Optimal. Leaf size=129 \[ \frac{24}{25} \sqrt{\frac{3}{11}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )+\frac{7 \sqrt{5 x+3} (3 x+2)^{3/2}}{11 \sqrt{1-2 x}}+\frac{69}{55} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}+\frac{1597}{50} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.0406841, 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 = {98, 154, 158, 113, 119} \[ \frac{7 \sqrt{5 x+3} (3 x+2)^{3/2}}{11 \sqrt{1-2 x}}+\frac{69}{55} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}+\frac{24}{25} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{1597}{50} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(2+3 x)^{5/2}}{(1-2 x)^{3/2} \sqrt{3+5 x}} \, dx &=\frac{7 (2+3 x)^{3/2} \sqrt{3+5 x}}{11 \sqrt{1-2 x}}-\frac{1}{11} \int \frac{\sqrt{2+3 x} \left (\frac{255}{2}+207 x\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{69}{55} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{7 (2+3 x)^{3/2} \sqrt{3+5 x}}{11 \sqrt{1-2 x}}+\frac{1}{165} \int \frac{-\frac{9099}{2}-\frac{14373 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{69}{55} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{7 (2+3 x)^{3/2} \sqrt{3+5 x}}{11 \sqrt{1-2 x}}-\frac{36}{25} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx-\frac{4791}{550} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{69}{55} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{7 (2+3 x)^{3/2} \sqrt{3+5 x}}{11 \sqrt{1-2 x}}+\frac{1597}{50} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{24}{25} \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 = 0.0883881, size = 105, normalized size = 0.81 \[ \frac{805 \sqrt{2-4 x} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+10 \sqrt{3 x+2} \sqrt{5 x+3} (139-33 x)-1597 \sqrt{2-4 x} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{550 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.017, size = 140, normalized size = 1.1 \begin{align*} -{\frac{1}{16500\,{x}^{3}+12650\,{x}^{2}-3850\,x-3300}\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 ) -4950\,{x}^{3}+14580\,{x}^{2}+24430\,x+8340 \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{5}{2}}}{\sqrt{5 \, x + 3}{\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{{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3}, 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{5}{2}}}{\sqrt{5 \, x + 3}{\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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