Optimal. Leaf size=160 \[ -\frac{942 \sqrt{\frac{3}{11}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{3125}-\frac{2 \sqrt{1-2 x} (3 x+2)^{5/2}}{55 \sqrt{5 x+3}}-\frac{69 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{1375}-\frac{2577 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{6875}-\frac{61151 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6250} \]
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Rubi [A] time = 0.052713, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {98, 154, 158, 113, 119} \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^{5/2}}{55 \sqrt{5 x+3}}-\frac{69 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{1375}-\frac{2577 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{6875}-\frac{942 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3125}-\frac{61151 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6250} \]
Antiderivative was successfully verified.
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Rule 98
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(2+3 x)^{7/2}}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{5/2}}{55 \sqrt{3+5 x}}-\frac{2}{55} \int \frac{\left (-\frac{81}{2}-\frac{69 x}{2}\right ) (2+3 x)^{3/2}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{5/2}}{55 \sqrt{3+5 x}}-\frac{69 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{1375}+\frac{2 \int \frac{\sqrt{2+3 x} \left (\frac{9825}{4}+\frac{7731 x}{2}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{1375}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{5/2}}{55 \sqrt{3+5 x}}-\frac{2577 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{6875}-\frac{69 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{1375}-\frac{2 \int \frac{-\frac{348867}{4}-\frac{550359 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{20625}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{5/2}}{55 \sqrt{3+5 x}}-\frac{2577 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{6875}-\frac{69 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{1375}+\frac{1413 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{3125}+\frac{183453 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{68750}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{5/2}}{55 \sqrt{3+5 x}}-\frac{2577 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{6875}-\frac{69 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{1375}-\frac{61151 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6250}-\frac{942 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3125}\\ \end{align*}
Mathematica [A] time = 0.209013, size = 122, normalized size = 0.76 \[ \frac{61151 \sqrt{2} (5 x+3) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (6013 \sqrt{2} (5 x+3) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+2 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (7425 x^2+22440 x+10801\right )\right )}{68750 (5 x+3)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.02, size = 145, normalized size = 0.9 \begin{align*}{\frac{1}{2062500\,{x}^{3}+1581250\,{x}^{2}-481250\,x-412500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 30065\,\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 ) -61151\,\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 ) -445500\,{x}^{4}-1420650\,{x}^{3}-723960\,{x}^{2}+340790\,x+216020 \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{{\left (3 \, x + 2\right )}^{\frac{7}{2}}}{{\left (5 \, x + 3\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{{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9}, 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{{\left (3 \, x + 2\right )}^{\frac{7}{2}}}{{\left (5 \, x + 3\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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