Optimal. Leaf size=96 \[ \frac{365}{14641 \sqrt{1-2 x}}-\frac{73}{1210 (1-2 x)^{3/2} (5 x+3)}+\frac{73}{3993 (1-2 x)^{3/2}}-\frac{1}{110 (1-2 x)^{3/2} (5 x+3)^2}-\frac{365 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{14641} \]
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Rubi [A] time = 0.0251085, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {78, 51, 63, 206} \[ \frac{365}{14641 \sqrt{1-2 x}}-\frac{73}{1210 (1-2 x)^{3/2} (5 x+3)}+\frac{73}{3993 (1-2 x)^{3/2}}-\frac{1}{110 (1-2 x)^{3/2} (5 x+3)^2}-\frac{365 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{14641} \]
Antiderivative was successfully verified.
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Rule 78
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{2+3 x}{(1-2 x)^{5/2} (3+5 x)^3} \, dx &=-\frac{1}{110 (1-2 x)^{3/2} (3+5 x)^2}+\frac{73}{110} \int \frac{1}{(1-2 x)^{5/2} (3+5 x)^2} \, dx\\ &=-\frac{1}{110 (1-2 x)^{3/2} (3+5 x)^2}-\frac{73}{1210 (1-2 x)^{3/2} (3+5 x)}+\frac{73}{242} \int \frac{1}{(1-2 x)^{5/2} (3+5 x)} \, dx\\ &=\frac{73}{3993 (1-2 x)^{3/2}}-\frac{1}{110 (1-2 x)^{3/2} (3+5 x)^2}-\frac{73}{1210 (1-2 x)^{3/2} (3+5 x)}+\frac{365 \int \frac{1}{(1-2 x)^{3/2} (3+5 x)} \, dx}{2662}\\ &=\frac{73}{3993 (1-2 x)^{3/2}}+\frac{365}{14641 \sqrt{1-2 x}}-\frac{1}{110 (1-2 x)^{3/2} (3+5 x)^2}-\frac{73}{1210 (1-2 x)^{3/2} (3+5 x)}+\frac{1825 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{29282}\\ &=\frac{73}{3993 (1-2 x)^{3/2}}+\frac{365}{14641 \sqrt{1-2 x}}-\frac{1}{110 (1-2 x)^{3/2} (3+5 x)^2}-\frac{73}{1210 (1-2 x)^{3/2} (3+5 x)}-\frac{1825 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{29282}\\ &=\frac{73}{3993 (1-2 x)^{3/2}}+\frac{365}{14641 \sqrt{1-2 x}}-\frac{1}{110 (1-2 x)^{3/2} (3+5 x)^2}-\frac{73}{1210 (1-2 x)^{3/2} (3+5 x)}-\frac{365 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{14641}\\ \end{align*}
Mathematica [C] time = 0.0119974, size = 48, normalized size = 0.5 \[ -\frac{363-292 (5 x+3)^2 \, _2F_1\left (-\frac{3}{2},2;-\frac{1}{2};\frac{5}{11} (1-2 x)\right )}{39930 (1-2 x)^{3/2} (5 x+3)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 66, normalized size = 0.7 \begin{align*}{\frac{28}{3993} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{288}{14641}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{500}{14641\, \left ( -10\,x-6 \right ) ^{2}} \left ({\frac{77}{20} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{869}{100}\sqrt{1-2\,x}} \right ) }-{\frac{365\,\sqrt{55}}{161051}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.06616, size = 124, normalized size = 1.29 \begin{align*} \frac{365}{322102} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{27375 \,{\left (2 \, x - 1\right )}^{3} + 100375 \,{\left (2 \, x - 1\right )}^{2} + 141328 \, x - 107932}{43923 \,{\left (25 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 110 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 121 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.05959, size = 313, normalized size = 3.26 \begin{align*} \frac{1095 \, \sqrt{11} \sqrt{5}{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 11 \,{\left (109500 \, x^{3} + 36500 \, x^{2} - 47961 \, x - 17466\right )} \sqrt{-2 \, x + 1}}{966306 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.7296, size = 120, normalized size = 1.25 \begin{align*} \frac{365}{322102} \, \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 (432 \, x - 293\right )}}{43923 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} + \frac{5 \,{\left (35 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 79 \, \sqrt{-2 \, x + 1}\right )}}{5324 \,{\left (5 \, x + 3\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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