Optimal. Leaf size=191 \[ \frac{40456 \sqrt{\frac{3}{11}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{1715}-\frac{1344984 \sqrt{1-2 x} \sqrt{3 x+2}}{3773 \sqrt{5 x+3}}+\frac{60684 \sqrt{1-2 x}}{1715 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{436 \sqrt{1-2 x}}{245 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{6 \sqrt{1-2 x}}{35 (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{1344984 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1715} \]
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Rubi [A] time = 0.0689001, antiderivative size = 191, 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} \[ -\frac{1344984 \sqrt{1-2 x} \sqrt{3 x+2}}{3773 \sqrt{5 x+3}}+\frac{60684 \sqrt{1-2 x}}{1715 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{436 \sqrt{1-2 x}}{245 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{6 \sqrt{1-2 x}}{35 (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{40456 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1715}+\frac{1344984 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1715} \]
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}{\sqrt{1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx &=\frac{6 \sqrt{1-2 x}}{35 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{2}{35} \int \frac{59-75 x}{\sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx\\ &=\frac{6 \sqrt{1-2 x}}{35 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{436 \sqrt{1-2 x}}{245 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{4}{735} \int \frac{\frac{8631}{2}-4905 x}{\sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=\frac{6 \sqrt{1-2 x}}{35 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{436 \sqrt{1-2 x}}{245 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{60684 \sqrt{1-2 x}}{1715 \sqrt{2+3 x} \sqrt{3+5 x}}+\frac{8 \int \frac{183915-\frac{227565 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx}{5145}\\ &=\frac{6 \sqrt{1-2 x}}{35 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{436 \sqrt{1-2 x}}{245 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{60684 \sqrt{1-2 x}}{1715 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{1344984 \sqrt{1-2 x} \sqrt{2+3 x}}{3773 \sqrt{3+5 x}}-\frac{16 \int \frac{\frac{9579285}{4}+\frac{7565535 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{56595}\\ &=\frac{6 \sqrt{1-2 x}}{35 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{436 \sqrt{1-2 x}}{245 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{60684 \sqrt{1-2 x}}{1715 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{1344984 \sqrt{1-2 x} \sqrt{2+3 x}}{3773 \sqrt{3+5 x}}-\frac{60684 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1715}-\frac{4034952 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{18865}\\ &=\frac{6 \sqrt{1-2 x}}{35 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{436 \sqrt{1-2 x}}{245 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{60684 \sqrt{1-2 x}}{1715 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{1344984 \sqrt{1-2 x} \sqrt{2+3 x}}{3773 \sqrt{3+5 x}}+\frac{1344984 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1715}+\frac{40456 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1715}\\ \end{align*}
Mathematica [A] time = 0.21588, size = 105, normalized size = 0.55 \[ \frac{2 \left (-6 \sqrt{2} \left (112082 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-56455 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )-\frac{\sqrt{1-2 x} \left (90786420 x^3+178568982 x^2+116993058 x+25529443\right )}{(3 x+2)^{5/2} \sqrt{5 x+3}}\right )}{18865} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.028, size = 314, normalized size = 1.6 \begin{align*} -{\frac{2}{188650\,{x}^{2}+18865\,x-56595}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 3048570\,\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}-6052428\,\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}+4064760\,\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}-8069904\,\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}+1354920\,\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 ) -2689968\,\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 ) +181572840\,{x}^{4}+266351544\,{x}^{3}+55417134\,{x}^{2}-65934172\,x-25529443 \right ) \left ( 2+3\,x \right ) ^{-{\frac{5}{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{1}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{2}} \sqrt{-2 \, x + 1}}\,{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}}{4050 \, x^{7} + 13635 \, x^{6} + 17388 \, x^{5} + 9039 \, x^{4} - 376 \, x^{3} - 2536 \, x^{2} - 1056 \, x - 144}, 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{3}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{2}} \sqrt{-2 \, x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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