Optimal. Leaf size=117 \[ \frac{4 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{\sqrt{33}}-\frac{20 \sqrt{1-2 x} \sqrt{3 x+2}}{\sqrt{5 x+3}}+\frac{2 \sqrt{1-2 x}}{\sqrt{3 x+2} \sqrt{5 x+3}}+4 \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.0381065, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 5, 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} \[ -\frac{20 \sqrt{1-2 x} \sqrt{3 x+2}}{\sqrt{5 x+3}}+\frac{2 \sqrt{1-2 x}}{\sqrt{3 x+2} \sqrt{5 x+3}}+\frac{4 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{\sqrt{33}}+4 \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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{1-2 x}}{(2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx &=\frac{2 \sqrt{1-2 x}}{\sqrt{2+3 x} \sqrt{3+5 x}}-2 \int \frac{-8+5 x}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac{2 \sqrt{1-2 x}}{\sqrt{2+3 x} \sqrt{3+5 x}}-\frac{20 \sqrt{1-2 x} \sqrt{2+3 x}}{\sqrt{3+5 x}}+\frac{4}{11} \int \frac{-\frac{209}{2}-165 x}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{2 \sqrt{1-2 x}}{\sqrt{2+3 x} \sqrt{3+5 x}}-\frac{20 \sqrt{1-2 x} \sqrt{2+3 x}}{\sqrt{3+5 x}}-2 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx-12 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{2 \sqrt{1-2 x}}{\sqrt{2+3 x} \sqrt{3+5 x}}-\frac{20 \sqrt{1-2 x} \sqrt{2+3 x}}{\sqrt{3+5 x}}+4 \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{4 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{\sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.182493, size = 128, normalized size = 1.09 \[ \frac{2 \sqrt{2} \left (15 x^2+19 x+6\right ) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-4 \sqrt{2} \left (15 x^2+19 x+6\right ) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-2 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3} (30 x+19)}{(3 x+2) (5 x+3)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.019, size = 134, normalized size = 1.2 \begin{align*} -2\,{\frac{\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( \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 ) -2\,\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 ) +60\,{x}^{2}+8\,x-19 \right ) }{30\,{x}^{3}+23\,{x}^{2}-7\,x-6}} \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{-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\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}}{225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36}, 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{-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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