Optimal. Leaf size=156 \[ -\frac{16 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{121 \sqrt{33}}-\frac{490 \sqrt{1-2 x} \sqrt{3 x+2}}{3993 \sqrt{5 x+3}}-\frac{40 \sqrt{1-2 x} \sqrt{3 x+2}}{363 (5 x+3)^{3/2}}+\frac{2 \sqrt{3 x+2}}{11 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{98 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{121 \sqrt{33}} \]
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Rubi [A] time = 0.0501328, 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} \[ -\frac{490 \sqrt{1-2 x} \sqrt{3 x+2}}{3993 \sqrt{5 x+3}}-\frac{40 \sqrt{1-2 x} \sqrt{3 x+2}}{363 (5 x+3)^{3/2}}+\frac{2 \sqrt{3 x+2}}{11 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{16 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{121 \sqrt{33}}+\frac{98 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{121 \sqrt{33}} \]
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)^{3/2} (3+5 x)^{5/2}} \, dx &=\frac{2 \sqrt{2+3 x}}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{2}{11} \int \frac{-\frac{31}{2}-\frac{45 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}} \, dx\\ &=\frac{2 \sqrt{2+3 x}}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{40 \sqrt{1-2 x} \sqrt{2+3 x}}{363 (3+5 x)^{3/2}}+\frac{4}{363} \int \frac{\frac{121}{4}+30 x}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac{2 \sqrt{2+3 x}}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{40 \sqrt{1-2 x} \sqrt{2+3 x}}{363 (3+5 x)^{3/2}}-\frac{490 \sqrt{1-2 x} \sqrt{2+3 x}}{3993 \sqrt{3+5 x}}-\frac{8 \int \frac{\frac{309}{4}+\frac{735 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{3993}\\ &=\frac{2 \sqrt{2+3 x}}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{40 \sqrt{1-2 x} \sqrt{2+3 x}}{363 (3+5 x)^{3/2}}-\frac{490 \sqrt{1-2 x} \sqrt{2+3 x}}{3993 \sqrt{3+5 x}}+\frac{8}{121} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx-\frac{98 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{1331}\\ &=\frac{2 \sqrt{2+3 x}}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{40 \sqrt{1-2 x} \sqrt{2+3 x}}{363 (3+5 x)^{3/2}}-\frac{490 \sqrt{1-2 x} \sqrt{2+3 x}}{3993 \sqrt{3+5 x}}+\frac{98 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{121 \sqrt{33}}-\frac{16 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{121 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.197473, size = 96, normalized size = 0.62 \[ \frac{\sqrt{2} \left (362 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{\sqrt{6 x+4} \left (2450 x^2+685 x-592\right )}{\sqrt{1-2 x} (5 x+3)^{3/2}}-98 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{3993} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.023, size = 219, normalized size = 1.4 \begin{align*} -{\frac{2}{23958\,{x}^{2}+3993\,x-7986}\sqrt{2+3\,x}\sqrt{1-2\,x} \left ( 905\,\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}-245\,\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}+543\,\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 ) -147\,\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 ) +7350\,{x}^{3}+6955\,{x}^{2}-406\,x-1184 \right ) \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \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}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{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}}{500 \, x^{5} + 400 \, x^{4} - 235 \, x^{3} - 207 \, x^{2} + 27 \, x + 27}, 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}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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