Optimal. Leaf size=107 \[ \frac{7 (3 x+2)^3}{11 \sqrt{1-2 x} (5 x+3)^2}-\frac{71 \sqrt{1-2 x} (3 x+2)^2}{1210 (5 x+3)^2}+\frac{9 \sqrt{1-2 x} (5093 x+3044)}{13310 (5 x+3)}-\frac{111 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331 \sqrt{55}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0289082, antiderivative size = 107, 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, 146, 63, 206} \[ \frac{7 (3 x+2)^3}{11 \sqrt{1-2 x} (5 x+3)^2}-\frac{71 \sqrt{1-2 x} (3 x+2)^2}{1210 (5 x+3)^2}+\frac{9 \sqrt{1-2 x} (5093 x+3044)}{13310 (5 x+3)}-\frac{111 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331 \sqrt{55}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 98
Rule 149
Rule 146
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(2+3 x)^4}{(1-2 x)^{3/2} (3+5 x)^3} \, dx &=\frac{7 (2+3 x)^3}{11 \sqrt{1-2 x} (3+5 x)^2}-\frac{1}{11} \int \frac{(2+3 x)^2 (47+102 x)}{\sqrt{1-2 x} (3+5 x)^3} \, dx\\ &=-\frac{71 \sqrt{1-2 x} (2+3 x)^2}{1210 (3+5 x)^2}+\frac{7 (2+3 x)^3}{11 \sqrt{1-2 x} (3+5 x)^2}-\frac{\int \frac{(2+3 x) (3636+6945 x)}{\sqrt{1-2 x} (3+5 x)^2} \, dx}{1210}\\ &=-\frac{71 \sqrt{1-2 x} (2+3 x)^2}{1210 (3+5 x)^2}+\frac{7 (2+3 x)^3}{11 \sqrt{1-2 x} (3+5 x)^2}+\frac{9 \sqrt{1-2 x} (3044+5093 x)}{13310 (3+5 x)}+\frac{111 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{2662}\\ &=-\frac{71 \sqrt{1-2 x} (2+3 x)^2}{1210 (3+5 x)^2}+\frac{7 (2+3 x)^3}{11 \sqrt{1-2 x} (3+5 x)^2}+\frac{9 \sqrt{1-2 x} (3044+5093 x)}{13310 (3+5 x)}-\frac{111 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{2662}\\ &=-\frac{71 \sqrt{1-2 x} (2+3 x)^2}{1210 (3+5 x)^2}+\frac{7 (2+3 x)^3}{11 \sqrt{1-2 x} (3+5 x)^2}+\frac{9 \sqrt{1-2 x} (3044+5093 x)}{13310 (3+5 x)}-\frac{111 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331 \sqrt{55}}\\ \end{align*}
Mathematica [C] time = 0.0923498, size = 91, normalized size = 0.85 \[ \frac{\frac{2184 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{5}{11} (1-2 x)\right )}{\sqrt{1-2 x}}+\frac{11 \left (-490050 x^3+334350 x^2+930205 x+331904\right )}{\sqrt{1-2 x} (5 x+3)^2}-306 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{332750} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.012, size = 66, normalized size = 0.6 \begin{align*}{\frac{81}{250}\sqrt{1-2\,x}}+{\frac{2401}{2662}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{4}{6655\, \left ( -10\,x-6 \right ) ^{2}} \left ({\frac{271}{20} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{3003}{100}\sqrt{1-2\,x}} \right ) }-{\frac{111\,\sqrt{55}}{73205}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.59487, size = 124, normalized size = 1.16 \begin{align*} \frac{111}{146410} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{81}{250} \, \sqrt{-2 \, x + 1} + \frac{7505835 \,{\left (2 \, x - 1\right )}^{2} + 66039512 \, x + 3295369}{332750 \,{\left (25 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 110 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 121 \, \sqrt{-2 \, x + 1}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.62296, size = 270, normalized size = 2.52 \begin{align*} \frac{111 \, \sqrt{55}{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 11 \,{\left (215622 \, x^{3} - 149298 \, x^{2} - 411911 \, x - 146824\right )} \sqrt{-2 \, x + 1}}{146410 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 2.49091, size = 116, normalized size = 1.08 \begin{align*} \frac{111}{146410} \, \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{81}{250} \, \sqrt{-2 \, x + 1} + \frac{2401}{2662 \, \sqrt{-2 \, x + 1}} + \frac{1355 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 3003 \, \sqrt{-2 \, x + 1}}{665500 \,{\left (5 \, x + 3\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]