Optimal. Leaf size=80 \[ -\frac{\sqrt{1-2 x} (3 x+2)^2}{110 (5 x+3)^2}-\frac{9 \sqrt{1-2 x} (715 x+432)}{6050 (5 x+3)}-\frac{1347 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3025 \sqrt{55}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0183321, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {98, 146, 63, 206} \[ -\frac{\sqrt{1-2 x} (3 x+2)^2}{110 (5 x+3)^2}-\frac{9 \sqrt{1-2 x} (715 x+432)}{6050 (5 x+3)}-\frac{1347 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3025 \sqrt{55}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 98
Rule 146
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(2+3 x)^3}{\sqrt{1-2 x} (3+5 x)^3} \, dx &=-\frac{\sqrt{1-2 x} (2+3 x)^2}{110 (3+5 x)^2}-\frac{1}{110} \int \frac{(-144-195 x) (2+3 x)}{\sqrt{1-2 x} (3+5 x)^2} \, dx\\ &=-\frac{\sqrt{1-2 x} (2+3 x)^2}{110 (3+5 x)^2}-\frac{9 \sqrt{1-2 x} (432+715 x)}{6050 (3+5 x)}+\frac{1347 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{6050}\\ &=-\frac{\sqrt{1-2 x} (2+3 x)^2}{110 (3+5 x)^2}-\frac{9 \sqrt{1-2 x} (432+715 x)}{6050 (3+5 x)}-\frac{1347 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{6050}\\ &=-\frac{\sqrt{1-2 x} (2+3 x)^2}{110 (3+5 x)^2}-\frac{9 \sqrt{1-2 x} (432+715 x)}{6050 (3+5 x)}-\frac{1347 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3025 \sqrt{55}}\\ \end{align*}
Mathematica [A] time = 0.0333752, size = 58, normalized size = 0.72 \[ \frac{-\frac{55 \sqrt{1-2 x} \left (32670 x^2+39405 x+11884\right )}{(5 x+3)^2}-2694 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{332750} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.01, size = 57, normalized size = 0.7 \begin{align*} -{\frac{27}{125}\sqrt{1-2\,x}}+{\frac{2}{5\, \left ( -10\,x-6 \right ) ^{2}} \left ({\frac{201}{1210} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{203}{550}\sqrt{1-2\,x}} \right ) }-{\frac{1347\,\sqrt{55}}{166375}{\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 = 1.9248, size = 112, normalized size = 1.4 \begin{align*} \frac{1347}{332750} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{27}{125} \, \sqrt{-2 \, x + 1} + \frac{1005 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2233 \, \sqrt{-2 \, x + 1}}{15125 \,{\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 = 1.60216, size = 225, normalized size = 2.81 \begin{align*} \frac{1347 \, \sqrt{55}{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \,{\left (32670 \, x^{2} + 39405 \, x + 11884\right )} \sqrt{-2 \, x + 1}}{332750 \,{\left (25 \, x^{2} + 30 \, 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 = 2.45958, size = 104, normalized size = 1.3 \begin{align*} \frac{1347}{332750} \, \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{27}{125} \, \sqrt{-2 \, x + 1} + \frac{1005 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2233 \, \sqrt{-2 \, x + 1}}{60500 \,{\left (5 \, x + 3\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]