Optimal. Leaf size=113 \[ -\frac {\sqrt {1-2 x} (3 x+2)^4}{5 (5 x+3)}+\frac {27}{175} \sqrt {1-2 x} (3 x+2)^3+\frac {12}{625} \sqrt {1-2 x} (3 x+2)^2-\frac {3 \sqrt {1-2 x} (375 x+1256)}{3125}-\frac {262 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125 \sqrt {55}} \]
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Rubi [A] time = 0.04, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {97, 153, 147, 63, 206} \begin {gather*} -\frac {\sqrt {1-2 x} (3 x+2)^4}{5 (5 x+3)}+\frac {27}{175} \sqrt {1-2 x} (3 x+2)^3+\frac {12}{625} \sqrt {1-2 x} (3 x+2)^2-\frac {3 \sqrt {1-2 x} (375 x+1256)}{3125}-\frac {262 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 97
Rule 147
Rule 153
Rule 206
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (2+3 x)^4}{(3+5 x)^2} \, dx &=-\frac {\sqrt {1-2 x} (2+3 x)^4}{5 (3+5 x)}+\frac {1}{5} \int \frac {(10-27 x) (2+3 x)^3}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {27}{175} \sqrt {1-2 x} (2+3 x)^3-\frac {\sqrt {1-2 x} (2+3 x)^4}{5 (3+5 x)}-\frac {1}{175} \int \frac {(2+3 x)^2 (-133+84 x)}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {12}{625} \sqrt {1-2 x} (2+3 x)^2+\frac {27}{175} \sqrt {1-2 x} (2+3 x)^3-\frac {\sqrt {1-2 x} (2+3 x)^4}{5 (3+5 x)}+\frac {\int \frac {(2+3 x) (5642+7875 x)}{\sqrt {1-2 x} (3+5 x)} \, dx}{4375}\\ &=\frac {12}{625} \sqrt {1-2 x} (2+3 x)^2+\frac {27}{175} \sqrt {1-2 x} (2+3 x)^3-\frac {\sqrt {1-2 x} (2+3 x)^4}{5 (3+5 x)}-\frac {3 \sqrt {1-2 x} (1256+375 x)}{3125}+\frac {131 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{3125}\\ &=\frac {12}{625} \sqrt {1-2 x} (2+3 x)^2+\frac {27}{175} \sqrt {1-2 x} (2+3 x)^3-\frac {\sqrt {1-2 x} (2+3 x)^4}{5 (3+5 x)}-\frac {3 \sqrt {1-2 x} (1256+375 x)}{3125}-\frac {131 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{3125}\\ &=\frac {12}{625} \sqrt {1-2 x} (2+3 x)^2+\frac {27}{175} \sqrt {1-2 x} (2+3 x)^3-\frac {\sqrt {1-2 x} (2+3 x)^4}{5 (3+5 x)}-\frac {3 \sqrt {1-2 x} (1256+375 x)}{3125}-\frac {262 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125 \sqrt {55}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 68, normalized size = 0.60 \begin {gather*} \frac {\sqrt {1-2 x} \left (101250 x^4+258525 x^3+206415 x^2-52485 x-63088\right )}{21875 (5 x+3)}-\frac {262 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 88, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {1-2 x} \left (50625 (1-2 x)^4-461025 (1-2 x)^3+1492155 (1-2 x)^2-1593795 (1-2 x)+7336\right )}{87500 (5 (1-2 x)-11)}-\frac {262 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.57, size = 74, normalized size = 0.65 \begin {gather*} \frac {917 \, \sqrt {55} {\left (5 \, x + 3\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \, {\left (101250 \, x^{4} + 258525 \, x^{3} + 206415 \, x^{2} - 52485 \, x - 63088\right )} \sqrt {-2 \, x + 1}}{1203125 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.31, size = 106, normalized size = 0.94 \begin {gather*} \frac {81}{700} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {999}{1250} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {4131}{2500} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {131}{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 {24}{3125} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{3125 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 72, normalized size = 0.64 \begin {gather*} -\frac {262 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{171875}-\frac {81 \left (-2 x +1\right )^{\frac {7}{2}}}{700}+\frac {999 \left (-2 x +1\right )^{\frac {5}{2}}}{1250}-\frac {4131 \left (-2 x +1\right )^{\frac {3}{2}}}{2500}+\frac {24 \sqrt {-2 x +1}}{3125}+\frac {2 \sqrt {-2 x +1}}{15625 \left (-2 x -\frac {6}{5}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.49, size = 89, normalized size = 0.79 \begin {gather*} -\frac {81}{700} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {999}{1250} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {4131}{2500} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {131}{171875} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {24}{3125} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{3125 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 73, normalized size = 0.65 \begin {gather*} \frac {24\,\sqrt {1-2\,x}}{3125}-\frac {2\,\sqrt {1-2\,x}}{15625\,\left (2\,x+\frac {6}{5}\right )}-\frac {4131\,{\left (1-2\,x\right )}^{3/2}}{2500}+\frac {999\,{\left (1-2\,x\right )}^{5/2}}{1250}-\frac {81\,{\left (1-2\,x\right )}^{7/2}}{700}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,262{}\mathrm {i}}{171875} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 162.62, size = 214, normalized size = 1.89 \begin {gather*} - \frac {81 \left (1 - 2 x\right )^{\frac {7}{2}}}{700} + \frac {999 \left (1 - 2 x\right )^{\frac {5}{2}}}{1250} - \frac {4131 \left (1 - 2 x\right )^{\frac {3}{2}}}{2500} + \frac {24 \sqrt {1 - 2 x}}{3125} - \frac {44 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{3125} + \frac {52 \left (\begin {cases} - \frac {\sqrt {55} \operatorname {acoth}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: 2 x - 1 < - \frac {11}{5} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: 2 x - 1 > - \frac {11}{5} \end {cases}\right )}{625} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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