Optimal. Leaf size=187 \[ \frac{10628 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{41503 \sqrt{33}}-\frac{2377960 \sqrt{1-2 x} \sqrt{3 x+2}}{1369599 \sqrt{5 x+3}}+\frac{5314 \sqrt{1-2 x}}{41503 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{1088}{17787 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{4}{231 (1-2 x)^{3/2} \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{475592 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{41503 \sqrt{33}} \]
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Rubi [A] time = 0.0681657, 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} \[ -\frac{2377960 \sqrt{1-2 x} \sqrt{3 x+2}}{1369599 \sqrt{5 x+3}}+\frac{5314 \sqrt{1-2 x}}{41503 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{1088}{17787 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{4}{231 (1-2 x)^{3/2} \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{10628 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{41503 \sqrt{33}}+\frac{475592 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{41503 \sqrt{33}} \]
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)^{3/2} (3+5 x)^{3/2}} \, dx &=\frac{4}{231 (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{2}{231} \int \frac{-\frac{197}{2}-75 x}{(1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=\frac{4}{231 (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}}+\frac{1088}{17787 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}+\frac{4 \int \frac{\frac{18977}{4}+6120 x}{\sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx}{17787}\\ &=\frac{4}{231 (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}}+\frac{1088}{17787 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}+\frac{5314 \sqrt{1-2 x}}{41503 \sqrt{2+3 x} \sqrt{3+5 x}}+\frac{8 \int \frac{\frac{94985}{4}-\frac{39855 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx}{124509}\\ &=\frac{4}{231 (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}}+\frac{1088}{17787 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}+\frac{5314 \sqrt{1-2 x}}{41503 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{2377960 \sqrt{1-2 x} \sqrt{2+3 x}}{1369599 \sqrt{3+5 x}}-\frac{16 \int \frac{\frac{2227845}{8}+\frac{891735 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1369599}\\ &=\frac{4}{231 (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}}+\frac{1088}{17787 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}+\frac{5314 \sqrt{1-2 x}}{41503 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{2377960 \sqrt{1-2 x} \sqrt{2+3 x}}{1369599 \sqrt{3+5 x}}-\frac{5314 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{41503}-\frac{475592 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{456533}\\ &=\frac{4}{231 (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}}+\frac{1088}{17787 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}+\frac{5314 \sqrt{1-2 x}}{41503 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{2377960 \sqrt{1-2 x} \sqrt{2+3 x}}{1369599 \sqrt{3+5 x}}+\frac{475592 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{41503 \sqrt{33}}+\frac{10628 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{41503 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.191142, size = 103, normalized size = 0.55 \[ \frac{2 \left (\sqrt{2} \left (150115 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-237796 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )+\frac{-14267760 x^3+5106644 x^2+5510400 x-2236533}{(1-2 x)^{3/2} \sqrt{3 x+2} \sqrt{5 x+3}}\right )}{1369599} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.025, size = 228, normalized size = 1.2 \begin{align*} -{\frac{2}{ \left ( 20543985\,{x}^{2}+26022381\,x+8217594 \right ) \left ( 2\,x-1 \right ) ^{2}}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 300230\,\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}-475592\,\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}-150115\,\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 ) +237796\,\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 ) +14267760\,{x}^{3}-5106644\,{x}^{2}-5510400\,x+2236533 \right ) } \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{3}{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}}{1800 \, x^{7} + 1860 \, x^{6} - 1162 \, x^{5} - 1473 \, x^{4} + 228 \, x^{3} + 395 \, x^{2} - 12 \, x - 36}, 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{3}{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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