Optimal. Leaf size=81 \[ \frac{142}{6655 \sqrt{1-2 x}}-\frac{1231}{3630 \sqrt{1-2 x} (5 x+3)}+\frac{49}{66 (1-2 x)^{3/2} (5 x+3)}-\frac{142 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331 \sqrt{55}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0200319, antiderivative size = 81, 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 = {89, 78, 51, 63, 206} \[ \frac{142}{6655 \sqrt{1-2 x}}-\frac{1231}{3630 \sqrt{1-2 x} (5 x+3)}+\frac{49}{66 (1-2 x)^{3/2} (5 x+3)}-\frac{142 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331 \sqrt{55}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 89
Rule 78
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(2+3 x)^2}{(1-2 x)^{5/2} (3+5 x)^2} \, dx &=\frac{49}{66 (1-2 x)^{3/2} (3+5 x)}-\frac{1}{66} \int \frac{-68+297 x}{(1-2 x)^{3/2} (3+5 x)^2} \, dx\\ &=\frac{49}{66 (1-2 x)^{3/2} (3+5 x)}-\frac{1231}{3630 \sqrt{1-2 x} (3+5 x)}+\frac{71}{605} \int \frac{1}{(1-2 x)^{3/2} (3+5 x)} \, dx\\ &=\frac{142}{6655 \sqrt{1-2 x}}+\frac{49}{66 (1-2 x)^{3/2} (3+5 x)}-\frac{1231}{3630 \sqrt{1-2 x} (3+5 x)}+\frac{71 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{1331}\\ &=\frac{142}{6655 \sqrt{1-2 x}}+\frac{49}{66 (1-2 x)^{3/2} (3+5 x)}-\frac{1231}{3630 \sqrt{1-2 x} (3+5 x)}-\frac{71 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{1331}\\ &=\frac{142}{6655 \sqrt{1-2 x}}+\frac{49}{66 (1-2 x)^{3/2} (3+5 x)}-\frac{1231}{3630 \sqrt{1-2 x} (3+5 x)}-\frac{142 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331 \sqrt{55}}\\ \end{align*}
Mathematica [C] time = 0.0135995, size = 55, normalized size = 0.68 \[ -\frac{426 \left (10 x^2+x-3\right ) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\frac{5}{11} (2 x-1)\right )-11 (1231 x+732)}{19965 (1-2 x)^{3/2} (5 x+3)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.011, size = 54, normalized size = 0.7 \begin{align*}{\frac{49}{363} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{28}{1331}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{2}{6655}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}-{\frac{142\,\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 = 2.86873, size = 100, normalized size = 1.23 \begin{align*} \frac{71}{73205} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2 \,{\left (213 \,{\left (2 \, x - 1\right )}^{2} - 1771 \, x - 2079\right )}}{3993 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 11 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.10727, size = 238, normalized size = 2.94 \begin{align*} \frac{213 \, \sqrt{55}{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \,{\left (852 \, x^{2} - 2623 \, x - 1866\right )} \sqrt{-2 \, x + 1}}{219615 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\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 = 2.24734, size = 104, normalized size = 1.28 \begin{align*} \frac{71}{73205} \, \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{7 \,{\left (24 \, x - 89\right )}}{3993 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} - \frac{\sqrt{-2 \, x + 1}}{1331 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]