Optimal. Leaf size=191 \[ \frac{23612 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{108045}-\frac{2 \sqrt{1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}-\frac{118 \sqrt{1-2 x} (5 x+3)^{3/2}}{735 (3 x+2)^{5/2}}+\frac{173482 \sqrt{1-2 x} \sqrt{5 x+3}}{108045 \sqrt{3 x+2}}-\frac{4282 \sqrt{1-2 x} \sqrt{5 x+3}}{15435 (3 x+2)^{3/2}}-\frac{173482 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{108045} \]
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Rubi [A] time = 0.0649354, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {97, 150, 152, 158, 113, 119} \[ -\frac{2 \sqrt{1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}-\frac{118 \sqrt{1-2 x} (5 x+3)^{3/2}}{735 (3 x+2)^{5/2}}+\frac{173482 \sqrt{1-2 x} \sqrt{5 x+3}}{108045 \sqrt{3 x+2}}-\frac{4282 \sqrt{1-2 x} \sqrt{5 x+3}}{15435 (3 x+2)^{3/2}}+\frac{23612 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{108045}-\frac{173482 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{108045} \]
Antiderivative was successfully verified.
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Rule 97
Rule 150
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{9/2}} \, dx &=-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac{2}{21} \int \frac{\left (\frac{19}{2}-30 x\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^{7/2}} \, dx\\ &=-\frac{118 \sqrt{1-2 x} (3+5 x)^{3/2}}{735 (2+3 x)^{5/2}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac{4 \int \frac{\left (\frac{603}{4}-\frac{4365 x}{2}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^{5/2}} \, dx}{2205}\\ &=-\frac{4282 \sqrt{1-2 x} \sqrt{3+5 x}}{15435 (2+3 x)^{3/2}}-\frac{118 \sqrt{1-2 x} (3+5 x)^{3/2}}{735 (2+3 x)^{5/2}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac{8 \int \frac{-\frac{222417}{8}-90495 x}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{138915}\\ &=-\frac{4282 \sqrt{1-2 x} \sqrt{3+5 x}}{15435 (2+3 x)^{3/2}}+\frac{173482 \sqrt{1-2 x} \sqrt{3+5 x}}{108045 \sqrt{2+3 x}}-\frac{118 \sqrt{1-2 x} (3+5 x)^{3/2}}{735 (2+3 x)^{5/2}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac{16 \int \frac{\frac{878805}{4}+\frac{3903345 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{972405}\\ &=-\frac{4282 \sqrt{1-2 x} \sqrt{3+5 x}}{15435 (2+3 x)^{3/2}}+\frac{173482 \sqrt{1-2 x} \sqrt{3+5 x}}{108045 \sqrt{2+3 x}}-\frac{118 \sqrt{1-2 x} (3+5 x)^{3/2}}{735 (2+3 x)^{5/2}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}-\frac{129866 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{108045}+\frac{173482 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{108045}\\ &=-\frac{4282 \sqrt{1-2 x} \sqrt{3+5 x}}{15435 (2+3 x)^{3/2}}+\frac{173482 \sqrt{1-2 x} \sqrt{3+5 x}}{108045 \sqrt{2+3 x}}-\frac{118 \sqrt{1-2 x} (3+5 x)^{3/2}}{735 (2+3 x)^{5/2}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}-\frac{173482 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{108045}+\frac{23612 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{108045}\\ \end{align*}
Mathematica [A] time = 0.250173, size = 104, normalized size = 0.54 \[ \frac{2 \left (\sqrt{2} \left (86741 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-281540 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (2342007 x^3+4290411 x^2+2623695 x+535637\right )}{(3 x+2)^{7/2}}\right )}{324135} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.02, size = 409, normalized size = 2.1 \begin{align*}{\frac{2}{3241350\,{x}^{2}+324135\,x-972405} \left ( 7601580\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{3}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-2342007\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{3}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+15203160\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-4684014\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+10135440\,\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}-3122676\,\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}+2252320\,\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 ) -693928\,\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 ) +70260210\,{x}^{5}+135738351\,{x}^{4}+70504020\,{x}^{3}-14673504\,{x}^{2}-22006344\,x-4820733 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2+3\,x \right ) ^{-{\frac{7}{2}}}} \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{-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac{9}{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}}{243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32}, 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{-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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