Optimal. Leaf size=125 \[ -\frac{2 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{77 \sqrt{33}}+\frac{74 \sqrt{3 x+2} \sqrt{5 x+3}}{2541 \sqrt{1-2 x}}+\frac{2 \sqrt{3 x+2} \sqrt{5 x+3}}{33 (1-2 x)^{3/2}}+\frac{37 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{77 \sqrt{33}} \]
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Rubi [A] time = 0.0399907, antiderivative size = 125, 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 = {99, 152, 158, 113, 119} \[ \frac{74 \sqrt{3 x+2} \sqrt{5 x+3}}{2541 \sqrt{1-2 x}}+\frac{2 \sqrt{3 x+2} \sqrt{5 x+3}}{33 (1-2 x)^{3/2}}-\frac{2 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{77 \sqrt{33}}+\frac{37 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{77 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 99
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{\sqrt{2+3 x}}{(1-2 x)^{5/2} \sqrt{3+5 x}} \, dx &=\frac{2 \sqrt{2+3 x} \sqrt{3+5 x}}{33 (1-2 x)^{3/2}}-\frac{2}{33} \int \frac{-\frac{11}{2}-\frac{15 x}{2}}{(1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{2 \sqrt{2+3 x} \sqrt{3+5 x}}{33 (1-2 x)^{3/2}}+\frac{74 \sqrt{2+3 x} \sqrt{3+5 x}}{2541 \sqrt{1-2 x}}+\frac{4 \int \frac{-75-\frac{555 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{2541}\\ &=\frac{2 \sqrt{2+3 x} \sqrt{3+5 x}}{33 (1-2 x)^{3/2}}+\frac{74 \sqrt{2+3 x} \sqrt{3+5 x}}{2541 \sqrt{1-2 x}}+\frac{1}{77} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx-\frac{37}{847} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{2 \sqrt{2+3 x} \sqrt{3+5 x}}{33 (1-2 x)^{3/2}}+\frac{74 \sqrt{2+3 x} \sqrt{3+5 x}}{2541 \sqrt{1-2 x}}+\frac{37 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{77 \sqrt{33}}-\frac{2 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{77 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.125434, size = 115, normalized size = 0.92 \[ -\frac{70 \sqrt{2-4 x} (2 x-1) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+4 \sqrt{3 x+2} \sqrt{5 x+3} (37 x-57)-37 \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{2541 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.02, size = 228, normalized size = 1.8 \begin{align*} -{\frac{1}{2541\, \left ( 2\,x-1 \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) } \left ( 140\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-74\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-70\,\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 ) +37\,\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 ) +2220\,{x}^{3}-608\,{x}^{2}-3444\,x-1368 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}\sqrt{2+3\,x}} \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{\sqrt{3 \, x + 2}}{\sqrt{5 \, x + 3}{\left (-2 \, x + 1\right )}^{\frac{5}{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{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{40 \, x^{4} - 36 \, x^{3} - 6 \, x^{2} + 13 \, 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{\sqrt{3 \, x + 2}}{\sqrt{5 \, x + 3}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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