Optimal. Leaf size=100 \[ \frac{7 (3 x+2)^3}{11 \sqrt{1-2 x} (5 x+3)}-\frac{36 \sqrt{1-2 x} (3 x+2)^2}{605 (5 x+3)}+\frac{27 \sqrt{1-2 x} (265 x+792)}{3025}-\frac{54 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3025 \sqrt{55}} \]
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Rubi [A] time = 0.0276161, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {98, 149, 147, 63, 206} \[ \frac{7 (3 x+2)^3}{11 \sqrt{1-2 x} (5 x+3)}-\frac{36 \sqrt{1-2 x} (3 x+2)^2}{605 (5 x+3)}+\frac{27 \sqrt{1-2 x} (265 x+792)}{3025}-\frac{54 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3025 \sqrt{55}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 149
Rule 147
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(2+3 x)^4}{(1-2 x)^{3/2} (3+5 x)^2} \, dx &=\frac{7 (2+3 x)^3}{11 \sqrt{1-2 x} (3+5 x)}-\frac{1}{11} \int \frac{(2+3 x)^2 (117+207 x)}{\sqrt{1-2 x} (3+5 x)^2} \, dx\\ &=-\frac{36 \sqrt{1-2 x} (2+3 x)^2}{605 (3+5 x)}+\frac{7 (2+3 x)^3}{11 \sqrt{1-2 x} (3+5 x)}-\frac{1}{605} \int \frac{(2+3 x) (4266+7155 x)}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{36 \sqrt{1-2 x} (2+3 x)^2}{605 (3+5 x)}+\frac{7 (2+3 x)^3}{11 \sqrt{1-2 x} (3+5 x)}+\frac{27 \sqrt{1-2 x} (792+265 x)}{3025}+\frac{27 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{3025}\\ &=-\frac{36 \sqrt{1-2 x} (2+3 x)^2}{605 (3+5 x)}+\frac{7 (2+3 x)^3}{11 \sqrt{1-2 x} (3+5 x)}+\frac{27 \sqrt{1-2 x} (792+265 x)}{3025}-\frac{27 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{3025}\\ &=-\frac{36 \sqrt{1-2 x} (2+3 x)^2}{605 (3+5 x)}+\frac{7 (2+3 x)^3}{11 \sqrt{1-2 x} (3+5 x)}+\frac{27 \sqrt{1-2 x} (792+265 x)}{3025}-\frac{54 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3025 \sqrt{55}}\\ \end{align*}
Mathematica [C] time = 0.0851272, size = 91, normalized size = 0.91 \[ \frac{\frac{252 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{5}{11} (1-2 x)\right )}{\sqrt{1-2 x}}+\frac{11 \left (-7425 x^3-51975 x^2+31095 x+35764\right )}{\sqrt{1-2 x} (5 x+3)}+18 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15125} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 63, normalized size = 0.6 \begin{align*} -{\frac{27}{100} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{999}{250}\sqrt{1-2\,x}}+{\frac{2401}{484}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{2}{75625}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}-{\frac{54\,\sqrt{55}}{166375}{\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 = 3.83863, size = 112, normalized size = 1.12 \begin{align*} -\frac{27}{100} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{27}{166375} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{999}{250} \, \sqrt{-2 \, x + 1} - \frac{1500633 \, x + 900371}{30250 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 11 \, \sqrt{-2 \, x + 1}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62611, size = 232, normalized size = 2.32 \begin{align*} \frac{27 \, \sqrt{55}{\left (10 \, x^{2} + x - 3\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \,{\left (16335 \, x^{3} + 114345 \, x^{2} - 68661 \, x - 78832\right )} \sqrt{-2 \, x + 1}}{166375 \,{\left (10 \, x^{2} + 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 = 2.40492, size = 116, normalized size = 1.16 \begin{align*} -\frac{27}{100} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{27}{166375} \, \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{999}{250} \, \sqrt{-2 \, x + 1} - \frac{1500633 \, x + 900371}{30250 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 11 \, \sqrt{-2 \, x + 1}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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