Optimal. Leaf size=100 \[ \frac{11 (5 x+3)^2}{7 \sqrt{1-2 x} (3 x+2)^3}+\frac{2 \sqrt{1-2 x} (470 x+297)}{441 (3 x+2)^3}-\frac{4660 \sqrt{1-2 x}}{3087 (3 x+2)}-\frac{9320 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3087 \sqrt{21}} \]
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Rubi [A] time = 0.0274599, 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, 145, 51, 63, 206} \[ \frac{11 (5 x+3)^2}{7 \sqrt{1-2 x} (3 x+2)^3}+\frac{2 \sqrt{1-2 x} (470 x+297)}{441 (3 x+2)^3}-\frac{4660 \sqrt{1-2 x}}{3087 (3 x+2)}-\frac{9320 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3087 \sqrt{21}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 145
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^4} \, dx &=\frac{11 (3+5 x)^2}{7 \sqrt{1-2 x} (2+3 x)^3}-\frac{1}{7} \int \frac{(-74-160 x) (3+5 x)}{\sqrt{1-2 x} (2+3 x)^4} \, dx\\ &=\frac{11 (3+5 x)^2}{7 \sqrt{1-2 x} (2+3 x)^3}+\frac{2 \sqrt{1-2 x} (297+470 x)}{441 (2+3 x)^3}+\frac{4660}{441} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=-\frac{4660 \sqrt{1-2 x}}{3087 (2+3 x)}+\frac{11 (3+5 x)^2}{7 \sqrt{1-2 x} (2+3 x)^3}+\frac{2 \sqrt{1-2 x} (297+470 x)}{441 (2+3 x)^3}+\frac{4660 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{3087}\\ &=-\frac{4660 \sqrt{1-2 x}}{3087 (2+3 x)}+\frac{11 (3+5 x)^2}{7 \sqrt{1-2 x} (2+3 x)^3}+\frac{2 \sqrt{1-2 x} (297+470 x)}{441 (2+3 x)^3}-\frac{4660 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{3087}\\ &=-\frac{4660 \sqrt{1-2 x}}{3087 (2+3 x)}+\frac{11 (3+5 x)^2}{7 \sqrt{1-2 x} (2+3 x)^3}+\frac{2 \sqrt{1-2 x} (297+470 x)}{441 (2+3 x)^3}-\frac{9320 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3087 \sqrt{21}}\\ \end{align*}
Mathematica [C] time = 0.0201878, size = 59, normalized size = 0.59 \[ \frac{18640 (3 x+2)^3 \, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )+49 \left (7875 x^2+10434 x+3457\right )}{27783 \sqrt{1-2 x} (3 x+2)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 66, normalized size = 0.7 \begin{align*}{\frac{54}{2401\, \left ( -6\,x-4 \right ) ^{3}} \left ( -{\frac{3317}{27} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{137186}{243} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{157633}{243}\sqrt{1-2\,x}} \right ) }-{\frac{9320\,\sqrt{21}}{64827}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{2662}{2401}{\frac{1}{\sqrt{1-2\,x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.17044, size = 136, normalized size = 1.36 \begin{align*} \frac{4660}{64827} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2 \,{\left (41940 \,{\left (2 \, x - 1\right )}^{3} + 303835 \,{\left (2 \, x - 1\right )}^{2} + 1464316 \, x - 145187\right )}}{3087 \,{\left (27 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 189 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 441 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 343 \, \sqrt{-2 \, x + 1}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57164, size = 292, normalized size = 2.92 \begin{align*} \frac{4660 \, \sqrt{21}{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \,{\left (83880 \, x^{3} + 178015 \, x^{2} + 125154 \, x + 29177\right )} \sqrt{-2 \, x + 1}}{64827 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\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 = 1.87378, size = 126, normalized size = 1.26 \begin{align*} \frac{4660}{64827} \, \sqrt{21} \log \left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{2662}{2401 \, \sqrt{-2 \, x + 1}} + \frac{29853 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 137186 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 157633 \, \sqrt{-2 \, x + 1}}{86436 \,{\left (3 \, x + 2\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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