Optimal. Leaf size=72 \[ -\frac{2 (1-2 x)^{3/2}}{55 \sqrt{5 x+3}}+\frac{29}{275} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{29 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{25 \sqrt{10}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0157993, antiderivative size = 72, 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 = {78, 50, 54, 216} \[ -\frac{2 (1-2 x)^{3/2}}{55 \sqrt{5 x+3}}+\frac{29}{275} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{29 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{25 \sqrt{10}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 78
Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{\sqrt{1-2 x} (2+3 x)}{(3+5 x)^{3/2}} \, dx &=-\frac{2 (1-2 x)^{3/2}}{55 \sqrt{3+5 x}}+\frac{29}{55} \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2}}{55 \sqrt{3+5 x}}+\frac{29}{275} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{29}{50} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2}}{55 \sqrt{3+5 x}}+\frac{29}{275} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{29 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{25 \sqrt{5}}\\ &=-\frac{2 (1-2 x)^{3/2}}{55 \sqrt{3+5 x}}+\frac{29}{275} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{29 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{25 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.0272735, size = 71, normalized size = 0.99 \[ \frac{10 \left (-30 x^2+x+7\right )-29 \sqrt{10-20 x} \sqrt{5 x+3} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{250 \sqrt{1-2 x} \sqrt{5 x+3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.009, size = 82, normalized size = 1.1 \begin{align*}{\frac{1}{500} \left ( 145\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+87\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +300\,x\sqrt{-10\,{x}^{2}-x+3}+140\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 2.18689, size = 68, normalized size = 0.94 \begin{align*} \frac{29}{500} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{3}{25} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{25 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.22202, size = 224, normalized size = 3.11 \begin{align*} -\frac{29 \, \sqrt{10}{\left (5 \, x + 3\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) - 20 \,{\left (15 \, x + 7\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{500 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{1 - 2 x} \left (3 x + 2\right )}{\left (5 x + 3\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.97671, size = 132, normalized size = 1.83 \begin{align*} \frac{3}{125} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{29}{250} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{250 \, \sqrt{5 \, x + 3}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{125 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]