Optimal. Leaf size=218 \[ -\frac{609304 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{456533 \sqrt{33}}+\frac{99425780 \sqrt{1-2 x} \sqrt{3 x+2}}{15065589 \sqrt{5 x+3}}-\frac{1523260 \sqrt{1-2 x} \sqrt{3 x+2}}{1369599 (5 x+3)^{3/2}}+\frac{5034 \sqrt{1-2 x}}{41503 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{456}{5929 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{4}{231 (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{3/2}}-\frac{19885156 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{456533 \sqrt{33}} \]
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Rubi [A] time = 0.0829383, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{99425780 \sqrt{1-2 x} \sqrt{3 x+2}}{15065589 \sqrt{5 x+3}}-\frac{1523260 \sqrt{1-2 x} \sqrt{3 x+2}}{1369599 (5 x+3)^{3/2}}+\frac{5034 \sqrt{1-2 x}}{41503 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{456}{5929 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{4}{231 (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{3/2}}-\frac{609304 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{456533 \sqrt{33}}-\frac{19885156 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{456533 \sqrt{33}} \]
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)^{5/2}} \, dx &=\frac{4}{231 (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{2}{231} \int \frac{-\frac{237}{2}-105 x}{(1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=\frac{4}{231 (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{3/2}}+\frac{456}{5929 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}+\frac{4 \int \frac{\frac{36717}{4}+12825 x}{\sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx}{17787}\\ &=\frac{4}{231 (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{3/2}}+\frac{456}{5929 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}+\frac{5034 \sqrt{1-2 x}}{41503 \sqrt{2+3 x} (3+5 x)^{3/2}}+\frac{8 \int \frac{\frac{80265}{2}-\frac{113265 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}} \, dx}{124509}\\ &=\frac{4}{231 (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{3/2}}+\frac{456}{5929 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}+\frac{5034 \sqrt{1-2 x}}{41503 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{1523260 \sqrt{1-2 x} \sqrt{2+3 x}}{1369599 (3+5 x)^{3/2}}-\frac{16 \int \frac{\frac{10801065}{8}-\frac{3427335 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx}{4108797}\\ &=\frac{4}{231 (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{3/2}}+\frac{456}{5929 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}+\frac{5034 \sqrt{1-2 x}}{41503 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{1523260 \sqrt{1-2 x} \sqrt{2+3 x}}{1369599 (3+5 x)^{3/2}}+\frac{99425780 \sqrt{1-2 x} \sqrt{2+3 x}}{15065589 \sqrt{3+5 x}}+\frac{32 \int \frac{\frac{35441235}{2}+\frac{223708005 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{45196767}\\ &=\frac{4}{231 (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{3/2}}+\frac{456}{5929 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}+\frac{5034 \sqrt{1-2 x}}{41503 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{1523260 \sqrt{1-2 x} \sqrt{2+3 x}}{1369599 (3+5 x)^{3/2}}+\frac{99425780 \sqrt{1-2 x} \sqrt{2+3 x}}{15065589 \sqrt{3+5 x}}+\frac{304652 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{456533}+\frac{19885156 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{5021863}\\ &=\frac{4}{231 (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{3/2}}+\frac{456}{5929 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}+\frac{5034 \sqrt{1-2 x}}{41503 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{1523260 \sqrt{1-2 x} \sqrt{2+3 x}}{1369599 (3+5 x)^{3/2}}+\frac{99425780 \sqrt{1-2 x} \sqrt{2+3 x}}{15065589 \sqrt{3+5 x}}-\frac{19885156 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{456533 \sqrt{33}}-\frac{609304 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{456533 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.194339, size = 109, normalized size = 0.5 \[ \frac{2 \left (2 \sqrt{2} \left (4971289 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-2457910 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{2982773400 x^4+694871080 x^3-1802210526 x^2-211488180 x+283144937}{(1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{3/2}}\right )}{15065589} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.027, size = 311, normalized size = 1.4 \begin{align*} -{\frac{2}{15065589\, \left ( 2\,x-1 \right ) ^{2}}\sqrt{1-2\,x} \left ( 99425780\,\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}-49158200\,\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}+9942578\,\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}-4915820\,\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}-29827734\,\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 ) +14747460\,\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 ) -2982773400\,{x}^{4}-694871080\,{x}^{3}+1802210526\,{x}^{2}+211488180\,x-283144937 \right ) \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{2+3\,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{5}{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}}{9000 \, x^{8} + 14700 \, x^{7} - 230 \, x^{6} - 10851 \, x^{5} - 3279 \, x^{4} + 2659 \, x^{3} + 1125 \, x^{2} - 216 \, 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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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