Optimal. Leaf size=127 \[ -\frac{857 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{625 \sqrt{33}}-\frac{3}{25} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}-\frac{74}{125} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}-\frac{5161 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1250} \]
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Rubi [A] time = 0.0438019, antiderivative size = 127, 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} \[ -\frac{3}{25} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}-\frac{74}{125} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}-\frac{857 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{625 \sqrt{33}}-\frac{5161 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1250} \]
Antiderivative was successfully verified.
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Rule 102
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(2+3 x)^{5/2}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx &=-\frac{3}{25} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}-\frac{1}{25} \int \frac{\left (-\frac{275}{2}-222 x\right ) \sqrt{2+3 x}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{74}{125} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{3}{25} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}+\frac{1}{375} \int \frac{4902+\frac{15483 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{74}{125} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{3}{25} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}+\frac{857 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1250}+\frac{5161 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{1250}\\ &=-\frac{74}{125} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{3}{25} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}-\frac{5161 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1250}-\frac{857 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{625 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.199519, size = 95, normalized size = 0.75 \[ \frac{5161 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (518 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+3 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} (45 x+104)\right )}{1875 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.014, size = 145, normalized size = 1.1 \begin{align*}{\frac{1}{112500\,{x}^{3}+86250\,{x}^{2}-26250\,x-22500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 2590\,\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 ) -5161\,\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 ) -40500\,{x}^{4}-124650\,{x}^{3}-62310\,{x}^{2}+29940\,x+18720 \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 (3 \, x + 2\right )}^{\frac{5}{2}}}{\sqrt{5 \, x + 3} \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 (9 \, x^{2} + 12 \, x + 4\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{10 \, x^{2} + x - 3}, 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 (3 \, x + 2\right )}^{\frac{5}{2}}}{\sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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