Optimal. Leaf size=129 \[ -\frac{31 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{1125}+\frac{2}{25} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{31}{225} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{1159 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1125} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0426397, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {101, 154, 158, 113, 119} \[ \frac{2}{25} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{31}{225} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{31 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1125}-\frac{1159 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1125} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 101
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x} \, dx &=\frac{2}{25} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{2}{25} \int \frac{\left (-\frac{23}{2}-\frac{31 x}{2}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=-\frac{31}{225} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{2}{25} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{2}{225} \int \frac{\frac{1459}{4}+\frac{1159 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{31}{225} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{2}{25} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{341 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{2250}+\frac{1159 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{1125}\\ &=-\frac{31}{225} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{2}{25} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{1159 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1125}-\frac{31 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1125}\\ \end{align*}
Mathematica [A] time = 0.175478, size = 97, normalized size = 0.75 \[ \frac{-1295 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+30 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3} (90 x+23)+2318 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{6750} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.01, size = 145, normalized size = 1.1 \begin{align*}{\frac{1}{202500\,{x}^{3}+155250\,{x}^{2}-47250\,x-40500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 1295\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) -2318\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) +81000\,{x}^{4}+82800\,{x}^{3}-3030\,{x}^{2}-21030\,x-4140 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}, x\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 \sqrt{1 - 2 x} \sqrt{3 x + 2} \sqrt{5 x + 3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]