Optimal. Leaf size=191 \[ -\frac{175111 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{1181250}+\frac{2}{45} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}-\frac{23 \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}}{1575}-\frac{1244 \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}}{13125}-\frac{175111 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{236250}-\frac{2911577 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{590625} \]
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Rubi [A] time = 0.0729185, 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} \[ \frac{2}{45} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}-\frac{23 \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}}{1575}-\frac{1244 \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}}{13125}-\frac{175111 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{236250}-\frac{175111 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1181250}-\frac{2911577 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{590625} \]
Antiderivative was successfully verified.
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Rule 101
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x} \, dx &=\frac{2}{45} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}-\frac{2}{45} \int \frac{\left (-\frac{27}{2}-\frac{23 x}{2}\right ) (2+3 x)^{3/2} \sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{23 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac{2}{45} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}+\frac{2 \int \frac{\sqrt{2+3 x} \sqrt{3+5 x} \left (\frac{4815}{4}+1866 x\right )}{\sqrt{1-2 x}} \, dx}{1575}\\ &=-\frac{1244 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{13125}-\frac{23 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac{2}{45} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}-\frac{2 \int \frac{\left (-\frac{170757}{2}-\frac{525333 x}{4}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{39375}\\ &=-\frac{175111 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{236250}-\frac{1244 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{13125}-\frac{23 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac{2}{45} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}+\frac{2 \int \frac{\frac{22119087}{8}+\frac{8734731 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{354375}\\ &=-\frac{175111 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{236250}-\frac{1244 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{13125}-\frac{23 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac{2}{45} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}+\frac{1926221 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{2362500}+\frac{2911577 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{590625}\\ &=-\frac{175111 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{236250}-\frac{1244 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{13125}-\frac{23 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac{2}{45} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}-\frac{2911577 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{590625}-\frac{175111 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1181250}\\ \end{align*}
Mathematica [A] time = 0.278869, size = 102, normalized size = 0.53 \[ \frac{-5867645 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (472500 x^3+861750 x^2+410490 x-136987\right )+11646308 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{3543750 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.038, size = 155, normalized size = 0.8 \begin{align*}{\frac{1}{212625000\,{x}^{3}+163012500\,{x}^{2}-49612500\,x-42525000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 425250000\,{x}^{6}+5867645\,\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 ) -11646308\,\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 ) +1101600000\,{x}^{5}+864823500\,{x}^{4}-106067700\,{x}^{3}-335838930\,{x}^{2}-45120930\,x+24657660 \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 \sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}\,{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 (9 \, x^{2} + 12 \, x + 4\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 \sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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