Optimal. Leaf size=218 \[ -\frac{2930159 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{1275750 \sqrt{33}}+\frac{2}{55} (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{7/2}+\frac{194 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}}{7425}-\frac{2377 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{155925}-\frac{22576 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{155925}-\frac{2930159 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{2806650}-\frac{97540001 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1275750 \sqrt{33}} \]
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Rubi [A] time = 0.0832043, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{2}{55} (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{7/2}+\frac{194 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}}{7425}-\frac{2377 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{155925}-\frac{22576 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{155925}-\frac{2930159 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{2806650}-\frac{2930159 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1275750 \sqrt{33}}-\frac{97540001 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1275750 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 101
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2} \, dx &=\frac{2}{55} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{7/2}-\frac{2}{55} \int \frac{\left (-\frac{67}{2}-\frac{97 x}{2}\right ) \sqrt{1-2 x} (3+5 x)^{5/2}}{\sqrt{2+3 x}} \, dx\\ &=\frac{194 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{7425}+\frac{2}{55} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{7/2}-\frac{4 \int \frac{\left (-\frac{1273}{2}-\frac{2377 x}{4}\right ) (3+5 x)^{5/2}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{7425}\\ &=-\frac{2377 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{155925}+\frac{194 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{7425}+\frac{2}{55} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{7/2}+\frac{4 \int \frac{(3+5 x)^{3/2} \left (\frac{432515}{8}+84660 x\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{155925}\\ &=-\frac{22576 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{155925}-\frac{2377 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{155925}+\frac{194 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{7425}+\frac{2}{55} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{7/2}-\frac{4 \int \frac{\left (-\frac{28606455}{8}-\frac{43952385 x}{8}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{2338875}\\ &=-\frac{2930159 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{2806650}-\frac{22576 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{155925}-\frac{2377 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{155925}+\frac{194 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{7425}+\frac{2}{55} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{7/2}+\frac{4 \int \frac{\frac{1852415265}{16}+\frac{1463100015 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{21049875}\\ &=-\frac{2930159 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{2806650}-\frac{22576 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{155925}-\frac{2377 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{155925}+\frac{194 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{7425}+\frac{2}{55} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{7/2}+\frac{2930159 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{2551500}+\frac{97540001 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{14033250}\\ &=-\frac{2930159 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{2806650}-\frac{22576 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{155925}-\frac{2377 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{155925}+\frac{194 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{7425}+\frac{2}{55} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{7/2}-\frac{97540001 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1275750 \sqrt{33}}-\frac{2930159 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1275750 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.256082, size = 107, normalized size = 0.49 \[ \frac{-98384755 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (25515000 x^4+24003000 x^3-10837350 x^2-14851260 x-201247\right )+195080002 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{42099750 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.01, size = 160, normalized size = 0.7 \begin{align*}{\frac{1}{2525985000\,{x}^{3}+1936588500\,{x}^{2}-589396500\,x-505197000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -22963500000\,{x}^{7}-39208050000\,{x}^{6}+98384755\,\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 ) -195080002\,\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 ) -1450305000\,{x}^{5}+30477235500\,{x}^{4}+12473188200\,{x}^{3}-4930627170\,{x}^{2}-2715488670\,x-36224460 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{3 \, x + 2}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \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.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{3 \, x + 2}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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