Optimal. Leaf size=189 \[ -\frac{523 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{15625}-\frac{2 \sqrt{1-2 x} (3 x+2)^{7/2}}{15 (5 x+3)^{3/2}}-\frac{458 \sqrt{1-2 x} (3 x+2)^{5/2}}{825 \sqrt{5 x+3}}+\frac{2818 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{6875}+\frac{2719 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{34375}-\frac{47342 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15625 \sqrt{33}} \]
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Rubi [A] time = 0.0649342, antiderivative size = 189, 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, 154, 158, 113, 119} \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^{7/2}}{15 (5 x+3)^{3/2}}-\frac{458 \sqrt{1-2 x} (3 x+2)^{5/2}}{825 \sqrt{5 x+3}}+\frac{2818 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{6875}+\frac{2719 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{34375}-\frac{523 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15625}-\frac{47342 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15625 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 97
Rule 150
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{\sqrt{1-2 x} (2+3 x)^{7/2}}{(3+5 x)^{5/2}} \, dx &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}+\frac{2}{15} \int \frac{\left (\frac{17}{2}-24 x\right ) (2+3 x)^{5/2}}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac{458 \sqrt{1-2 x} (2+3 x)^{5/2}}{825 \sqrt{3+5 x}}+\frac{4}{825} \int \frac{\left (\frac{2379}{4}-\frac{4227 x}{2}\right ) (2+3 x)^{3/2}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac{458 \sqrt{1-2 x} (2+3 x)^{5/2}}{825 \sqrt{3+5 x}}+\frac{2818 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{6875}-\frac{4 \int \frac{\sqrt{2+3 x} \left (-\frac{13275}{4}+\frac{24471 x}{4}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{20625}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac{458 \sqrt{1-2 x} (2+3 x)^{5/2}}{825 \sqrt{3+5 x}}+\frac{2719 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{34375}+\frac{2818 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{6875}+\frac{4 \int \frac{\frac{625203}{8}+\frac{213039 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{309375}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac{458 \sqrt{1-2 x} (2+3 x)^{5/2}}{825 \sqrt{3+5 x}}+\frac{2719 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{34375}+\frac{2818 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{6875}+\frac{5753 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{31250}+\frac{47342 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{171875}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac{458 \sqrt{1-2 x} (2+3 x)^{5/2}}{825 \sqrt{3+5 x}}+\frac{2719 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{34375}+\frac{2818 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{6875}-\frac{47342 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15625 \sqrt{33}}-\frac{523 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15625}\\ \end{align*}
Mathematica [A] time = 0.278416, size = 107, normalized size = 0.57 \[ \frac{95165 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{10 \sqrt{1-2 x} \sqrt{3 x+2} \left (222750 x^3+398475 x^2+221200 x+37273\right )}{(5 x+3)^{3/2}}+94684 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{1031250} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.018, size = 229, normalized size = 1.2 \begin{align*} -{\frac{1}{6187500\,{x}^{2}+1031250\,x-2062500} \left ( 475825\,\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}+473420\,\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}+285495\,\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 ) +284052\,\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 ) -13365000\,{x}^{5}-26136000\,{x}^{4}-12801750\,{x}^{3}+3521120\,{x}^{2}+4051270\,x+745460 \right ) \sqrt{1-2\,x}\sqrt{2+3\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{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 (3 \, x + 2\right )}^{\frac{7}{2}} \sqrt{-2 \, x + 1}}{{\left (5 \, x + 3\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 (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27}, 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 (3 \, x + 2\right )}^{\frac{7}{2}} \sqrt{-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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