Optimal. Leaf size=120 \[ \frac{7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)}-\frac{38 (3 x+2)^3}{1815 \sqrt{1-2 x} (5 x+3)}-\frac{7588 (3 x+2)^2}{6655 \sqrt{1-2 x}}-\frac{6 \sqrt{1-2 x} (38025 x+114092)}{33275}-\frac{68 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{33275 \sqrt{55}} \]
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Rubi [A] time = 0.0395721, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {98, 149, 150, 147, 63, 206} \[ \frac{7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)}-\frac{38 (3 x+2)^3}{1815 \sqrt{1-2 x} (5 x+3)}-\frac{7588 (3 x+2)^2}{6655 \sqrt{1-2 x}}-\frac{6 \sqrt{1-2 x} (38025 x+114092)}{33275}-\frac{68 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{33275 \sqrt{55}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 149
Rule 150
Rule 147
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^2} \, dx &=\frac{7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)}-\frac{1}{33} \int \frac{(2+3 x)^3 (176+306 x)}{(1-2 x)^{3/2} (3+5 x)^2} \, dx\\ &=-\frac{38 (2+3 x)^3}{1815 \sqrt{1-2 x} (3+5 x)}+\frac{7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)}-\frac{\int \frac{(2+3 x)^2 (6162+10440 x)}{(1-2 x)^{3/2} (3+5 x)} \, dx}{1815}\\ &=-\frac{7588 (2+3 x)^2}{6655 \sqrt{1-2 x}}-\frac{38 (2+3 x)^3}{1815 \sqrt{1-2 x} (3+5 x)}+\frac{7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)}-\frac{\int \frac{(-410772-684450 x) (2+3 x)}{\sqrt{1-2 x} (3+5 x)} \, dx}{19965}\\ &=-\frac{7588 (2+3 x)^2}{6655 \sqrt{1-2 x}}-\frac{38 (2+3 x)^3}{1815 \sqrt{1-2 x} (3+5 x)}+\frac{7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)}-\frac{6 \sqrt{1-2 x} (114092+38025 x)}{33275}+\frac{34 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{33275}\\ &=-\frac{7588 (2+3 x)^2}{6655 \sqrt{1-2 x}}-\frac{38 (2+3 x)^3}{1815 \sqrt{1-2 x} (3+5 x)}+\frac{7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)}-\frac{6 \sqrt{1-2 x} (114092+38025 x)}{33275}-\frac{34 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{33275}\\ &=-\frac{7588 (2+3 x)^2}{6655 \sqrt{1-2 x}}-\frac{38 (2+3 x)^3}{1815 \sqrt{1-2 x} (3+5 x)}+\frac{7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)}-\frac{6 \sqrt{1-2 x} (114092+38025 x)}{33275}-\frac{68 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{33275 \sqrt{55}}\\ \end{align*}
Mathematica [C] time = 0.0472507, size = 94, normalized size = 0.78 \[ -\frac{-270 \left (10 x^2+x-3\right ) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{5}{11} (1-2 x)\right )-266 (5 x+3) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{5}{11} (1-2 x)\right )+33 \left (111375 x^4+1113750 x^3-1975050 x^2-734880 x+496226\right )}{226875 (1-2 x)^{3/2} (5 x+3)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 72, normalized size = 0.6 \begin{align*}{\frac{81}{200} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{8829}{1000}\sqrt{1-2\,x}}+{\frac{16807}{2904} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}-{\frac{228095}{10648}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{2}{831875}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}-{\frac{68\,\sqrt{55}}{1830125}{\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 = 1.82118, size = 124, normalized size = 1.03 \begin{align*} \frac{81}{200} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{34}{1830125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{8829}{1000} \, \sqrt{-2 \, x + 1} - \frac{427678077 \,{\left (2 \, x - 1\right )}^{2} + 2112880000 \, x - 802234125}{3993000 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 11 \,{\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 = 1.37743, size = 294, normalized size = 2.45 \begin{align*} \frac{102 \, \sqrt{55}{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \,{\left (1617165 \, x^{4} + 16171650 \, x^{3} - 28677318 \, x^{2} - 10671002 \, x + 7204728\right )} \sqrt{-2 \, x + 1}}{5490375 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\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 = 2.5799, size = 128, normalized size = 1.07 \begin{align*} \frac{81}{200} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{34}{1830125} \, \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{8829}{1000} \, \sqrt{-2 \, x + 1} - \frac{2401 \,{\left (285 \, x - 104\right )}}{15972 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} - \frac{\sqrt{-2 \, x + 1}}{166375 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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