Optimal. Leaf size=187 \[ -\frac{7388 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{65219 \sqrt{33}}+\frac{598660 \sqrt{1-2 x} \sqrt{3 x+2}}{2152227 \sqrt{5 x+3}}-\frac{18470 \sqrt{1-2 x} \sqrt{3 x+2}}{195657 (5 x+3)^{3/2}}+\frac{368 \sqrt{3 x+2}}{5929 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{4 \sqrt{3 x+2}}{231 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{119732 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{65219 \sqrt{33}} \]
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Rubi [A] time = 0.067809, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {104, 152, 158, 113, 119} \[ \frac{598660 \sqrt{1-2 x} \sqrt{3 x+2}}{2152227 \sqrt{5 x+3}}-\frac{18470 \sqrt{1-2 x} \sqrt{3 x+2}}{195657 (5 x+3)^{3/2}}+\frac{368 \sqrt{3 x+2}}{5929 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{4 \sqrt{3 x+2}}{231 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{7388 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{65219 \sqrt{33}}-\frac{119732 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{65219 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 104
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{1}{(1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{5/2}} \, dx &=\frac{4 \sqrt{2+3 x}}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{2}{231} \int \frac{-\frac{201}{2}-75 x}{(1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}} \, dx\\ &=\frac{4 \sqrt{2+3 x}}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{368 \sqrt{2+3 x}}{5929 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{4 \int \frac{\frac{20445}{4}+6210 x}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}} \, dx}{17787}\\ &=\frac{4 \sqrt{2+3 x}}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{368 \sqrt{2+3 x}}{5929 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{18470 \sqrt{1-2 x} \sqrt{2+3 x}}{195657 (3+5 x)^{3/2}}-\frac{8 \int \frac{\frac{19965}{2}-\frac{83115 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx}{586971}\\ &=\frac{4 \sqrt{2+3 x}}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{368 \sqrt{2+3 x}}{5929 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{18470 \sqrt{1-2 x} \sqrt{2+3 x}}{195657 (3+5 x)^{3/2}}+\frac{598660 \sqrt{1-2 x} \sqrt{2+3 x}}{2152227 \sqrt{3+5 x}}+\frac{16 \int \frac{\frac{1799235}{8}+\frac{1346985 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{6456681}\\ &=\frac{4 \sqrt{2+3 x}}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{368 \sqrt{2+3 x}}{5929 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{18470 \sqrt{1-2 x} \sqrt{2+3 x}}{195657 (3+5 x)^{3/2}}+\frac{598660 \sqrt{1-2 x} \sqrt{2+3 x}}{2152227 \sqrt{3+5 x}}+\frac{3694 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{65219}+\frac{119732 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{717409}\\ &=\frac{4 \sqrt{2+3 x}}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{368 \sqrt{2+3 x}}{5929 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{18470 \sqrt{1-2 x} \sqrt{2+3 x}}{195657 (3+5 x)^{3/2}}+\frac{598660 \sqrt{1-2 x} \sqrt{2+3 x}}{2152227 \sqrt{3+5 x}}-\frac{119732 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{65219 \sqrt{33}}-\frac{7388 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{65219 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.15124, size = 103, normalized size = 0.55 \[ \frac{2 \left (\sqrt{2} \left (1085 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+59866 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )+\frac{\sqrt{3 x+2} \left (5986600 x^3-2800980 x^2-1822554 x+881831\right )}{(1-2 x)^{3/2} (5 x+3)^{3/2}}\right )}{2152227} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.025, size = 311, normalized size = 1.7 \begin{align*} -{\frac{2}{2152227\, \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}-3570260\,{x}^{3}+11069622\,{x}^{2}+999615\,x-1763662 \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{1}{{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{3 \, x + 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}}{3000 \, x^{7} + 2900 \, x^{6} - 2010 \, x^{5} - 2277 \, x^{4} + 425 \, x^{3} + 603 \, x^{2} - 27 \, x - 54}, 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{1}{{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{3 \, x + 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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