Optimal. Leaf size=160 \[ -\frac{4721 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{5250}-\frac{1}{7} \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}-\frac{102}{175} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{4721 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{1050}-\frac{78472 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2625} \]
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Rubi [A] time = 0.0505249, 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} \[ -\frac{1}{7} \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}-\frac{102}{175} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{4721 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{1050}-\frac{4721 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5250}-\frac{78472 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2625} \]
Antiderivative was successfully verified.
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Rule 101
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(2+3 x)^{3/2} (3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx &=-\frac{1}{7} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac{1}{7} \int \frac{\sqrt{2+3 x} \sqrt{3+5 x} \left (\frac{129}{2}+102 x\right )}{\sqrt{1-2 x}} \, dx\\ &=-\frac{102}{175} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{1}{7} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac{1}{175} \int \frac{\left (-4602-\frac{14163 x}{2}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=-\frac{4721 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{1050}-\frac{102}{175} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{1}{7} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac{\int \frac{\frac{596157}{4}+235416 x}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1575}\\ &=-\frac{4721 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{1050}-\frac{102}{175} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{1}{7} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac{51931 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{10500}+\frac{78472 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{2625}\\ &=-\frac{4721 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{1050}-\frac{102}{175} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{1}{7} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac{78472 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2625}-\frac{4721 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5250}\\ \end{align*}
Mathematica [A] time = 0.200589, size = 100, normalized size = 0.62 \[ \frac{313888 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (31619 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+3 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (2250 x^2+5910 x+7457\right )\right )}{15750 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.011, size = 150, normalized size = 0.9 \begin{align*}{\frac{1}{945000\,{x}^{3}+724500\,{x}^{2}-220500\,x-189000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 158095\,\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 ) -313888\,\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}-6871500\,{x}^{4}-10316700\,{x}^{3}-3499230\,{x}^{2}+2629770\,x+1342260 \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 \frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\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 (-\frac{{\left (15 \, x^{2} + 19 \, x + 6\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{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 \frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\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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