Optimal. Leaf size=61 \[ \frac {72}{605 \sqrt {1-2 x}}-\frac {1}{55 \sqrt {1-2 x} (5 x+3)}-\frac {72 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{121 \sqrt {55}} \]
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Rubi [A] time = 0.01, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {78, 51, 63, 206} \begin {gather*} \frac {72}{605 \sqrt {1-2 x}}-\frac {1}{55 \sqrt {1-2 x} (5 x+3)}-\frac {72 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{121 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 206
Rubi steps
\begin {align*} \int \frac {2+3 x}{(1-2 x)^{3/2} (3+5 x)^2} \, dx &=-\frac {1}{55 \sqrt {1-2 x} (3+5 x)}+\frac {36}{55} \int \frac {1}{(1-2 x)^{3/2} (3+5 x)} \, dx\\ &=\frac {72}{605 \sqrt {1-2 x}}-\frac {1}{55 \sqrt {1-2 x} (3+5 x)}+\frac {36}{121} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {72}{605 \sqrt {1-2 x}}-\frac {1}{55 \sqrt {1-2 x} (3+5 x)}-\frac {36}{121} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {72}{605 \sqrt {1-2 x}}-\frac {1}{55 \sqrt {1-2 x} (3+5 x)}-\frac {72 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{121 \sqrt {55}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 46, normalized size = 0.75 \begin {gather*} \frac {72 (5 x+3) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\frac {5}{11} (2 x-1)\right )-11}{605 \sqrt {1-2 x} (5 x+3)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 61, normalized size = 1.00 \begin {gather*} \frac {2 (36 (1-2 x)-77)}{121 (5 (1-2 x)-11) \sqrt {1-2 x}}-\frac {72 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{121 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.31, size = 65, normalized size = 1.07 \begin {gather*} \frac {36 \, \sqrt {55} {\left (10 \, x^{2} + x - 3\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (72 \, x + 41\right )} \sqrt {-2 \, x + 1}}{6655 \, {\left (10 \, x^{2} + x - 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.21, size = 68, normalized size = 1.11 \begin {gather*} \frac {36}{6655} \, \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 {2 \, {\left (72 \, x + 41\right )}}{121 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 45, normalized size = 0.74 \begin {gather*} -\frac {72 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{6655}+\frac {14}{121 \sqrt {-2 x +1}}+\frac {2 \sqrt {-2 x +1}}{605 \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 = 1.20, size = 65, normalized size = 1.07 \begin {gather*} \frac {36}{6655} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (72 \, x + 41\right )}}{121 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 46, normalized size = 0.75 \begin {gather*} \frac {\frac {144\,x}{605}+\frac {82}{605}}{\frac {11\,\sqrt {1-2\,x}}{5}-{\left (1-2\,x\right )}^{3/2}}-\frac {72\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{6655} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] 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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