Optimal. Leaf size=56 \[ \frac {2}{121 \sqrt {1-2 x}}+\frac {7}{33 (1-2 x)^{3/2}}-\frac {2}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 56, 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} \[ \frac {2}{121 \sqrt {1-2 x}}+\frac {7}{33 (1-2 x)^{3/2}}-\frac {2}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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)^{5/2} (3+5 x)} \, dx &=\frac {7}{33 (1-2 x)^{3/2}}+\frac {1}{11} \int \frac {1}{(1-2 x)^{3/2} (3+5 x)} \, dx\\ &=\frac {7}{33 (1-2 x)^{3/2}}+\frac {2}{121 \sqrt {1-2 x}}+\frac {5}{121} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {7}{33 (1-2 x)^{3/2}}+\frac {2}{121 \sqrt {1-2 x}}-\frac {5}{121} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {7}{33 (1-2 x)^{3/2}}+\frac {2}{121 \sqrt {1-2 x}}-\frac {2}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [C] time = 0.02, size = 38, normalized size = 0.68 \[ \frac {(6-12 x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\frac {5}{11} (2 x-1)\right )+77}{363 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.85, size = 75, normalized size = 1.34 \[ \frac {3 \, \sqrt {11} \sqrt {5} {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 11 \, {\left (12 \, x - 83\right )} \sqrt {-2 \, x + 1}}{3993 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.25, size = 61, normalized size = 1.09 \[ \frac {1}{1331} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {12 \, x - 83}{363 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 38, normalized size = 0.68 \[ -\frac {2 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{1331}+\frac {7}{33 \left (-2 x +1\right )^{\frac {3}{2}}}+\frac {2}{121 \sqrt {-2 x +1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 51, normalized size = 0.91 \[ \frac {1}{1331} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {12 \, x - 83}{363 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 33, normalized size = 0.59 \[ -\frac {\frac {4\,x}{121}-\frac {83}{363}}{{\left (1-2\,x\right )}^{3/2}}-\frac {2\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 27.95, size = 90, normalized size = 1.61 \[ \frac {10 \left (\begin {cases} - \frac {\sqrt {55} \operatorname {acoth}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: 2 x - 1 < - \frac {11}{5} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: 2 x - 1 > - \frac {11}{5} \end {cases}\right )}{121} + \frac {2}{121 \sqrt {1 - 2 x}} + \frac {7}{33 \left (1 - 2 x\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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