\frac {1}{315} \int \frac {(1-2 x)^{3/2} (1405+2625 x)}{(2+3 x)^5} \, dx\\ &=-\frac {(1-2 x)^{5/2}}{315 (2+3 x)^5}+\frac {23 (1-2 x)^{5/2}}{588 (2+3 x)^4}+\frac {4693 \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4} \, dx}{1764}\\ &=-\frac {(1-2 x)^{5/2}}{315 (2+3 x)^5}+\frac {23 (1-2 x)^{5/2}}{588 (2+3 x)^4}-\frac {4693 (1-2 x)^{3/2}}{15876 (2+3 x)^3}-\frac {4693 \int \frac {\sqrt {1-2 x}}{(2+3 x)^3} \, dx}{5292}\\ &=-\frac {(1-2 x)^{5/2}}{315 (2+3 x)^5}+\frac {23 (1-2 x)^{5/2}}{588 (2+3 x)^4}-\frac {4693 (1-2 x)^{3/2}}{15876 (2+3 x)^3}+\frac {4693 \sqrt {1-2 x}}{31752 (2+3 x)^2}+\frac {4693 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{31752}\\ &=-\frac {(1-2 x)^{5/2}}{315 (2+3 x)^5}+\frac {23 (1-2 x)^{5/2}}{588 (2+3 x)^4}-\frac {4693 (1-2 x)^{3/2}}{15876 (2+3 x)^3}+\frac {4693 \sqrt {1-2 x}}{31752 (2+3 x)^2}-\frac {4693 \sqrt {1-2 x}}{222264 (2+3 x)}+\frac {4693 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{222264}\\ &=-\frac {(1-2 x)^{5/2}}{315 (2+3 x)^5}+\frac {23 (1-2 x)^{5/2}}{588 (2+3 x)^4}-\frac {4693 (1-2 x)^{3/2}}{15876 (2+3 x)^3}+\frac {4693 \sqrt {1-2 x}}{31752 (2+3 x)^2}-\frac {4693 \sqrt {1-2 x}}{222264 (2+3 x)}-\frac {4693 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{222264}\\ &=-\frac {(1-2 x)^{5/2}}{315 (2+3 x)^5}+\frac {23 (1-2 x)^{5/2}}{588 (2+3 x)^4}-\frac {4693 (1-2 x)^{3/2}}{15876 (2+3 x)^3}+\frac {4693 \sqrt {1-2 x}}{31752 (2+3 x)^2}-\frac {4693 \sqrt {1-2 x}}{222264 (2+3 x)}-\frac {4693 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{111132 \sqrt {21}}\\ \end {align*}
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Mathematica [A]
time = 0.29, size = 70, normalized size = 0.55 \begin {gather*} \frac {-\frac {21 \sqrt {1-2 x} \left (292028-2143262 x-8540988 x^2-5801265 x^3+1900665 x^4\right )}{2 (2+3 x)^5}-23465 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{11668860} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 75, normalized size = 0.59
method | result | size |
risch | \(\frac {3801330 x^{5}-13503195 x^{4}-11280711 x^{3}+4254464 x^{2}+2727318 x -292028}{1111320 \left (2+3 x \right )^{5} \sqrt {1-2 x}}-\frac {4693 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2333772}\) | \(61\) |
derivativedivides | \(-\frac {3888 \left (-\frac {4693 \left (1-2 x \right )^{\frac {9}{2}}}{5334336}-\frac {907 \left (1-2 x \right )^{\frac {7}{2}}}{489888}+\frac {6119 \left (1-2 x \right )^{\frac {5}{2}}}{229635}-\frac {32851 \left (1-2 x \right )^{\frac {3}{2}}}{629856}+\frac {32851 \sqrt {1-2 x}}{1259712}\right )}{\left (-4-6 x \right )^{5}}-\frac {4693 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2333772}\) | \(75\) |
default | \(-\frac {3888 \left (-\frac {4693 \left (1-2 x \right )^{\frac {9}{2}}}{5334336}-\frac {907 \left (1-2 x \right )^{\frac {7}{2}}}{489888}+\frac {6119 \left (1-2 x \right )^{\frac {5}{2}}}{229635}-\frac {32851 \left (1-2 x \right )^{\frac {3}{2}}}{629856}+\frac {32851 \sqrt {1-2 x}}{1259712}\right )}{\left (-4-6 x \right )^{5}}-\frac {4693 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2333772}\) | \(75\) |
trager | \(-\frac {\left (1900665 x^{4}-5801265 x^{3}-8540988 x^{2}-2143262 x +292028\right ) \sqrt {1-2 x}}{1111320 \left (2+3 x \right )^{5}}+\frac {4693 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{4667544}\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 128, normalized size = 1.00 \begin {gather*} \frac {4693}{4667544} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {1900665 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 3999870 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 57567552 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 112678930 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 56339465 \, \sqrt {-2 \, x + 1}}{555660 \, {\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 = 0.99, size = 114, normalized size = 0.89 \begin {gather*} \frac {23465 \, \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 (1900665 \, x^{4} - 5801265 \, x^{3} - 8540988 \, x^{2} - 2143262 \, x + 292028\right )} \sqrt {-2 \, x + 1}}{23337720 \, {\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 = 0.84, size = 116, normalized size = 0.91 \begin {gather*} \frac {4693}{4667544} \, \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 {1900665 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - 3999870 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - 57567552 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 112678930 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 56339465 \, \sqrt {-2 \, x + 1}}{17781120 \, {\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 {4693\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2333772}-\frac {\frac {32851\,{\left (1-2\,x\right )}^{3/2}}{39366}-\frac {32851\,\sqrt {1-2\,x}}{78732}-\frac {97904\,{\left (1-2\,x\right )}^{5/2}}{229635}+\frac {907\,{\left (1-2\,x\right )}^{7/2}}{30618}+\frac {4693\,{\left (1-2\,x\right )}^{9/2}}{333396}}{\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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