Optimal. Leaf size=228 \[ -\frac{4420 \left (3 x^2+5 x+2\right )^{3/2} \sqrt{x}}{6237}+\frac{8 (74313 x+57860) \sqrt{3 x^2+5 x+2} \sqrt{x}}{280665}-\frac{261784 (3 x+2) \sqrt{x}}{841995 \sqrt{3 x^2+5 x+2}}-\frac{13016 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{56133 \sqrt{3 x^2+5 x+2}}+\frac{261784 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{841995 \sqrt{3 x^2+5 x+2}}-\frac{10}{33} \left (3 x^2+5 x+2\right )^{3/2} x^{5/2}+\frac{532}{891} \left (3 x^2+5 x+2\right )^{3/2} x^{3/2} \]
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Rubi [A] time = 0.410852, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ -\frac{4420 \left (3 x^2+5 x+2\right )^{3/2} \sqrt{x}}{6237}+\frac{8 (74313 x+57860) \sqrt{3 x^2+5 x+2} \sqrt{x}}{280665}-\frac{261784 (3 x+2) \sqrt{x}}{841995 \sqrt{3 x^2+5 x+2}}-\frac{13016 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{56133 \sqrt{3 x^2+5 x+2}}+\frac{261784 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{841995 \sqrt{3 x^2+5 x+2}}-\frac{10}{33} \left (3 x^2+5 x+2\right )^{3/2} x^{5/2}+\frac{532}{891} \left (3 x^2+5 x+2\right )^{3/2} x^{3/2} \]
Antiderivative was successfully verified.
[In] Int[(2 - 5*x)*x^(5/2)*Sqrt[2 + 5*x + 3*x^2],x]
[Out]
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Rubi in Sympy [A] time = 42.6801, size = 212, normalized size = 0.93 \[ - \frac{10 x^{\frac{5}{2}} \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{33} + \frac{532 x^{\frac{3}{2}} \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{891} - \frac{130892 \sqrt{x} \left (6 x + 4\right )}{841995 \sqrt{3 x^{2} + 5 x + 2}} + \frac{32 \sqrt{x} \left (\frac{222939 x}{4} + 43395\right ) \sqrt{3 x^{2} + 5 x + 2}}{841995} - \frac{4420 \sqrt{x} \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{6237} + \frac{65446 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{841995 \sqrt{3 x^{2} + 5 x + 2}} - \frac{3254 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{56133 \sqrt{3 x^{2} + 5 x + 2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2-5*x)*x**(5/2)*(3*x**2+5*x+2)**(1/2),x)
[Out]
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Mathematica [C] time = 0.242404, size = 170, normalized size = 0.75 \[ \frac{66544 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-261784 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-2296350 x^7-3129840 x^6+271350 x^5+947916 x^4+39780 x^3-198168 x^2-918440 x-523568}{841995 \sqrt{x} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
[In] Integrate[(2 - 5*x)*x^(5/2)*Sqrt[2 + 5*x + 3*x^2],x]
[Out]
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Maple [A] time = 0.028, size = 138, normalized size = 0.6 \[{\frac{2}{2525985} \left ( -3444525\,{x}^{7}-4694760\,{x}^{6}+407025\,{x}^{5}+98718\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -65446\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) +1421874\,{x}^{4}+59670\,{x}^{3}+880776\,{x}^{2}+585720\,x \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2-5*x)*x^(5/2)*(3*x^2+5*x+2)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (5 \, x - 2\right )} x^{\frac{5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(3*x^2 + 5*x + 2)*(5*x - 2)*x^(5/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-{\left (5 \, x^{3} - 2 \, x^{2}\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(3*x^2 + 5*x + 2)*(5*x - 2)*x^(5/2),x, algorithm="fricas")
[Out]
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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((2-5*x)*x**(5/2)*(3*x**2+5*x+2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (5 \, x - 2\right )} x^{\frac{5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(3*x^2 + 5*x + 2)*(5*x - 2)*x^(5/2),x, algorithm="giac")
[Out]