Optimal. Leaf size=101 \[ \frac{2873}{73205 \sqrt{1-2 x}}-\frac{2873}{39930 \sqrt{1-2 x} (5 x+3)}-\frac{614}{1815 \sqrt{1-2 x} (5 x+3)^2}+\frac{49}{66 (1-2 x)^{3/2} (5 x+3)^2}-\frac{2873 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{14641 \sqrt{55}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0296638, antiderivative size = 108, normalized size of antiderivative = 1.07, 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 = {89, 78, 51, 63, 206} \[ -\frac{2873 \sqrt{1-2 x}}{29282 (5 x+3)}+\frac{2873}{19965 \sqrt{1-2 x} (5 x+3)}-\frac{614}{1815 \sqrt{1-2 x} (5 x+3)^2}+\frac{49}{66 (1-2 x)^{3/2} (5 x+3)^2}-\frac{2873 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{14641 \sqrt{55}} \]
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)^3} \, dx &=\frac{49}{66 (1-2 x)^{3/2} (3+5 x)^2}-\frac{1}{66} \int \frac{-313+297 x}{(1-2 x)^{3/2} (3+5 x)^3} \, dx\\ &=\frac{49}{66 (1-2 x)^{3/2} (3+5 x)^2}-\frac{614}{1815 \sqrt{1-2 x} (3+5 x)^2}+\frac{2873 \int \frac{1}{(1-2 x)^{3/2} (3+5 x)^2} \, dx}{3630}\\ &=\frac{49}{66 (1-2 x)^{3/2} (3+5 x)^2}-\frac{614}{1815 \sqrt{1-2 x} (3+5 x)^2}+\frac{2873}{19965 \sqrt{1-2 x} (3+5 x)}+\frac{2873 \int \frac{1}{\sqrt{1-2 x} (3+5 x)^2} \, dx}{2662}\\ &=\frac{49}{66 (1-2 x)^{3/2} (3+5 x)^2}-\frac{614}{1815 \sqrt{1-2 x} (3+5 x)^2}+\frac{2873}{19965 \sqrt{1-2 x} (3+5 x)}-\frac{2873 \sqrt{1-2 x}}{29282 (3+5 x)}+\frac{2873 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{29282}\\ &=\frac{49}{66 (1-2 x)^{3/2} (3+5 x)^2}-\frac{614}{1815 \sqrt{1-2 x} (3+5 x)^2}+\frac{2873}{19965 \sqrt{1-2 x} (3+5 x)}-\frac{2873 \sqrt{1-2 x}}{29282 (3+5 x)}-\frac{2873 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{29282}\\ &=\frac{49}{66 (1-2 x)^{3/2} (3+5 x)^2}-\frac{614}{1815 \sqrt{1-2 x} (3+5 x)^2}+\frac{2873}{19965 \sqrt{1-2 x} (3+5 x)}-\frac{2873 \sqrt{1-2 x}}{29282 (3+5 x)}-\frac{2873 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{14641 \sqrt{55}}\\ \end{align*}
Mathematica [C] time = 0.0167994, size = 59, normalized size = 0.58 \[ -\frac{11492 (2 x-1) (5 x+3)^2 \, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};\frac{5}{11} (1-2 x)\right )-121 (2456 x+1467)}{439230 (1-2 x)^{3/2} (5 x+3)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.012, size = 66, normalized size = 0.7 \begin{align*}{\frac{98}{3993} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{546}{14641}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{50}{14641\, \left ( -10\,x-6 \right ) ^{2}} \left ({\frac{143}{10} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{319}{10}\sqrt{1-2\,x}} \right ) }-{\frac{2873\,\sqrt{55}}{805255}{\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 = 3.57306, size = 124, normalized size = 1.23 \begin{align*} \frac{2873}{1610510} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{43095 \,{\left (2 \, x - 1\right )}^{3} + 158015 \,{\left (2 \, x - 1\right )}^{2} + 159236 \, x - 210056}{43923 \,{\left (25 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 110 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 121 \,{\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 = 1.07181, size = 294, normalized size = 2.91 \begin{align*} \frac{8619 \, \sqrt{55}{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \,{\left (172380 \, x^{3} + 57460 \, x^{2} - 107127 \, x - 47568\right )} \sqrt{-2 \, x + 1}}{4831530 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, 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 = 2.09314, size = 120, normalized size = 1.19 \begin{align*} \frac{2873}{1610510} \, \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{28 \,{\left (117 \, x - 97\right )}}{43923 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} + \frac{5 \,{\left (13 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 29 \, \sqrt{-2 \, x + 1}\right )}}{5324 \,{\left (5 \, x + 3\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]