Optimal. Leaf size=187 \[ -\frac{18016 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{5929 \sqrt{33}}+\frac{2976620 \sqrt{1-2 x} \sqrt{3 x+2}}{195657 \sqrt{5 x+3}}-\frac{45040 \sqrt{1-2 x} \sqrt{3 x+2}}{17787 (5 x+3)^{3/2}}+\frac{186 \sqrt{1-2 x}}{539 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{4}{77 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}-\frac{595324 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5929 \sqrt{33}} \]
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Rubi [A] time = 0.0661951, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{2976620 \sqrt{1-2 x} \sqrt{3 x+2}}{195657 \sqrt{5 x+3}}-\frac{45040 \sqrt{1-2 x} \sqrt{3 x+2}}{17787 (5 x+3)^{3/2}}+\frac{186 \sqrt{1-2 x}}{539 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{4}{77 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}-\frac{18016 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5929 \sqrt{33}}-\frac{595324 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5929 \sqrt{33}} \]
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)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx &=\frac{4}{77 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{2}{77} \int \frac{-\frac{131}{2}-75 x}{\sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=\frac{4}{77 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}+\frac{186 \sqrt{1-2 x}}{539 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{4}{539} \int \frac{-\frac{1415}{2}+\frac{1395 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}} \, dx\\ &=\frac{4}{77 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}+\frac{186 \sqrt{1-2 x}}{539 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{45040 \sqrt{1-2 x} \sqrt{2+3 x}}{17787 (3+5 x)^{3/2}}+\frac{8 \int \frac{-\frac{108295}{4}+16890 x}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx}{17787}\\ &=\frac{4}{77 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}+\frac{186 \sqrt{1-2 x}}{539 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{45040 \sqrt{1-2 x} \sqrt{2+3 x}}{17787 (3+5 x)^{3/2}}+\frac{2976620 \sqrt{1-2 x} \sqrt{2+3 x}}{195657 \sqrt{3+5 x}}-\frac{16 \int \frac{-\frac{1413795}{4}-\frac{2232465 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{195657}\\ &=\frac{4}{77 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}+\frac{186 \sqrt{1-2 x}}{539 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{45040 \sqrt{1-2 x} \sqrt{2+3 x}}{17787 (3+5 x)^{3/2}}+\frac{2976620 \sqrt{1-2 x} \sqrt{2+3 x}}{195657 \sqrt{3+5 x}}+\frac{9008 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{5929}+\frac{595324 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{65219}\\ &=\frac{4}{77 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}+\frac{186 \sqrt{1-2 x}}{539 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{45040 \sqrt{1-2 x} \sqrt{2+3 x}}{17787 (3+5 x)^{3/2}}+\frac{2976620 \sqrt{1-2 x} \sqrt{2+3 x}}{195657 \sqrt{3+5 x}}-\frac{595324 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5929 \sqrt{33}}-\frac{18016 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5929 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.162832, size = 104, normalized size = 0.56 \[ \frac{2 \left (2 \sqrt{2} \left (148831 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-74515 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{-44649300 x^3-32744810 x^2+10598372 x+8473261}{\sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}\right )}{195657} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.026, size = 219, normalized size = 1.2 \begin{align*}{\frac{2}{1173942\,{x}^{2}+195657\,x-391314}\sqrt{1-2\,x}\sqrt{2+3\,x} \left ( 745150\,\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}-1488310\,\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}+447090\,\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 ) -892986\,\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 ) +44649300\,{x}^{3}+32744810\,{x}^{2}-10598372\,x-8473261 \right ) \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{1}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{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}}{4500 \, x^{7} + 9600 \, x^{6} + 4685 \, x^{5} - 3083 \, x^{4} - 3181 \, x^{3} - 261 \, x^{2} + 432 \, x + 108}, 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{5}{2}}{\left (3 \, x + 2\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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