Optimal. Leaf size=191 \[ -\frac{10628 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{324135}-\frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{21 (3 x+2)^{7/2}}+\frac{475592 \sqrt{1-2 x} \sqrt{5 x+3}}{324135 \sqrt{3 x+2}}+\frac{8578 \sqrt{1-2 x} \sqrt{5 x+3}}{46305 (3 x+2)^{3/2}}-\frac{214 \sqrt{1-2 x} \sqrt{5 x+3}}{2205 (3 x+2)^{5/2}}-\frac{475592 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{324135} \]
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Rubi [A] time = 0.0674905, 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} \[ -\frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{21 (3 x+2)^{7/2}}+\frac{475592 \sqrt{1-2 x} \sqrt{5 x+3}}{324135 \sqrt{3 x+2}}+\frac{8578 \sqrt{1-2 x} \sqrt{5 x+3}}{46305 (3 x+2)^{3/2}}-\frac{214 \sqrt{1-2 x} \sqrt{5 x+3}}{2205 (3 x+2)^{5/2}}-\frac{10628 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{324135}-\frac{475592 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{324135} \]
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)^{3/2}}{(2+3 x)^{9/2}} \, dx &=-\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac{2}{21} \int \frac{\left (\frac{9}{2}-20 x\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^{7/2}} \, dx\\ &=-\frac{214 \sqrt{1-2 x} \sqrt{3+5 x}}{2205 (2+3 x)^{5/2}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac{4 \int \frac{-\frac{1097}{4}-\frac{1895 x}{2}}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx}{2205}\\ &=-\frac{214 \sqrt{1-2 x} \sqrt{3+5 x}}{2205 (2+3 x)^{5/2}}+\frac{8578 \sqrt{1-2 x} \sqrt{3+5 x}}{46305 (2+3 x)^{3/2}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac{8 \int \frac{6334-\frac{21445 x}{4}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{46305}\\ &=-\frac{214 \sqrt{1-2 x} \sqrt{3+5 x}}{2205 (2+3 x)^{5/2}}+\frac{8578 \sqrt{1-2 x} \sqrt{3+5 x}}{46305 (2+3 x)^{3/2}}+\frac{475592 \sqrt{1-2 x} \sqrt{3+5 x}}{324135 \sqrt{2+3 x}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac{16 \int \frac{\frac{742615}{8}+\frac{297245 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{324135}\\ &=-\frac{214 \sqrt{1-2 x} \sqrt{3+5 x}}{2205 (2+3 x)^{5/2}}+\frac{8578 \sqrt{1-2 x} \sqrt{3+5 x}}{46305 (2+3 x)^{3/2}}+\frac{475592 \sqrt{1-2 x} \sqrt{3+5 x}}{324135 \sqrt{2+3 x}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac{58454 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{324135}+\frac{475592 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{324135}\\ &=-\frac{214 \sqrt{1-2 x} \sqrt{3+5 x}}{2205 (2+3 x)^{5/2}}+\frac{8578 \sqrt{1-2 x} \sqrt{3+5 x}}{46305 (2+3 x)^{3/2}}+\frac{475592 \sqrt{1-2 x} \sqrt{3+5 x}}{324135 \sqrt{2+3 x}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}-\frac{475592 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{324135}-\frac{10628 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{324135}\\ \end{align*}
Mathematica [A] time = 0.244907, size = 104, normalized size = 0.54 \[ \frac{2 \left (\sqrt{2} \left (237796 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-150115 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (6420492 x^3+13111191 x^2+8796570 x+1944697\right )}{(3 x+2)^{7/2}}\right )}{972405} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.02, size = 409, normalized size = 2.1 \begin{align*}{\frac{2}{9724050\,{x}^{2}+972405\,x-2917215} \left ( 4053105\,\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}-6420492\,\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}+8106210\,\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}-12840984\,\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}+5404140\,\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}-8560656\,\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}+1200920\,\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 ) -1902368\,\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 ) +192614760\,{x}^{5}+412597206\,{x}^{4}+245446245\,{x}^{3}-33270099\,{x}^{2}-73335039\,x-17502273 \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{3}{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 (5 \, x + 3\right )}^{\frac{3}{2}} \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{3}{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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