Optimal. Leaf size=187 \[ -\frac{1847 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{33275 \sqrt{33}}+\frac{7 (3 x+2)^{5/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{63 (3 x+2)^{3/2}}{121 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{29933 \sqrt{1-2 x} \sqrt{3 x+2}}{219615 \sqrt{5 x+3}}+\frac{908 \sqrt{1-2 x} \sqrt{3 x+2}}{19965 (5 x+3)^{3/2}}-\frac{29933 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{33275 \sqrt{33}} \]
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Rubi [A] time = 0.06667, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {98, 150, 152, 158, 113, 119} \[ \frac{7 (3 x+2)^{5/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{63 (3 x+2)^{3/2}}{121 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{29933 \sqrt{1-2 x} \sqrt{3 x+2}}{219615 \sqrt{5 x+3}}+\frac{908 \sqrt{1-2 x} \sqrt{3 x+2}}{19965 (5 x+3)^{3/2}}-\frac{1847 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{33275 \sqrt{33}}-\frac{29933 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{33275 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 150
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(2+3 x)^{7/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx &=\frac{7 (2+3 x)^{5/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{1}{33} \int \frac{(2+3 x)^{3/2} \left (\frac{93}{2}+96 x\right )}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=-\frac{63 (2+3 x)^{3/2}}{121 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{5/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{1}{363} \int \frac{\left (\frac{345}{2}-\frac{333 x}{2}\right ) \sqrt{2+3 x}}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx\\ &=\frac{908 \sqrt{1-2 x} \sqrt{2+3 x}}{19965 (3+5 x)^{3/2}}-\frac{63 (2+3 x)^{3/2}}{121 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{5/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{2 \int \frac{\frac{3993}{2}-\frac{16623 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx}{59895}\\ &=\frac{908 \sqrt{1-2 x} \sqrt{2+3 x}}{19965 (3+5 x)^{3/2}}-\frac{63 (2+3 x)^{3/2}}{121 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{5/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{29933 \sqrt{1-2 x} \sqrt{2+3 x}}{219615 \sqrt{3+5 x}}+\frac{4 \int \frac{\frac{359847}{8}+\frac{269397 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{658845}\\ &=\frac{908 \sqrt{1-2 x} \sqrt{2+3 x}}{19965 (3+5 x)^{3/2}}-\frac{63 (2+3 x)^{3/2}}{121 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{5/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{29933 \sqrt{1-2 x} \sqrt{2+3 x}}{219615 \sqrt{3+5 x}}+\frac{1847 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{66550}+\frac{29933 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{366025}\\ &=\frac{908 \sqrt{1-2 x} \sqrt{2+3 x}}{19965 (3+5 x)^{3/2}}-\frac{63 (2+3 x)^{3/2}}{121 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{5/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{29933 \sqrt{1-2 x} \sqrt{2+3 x}}{219615 \sqrt{3+5 x}}-\frac{29933 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{33275 \sqrt{33}}-\frac{1847 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{33275 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.178998, size = 107, normalized size = 0.57 \[ \frac{1085 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{10 \sqrt{3 x+2} \left (598660 x^3+905823 x^2+423882 x+57437\right )}{(1-2 x)^{3/2} (5 x+3)^{3/2}}+59866 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{2196150} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.023, size = 311, normalized size = 1.7 \begin{align*} -{\frac{1}{2196150\, \left ( 2\,x-1 \right ) ^{2}}\sqrt{1-2\,x} \left ( 10850\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+598660\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+1085\,\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}+59866\,\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}-3255\,\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 ) -179598\,\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 ) -17959800\,{x}^{4}-39147890\,{x}^{3}-30832920\,{x}^{2}-10200750\,x-1148740 \right ) \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\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{{\left (3 \, x + 2\right )}^{\frac{7}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{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{{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{1000 \, x^{6} + 300 \, x^{5} - 870 \, x^{4} - 179 \, x^{3} + 261 \, 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{{\left (3 \, x + 2\right )}^{\frac{7}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{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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