Optimal. Leaf size=93 \[ -\frac {27}{80} (1-2 x)^{9/2}+\frac {5751 (1-2 x)^{7/2}}{1400}-\frac {51057 (1-2 x)^{5/2}}{2500}+\frac {268707 (1-2 x)^{3/2}}{5000}-\frac {4774713 \sqrt {1-2 x}}{50000}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125 \sqrt {55}} \]
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Rubi [A] time = 0.03, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {88, 63, 206} \[ -\frac {27}{80} (1-2 x)^{9/2}+\frac {5751 (1-2 x)^{7/2}}{1400}-\frac {51057 (1-2 x)^{5/2}}{2500}+\frac {268707 (1-2 x)^{3/2}}{5000}-\frac {4774713 \sqrt {1-2 x}}{50000}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125 \sqrt {55}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 88
Rule 206
Rubi steps
\begin {align*} \int \frac {(2+3 x)^5}{\sqrt {1-2 x} (3+5 x)} \, dx &=\int \left (\frac {4774713}{50000 \sqrt {1-2 x}}-\frac {806121 \sqrt {1-2 x}}{5000}+\frac {51057}{500} (1-2 x)^{3/2}-\frac {5751}{200} (1-2 x)^{5/2}+\frac {243}{80} (1-2 x)^{7/2}+\frac {1}{3125 \sqrt {1-2 x} (3+5 x)}\right ) \, dx\\ &=-\frac {4774713 \sqrt {1-2 x}}{50000}+\frac {268707 (1-2 x)^{3/2}}{5000}-\frac {51057 (1-2 x)^{5/2}}{2500}+\frac {5751 (1-2 x)^{7/2}}{1400}-\frac {27}{80} (1-2 x)^{9/2}+\frac {\int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{3125}\\ &=-\frac {4774713 \sqrt {1-2 x}}{50000}+\frac {268707 (1-2 x)^{3/2}}{5000}-\frac {51057 (1-2 x)^{5/2}}{2500}+\frac {5751 (1-2 x)^{7/2}}{1400}-\frac {27}{80} (1-2 x)^{9/2}-\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{3125}\\ &=-\frac {4774713 \sqrt {1-2 x}}{50000}+\frac {268707 (1-2 x)^{3/2}}{5000}-\frac {51057 (1-2 x)^{5/2}}{2500}+\frac {5751 (1-2 x)^{7/2}}{1400}-\frac {27}{80} (1-2 x)^{9/2}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125 \sqrt {55}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 61, normalized size = 0.66 \[ -\frac {3 \sqrt {1-2 x} \left (39375 x^4+160875 x^3+295290 x^2+348095 x+425872\right )}{21875}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125 \sqrt {55}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 60, normalized size = 0.65 \[ -\frac {3}{21875} \, {\left (39375 \, x^{4} + 160875 \, x^{3} + 295290 \, x^{2} + 348095 \, x + 425872\right )} \sqrt {-2 \, x + 1} + \frac {1}{171875} \, \sqrt {55} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.31, size = 106, normalized size = 1.14 \[ -\frac {27}{80} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {5751}{1400} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {51057}{2500} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {268707}{5000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{171875} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {4774713}{50000} \, \sqrt {-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 65, normalized size = 0.70 \[ -\frac {2 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{171875}+\frac {268707 \left (-2 x +1\right )^{\frac {3}{2}}}{5000}-\frac {51057 \left (-2 x +1\right )^{\frac {5}{2}}}{2500}+\frac {5751 \left (-2 x +1\right )^{\frac {7}{2}}}{1400}-\frac {27 \left (-2 x +1\right )^{\frac {9}{2}}}{80}-\frac {4774713 \sqrt {-2 x +1}}{50000} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 82, normalized size = 0.88 \[ -\frac {27}{80} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {5751}{1400} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {51057}{2500} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {268707}{5000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{171875} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {4774713}{50000} \, \sqrt {-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 66, normalized size = 0.71 \[ \frac {268707\,{\left (1-2\,x\right )}^{3/2}}{5000}-\frac {4774713\,\sqrt {1-2\,x}}{50000}-\frac {51057\,{\left (1-2\,x\right )}^{5/2}}{2500}+\frac {5751\,{\left (1-2\,x\right )}^{7/2}}{1400}-\frac {27\,{\left (1-2\,x\right )}^{9/2}}{80}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,2{}\mathrm {i}}{171875} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 78.71, size = 126, normalized size = 1.35 \[ - \frac {27 \left (1 - 2 x\right )^{\frac {9}{2}}}{80} + \frac {5751 \left (1 - 2 x\right )^{\frac {7}{2}}}{1400} - \frac {51057 \left (1 - 2 x\right )^{\frac {5}{2}}}{2500} + \frac {268707 \left (1 - 2 x\right )^{\frac {3}{2}}}{5000} - \frac {4774713 \sqrt {1 - 2 x}}{50000} + \frac {2 \left (\begin {cases} - \frac {\sqrt {55} \operatorname {acoth}{\left (\frac {\sqrt {55}}{5 \sqrt {1 - 2 x}} \right )}}{55} & \text {for}\: \frac {1}{1 - 2 x} > \frac {5}{11} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55}}{5 \sqrt {1 - 2 x}} \right )}}{55} & \text {for}\: \frac {1}{1 - 2 x} < \frac {5}{11} \end {cases}\right )}{3125} \]
Verification of antiderivative is not currently implemented for this CAS.
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