Optimal. Leaf size=125 \[ -\frac{62 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{49 \sqrt{33}}-\frac{12 \sqrt{1-2 x} \sqrt{5 x+3}}{49 \sqrt{3 x+2}}+\frac{2 \sqrt{5 x+3}}{7 \sqrt{1-2 x} \sqrt{3 x+2}}+\frac{4}{49} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.0394081, antiderivative size = 125, 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} \[ -\frac{12 \sqrt{1-2 x} \sqrt{5 x+3}}{49 \sqrt{3 x+2}}+\frac{2 \sqrt{5 x+3}}{7 \sqrt{1-2 x} \sqrt{3 x+2}}-\frac{62 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{49 \sqrt{33}}+\frac{4}{49} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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{3+5 x}}{(1-2 x)^{3/2} (2+3 x)^{3/2}} \, dx &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} \sqrt{2+3 x}}-\frac{2}{7} \int \frac{-4-\frac{15 x}{2}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx\\ &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} \sqrt{2+3 x}}-\frac{12 \sqrt{1-2 x} \sqrt{3+5 x}}{49 \sqrt{2+3 x}}-\frac{4}{49} \int \frac{\frac{5}{4}+15 x}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} \sqrt{2+3 x}}-\frac{12 \sqrt{1-2 x} \sqrt{3+5 x}}{49 \sqrt{2+3 x}}-\frac{12}{49} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx+\frac{31}{49} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} \sqrt{2+3 x}}-\frac{12 \sqrt{1-2 x} \sqrt{3+5 x}}{49 \sqrt{2+3 x}}+\frac{4}{49} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{62 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{49 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.112119, size = 122, normalized size = 0.98 \[ \frac{35 \sqrt{2-4 x} (3 x+2) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+2 \sqrt{3 x+2} \sqrt{5 x+3} (12 x+1)-4 \sqrt{2-4 x} (3 x+2) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{49 \sqrt{1-2 x} (3 x+2)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.019, size = 135, normalized size = 1.1 \begin{align*} -{\frac{1}{1470\,{x}^{3}+1127\,{x}^{2}-343\,x-294}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 35\,\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 ) -4\,\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 ) +120\,{x}^{2}+82\,x+6 \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{\sqrt{5 \, x + 3}}{{\left (3 \, x + 2\right )}^{\frac{3}{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}}{36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4}, 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{5 \, x + 3}}{{\left (3 \, x + 2\right )}^{\frac{3}{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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