Optimal. Leaf size=160 \[ -\frac{823 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{13125}+\frac{2}{35} \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}-\frac{27}{875} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{823 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{2625}-\frac{55019 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{26250} \]
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Rubi [A] time = 0.0534294, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ \frac{2}{35} \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}-\frac{27}{875} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{823 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{2625}-\frac{823 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{13125}-\frac{55019 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{26250} \]
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)^{3/2} \sqrt{3+5 x} \, dx &=\frac{2}{35} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac{2}{35} \int \frac{\left (-\frac{25}{2}-\frac{27 x}{2}\right ) \sqrt{2+3 x} \sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{27}{875} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{2}{35} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac{2}{875} \int \frac{\sqrt{3+5 x} \left (\frac{3229}{4}+\frac{2469 x}{2}\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=-\frac{823 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{2625}-\frac{27}{875} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{2}{35} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac{2 \int \frac{-\frac{52233}{2}-\frac{165057 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{7875}\\ &=-\frac{823 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{2625}-\frac{27}{875} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{2}{35} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac{9053 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{26250}+\frac{55019 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{26250}\\ &=-\frac{823 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{2625}-\frac{27}{875} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{2}{35} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac{55019 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{26250}-\frac{823 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{13125}\\ \end{align*}
Mathematica [A] time = 0.216544, size = 97, normalized size = 0.61 \[ \frac{-27860 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (2250 x^2+2445 x-166\right )+55019 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{39375 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.01, size = 150, normalized size = 0.9 \begin{align*}{\frac{1}{2362500\,{x}^{3}+1811250\,{x}^{2}-551250\,x-472500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 27860\,\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 ) -55019\,\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 ) +2025000\,{x}^{5}+3753000\,{x}^{4}+1065150\,{x}^{3}-1032990\,{x}^{2}-405240\,x+29880 \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{3}{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 (\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{3}{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{3}{2}} \sqrt{-2 \, x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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