Optimal. Leaf size=160 \[ -\frac{34154 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{46305}+\frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{105 (3 x+2)^{5/2}}-\frac{53194 \sqrt{1-2 x} \sqrt{5 x+3}}{46305 \sqrt{3 x+2}}+\frac{544 \sqrt{1-2 x} \sqrt{5 x+3}}{6615 (3 x+2)^{3/2}}+\frac{53194 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{46305} \]
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Rubi [A] time = 0.0530126, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {98, 150, 152, 158, 113, 119} \[ \frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{105 (3 x+2)^{5/2}}-\frac{53194 \sqrt{1-2 x} \sqrt{5 x+3}}{46305 \sqrt{3 x+2}}+\frac{544 \sqrt{1-2 x} \sqrt{5 x+3}}{6615 (3 x+2)^{3/2}}-\frac{34154 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{46305}+\frac{53194 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{46305} \]
Antiderivative was successfully verified.
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Rule 98
Rule 150
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x} (2+3 x)^{7/2}} \, dx &=\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^{5/2}}-\frac{2}{105} \int \frac{\left (-243-\frac{865 x}{2}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^{5/2}} \, dx\\ &=\frac{544 \sqrt{1-2 x} \sqrt{3+5 x}}{6615 (2+3 x)^{3/2}}+\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^{5/2}}-\frac{4 \int \frac{-\frac{49871}{4}-\frac{88105 x}{4}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{6615}\\ &=\frac{544 \sqrt{1-2 x} \sqrt{3+5 x}}{6615 (2+3 x)^{3/2}}-\frac{53194 \sqrt{1-2 x} \sqrt{3+5 x}}{46305 \sqrt{2+3 x}}+\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^{5/2}}-\frac{8 \int \frac{-\frac{28265}{8}+\frac{132985 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{46305}\\ &=\frac{544 \sqrt{1-2 x} \sqrt{3+5 x}}{6615 (2+3 x)^{3/2}}-\frac{53194 \sqrt{1-2 x} \sqrt{3+5 x}}{46305 \sqrt{2+3 x}}+\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^{5/2}}-\frac{53194 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{46305}+\frac{187847 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{46305}\\ &=\frac{544 \sqrt{1-2 x} \sqrt{3+5 x}}{6615 (2+3 x)^{3/2}}-\frac{53194 \sqrt{1-2 x} \sqrt{3+5 x}}{46305 \sqrt{2+3 x}}+\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^{5/2}}+\frac{53194 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{46305}-\frac{34154 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{46305}\\ \end{align*}
Mathematica [A] time = 0.141863, size = 99, normalized size = 0.62 \[ \frac{\sqrt{2} \left (616735 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-53194 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )-\frac{6 \sqrt{1-2 x} \sqrt{5 x+3} \left (239373 x^2+311247 x+101257\right )}{(3 x+2)^{5/2}}}{138915} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.023, size = 314, normalized size = 2. \begin{align*} -{\frac{1}{1389150\,{x}^{2}+138915\,x-416745} \left ( 5550615\,\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}-478746\,\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}+7400820\,\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}-638328\,\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}+2466940\,\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 ) -212776\,\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 ) +14362380\,{x}^{4}+20111058\,{x}^{3}+3634188\,{x}^{2}-4994904\,x-1822626 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2+3\,x \right ) ^{-{\frac{5}{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}}}{{\left (3 \, x + 2\right )}^{\frac{7}{2}} \sqrt{-2 \, x + 1}}\,{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}}{162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16}, 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}}}{{\left (3 \, x + 2\right )}^{\frac{7}{2}} \sqrt{-2 \, x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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