Optimal. Leaf size=218 \[ -\frac{11346991 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{1771875 \sqrt{33}}+\frac{2}{55} \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}-\frac{23 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}{2475}-\frac{543 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{9625}-\frac{342971 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{866250}-\frac{11346991 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{3898125}-\frac{1508889271 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7087500 \sqrt{33}} \]
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Rubi [A] time = 0.0811409, 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} \[ \frac{2}{55} \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}-\frac{23 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}{2475}-\frac{543 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{9625}-\frac{342971 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{866250}-\frac{11346991 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{3898125}-\frac{11346991 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1771875 \sqrt{33}}-\frac{1508889271 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7087500 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 101
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2} \, dx &=\frac{2}{55} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac{2}{55} \int \frac{\left (-\frac{27}{2}-\frac{23 x}{2}\right ) (2+3 x)^{3/2} (3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{23 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{2475}+\frac{2}{55} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac{2 \int \frac{\sqrt{2+3 x} (3+5 x)^{3/2} \left (\frac{6355}{4}+\frac{4887 x}{2}\right )}{\sqrt{1-2 x}} \, dx}{2475}\\ &=-\frac{543 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{9625}-\frac{23 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{2475}+\frac{2}{55} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac{2 \int \frac{\left (-\frac{674539}{4}-\frac{1028913 x}{4}\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{86625}\\ &=-\frac{342971 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{866250}-\frac{543 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{9625}-\frac{23 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{2475}+\frac{2}{55} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac{2 \int \frac{\sqrt{3+5 x} \left (\frac{88489161}{8}+\frac{34040973 x}{2}\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{1299375}\\ &=-\frac{11346991 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{3898125}-\frac{342971 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{866250}-\frac{543 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{9625}-\frac{23 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{2475}+\frac{2}{55} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac{2 \int \frac{-\frac{2865780969}{8}-\frac{4526667813 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{11694375}\\ &=-\frac{11346991 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{3898125}-\frac{342971 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{866250}-\frac{543 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{9625}-\frac{23 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{2475}+\frac{2}{55} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac{11346991 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{3543750}+\frac{1508889271 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{77962500}\\ &=-\frac{11346991 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{3898125}-\frac{342971 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{866250}-\frac{543 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{9625}-\frac{23 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{2475}+\frac{2}{55} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac{1508889271 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7087500 \sqrt{33}}-\frac{11346991 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1771875 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.255125, size = 107, normalized size = 0.49 \[ \frac{-759987865 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (63787500 x^4+156161250 x^3+132234750 x^2+29706255 x-27010769\right )+1508889271 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{116943750 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.02, size = 160, normalized size = 0.7 \begin{align*}{\frac{1}{7016625000\,{x}^{3}+5379412500\,{x}^{2}-1637212500\,x-1403325000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 57408750000\,{x}^{7}+184558500000\,{x}^{6}+759987865\,\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 ) -1508889271\,\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 ) +213367162500\,{x}^{5}+73701994500\,{x}^{4}-59690698650\,{x}^{3}-48677999160\,{x}^{2}+325135590\,x+4861938420 \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{\left (5 \, x + 3\right )}^{\frac{3}{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 ({\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-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{\left (5 \, x + 3\right )}^{\frac{3}{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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