Optimal. Leaf size=218 \[ -\frac{43624697 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{283500 \sqrt{33}}-\frac{1}{11} \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}-\frac{34}{99} \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}-\frac{1053}{770} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{329683 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{34650}-\frac{43624697 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{623700}-\frac{725140729 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{141750 \sqrt{33}} \]
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Rubi [A] time = 0.0807899, 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 = {101, 154, 158, 113, 119} \[ -\frac{1}{11} \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}-\frac{34}{99} \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}-\frac{1053}{770} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{329683 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{34650}-\frac{43624697 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{623700}-\frac{43624697 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{283500 \sqrt{33}}-\frac{725140729 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{141750 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 101
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(2+3 x)^{5/2} (3+5 x)^{5/2}}{\sqrt{1-2 x}} \, dx &=-\frac{1}{11} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac{1}{11} \int \frac{(2+3 x)^{3/2} (3+5 x)^{3/2} \left (\frac{215}{2}+170 x\right )}{\sqrt{1-2 x}} \, dx\\ &=-\frac{34}{99} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac{1}{11} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac{1}{495} \int \frac{\left (-15200-\frac{47385 x}{2}\right ) \sqrt{2+3 x} (3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{1053}{770} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{34}{99} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac{1}{11} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac{\int \frac{(3+5 x)^{3/2} \left (\frac{6483095}{4}+\frac{4945245 x}{2}\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{17325}\\ &=-\frac{329683 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{34650}-\frac{1053}{770} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{34}{99} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac{1}{11} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac{\int \frac{\left (-\frac{212630445}{2}-\frac{654370455 x}{4}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{259875}\\ &=-\frac{43624697 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{623700}-\frac{329683 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{34650}-\frac{1053}{770} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{34}{99} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac{1}{11} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac{\int \frac{\frac{27544681245}{8}+\frac{10877110935 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{2338875}\\ &=-\frac{43624697 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{623700}-\frac{329683 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{34650}-\frac{1053}{770} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{34}{99} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac{1}{11} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac{43624697 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{567000}+\frac{725140729 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{1559250}\\ &=-\frac{43624697 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{623700}-\frac{329683 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{34650}-\frac{1053}{770} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{34}{99} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac{1}{11} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac{725140729 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{141750 \sqrt{33}}-\frac{43624697 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{283500 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.253603, size = 110, normalized size = 0.5 \[ \frac{2900562916 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (292189583 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+3 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (12757500 x^4+48384000 x^3+81985950 x^2+86822370 x+75000749\right )\right )}{9355500 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.012, size = 160, normalized size = 0.7 \begin{align*}{\frac{1}{561330000\,{x}^{3}+430353000\,{x}^{2}-130977000\,x-112266000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -11481750000\,{x}^{7}-52348275000\,{x}^{6}+1460947915\,\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 ) -2900562916\,\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 ) -104493240000\,{x}^{5}-122253448500\,{x}^{4}-101481939900\,{x}^{3}-18760348110\,{x}^{2}+31378183890\,x+13500134820 \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{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{5}{2}}}{\sqrt{-2 \, x + 1}}\,{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{{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{2 \, x - 1}, 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{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{5}{2}}}{\sqrt{-2 \, x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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