Optimal. Leaf size=108 \[ -\frac {(1-2 x)^{3/2}}{252 (2+3 x)^4}+\frac {275 (1-2 x)^{3/2}}{5292 (2+3 x)^3}-\frac {4625 \sqrt {1-2 x}}{10584 (2+3 x)^2}+\frac {4625 \sqrt {1-2 x}}{74088 (2+3 x)}+\frac {4625 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{37044 \sqrt {21}} \]
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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 = 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 {275 (1-2 x)^{3/2}}{5292 (3 x+2)^3}-\frac {(1-2 x)^{3/2}}{252 (3 x+2)^4}+\frac {4625 \sqrt {1-2 x}}{74088 (3 x+2)}-\frac {4625 \sqrt {1-2 x}}{10584 (3 x+2)^2}+\frac {4625 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{37044 \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)^5} \, dx &=-\frac {(1-2 x)^{3/2}}{252 (2+3 x)^4}+\frac {1}{252} \int \frac {\sqrt {1-2 x} (1125+2100 x)}{(2+3 x)^4} \, dx\\ &=-\frac {(1-2 x)^{3/2}}{252 (2+3 x)^4}+\frac {275 (1-2 x)^{3/2}}{5292 (2+3 x)^3}+\frac {4625 \int \frac {\sqrt {1-2 x}}{(2+3 x)^3} \, dx}{1764}\\ &=-\frac {(1-2 x)^{3/2}}{252 (2+3 x)^4}+\frac {275 (1-2 x)^{3/2}}{5292 (2+3 x)^3}-\frac {4625 \sqrt {1-2 x}}{10584 (2+3 x)^2}-\frac {4625 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{10584}\\ &=-\frac {(1-2 x)^{3/2}}{252 (2+3 x)^4}+\frac {275 (1-2 x)^{3/2}}{5292 (2+3 x)^3}-\frac {4625 \sqrt {1-2 x}}{10584 (2+3 x)^2}+\frac {4625 \sqrt {1-2 x}}{74088 (2+3 x)}-\frac {4625 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{74088}\\ &=-\frac {(1-2 x)^{3/2}}{252 (2+3 x)^4}+\frac {275 (1-2 x)^{3/2}}{5292 (2+3 x)^3}-\frac {4625 \sqrt {1-2 x}}{10584 (2+3 x)^2}+\frac {4625 \sqrt {1-2 x}}{74088 (2+3 x)}+\frac {4625 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{74088}\\ &=-\frac {(1-2 x)^{3/2}}{252 (2+3 x)^4}+\frac {275 (1-2 x)^{3/2}}{5292 (2+3 x)^3}-\frac {4625 \sqrt {1-2 x}}{10584 (2+3 x)^2}+\frac {4625 \sqrt {1-2 x}}{74088 (2+3 x)}+\frac {4625 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{37044 \sqrt {21}}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 65, normalized size = 0.60 \begin {gather*} \frac {\frac {21 \sqrt {1-2 x} \left (-85094-225262 x-64725 x^2+124875 x^3\right )}{2 (2+3 x)^4}+4625 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{777924} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 66, normalized size = 0.61
method | result | size |
risch | \(-\frac {249750 x^{4}-254325 x^{3}-385799 x^{2}+55074 x +85094}{74088 \left (2+3 x \right )^{4} \sqrt {1-2 x}}+\frac {4625 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{777924}\) | \(56\) |
derivativedivides | \(\frac {-\frac {4625 \left (1-2 x \right )^{\frac {7}{2}}}{1372}+\frac {11675 \left (1-2 x \right )^{\frac {5}{2}}}{1764}+\frac {16027 \left (1-2 x \right )^{\frac {3}{2}}}{756}-\frac {4625 \sqrt {1-2 x}}{108}}{\left (-4-6 x \right )^{4}}+\frac {4625 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{777924}\) | \(66\) |
default | \(\frac {-\frac {4625 \left (1-2 x \right )^{\frac {7}{2}}}{1372}+\frac {11675 \left (1-2 x \right )^{\frac {5}{2}}}{1764}+\frac {16027 \left (1-2 x \right )^{\frac {3}{2}}}{756}-\frac {4625 \sqrt {1-2 x}}{108}}{\left (-4-6 x \right )^{4}}+\frac {4625 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{777924}\) | \(66\) |
trager | \(\frac {\left (124875 x^{3}-64725 x^{2}-225262 x -85094\right ) \sqrt {1-2 x}}{74088 \left (2+3 x \right )^{4}}+\frac {4625 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{1555848}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 110, normalized size = 1.02 \begin {gather*} -\frac {4625}{1555848} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {124875 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 245175 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 785323 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 1586375 \, \sqrt {-2 \, x + 1}}{37044 \, {\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 = 0.88, size = 100, normalized size = 0.93 \begin {gather*} \frac {4625 \, \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 (124875 \, x^{3} - 64725 \, x^{2} - 225262 \, x - 85094\right )} \sqrt {-2 \, x + 1}}{1555848 \, {\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 = 0.54, size = 100, normalized size = 0.93 \begin {gather*} -\frac {4625}{1555848} \, \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 {124875 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 245175 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 785323 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1586375 \, \sqrt {-2 \, x + 1}}{592704 \, {\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 {4625\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{777924}-\frac {\frac {4625\,\sqrt {1-2\,x}}{8748}-\frac {16027\,{\left (1-2\,x\right )}^{3/2}}{61236}-\frac {11675\,{\left (1-2\,x\right )}^{5/2}}{142884}+\frac {4625\,{\left (1-2\,x\right )}^{7/2}}{111132}}{\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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