Optimal. Leaf size=127 \[ -\frac{8}{49} \sqrt{33} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )+\frac{11 \sqrt{3 x+2} (5 x+3)^{3/2}}{21 (1-2 x)^{3/2}}-\frac{264 \sqrt{3 x+2} \sqrt{5 x+3}}{49 \sqrt{1-2 x}}-\frac{1597}{98} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.0379025, 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 = {98, 150, 158, 113, 119} \[ \frac{11 \sqrt{3 x+2} (5 x+3)^{3/2}}{21 (1-2 x)^{3/2}}-\frac{264 \sqrt{3 x+2} \sqrt{5 x+3}}{49 \sqrt{1-2 x}}-\frac{8}{49} \sqrt{33} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{1597}{98} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 150
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(3+5 x)^{5/2}}{(1-2 x)^{5/2} \sqrt{2+3 x}} \, dx &=\frac{11 \sqrt{2+3 x} (3+5 x)^{3/2}}{21 (1-2 x)^{3/2}}-\frac{1}{21} \int \frac{\sqrt{3+5 x} \left (\frac{447}{2}+345 x\right )}{(1-2 x)^{3/2} \sqrt{2+3 x}} \, dx\\ &=-\frac{264 \sqrt{2+3 x} \sqrt{3+5 x}}{49 \sqrt{1-2 x}}+\frac{11 \sqrt{2+3 x} (3+5 x)^{3/2}}{21 (1-2 x)^{3/2}}-\frac{1}{147} \int \frac{-\frac{15165}{2}-\frac{23955 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{264 \sqrt{2+3 x} \sqrt{3+5 x}}{49 \sqrt{1-2 x}}+\frac{11 \sqrt{2+3 x} (3+5 x)^{3/2}}{21 (1-2 x)^{3/2}}+\frac{132}{49} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx+\frac{1597}{98} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=-\frac{264 \sqrt{2+3 x} \sqrt{3+5 x}}{49 \sqrt{1-2 x}}+\frac{11 \sqrt{2+3 x} (3+5 x)^{3/2}}{21 (1-2 x)^{3/2}}-\frac{1597}{98} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{8}{49} \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 = 0.122262, size = 115, normalized size = 0.91 \[ -\frac{-805 \sqrt{2-4 x} (2 x-1) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+22 \sqrt{3 x+2} \sqrt{5 x+3} (51-179 x)+1597 \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{294 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.021, size = 228, normalized size = 1.8 \begin{align*}{\frac{1}{294\, \left ( 2\,x-1 \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) } \left ( 1610\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-3194\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-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 ) +59070\,{x}^{3}+57992\,{x}^{2}+2310\,x-6732 \right ) \sqrt{2+3\,x}\sqrt{1-2\,x}\sqrt{3+5\,x}} \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 (5 \, x + 3\right )}^{\frac{5}{2}}}{\sqrt{3 \, x + 2}{\left (-2 \, x + 1\right )}^{\frac{5}{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 (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{24 \, x^{4} - 20 \, x^{3} - 6 \, x^{2} + 9 \, x - 2}, 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 (5 \, x + 3\right )}^{\frac{5}{2}}}{\sqrt{3 \, x + 2}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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