Optimal. Leaf size=125 \[ \frac{2 \sqrt{1-2 x} (5 x+3)^3}{(3 x+2)^2}-\frac{(1-2 x)^{3/2} (5 x+3)^3}{9 (3 x+2)^3}+\frac{251 \sqrt{1-2 x} (5 x+3)^2}{63 (3 x+2)}-\frac{5}{567} \sqrt{1-2 x} (7265 x+2323)-\frac{36038 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{567 \sqrt{21}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0394717, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {97, 149, 147, 63, 206} \[ \frac{2 \sqrt{1-2 x} (5 x+3)^3}{(3 x+2)^2}-\frac{(1-2 x)^{3/2} (5 x+3)^3}{9 (3 x+2)^3}+\frac{251 \sqrt{1-2 x} (5 x+3)^2}{63 (3 x+2)}-\frac{5}{567} \sqrt{1-2 x} (7265 x+2323)-\frac{36038 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{567 \sqrt{21}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 97
Rule 149
Rule 147
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^4} \, dx &=-\frac{(1-2 x)^{3/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac{1}{9} \int \frac{(6-45 x) \sqrt{1-2 x} (3+5 x)^2}{(2+3 x)^3} \, dx\\ &=-\frac{(1-2 x)^{3/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac{2 \sqrt{1-2 x} (3+5 x)^3}{(2+3 x)^2}-\frac{1}{54} \int \frac{(306-1800 x) (3+5 x)^2}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=\frac{251 \sqrt{1-2 x} (3+5 x)^2}{63 (2+3 x)}-\frac{(1-2 x)^{3/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac{2 \sqrt{1-2 x} (3+5 x)^3}{(2+3 x)^2}-\frac{\int \frac{(20934-130770 x) (3+5 x)}{\sqrt{1-2 x} (2+3 x)} \, dx}{1134}\\ &=\frac{251 \sqrt{1-2 x} (3+5 x)^2}{63 (2+3 x)}-\frac{(1-2 x)^{3/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac{2 \sqrt{1-2 x} (3+5 x)^3}{(2+3 x)^2}-\frac{5}{567} \sqrt{1-2 x} (2323+7265 x)+\frac{18019}{567} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{251 \sqrt{1-2 x} (3+5 x)^2}{63 (2+3 x)}-\frac{(1-2 x)^{3/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac{2 \sqrt{1-2 x} (3+5 x)^3}{(2+3 x)^2}-\frac{5}{567} \sqrt{1-2 x} (2323+7265 x)-\frac{18019}{567} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{251 \sqrt{1-2 x} (3+5 x)^2}{63 (2+3 x)}-\frac{(1-2 x)^{3/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac{2 \sqrt{1-2 x} (3+5 x)^3}{(2+3 x)^2}-\frac{5}{567} \sqrt{1-2 x} (2323+7265 x)-\frac{36038 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{567 \sqrt{21}}\\ \end{align*}
Mathematica [C] time = 0.0201029, size = 59, normalized size = 0.47 \[ \frac{(1-2 x)^{5/2} \left (72076 (3 x+2)^3 \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};\frac{3}{7}-\frac{6 x}{7}\right )-245 \left (18375 x^2+24657 x+8269\right )\right )}{324135 (3 x+2)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.01, size = 75, normalized size = 0.6 \begin{align*}{\frac{250}{243} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{2050}{243}\sqrt{1-2\,x}}+{\frac{2}{9\, \left ( -6\,x-4 \right ) ^{3}} \left ( -{\frac{3938}{21} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{23306}{27} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{26824}{27}\sqrt{1-2\,x}} \right ) }-{\frac{36038\,\sqrt{21}}{11907}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 3.69752, size = 149, normalized size = 1.19 \begin{align*} \frac{250}{243} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{18019}{11907} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2050}{243} \, \sqrt{-2 \, x + 1} + \frac{4 \,{\left (17721 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 81571 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 93884 \, \sqrt{-2 \, x + 1}\right )}}{1701 \,{\left (27 \,{\left (2 \, x - 1\right )}^{3} + 189 \,{\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.33996, size = 285, normalized size = 2.28 \begin{align*} \frac{18019 \, \sqrt{21}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \,{\left (31500 \, x^{4} - 81900 \, x^{3} - 259614 \, x^{2} - 199243 \, x - 47939\right )} \sqrt{-2 \, x + 1}}{11907 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\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 = 1.46674, size = 138, normalized size = 1.1 \begin{align*} \frac{250}{243} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{18019}{11907} \, \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{2050}{243} \, \sqrt{-2 \, x + 1} + \frac{17721 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 81571 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 93884 \, \sqrt{-2 \, x + 1}}{3402 \,{\left (3 \, x + 2\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]