Optimal. Leaf size=222 \[ -\frac{673072 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{6806835}-\frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}+\frac{22738708 \sqrt{1-2 x} \sqrt{5 x+3}}{6806835 \sqrt{3 x+2}}+\frac{332372 \sqrt{1-2 x} \sqrt{5 x+3}}{972405 (3 x+2)^{3/2}}+\frac{8842 \sqrt{1-2 x} \sqrt{5 x+3}}{138915 (3 x+2)^{5/2}}-\frac{214 \sqrt{1-2 x} \sqrt{5 x+3}}{3969 (3 x+2)^{7/2}}-\frac{22738708 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6806835} \]
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Rubi [A] time = 0.0792006, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ -\frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}+\frac{22738708 \sqrt{1-2 x} \sqrt{5 x+3}}{6806835 \sqrt{3 x+2}}+\frac{332372 \sqrt{1-2 x} \sqrt{5 x+3}}{972405 (3 x+2)^{3/2}}+\frac{8842 \sqrt{1-2 x} \sqrt{5 x+3}}{138915 (3 x+2)^{5/2}}-\frac{214 \sqrt{1-2 x} \sqrt{5 x+3}}{3969 (3 x+2)^{7/2}}-\frac{673072 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6806835}-\frac{22738708 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6806835} \]
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)^{11/2}} \, dx &=-\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac{2}{27} \int \frac{\left (\frac{9}{2}-20 x\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^{9/2}} \, dx\\ &=-\frac{214 \sqrt{1-2 x} \sqrt{3+5 x}}{3969 (2+3 x)^{7/2}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac{4 \int \frac{-\frac{1493}{4}-\frac{2225 x}{2}}{\sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}} \, dx}{3969}\\ &=-\frac{214 \sqrt{1-2 x} \sqrt{3+5 x}}{3969 (2+3 x)^{7/2}}+\frac{8842 \sqrt{1-2 x} \sqrt{3+5 x}}{138915 (2+3 x)^{5/2}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac{8 \int \frac{\frac{38883}{4}-\frac{66315 x}{4}}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx}{138915}\\ &=-\frac{214 \sqrt{1-2 x} \sqrt{3+5 x}}{3969 (2+3 x)^{7/2}}+\frac{8842 \sqrt{1-2 x} \sqrt{3+5 x}}{138915 (2+3 x)^{5/2}}+\frac{332372 \sqrt{1-2 x} \sqrt{3+5 x}}{972405 (2+3 x)^{3/2}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac{16 \int \frac{\frac{4022817}{8}-\frac{1246395 x}{4}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{2917215}\\ &=-\frac{214 \sqrt{1-2 x} \sqrt{3+5 x}}{3969 (2+3 x)^{7/2}}+\frac{8842 \sqrt{1-2 x} \sqrt{3+5 x}}{138915 (2+3 x)^{5/2}}+\frac{332372 \sqrt{1-2 x} \sqrt{3+5 x}}{972405 (2+3 x)^{3/2}}+\frac{22738708 \sqrt{1-2 x} \sqrt{3+5 x}}{6806835 \sqrt{2+3 x}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac{32 \int \frac{\frac{53938515}{8}+\frac{85270155 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{20420505}\\ &=-\frac{214 \sqrt{1-2 x} \sqrt{3+5 x}}{3969 (2+3 x)^{7/2}}+\frac{8842 \sqrt{1-2 x} \sqrt{3+5 x}}{138915 (2+3 x)^{5/2}}+\frac{332372 \sqrt{1-2 x} \sqrt{3+5 x}}{972405 (2+3 x)^{3/2}}+\frac{22738708 \sqrt{1-2 x} \sqrt{3+5 x}}{6806835 \sqrt{2+3 x}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac{3701896 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{6806835}+\frac{22738708 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{6806835}\\ &=-\frac{214 \sqrt{1-2 x} \sqrt{3+5 x}}{3969 (2+3 x)^{7/2}}+\frac{8842 \sqrt{1-2 x} \sqrt{3+5 x}}{138915 (2+3 x)^{5/2}}+\frac{332372 \sqrt{1-2 x} \sqrt{3+5 x}}{972405 (2+3 x)^{3/2}}+\frac{22738708 \sqrt{1-2 x} \sqrt{3+5 x}}{6806835 \sqrt{2+3 x}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}-\frac{22738708 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6806835}-\frac{673072 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6806835}\\ \end{align*}
Mathematica [A] time = 0.278058, size = 107, normalized size = 0.48 \[ \frac{-93064160 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{24 \sqrt{2-4 x} \sqrt{5 x+3} \left (920917674 x^4+2487189618 x^3+2520548433 x^2+1134125364 x+190959271\right )}{(3 x+2)^{9/2}}+181909664 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{81682020 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.035, size = 504, normalized size = 2.3 \begin{align*}{\frac{2}{204205050\,{x}^{2}+20420505\,x-61261515} \left ( 471137310\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{4}\sqrt{2+3\,x}\sqrt{1-2\,x}\sqrt{3+5\,x}-920917674\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{4}\sqrt{2+3\,x}\sqrt{1-2\,x}\sqrt{3+5\,x}+1256366160\,\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}-2455780464\,\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}+1256366160\,\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}-2455780464\,\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}+558384960\,\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}-1091457984\,\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}+27627530220\,{x}^{6}+93064160\,\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 ) -181909664\,\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 ) +77378441562\,{x}^{5}+74789762778\,{x}^{4}+19200699657\,{x}^{3}-13553781675\,{x}^{2}-9634250463\,x-1718633439 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2+3\,x \right ) ^{-{\frac{9}{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{11}{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}}{729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64}, 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{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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