Optimal. Leaf size=76 \[ \frac {76}{1331 \sqrt {1-2 x}}-\frac {1}{55 (1-2 x)^{3/2} (5 x+3)}+\frac {76}{1815 (1-2 x)^{3/2}}-\frac {76 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331} \]
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Rubi [A] time = 0.02, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {78, 51, 63, 206} \[ \frac {76}{1331 \sqrt {1-2 x}}-\frac {1}{55 (1-2 x)^{3/2} (5 x+3)}+\frac {76}{1815 (1-2 x)^{3/2}}-\frac {76 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331} \]
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)^2} \, dx &=-\frac {1}{55 (1-2 x)^{3/2} (3+5 x)}+\frac {38}{55} \int \frac {1}{(1-2 x)^{5/2} (3+5 x)} \, dx\\ &=\frac {76}{1815 (1-2 x)^{3/2}}-\frac {1}{55 (1-2 x)^{3/2} (3+5 x)}+\frac {38}{121} \int \frac {1}{(1-2 x)^{3/2} (3+5 x)} \, dx\\ &=\frac {76}{1815 (1-2 x)^{3/2}}+\frac {76}{1331 \sqrt {1-2 x}}-\frac {1}{55 (1-2 x)^{3/2} (3+5 x)}+\frac {190 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{1331}\\ &=\frac {76}{1815 (1-2 x)^{3/2}}+\frac {76}{1331 \sqrt {1-2 x}}-\frac {1}{55 (1-2 x)^{3/2} (3+5 x)}-\frac {190 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{1331}\\ &=\frac {76}{1815 (1-2 x)^{3/2}}+\frac {76}{1331 \sqrt {1-2 x}}-\frac {1}{55 (1-2 x)^{3/2} (3+5 x)}-\frac {76 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 46, normalized size = 0.61 \[ -\frac {33-76 (5 x+3) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\frac {5}{11} (2 x-1)\right )}{1815 (1-2 x)^{3/2} (5 x+3)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 90, normalized size = 1.18 \[ \frac {114 \, \sqrt {11} \sqrt {5} {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 11 \, {\left (2280 \, x^{2} - 608 \, x - 1113\right )} \sqrt {-2 \, x + 1}}{43923 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.28, size = 77, normalized size = 1.01 \[ \frac {38}{14641} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {4 \, {\left (111 \, x - 94\right )}}{3993 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {5 \, \sqrt {-2 \, x + 1}}{1331 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 54, normalized size = 0.71 \[ -\frac {76 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{14641}+\frac {14}{363 \left (-2 x +1\right )^{\frac {3}{2}}}+\frac {74}{1331 \sqrt {-2 x +1}}+\frac {2 \sqrt {-2 x +1}}{1331 \left (-2 x -\frac {6}{5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.16, size = 74, normalized size = 0.97 \[ \frac {38}{14641} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (570 \, {\left (2 \, x - 1\right )}^{2} + 1672 \, x - 1683\right )}}{3993 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 11 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 56, normalized size = 0.74 \[ -\frac {\frac {304\,x}{1815}+\frac {76\,{\left (2\,x-1\right )}^2}{1331}-\frac {102}{605}}{\frac {11\,{\left (1-2\,x\right )}^{3/2}}{5}-{\left (1-2\,x\right )}^{5/2}}-\frac {76\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{14641} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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