Optimal. Leaf size=129 \[ -\frac{4}{77} \sqrt{\frac{3}{11}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )-\frac{370 \sqrt{1-2 x} \sqrt{3 x+2}}{847 \sqrt{5 x+3}}+\frac{4 \sqrt{3 x+2}}{77 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{74}{77} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.0383129, antiderivative size = 129, 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 = {104, 152, 158, 113, 119} \[ -\frac{370 \sqrt{1-2 x} \sqrt{3 x+2}}{847 \sqrt{5 x+3}}+\frac{4 \sqrt{3 x+2}}{77 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{4}{77} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{74}{77} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 104
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{1}{(1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx &=\frac{4 \sqrt{2+3 x}}{77 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{2}{77} \int \frac{-\frac{55}{2}-15 x}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac{4 \sqrt{2+3 x}}{77 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{370 \sqrt{1-2 x} \sqrt{2+3 x}}{847 \sqrt{3+5 x}}+\frac{4}{847} \int \frac{-150-\frac{555 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{4 \sqrt{2+3 x}}{77 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{370 \sqrt{1-2 x} \sqrt{2+3 x}}{847 \sqrt{3+5 x}}+\frac{6}{77} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx-\frac{222}{847} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{4 \sqrt{2+3 x}}{77 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{370 \sqrt{1-2 x} \sqrt{2+3 x}}{847 \sqrt{3+5 x}}+\frac{74}{77} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{4}{77} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )\\ \end{align*}
Mathematica [A] time = 0.125425, size = 122, normalized size = 0.95 \[ \frac{140 \sqrt{2-4 x} (5 x+3) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+2 \sqrt{3 x+2} \sqrt{5 x+3} (370 x-163)-74 \sqrt{2-4 x} (5 x+3) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{847 \sqrt{1-2 x} (5 x+3)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.02, size = 135, normalized size = 1.1 \begin{align*} -{\frac{2}{25410\,{x}^{3}+19481\,{x}^{2}-5929\,x-5082}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 70\,\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 ) -37\,\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 ) +1110\,{x}^{2}+251\,x-326 \right ) } \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}} \sqrt{3 \, x + 2}{\left (-2 \, x + 1\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}}{300 \, x^{5} + 260 \, x^{4} - 137 \, x^{3} - 136 \, x^{2} + 15 \, x + 18}, 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}} \sqrt{3 \, x + 2}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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