Optimal. Leaf size=127 \[ -\frac{8}{125} \sqrt{33} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )-\frac{1}{5} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}-\frac{23}{25} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}-\frac{1597}{250} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.0378338, antiderivative size = 127, 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 = {101, 154, 158, 113, 119} \[ -\frac{1}{5} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}-\frac{23}{25} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}-\frac{8}{125} \sqrt{33} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{1597}{250} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 101
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(2+3 x)^{3/2} \sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx &=-\frac{1}{5} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}+\frac{1}{5} \int \frac{\sqrt{2+3 x} \left (\frac{85}{2}+69 x\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{23}{25} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{1}{5} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}-\frac{1}{75} \int \frac{-\frac{3033}{2}-\frac{4791 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{23}{25} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{1}{5} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}+\frac{132}{125} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx+\frac{1597}{250} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=-\frac{23}{25} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{1}{5} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}-\frac{1597}{250} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{8}{125} \sqrt{33} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )\\ \end{align*}
Mathematica [A] time = 0.174669, size = 92, normalized size = 0.72 \[ \frac{-805 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-45 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} (5 x+11)+1597 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{375 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.012, size = 145, normalized size = 1.1 \begin{align*}{\frac{1}{22500\,{x}^{3}+17250\,{x}^{2}-5250\,x-4500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 805\,\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 ) -1597\,\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 ) -13500\,{x}^{4}-40050\,{x}^{3}-19620\,{x}^{2}+9630\,x+5940 \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}}}{\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}{\left (3 \, x + 2\right )}^{\frac{3}{2}} \sqrt{-2 \, x + 1}}{2 \, x - 1}, 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}}}{\sqrt{-2 \, x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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