Optimal. Leaf size=191 \[ -\frac{104663 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{2953125}+\frac{2}{45} (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^{5/2}+\frac{178 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}}{4725}+\frac{403 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{118125}-\frac{87476 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{590625}-\frac{6515539 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5906250} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0689408, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{2}{45} (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^{5/2}+\frac{178 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}}{4725}+\frac{403 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{118125}-\frac{87476 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{590625}-\frac{104663 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2953125}-\frac{6515539 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5906250} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 101
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2} (2+3 x)^{5/2}}{\sqrt{3+5 x}} \, dx &=\frac{2}{45} (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt{3+5 x}-\frac{2}{45} \int \frac{\left (-\frac{71}{2}-\frac{89 x}{2}\right ) \sqrt{1-2 x} (2+3 x)^{3/2}}{\sqrt{3+5 x}} \, dx\\ &=\frac{178 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}{4725}+\frac{2}{45} (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt{3+5 x}-\frac{4 \int \frac{(2+3 x)^{3/2} \left (-907+\frac{403 x}{4}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{4725}\\ &=\frac{403 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{118125}+\frac{178 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}{4725}+\frac{2}{45} (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt{3+5 x}+\frac{4 \int \frac{\sqrt{2+3 x} \left (\frac{352725}{8}+65607 x\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{118125}\\ &=-\frac{87476 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{590625}+\frac{403 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{118125}+\frac{178 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}{4725}+\frac{2}{45} (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt{3+5 x}-\frac{4 \int \frac{-\frac{6209373}{4}-\frac{19546617 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1771875}\\ &=-\frac{87476 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{590625}+\frac{403 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{118125}+\frac{178 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}{4725}+\frac{2}{45} (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt{3+5 x}+\frac{1151293 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{5906250}+\frac{6515539 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{5906250}\\ &=-\frac{87476 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{590625}+\frac{403 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{118125}+\frac{178 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}{4725}+\frac{2}{45} (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt{3+5 x}-\frac{6515539 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5906250}-\frac{104663 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2953125}\\ \end{align*}
Mathematica [A] time = 0.251891, size = 105, normalized size = 0.55 \[ \frac{6515539 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (612332 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+3 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (472500 x^3+193500 x^2-378045 x-110554\right )\right )}{8859375 \sqrt{2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.011, size = 155, normalized size = 0.8 \begin{align*}{\frac{1}{531562500\,{x}^{3}+407531250\,{x}^{2}-124031250\,x-106312500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -425250000\,{x}^{6}+3061660\,\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 ) -6515539\,\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 ) -500175000\,{x}^{5}+305950500\,{x}^{4}+486034650\,{x}^{3}+31722810\,{x}^{2}-91264440\,x-19899720 \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 \frac{{\left (3 \, x + 2\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{\sqrt{5 \, x + 3}}\,{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 (-\frac{{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{\sqrt{5 \, x + 3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \frac{{\left (3 \, x + 2\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{\sqrt{5 \, x + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]