Optimal. Leaf size=218 \[ -\frac{310208 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{245 \sqrt{33}}+\frac{10312712 \sqrt{1-2 x} \sqrt{3 x+2}}{1617 \sqrt{5 x+3}}-\frac{155104 \sqrt{1-2 x} \sqrt{3 x+2}}{147 (5 x+3)^{3/2}}+\frac{116044 \sqrt{1-2 x}}{735 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{556 \sqrt{1-2 x}}{105 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac{2 \sqrt{1-2 x}}{5 (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac{10312712 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{245 \sqrt{33}} \]
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Rubi [A] time = 0.0834373, 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 = {99, 152, 158, 113, 119} \[ \frac{10312712 \sqrt{1-2 x} \sqrt{3 x+2}}{1617 \sqrt{5 x+3}}-\frac{155104 \sqrt{1-2 x} \sqrt{3 x+2}}{147 (5 x+3)^{3/2}}+\frac{116044 \sqrt{1-2 x}}{735 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{556 \sqrt{1-2 x}}{105 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac{2 \sqrt{1-2 x}}{5 (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac{310208 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{245 \sqrt{33}}-\frac{10312712 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{245 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 99
Rule 152
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}}{5 (2+3 x)^{5/2} (3+5 x)^{3/2}}-\frac{2}{5} \int \frac{-23+35 x}{\sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx\\ &=\frac{2 \sqrt{1-2 x}}{5 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{556 \sqrt{1-2 x}}{105 (2+3 x)^{3/2} (3+5 x)^{3/2}}-\frac{4}{105} \int \frac{-\frac{5037}{2}+3475 x}{\sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=\frac{2 \sqrt{1-2 x}}{5 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{556 \sqrt{1-2 x}}{105 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{116044 \sqrt{1-2 x}}{735 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{8}{735} \int \frac{-\frac{378705}{2}+\frac{435165 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}} \, dx\\ &=\frac{2 \sqrt{1-2 x}}{5 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{556 \sqrt{1-2 x}}{105 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{116044 \sqrt{1-2 x}}{735 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{155104 \sqrt{1-2 x} \sqrt{2+3 x}}{147 (3+5 x)^{3/2}}+\frac{16 \int \frac{-\frac{31023465}{4}+4798530 x}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx}{24255}\\ &=\frac{2 \sqrt{1-2 x}}{5 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{556 \sqrt{1-2 x}}{105 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{116044 \sqrt{1-2 x}}{735 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{155104 \sqrt{1-2 x} \sqrt{2+3 x}}{147 (3+5 x)^{3/2}}+\frac{10312712 \sqrt{1-2 x} \sqrt{2+3 x}}{1617 \sqrt{3+5 x}}-\frac{32 \int \frac{-\frac{403972965}{4}-\frac{638099055 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{266805}\\ &=\frac{2 \sqrt{1-2 x}}{5 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{556 \sqrt{1-2 x}}{105 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{116044 \sqrt{1-2 x}}{735 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{155104 \sqrt{1-2 x} \sqrt{2+3 x}}{147 (3+5 x)^{3/2}}+\frac{10312712 \sqrt{1-2 x} \sqrt{2+3 x}}{1617 \sqrt{3+5 x}}+\frac{155104}{245} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx+\frac{10312712 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{2695}\\ &=\frac{2 \sqrt{1-2 x}}{5 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{556 \sqrt{1-2 x}}{105 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{116044 \sqrt{1-2 x}}{735 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{155104 \sqrt{1-2 x} \sqrt{2+3 x}}{147 (3+5 x)^{3/2}}+\frac{10312712 \sqrt{1-2 x} \sqrt{2+3 x}}{1617 \sqrt{3+5 x}}-\frac{10312712 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{245 \sqrt{33}}-\frac{310208 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{245 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.253308, size = 109, normalized size = 0.5 \[ \frac{2 \left (4 \sqrt{2} \left (1289089 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-649285 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{\sqrt{1-2 x} \left (3480540300 x^4+8934240060 x^3+8592783498 x^2+3669873602 x+587237237\right )}{(3 x+2)^{5/2} (5 x+3)^{3/2}}\right )}{8085} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.024, size = 406, normalized size = 1.9 \begin{align*} -{\frac{2}{16170\,x-8085}\sqrt{1-2\,x} \left ( 232036020\,\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}-116871300\,\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}+448602972\,\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}-225951180\,\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}+288755936\,\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}-145439840\,\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}+61876272\,\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 ) -31165680\,\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 ) -6961080600\,{x}^{5}-14387939820\,{x}^{4}-8251326936\,{x}^{3}+1253036294\,{x}^{2}+2495399128\,x+587237237 \right ) \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}} \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{\sqrt{-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{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}}{10125 \, x^{7} + 45225 \, x^{6} + 86535 \, x^{5} + 91947 \, x^{4} + 58592 \, x^{3} + 22392 \, x^{2} + 4752 \, x + 432}, 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{\sqrt{-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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