Optimal. Leaf size=249 \[ \frac{145418632 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{10168235 \sqrt{33}}-\frac{4839325048 \sqrt{1-2 x} \sqrt{3 x+2}}{67110351 \sqrt{5 x+3}}+\frac{72709316 \sqrt{1-2 x}}{10168235 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{499564 \sqrt{1-2 x}}{1452605 (3 x+2)^{3/2} \sqrt{5 x+3}}-\frac{2206 \sqrt{1-2 x}}{207515 (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{1616}{17787 \sqrt{1-2 x} (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{4839325048 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{10168235 \sqrt{33}} \]
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Rubi [A] time = 0.100031, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {104, 152, 158, 113, 119} \[ -\frac{4839325048 \sqrt{1-2 x} \sqrt{3 x+2}}{67110351 \sqrt{5 x+3}}+\frac{72709316 \sqrt{1-2 x}}{10168235 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{499564 \sqrt{1-2 x}}{1452605 (3 x+2)^{3/2} \sqrt{5 x+3}}-\frac{2206 \sqrt{1-2 x}}{207515 (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{1616}{17787 \sqrt{1-2 x} (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{145418632 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{10168235 \sqrt{33}}+\frac{4839325048 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{10168235 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 104
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx &=\frac{4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt{3+5 x}}-\frac{2}{231} \int \frac{-\frac{269}{2}-135 x}{(1-2 x)^{3/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx\\ &=\frac{4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{1616}{17787 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{4 \int \frac{\frac{55457}{4}+21210 x}{\sqrt{1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx}{17787}\\ &=\frac{4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{1616}{17787 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}-\frac{2206 \sqrt{1-2 x}}{207515 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{8 \int \frac{\frac{429823}{4}+\frac{82725 x}{4}}{\sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx}{622545}\\ &=\frac{4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{1616}{17787 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}-\frac{2206 \sqrt{1-2 x}}{207515 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{499564 \sqrt{1-2 x}}{1452605 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{16 \int \frac{\frac{32051607}{8}-\frac{16860285 x}{4}}{\sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx}{13073445}\\ &=\frac{4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{1616}{17787 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}-\frac{2206 \sqrt{1-2 x}}{207515 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{499564 \sqrt{1-2 x}}{1452605 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{72709316 \sqrt{1-2 x}}{10168235 \sqrt{2+3 x} \sqrt{3+5 x}}+\frac{32 \int \frac{\frac{661979505}{4}-\frac{817979805 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx}{91514115}\\ &=\frac{4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{1616}{17787 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}-\frac{2206 \sqrt{1-2 x}}{207515 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{499564 \sqrt{1-2 x}}{1452605 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{72709316 \sqrt{1-2 x}}{10168235 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{4839325048 \sqrt{1-2 x} \sqrt{2+3 x}}{67110351 \sqrt{3+5 x}}-\frac{64 \int \frac{\frac{34464999645}{16}+\frac{27221203395 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1006655265}\\ &=\frac{4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{1616}{17787 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}-\frac{2206 \sqrt{1-2 x}}{207515 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{499564 \sqrt{1-2 x}}{1452605 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{72709316 \sqrt{1-2 x}}{10168235 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{4839325048 \sqrt{1-2 x} \sqrt{2+3 x}}{67110351 \sqrt{3+5 x}}-\frac{72709316 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{10168235}-\frac{4839325048 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{111850585}\\ &=\frac{4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{1616}{17787 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}-\frac{2206 \sqrt{1-2 x}}{207515 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{499564 \sqrt{1-2 x}}{1452605 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{72709316 \sqrt{1-2 x}}{10168235 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{4839325048 \sqrt{1-2 x} \sqrt{2+3 x}}{67110351 \sqrt{3+5 x}}+\frac{4839325048 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{10168235 \sqrt{33}}+\frac{145418632 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{10168235 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.191305, size = 115, normalized size = 0.46 \[ \frac{2 \left (-2 \sqrt{2} \left (1209831262 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-609979405 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )-\frac{1306617762960 x^5+1263428429256 x^4-559512908172 x^3-673871013766 x^2+53503915182 x+91855922241}{(1-2 x)^{3/2} (3 x+2)^{5/2} \sqrt{5 x+3}}\right )}{335551755} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.027, size = 406, normalized size = 1.6 \begin{align*} -{\frac{2}{335551755\, \left ( 2\,x-1 \right ) ^{2}}\sqrt{1-2\,x} \left ( 21959258580\,\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}-43553925432\,\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}+18299382150\,\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}-36294937860\,\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}-4879835240\,\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}+9678650096\,\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}-4879835240\,\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 ) +9678650096\,\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 ) +1306617762960\,{x}^{5}+1263428429256\,{x}^{4}-559512908172\,{x}^{3}-673871013766\,{x}^{2}+53503915182\,x+91855922241 \right ) \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{3+5\,x}}}} \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{1}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{2}}{\left (-2 \, x + 1\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{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{16200 \, x^{9} + 38340 \, x^{8} + 19062 \, x^{7} - 19761 \, x^{6} - 20272 \, x^{5} + 399 \, x^{4} + 5544 \, x^{3} + 1112 \, x^{2} - 480 \, x - 144}, 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{1}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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