Optimal. Leaf size=84 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^2}{33 (1-2 x)^{3/2}}-\frac{(95621-33462 x) \sqrt{5 x+3}}{14520 \sqrt{1-2 x}}+\frac{1593 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{40 \sqrt{10}} \]
[Out]
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Rubi [A] time = 0.123918, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^2}{33 (1-2 x)^{3/2}}-\frac{(95621-33462 x) \sqrt{5 x+3}}{14520 \sqrt{1-2 x}}+\frac{1593 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{40 \sqrt{10}} \]
Antiderivative was successfully verified.
[In] Int[(2 + 3*x)^3/((1 - 2*x)^(5/2)*Sqrt[3 + 5*x]),x]
[Out]
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Rubi in Sympy [A] time = 11.6481, size = 78, normalized size = 0.93 \[ - \frac{\left (- \frac{16731 x}{2} + \frac{95621}{4}\right ) \sqrt{5 x + 3}}{3630 \sqrt{- 2 x + 1}} + \frac{1593 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{400} + \frac{7 \left (3 x + 2\right )^{2} \sqrt{5 x + 3}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2+3*x)**3/(1-2*x)**(5/2)/(3+5*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.154134, size = 69, normalized size = 0.82 \[ \frac{578259 \sqrt{10-20 x} (2 x-1) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (39204 x^2-261664 x+83301\right )}{145200 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(2 + 3*x)^3/((1 - 2*x)^(5/2)*Sqrt[3 + 5*x]),x]
[Out]
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Maple [A] time = 0.02, size = 120, normalized size = 1.4 \[{\frac{1}{290400\, \left ( -1+2\,x \right ) ^{2}} \left ( 2313036\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-2313036\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-784080\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+578259\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +5233280\,x\sqrt{-10\,{x}^{2}-x+3}-1666020\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2+3*x)^3/(1-2*x)^(5/2)/(3+5*x)^(1/2),x)
[Out]
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Maxima [A] time = 1.48652, size = 103, normalized size = 1.23 \[ \frac{1593}{800} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{27}{40} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{343 \, \sqrt{-10 \, x^{2} - x + 3}}{132 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{11123 \, \sqrt{-10 \, x^{2} - x + 3}}{1452 \,{\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^3/(sqrt(5*x + 3)*(-2*x + 1)^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.224416, size = 113, normalized size = 1.35 \[ -\frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (39204 \, x^{2} - 261664 \, x + 83301\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 578259 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{290400 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^3/(sqrt(5*x + 3)*(-2*x + 1)^(5/2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (3 x + 2\right )^{3}}{\left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2+3*x)**3/(1-2*x)**(5/2)/(3+5*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.265011, size = 96, normalized size = 1.14 \[ \frac{1593}{400} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{{\left (4 \,{\left (9801 \, \sqrt{5}{\left (5 \, x + 3\right )} - 385886 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 6360321 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{1815000 \,{\left (2 \, x - 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^3/(sqrt(5*x + 3)*(-2*x + 1)^(5/2)),x, algorithm="giac")
[Out]