Optimal. Leaf size=93 \[ -\frac{243}{560} (1-2 x)^{7/2}+\frac{5751 (1-2 x)^{5/2}}{1000}-\frac{17019}{500} (1-2 x)^{3/2}+\frac{806121 \sqrt{1-2 x}}{5000}+\frac{16807}{176 \sqrt{1-2 x}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{6875 \sqrt{55}} \]
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Rubi [A] time = 0.0629162, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {87, 43, 63, 206} \[ -\frac{243}{560} (1-2 x)^{7/2}+\frac{5751 (1-2 x)^{5/2}}{1000}-\frac{17019}{500} (1-2 x)^{3/2}+\frac{806121 \sqrt{1-2 x}}{5000}+\frac{16807}{176 \sqrt{1-2 x}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{6875 \sqrt{55}} \]
Antiderivative was successfully verified.
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Rule 87
Rule 43
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)} \, dx &=\int \left (\frac{16807}{176 (1-2 x)^{3/2}}-\frac{848277}{10000 \sqrt{1-2 x}}-\frac{107433 x}{1000 \sqrt{1-2 x}}-\frac{7857 x^2}{100 \sqrt{1-2 x}}-\frac{243 x^3}{10 \sqrt{1-2 x}}+\frac{1}{6875 \sqrt{1-2 x} (3+5 x)}\right ) \, dx\\ &=\frac{16807}{176 \sqrt{1-2 x}}+\frac{848277 \sqrt{1-2 x}}{10000}+\frac{\int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{6875}-\frac{243}{10} \int \frac{x^3}{\sqrt{1-2 x}} \, dx-\frac{7857}{100} \int \frac{x^2}{\sqrt{1-2 x}} \, dx-\frac{107433 \int \frac{x}{\sqrt{1-2 x}} \, dx}{1000}\\ &=\frac{16807}{176 \sqrt{1-2 x}}+\frac{848277 \sqrt{1-2 x}}{10000}-\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{6875}-\frac{243}{10} \int \left (\frac{1}{8 \sqrt{1-2 x}}-\frac{3}{8} \sqrt{1-2 x}+\frac{3}{8} (1-2 x)^{3/2}-\frac{1}{8} (1-2 x)^{5/2}\right ) \, dx-\frac{7857}{100} \int \left (\frac{1}{4 \sqrt{1-2 x}}-\frac{1}{2} \sqrt{1-2 x}+\frac{1}{4} (1-2 x)^{3/2}\right ) \, dx-\frac{107433 \int \left (\frac{1}{2 \sqrt{1-2 x}}-\frac{1}{2} \sqrt{1-2 x}\right ) \, dx}{1000}\\ &=\frac{16807}{176 \sqrt{1-2 x}}+\frac{806121 \sqrt{1-2 x}}{5000}-\frac{17019}{500} (1-2 x)^{3/2}+\frac{5751 (1-2 x)^{5/2}}{1000}-\frac{243}{560} (1-2 x)^{7/2}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{6875 \sqrt{55}}\\ \end{align*}
Mathematica [C] time = 0.0248353, size = 55, normalized size = 0.59 \[ \frac{14 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{5}{11} (1-2 x)\right )-33 \left (50625 x^4+234225 x^3+565500 x^2+1584705 x-1662482\right )}{240625 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 65, normalized size = 0.7 \begin{align*} -{\frac{17019}{500} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{5751}{1000} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{243}{560} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{2\,\sqrt{55}}{378125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{16807}{176}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{806121}{5000}\sqrt{1-2\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.76164, size = 111, normalized size = 1.19 \begin{align*} -\frac{243}{560} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{5751}{1000} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{17019}{500} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{378125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{806121}{5000} \, \sqrt{-2 \, x + 1} + \frac{16807}{176 \, \sqrt{-2 \, x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66153, size = 243, normalized size = 2.61 \begin{align*} \frac{7 \, \sqrt{55}{\left (2 \, x - 1\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \,{\left (334125 \, x^{4} + 1545885 \, x^{3} + 3732300 \, x^{2} + 10459053 \, x - 10972384\right )} \sqrt{-2 \, x + 1}}{2646875 \,{\left (2 \, x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 80.9808, size = 126, normalized size = 1.35 \begin{align*} - \frac{243 \left (1 - 2 x\right )^{\frac{7}{2}}}{560} + \frac{5751 \left (1 - 2 x\right )^{\frac{5}{2}}}{1000} - \frac{17019 \left (1 - 2 x\right )^{\frac{3}{2}}}{500} + \frac{806121 \sqrt{1 - 2 x}}{5000} + \frac{2 \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 )}{6875} + \frac{16807}{176 \sqrt{1 - 2 x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.93945, size = 134, normalized size = 1.44 \begin{align*} \frac{243}{560} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{5751}{1000} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{17019}{500} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{378125} \, \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{806121}{5000} \, \sqrt{-2 \, x + 1} + \frac{16807}{176 \, \sqrt{-2 \, x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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