Optimal. Leaf size=41 \[ \frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3 \sqrt{21}}-\frac{5}{3} \sqrt{1-2 x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0084517, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {80, 63, 206} \[ \frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3 \sqrt{21}}-\frac{5}{3} \sqrt{1-2 x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 80
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{3+5 x}{\sqrt{1-2 x} (2+3 x)} \, dx &=-\frac{5}{3} \sqrt{1-2 x}-\frac{1}{3} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=-\frac{5}{3} \sqrt{1-2 x}+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{5}{3} \sqrt{1-2 x}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3 \sqrt{21}}\\ \end{align*}
Mathematica [A] time = 0.0122914, size = 41, normalized size = 1. \[ \frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3 \sqrt{21}}-\frac{5}{3} \sqrt{1-2 x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 29, normalized size = 0.7 \begin{align*}{\frac{2\,\sqrt{21}}{63}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{5}{3}\sqrt{1-2\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.71474, size = 62, normalized size = 1.51 \begin{align*} -\frac{1}{63} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{5}{3} \, \sqrt{-2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.39596, size = 117, normalized size = 2.85 \begin{align*} \frac{1}{63} \, \sqrt{21} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - \frac{5}{3} \, \sqrt{-2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 7.44624, size = 80, normalized size = 1.95 \begin{align*} - \frac{5 \sqrt{1 - 2 x}}{3} - \frac{2 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21}}{3 \sqrt{1 - 2 x}} \right )}}{21} & \text{for}\: \frac{1}{1 - 2 x} > \frac{3}{7} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21}}{3 \sqrt{1 - 2 x}} \right )}}{21} & \text{for}\: \frac{1}{1 - 2 x} < \frac{3}{7} \end{cases}\right )}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.83859, size = 66, normalized size = 1.61 \begin{align*} -\frac{1}{63} \, \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{5}{3} \, \sqrt{-2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]