Optimal. Leaf size=155 \[ -\frac{127 (1-2 x)^{3/2} (3 x+2)^4}{50 (5 x+3)}-\frac{(1-2 x)^{5/2} (3 x+2)^4}{10 (5 x+3)^2}+\frac{1117}{750} (1-2 x)^{3/2} (3 x+2)^3+\frac{1903 (1-2 x)^{3/2} (3 x+2)^2}{4375}+\frac{(1-2 x)^{3/2} (24939 x+734)}{93750}+\frac{11763 \sqrt{1-2 x}}{78125}-\frac{11763 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{78125} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.056334, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {97, 149, 153, 147, 50, 63, 206} \[ -\frac{127 (1-2 x)^{3/2} (3 x+2)^4}{50 (5 x+3)}-\frac{(1-2 x)^{5/2} (3 x+2)^4}{10 (5 x+3)^2}+\frac{1117}{750} (1-2 x)^{3/2} (3 x+2)^3+\frac{1903 (1-2 x)^{3/2} (3 x+2)^2}{4375}+\frac{(1-2 x)^{3/2} (24939 x+734)}{93750}+\frac{11763 \sqrt{1-2 x}}{78125}-\frac{11763 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{78125} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 97
Rule 149
Rule 153
Rule 147
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^3} \, dx &=-\frac{(1-2 x)^{5/2} (2+3 x)^4}{10 (3+5 x)^2}+\frac{1}{10} \int \frac{(2-39 x) (1-2 x)^{3/2} (2+3 x)^3}{(3+5 x)^2} \, dx\\ &=-\frac{(1-2 x)^{5/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac{127 (1-2 x)^{3/2} (2+3 x)^4}{50 (3+5 x)}+\frac{1}{50} \int \frac{(342-3351 x) \sqrt{1-2 x} (2+3 x)^3}{3+5 x} \, dx\\ &=\frac{1117}{750} (1-2 x)^{3/2} (2+3 x)^3-\frac{(1-2 x)^{5/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac{127 (1-2 x)^{3/2} (2+3 x)^4}{50 (3+5 x)}-\frac{\int \frac{\sqrt{1-2 x} (2+3 x)^2 (-621+34254 x)}{3+5 x} \, dx}{2250}\\ &=\frac{1903 (1-2 x)^{3/2} (2+3 x)^2}{4375}+\frac{1117}{750} (1-2 x)^{3/2} (2+3 x)^3-\frac{(1-2 x)^{5/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac{127 (1-2 x)^{3/2} (2+3 x)^4}{50 (3+5 x)}+\frac{\int \frac{(43470-174573 x) \sqrt{1-2 x} (2+3 x)}{3+5 x} \, dx}{78750}\\ &=\frac{1903 (1-2 x)^{3/2} (2+3 x)^2}{4375}+\frac{1117}{750} (1-2 x)^{3/2} (2+3 x)^3-\frac{(1-2 x)^{5/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac{127 (1-2 x)^{3/2} (2+3 x)^4}{50 (3+5 x)}+\frac{(1-2 x)^{3/2} (734+24939 x)}{93750}+\frac{11763 \int \frac{\sqrt{1-2 x}}{3+5 x} \, dx}{31250}\\ &=\frac{11763 \sqrt{1-2 x}}{78125}+\frac{1903 (1-2 x)^{3/2} (2+3 x)^2}{4375}+\frac{1117}{750} (1-2 x)^{3/2} (2+3 x)^3-\frac{(1-2 x)^{5/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac{127 (1-2 x)^{3/2} (2+3 x)^4}{50 (3+5 x)}+\frac{(1-2 x)^{3/2} (734+24939 x)}{93750}+\frac{129393 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{156250}\\ &=\frac{11763 \sqrt{1-2 x}}{78125}+\frac{1903 (1-2 x)^{3/2} (2+3 x)^2}{4375}+\frac{1117}{750} (1-2 x)^{3/2} (2+3 x)^3-\frac{(1-2 x)^{5/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac{127 (1-2 x)^{3/2} (2+3 x)^4}{50 (3+5 x)}+\frac{(1-2 x)^{3/2} (734+24939 x)}{93750}-\frac{129393 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{156250}\\ &=\frac{11763 \sqrt{1-2 x}}{78125}+\frac{1903 (1-2 x)^{3/2} (2+3 x)^2}{4375}+\frac{1117}{750} (1-2 x)^{3/2} (2+3 x)^3-\frac{(1-2 x)^{5/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac{127 (1-2 x)^{3/2} (2+3 x)^4}{50 (3+5 x)}+\frac{(1-2 x)^{3/2} (734+24939 x)}{93750}-\frac{11763 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{78125}\\ \end{align*}
Mathematica [A] time = 0.0904048, size = 78, normalized size = 0.5 \[ \frac{\frac{5 \sqrt{1-2 x} \left (15750000 x^6+15075000 x^5-16051500 x^4-11139550 x^3+9372960 x^2+6891315 x+871208\right )}{(5 x+3)^2}-164682 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{5468750} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.011, size = 93, normalized size = 0.6 \begin{align*}{\frac{9}{250} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}-{\frac{1107}{8750} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{108}{15625} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{76}{3125} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{2404}{15625}\sqrt{1-2\,x}}+{\frac{44}{625\, \left ( -10\,x-6 \right ) ^{2}} \left ({\frac{51}{20} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{2827}{500}\sqrt{1-2\,x}} \right ) }-{\frac{11763\,\sqrt{55}}{390625}{\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 = 2.10443, size = 161, normalized size = 1.04 \begin{align*} \frac{9}{250} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{1107}{8750} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{108}{15625} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{76}{3125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{11763}{781250} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2404}{15625} \, \sqrt{-2 \, x + 1} + \frac{11 \,{\left (1275 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2827 \, \sqrt{-2 \, x + 1}\right )}}{78125 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.35735, size = 336, normalized size = 2.17 \begin{align*} \frac{82341 \, \sqrt{11} \sqrt{5}{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 5 \,{\left (15750000 \, x^{6} + 15075000 \, x^{5} - 16051500 \, x^{4} - 11139550 \, x^{3} + 9372960 \, x^{2} + 6891315 \, x + 871208\right )} \sqrt{-2 \, x + 1}}{5468750 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 2.60968, size = 181, normalized size = 1.17 \begin{align*} \frac{9}{250} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{1107}{8750} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{108}{15625} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{76}{3125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{11763}{781250} \, \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{2404}{15625} \, \sqrt{-2 \, x + 1} + \frac{11 \,{\left (1275 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2827 \, \sqrt{-2 \, x + 1}\right )}}{312500 \,{\left (5 \, x + 3\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]