Optimal. Leaf size=140 \[ \frac{7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)}-\frac{38 (3 x+2)^4}{1815 \sqrt{1-2 x} (5 x+3)}-\frac{10283 (3 x+2)^3}{6655 \sqrt{1-2 x}}-\frac{463344 \sqrt{1-2 x} (3 x+2)^2}{166375}-\frac{21 \sqrt{1-2 x} (1544625 x+4633904)}{831875}-\frac{406 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{831875 \sqrt{55}} \]
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Rubi [A] time = 0.0535995, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {98, 149, 150, 153, 147, 63, 206} \[ \frac{7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)}-\frac{38 (3 x+2)^4}{1815 \sqrt{1-2 x} (5 x+3)}-\frac{10283 (3 x+2)^3}{6655 \sqrt{1-2 x}}-\frac{463344 \sqrt{1-2 x} (3 x+2)^2}{166375}-\frac{21 \sqrt{1-2 x} (1544625 x+4633904)}{831875}-\frac{406 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{831875 \sqrt{55}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 149
Rule 150
Rule 153
Rule 147
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^2} \, dx &=\frac{7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)}-\frac{1}{33} \int \frac{(2+3 x)^4 (239+411 x)}{(1-2 x)^{3/2} (3+5 x)^2} \, dx\\ &=-\frac{38 (2+3 x)^4}{1815 \sqrt{1-2 x} (3+5 x)}+\frac{7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)}-\frac{\int \frac{(2+3 x)^3 (8358+14133 x)}{(1-2 x)^{3/2} (3+5 x)} \, dx}{1815}\\ &=-\frac{10283 (2+3 x)^3}{6655 \sqrt{1-2 x}}-\frac{38 (2+3 x)^4}{1815 \sqrt{1-2 x} (3+5 x)}+\frac{7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)}-\frac{\int \frac{(-834141-1390032 x) (2+3 x)^2}{\sqrt{1-2 x} (3+5 x)} \, dx}{19965}\\ &=-\frac{463344 \sqrt{1-2 x} (2+3 x)^2}{166375}-\frac{10283 (2+3 x)^3}{6655 \sqrt{1-2 x}}-\frac{38 (2+3 x)^4}{1815 \sqrt{1-2 x} (3+5 x)}+\frac{7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)}+\frac{\int \frac{(2+3 x) (58387434+97311375 x)}{\sqrt{1-2 x} (3+5 x)} \, dx}{499125}\\ &=-\frac{463344 \sqrt{1-2 x} (2+3 x)^2}{166375}-\frac{10283 (2+3 x)^3}{6655 \sqrt{1-2 x}}-\frac{38 (2+3 x)^4}{1815 \sqrt{1-2 x} (3+5 x)}+\frac{7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)}-\frac{21 \sqrt{1-2 x} (4633904+1544625 x)}{831875}+\frac{203 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{831875}\\ &=-\frac{463344 \sqrt{1-2 x} (2+3 x)^2}{166375}-\frac{10283 (2+3 x)^3}{6655 \sqrt{1-2 x}}-\frac{38 (2+3 x)^4}{1815 \sqrt{1-2 x} (3+5 x)}+\frac{7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)}-\frac{21 \sqrt{1-2 x} (4633904+1544625 x)}{831875}-\frac{203 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{831875}\\ &=-\frac{463344 \sqrt{1-2 x} (2+3 x)^2}{166375}-\frac{10283 (2+3 x)^3}{6655 \sqrt{1-2 x}}-\frac{38 (2+3 x)^4}{1815 \sqrt{1-2 x} (3+5 x)}+\frac{7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)}-\frac{21 \sqrt{1-2 x} (4633904+1544625 x)}{831875}-\frac{406 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{831875 \sqrt{55}}\\ \end{align*}
Mathematica [C] time = 0.0558326, size = 99, normalized size = 0.71 \[ -\frac{-252 \left (10 x^2+x-3\right ) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{5}{11} (1-2 x)\right )-266 (5 x+3) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{5}{11} (1-2 x)\right )+33 \left (1002375 x^5+6615675 x^4+36419625 x^3-52861545 x^2-19753541 x+14265224\right )}{1134375 (1-2 x)^{3/2} (5 x+3)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 81, normalized size = 0.6 \begin{align*} -{\frac{729}{2000} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{729}{125} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{315171}{5000}\sqrt{1-2\,x}}+{\frac{117649}{5808} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}-{\frac{134456}{1331}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{2}{4159375}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}-{\frac{406\,\sqrt{55}}{45753125}{\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 = 1.655, size = 136, normalized size = 0.97 \begin{align*} -\frac{729}{2000} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{729}{125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{203}{45753125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{315171}{5000} \, \sqrt{-2 \, x + 1} - \frac{10084199952 \,{\left (2 \, x - 1\right )}^{2} + 48414664375 \, x - 19758729375}{19965000 \,{\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 = 1.38111, size = 332, normalized size = 2.37 \begin{align*} \frac{609 \, \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 (72772425 \, x^{5} + 480298005 \, x^{4} + 2644064775 \, x^{3} - 3837745731 \, x^{2} - 1434109759 \, x + 1035652776\right )} \sqrt{-2 \, x + 1}}{137259375 \,{\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 = 1.87171, size = 150, normalized size = 1.07 \begin{align*} -\frac{729}{2000} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{729}{125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{203}{45753125} \, \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{315171}{5000} \, \sqrt{-2 \, x + 1} - \frac{16807 \,{\left (768 \, x - 307\right )}}{63888 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} - \frac{\sqrt{-2 \, x + 1}}{831875 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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