Optimal. Leaf size=222 \[ -\frac{8}{45} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{7/2}-\frac{2 (1-2 x)^{3/2} (3 x+2)^{7/2}}{5 \sqrt{5 x+3}}+\frac{958 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}}{1575}+\frac{5153 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{39375}-\frac{12601 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{196875}-\frac{31288 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{984375}-\frac{1473539 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1968750} \]
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Rubi [A] time = 0.484417, antiderivative size = 222, 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 \[ -\frac{8}{45} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{7/2}-\frac{2 (1-2 x)^{3/2} (3 x+2)^{7/2}}{5 \sqrt{5 x+3}}+\frac{958 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}}{1575}+\frac{5153 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{39375}-\frac{12601 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{196875}-\frac{31288 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{984375}-\frac{1473539 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1968750} \]
Antiderivative was successfully verified.
[In] Int[((1 - 2*x)^(3/2)*(2 + 3*x)^(7/2))/(3 + 5*x)^(3/2),x]
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Rubi in Sympy [A] time = 47.4585, size = 201, normalized size = 0.91 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{7}{2}}}{5 \sqrt{5 x + 3}} - \frac{8 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{7}{2}} \sqrt{5 x + 3}}{45} + \frac{958 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{1575} + \frac{5153 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{39375} - \frac{12601 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{196875} - \frac{1473539 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{5906250} - \frac{31288 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{2953125} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(3/2)*(2+3*x)**(7/2)/(3+5*x)**(3/2),x)
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Mathematica [A] time = 0.233998, size = 132, normalized size = 0.59 \[ \frac{1473539 \sqrt{2} (5 x+3) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (6 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (472500 x^4+517500 x^3-252225 x^2-377530 x-83787\right )+88207 \sqrt{2} (5 x+3) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{5906250 (5 x+3)} \]
Antiderivative was successfully verified.
[In] Integrate[((1 - 2*x)^(3/2)*(2 + 3*x)^(7/2))/(3 + 5*x)^(3/2),x]
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Maple [C] time = 0.026, size = 179, normalized size = 0.8 \[{\frac{1}{177187500\,{x}^{3}+135843750\,{x}^{2}-41343750\,x-35437500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -85050000\,{x}^{6}+441035\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -1473539\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -107325000\,{x}^{5}+58225500\,{x}^{4}+106572150\,{x}^{3}+11274060\,{x}^{2}-20138190\,x-5027220 \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(3/2)*(2+3*x)^(7/2)/(3+5*x)^(3/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{7}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^(7/2)*(-2*x + 1)^(3/2)/(5*x + 3)^(3/2),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^(7/2)*(-2*x + 1)^(3/2)/(5*x + 3)^(3/2),x, algorithm="fricas")
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-2*x)**(3/2)*(2+3*x)**(7/2)/(3+5*x)**(3/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{7}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^(7/2)*(-2*x + 1)^(3/2)/(5*x + 3)^(3/2),x, algorithm="giac")
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