Optimal. Leaf size=156 \[ -\frac{178 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{343 \sqrt{33}}-\frac{458 \sqrt{1-2 x} \sqrt{5 x+3}}{3773 \sqrt{3 x+2}}+\frac{194 \sqrt{5 x+3}}{1617 \sqrt{1-2 x} \sqrt{3 x+2}}+\frac{2 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} \sqrt{3 x+2}}+\frac{458 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{343 \sqrt{33}} \]
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Rubi [A] time = 0.0492778, antiderivative size = 156, 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 = {99, 152, 158, 113, 119} \[ -\frac{458 \sqrt{1-2 x} \sqrt{5 x+3}}{3773 \sqrt{3 x+2}}+\frac{194 \sqrt{5 x+3}}{1617 \sqrt{1-2 x} \sqrt{3 x+2}}+\frac{2 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} \sqrt{3 x+2}}-\frac{178 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{343 \sqrt{33}}+\frac{458 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{343 \sqrt{33}} \]
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{3+5 x}}{(1-2 x)^{5/2} (2+3 x)^{3/2}} \, dx &=\frac{2 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} \sqrt{2+3 x}}-\frac{2}{21} \int \frac{-13-\frac{45 x}{2}}{(1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx\\ &=\frac{2 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} \sqrt{2+3 x}}+\frac{194 \sqrt{3+5 x}}{1617 \sqrt{1-2 x} \sqrt{2+3 x}}+\frac{4 \int \frac{\frac{741}{4}+\frac{1455 x}{4}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{1617}\\ &=\frac{2 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} \sqrt{2+3 x}}+\frac{194 \sqrt{3+5 x}}{1617 \sqrt{1-2 x} \sqrt{2+3 x}}-\frac{458 \sqrt{1-2 x} \sqrt{3+5 x}}{3773 \sqrt{2+3 x}}+\frac{8 \int \frac{-\frac{1185}{8}-\frac{3435 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{11319}\\ &=\frac{2 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} \sqrt{2+3 x}}+\frac{194 \sqrt{3+5 x}}{1617 \sqrt{1-2 x} \sqrt{2+3 x}}-\frac{458 \sqrt{1-2 x} \sqrt{3+5 x}}{3773 \sqrt{2+3 x}}-\frac{458 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{3773}+\frac{89}{343} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{2 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} \sqrt{2+3 x}}+\frac{194 \sqrt{3+5 x}}{1617 \sqrt{1-2 x} \sqrt{2+3 x}}-\frac{458 \sqrt{1-2 x} \sqrt{3+5 x}}{3773 \sqrt{2+3 x}}+\frac{458 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{343 \sqrt{33}}-\frac{178 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{343 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.127694, size = 99, normalized size = 0.63 \[ \frac{\sqrt{2} \left (3395 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-458 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )-\frac{2 \sqrt{5 x+3} \left (2748 x^2-1390 x-531\right )}{(1-2 x)^{3/2} \sqrt{3 x+2}}}{11319} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.022, size = 228, normalized size = 1.5 \begin{align*} -{\frac{1}{11319\, \left ( 2\,x-1 \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) }\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 6790\,\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}-916\,\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}-3395\,\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 ) +458\,\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 ) +27480\,{x}^{3}+2588\,{x}^{2}-13650\,x-3186 \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{\sqrt{5 \, x + 3}}{{\left (3 \, x + 2\right )}^{\frac{3}{2}}{\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}}{72 \, x^{5} - 12 \, x^{4} - 58 \, x^{3} + 15 \, x^{2} + 12 \, x - 4}, 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{5 \, x + 3}}{{\left (3 \, x + 2\right )}^{\frac{3}{2}}{\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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