Optimal. Leaf size=129 \[ -\frac{34154 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{16875}-\frac{4}{75} \sqrt{3 x+2} \sqrt{5 x+3} (1-2 x)^{3/2}-\frac{1088 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{3375}+\frac{53194 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{16875} \]
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Rubi [A] time = 0.0383569, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {102, 154, 158, 113, 119} \[ -\frac{4}{75} \sqrt{3 x+2} \sqrt{5 x+3} (1-2 x)^{3/2}-\frac{1088 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{3375}-\frac{34154 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{16875}+\frac{53194 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{16875} \]
Antiderivative was successfully verified.
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Rule 102
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2}}{\sqrt{2+3 x} \sqrt{3+5 x}} \, dx &=-\frac{4}{75} (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{2}{75} \int \frac{\left (\frac{41}{2}-272 x\right ) \sqrt{1-2 x}}{\sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{1088 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{3375}-\frac{4}{75} (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{4 \int \frac{\frac{5653}{4}-\frac{26597 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{3375}\\ &=-\frac{1088 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{3375}-\frac{4}{75} (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{53194 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{16875}+\frac{187847 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{16875}\\ &=-\frac{1088 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{3375}-\frac{4}{75} (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{53194 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{16875}-\frac{34154 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{16875}\\ \end{align*}
Mathematica [A] time = 0.166392, size = 97, normalized size = 0.75 \[ \frac{616735 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+60 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3} (90 x-317)-53194 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{50625} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.013, size = 145, normalized size = 1.1 \begin{align*}{\frac{1}{1518750\,{x}^{3}+1164375\,{x}^{2}-354375\,x-303750}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 53194\,\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 ) -616735\,\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 ) +162000\,{x}^{4}-446400\,{x}^{3}-475260\,{x}^{2}+100740\,x+114120 \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 (-2 \, x + 1\right )}^{\frac{5}{2}}}{\sqrt{5 \, x + 3} \sqrt{3 \, x + 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 (\frac{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{15 \, x^{2} + 19 \, x + 6}, 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 (-2 \, x + 1\right )}^{\frac{5}{2}}}{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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