Optimal. Leaf size=187 \[ -\frac{37768 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{539 \sqrt{33}}+\frac{6277760 \sqrt{1-2 x} \sqrt{3 x+2}}{17787 \sqrt{5 x+3}}-\frac{94420 \sqrt{1-2 x} \sqrt{3 x+2}}{1617 (5 x+3)^{3/2}}+\frac{428 \sqrt{1-2 x}}{49 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{2 \sqrt{1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}-\frac{1255552 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{539 \sqrt{33}} \]
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Rubi [A] time = 0.0655837, 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} \[ \frac{6277760 \sqrt{1-2 x} \sqrt{3 x+2}}{17787 \sqrt{5 x+3}}-\frac{94420 \sqrt{1-2 x} \sqrt{3 x+2}}{1617 (5 x+3)^{3/2}}+\frac{428 \sqrt{1-2 x}}{49 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{2 \sqrt{1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}-\frac{37768 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{539 \sqrt{33}}-\frac{1255552 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{539 \sqrt{33}} \]
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}{\sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx &=\frac{2 \sqrt{1-2 x}}{7 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{2}{21} \int \frac{57-75 x}{\sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=\frac{2 \sqrt{1-2 x}}{7 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{428 \sqrt{1-2 x}}{49 \sqrt{2+3 x} (3+5 x)^{3/2}}+\frac{4}{147} \int \frac{\frac{8385}{2}-4815 x}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}} \, dx\\ &=\frac{2 \sqrt{1-2 x}}{7 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{428 \sqrt{1-2 x}}{49 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{94420 \sqrt{1-2 x} \sqrt{2+3 x}}{1617 (3+5 x)^{3/2}}-\frac{8 \int \frac{\frac{343365}{2}-\frac{212445 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx}{4851}\\ &=\frac{2 \sqrt{1-2 x}}{7 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{428 \sqrt{1-2 x}}{49 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{94420 \sqrt{1-2 x} \sqrt{2+3 x}}{1617 (3+5 x)^{3/2}}+\frac{6277760 \sqrt{1-2 x} \sqrt{2+3 x}}{17787 \sqrt{3+5 x}}+\frac{16 \int \frac{\frac{8942355}{4}+3531240 x}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{53361}\\ &=\frac{2 \sqrt{1-2 x}}{7 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{428 \sqrt{1-2 x}}{49 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{94420 \sqrt{1-2 x} \sqrt{2+3 x}}{1617 (3+5 x)^{3/2}}+\frac{6277760 \sqrt{1-2 x} \sqrt{2+3 x}}{17787 \sqrt{3+5 x}}+\frac{18884}{539} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx+\frac{1255552 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{5929}\\ &=\frac{2 \sqrt{1-2 x}}{7 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{428 \sqrt{1-2 x}}{49 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{94420 \sqrt{1-2 x} \sqrt{2+3 x}}{1617 (3+5 x)^{3/2}}+\frac{6277760 \sqrt{1-2 x} \sqrt{2+3 x}}{17787 \sqrt{3+5 x}}-\frac{1255552 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{539 \sqrt{33}}-\frac{37768 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{539 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.144957, size = 104, normalized size = 0.56 \[ \frac{2 \left (2 \sqrt{2} \left (313888 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-158095 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{\sqrt{1-2 x} \left (141249600 x^3+268408770 x^2+169778606 x+35747225\right )}{(3 x+2)^{3/2} (5 x+3)^{3/2}}\right )}{17787} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.024, size = 311, normalized size = 1.7 \begin{align*} -{\frac{2}{35574\,x-17787}\sqrt{1-2\,x} \left ( 9416640\,\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}-4742850\,\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}+11927744\,\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}-6007610\,\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}+3766656\,\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 ) -1897140\,\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 ) -282499200\,{x}^{4}-395567940\,{x}^{3}-71148442\,{x}^{2}+98284156\,x+35747225 \right ) \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}} \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{1}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{5}{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}}{6750 \, x^{7} + 22275 \, x^{6} + 27765 \, x^{5} + 13943 \, x^{4} - 883 \, x^{3} - 4014 \, x^{2} - 1620 \, x - 216}, 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}}{\left (3 \, x + 2\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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