Optimal. Leaf size=99 \[ -\frac{1}{10} \sqrt{1-2 x} (3 x+2) (5 x+3)^{3/2}-\frac{181}{400} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{6269 \sqrt{1-2 x} \sqrt{5 x+3}}{1600}+\frac{68959 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1600 \sqrt{10}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0236073, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {90, 80, 50, 54, 216} \[ -\frac{1}{10} \sqrt{1-2 x} (3 x+2) (5 x+3)^{3/2}-\frac{181}{400} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{6269 \sqrt{1-2 x} \sqrt{5 x+3}}{1600}+\frac{68959 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1600 \sqrt{10}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 90
Rule 80
Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{(2+3 x)^2 \sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx &=-\frac{1}{10} \sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}-\frac{1}{30} \int \frac{\left (-174-\frac{543 x}{2}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{181}{400} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{1}{10} \sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}+\frac{6269}{800} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{6269 \sqrt{1-2 x} \sqrt{3+5 x}}{1600}-\frac{181}{400} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{1}{10} \sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}+\frac{68959 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{3200}\\ &=-\frac{6269 \sqrt{1-2 x} \sqrt{3+5 x}}{1600}-\frac{181}{400} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{1}{10} \sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}+\frac{68959 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{1600 \sqrt{5}}\\ &=-\frac{6269 \sqrt{1-2 x} \sqrt{3+5 x}}{1600}-\frac{181}{400} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{1}{10} \sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}+\frac{68959 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{1600 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.0301972, size = 60, normalized size = 0.61 \[ \frac{-10 \sqrt{1-2 x} \sqrt{5 x+3} \left (2400 x^2+6660 x+9401\right )-68959 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{16000} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.009, size = 87, normalized size = 0.9 \begin{align*}{\frac{1}{32000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -48000\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+68959\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -133200\,x\sqrt{-10\,{x}^{2}-x+3}-188020\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 4.22554, size = 78, normalized size = 0.79 \begin{align*} \frac{68959}{32000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{3}{20} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{321}{80} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{10121}{1600} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.79808, size = 225, normalized size = 2.27 \begin{align*} -\frac{1}{1600} \,{\left (2400 \, x^{2} + 6660 \, x + 9401\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{68959}{32000} \, \sqrt{10} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 14.9652, size = 292, normalized size = 2.95 \begin{align*} \frac{2 \sqrt{5} \left (\begin{cases} \frac{11 \sqrt{2} \left (- \frac{\sqrt{2} \sqrt{5 - 10 x} \sqrt{5 x + 3}}{22} + \frac{\operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{2}\right )}{4} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{125} + \frac{12 \sqrt{5} \left (\begin{cases} \frac{121 \sqrt{2} \left (\frac{\sqrt{2} \sqrt{5 - 10 x} \left (- 20 x - 1\right ) \sqrt{5 x + 3}}{968} - \frac{\sqrt{2} \sqrt{5 - 10 x} \sqrt{5 x + 3}}{22} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{8}\right )}{8} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{125} + \frac{18 \sqrt{5} \left (\begin{cases} \frac{1331 \sqrt{2} \left (\frac{\sqrt{2} \left (5 - 10 x\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3993} + \frac{3 \sqrt{2} \sqrt{5 - 10 x} \left (- 20 x - 1\right ) \sqrt{5 x + 3}}{1936} - \frac{\sqrt{2} \sqrt{5 - 10 x} \sqrt{5 x + 3}}{22} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{16}\right )}{16} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{125} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 2.73681, size = 73, normalized size = 0.74 \begin{align*} -\frac{1}{16000} \, \sqrt{5}{\left (2 \,{\left (12 \,{\left (40 \, x + 87\right )}{\left (5 \, x + 3\right )} + 6269\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 68959 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]