Optimal. Leaf size=69 \[ \frac{2}{25} (1-2 x)^{5/2}+\frac{22}{75} (1-2 x)^{3/2}+\frac{242}{125} \sqrt{1-2 x}-\frac{242}{125} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0169833, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {50, 63, 206} \[ \frac{2}{25} (1-2 x)^{5/2}+\frac{22}{75} (1-2 x)^{3/2}+\frac{242}{125} \sqrt{1-2 x}-\frac{242}{125} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2}}{3+5 x} \, dx &=\frac{2}{25} (1-2 x)^{5/2}+\frac{11}{5} \int \frac{(1-2 x)^{3/2}}{3+5 x} \, dx\\ &=\frac{22}{75} (1-2 x)^{3/2}+\frac{2}{25} (1-2 x)^{5/2}+\frac{121}{25} \int \frac{\sqrt{1-2 x}}{3+5 x} \, dx\\ &=\frac{242}{125} \sqrt{1-2 x}+\frac{22}{75} (1-2 x)^{3/2}+\frac{2}{25} (1-2 x)^{5/2}+\frac{1331}{125} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{242}{125} \sqrt{1-2 x}+\frac{22}{75} (1-2 x)^{3/2}+\frac{2}{25} (1-2 x)^{5/2}-\frac{1331}{125} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{242}{125} \sqrt{1-2 x}+\frac{22}{75} (1-2 x)^{3/2}+\frac{2}{25} (1-2 x)^{5/2}-\frac{242}{125} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0252343, size = 51, normalized size = 0.74 \[ \frac{2 \left (5 \sqrt{1-2 x} \left (60 x^2-170 x+433\right )-363 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right )}{1875} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 47, normalized size = 0.7 \begin{align*}{\frac{22}{75} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{2}{25} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{242\,\sqrt{55}}{625}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{242}{125}\sqrt{1-2\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 3.14104, size = 86, normalized size = 1.25 \begin{align*} \frac{2}{25} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{22}{75} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{121}{625} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{242}{125} \, \sqrt{-2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.28904, size = 177, normalized size = 2.57 \begin{align*} \frac{121}{625} \, \sqrt{11} \sqrt{5} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + \frac{2}{375} \,{\left (60 \, x^{2} - 170 \, x + 433\right )} \sqrt{-2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.67292, size = 204, normalized size = 2.96 \begin{align*} \begin{cases} \frac{8 \sqrt{5} i \left (x + \frac{3}{5}\right )^{2} \sqrt{10 x - 5}}{125} - \frac{484 \sqrt{5} i \left (x + \frac{3}{5}\right ) \sqrt{10 x - 5}}{1875} + \frac{5566 \sqrt{5} i \sqrt{10 x - 5}}{9375} + \frac{242 \sqrt{55} i \operatorname{asin}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{625} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{8 \sqrt{5} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )^{2}}{125} - \frac{484 \sqrt{5} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )}{1875} + \frac{5566 \sqrt{5} \sqrt{5 - 10 x}}{9375} + \frac{121 \sqrt{55} \log{\left (x + \frac{3}{5} \right )}}{625} - \frac{242 \sqrt{55} \log{\left (\sqrt{\frac{5}{11} - \frac{10 x}{11}} + 1 \right )}}{625} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 2.29949, size = 100, normalized size = 1.45 \begin{align*} \frac{2}{25} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{22}{75} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{121}{625} \, \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{242}{125} \, \sqrt{-2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]