Optimal. Leaf size=180 \[ \frac{139 \sqrt{3 x^2+5 x+2}}{15 \sqrt{x}}-\frac{139 \sqrt{x} (3 x+2)}{15 \sqrt{3 x^2+5 x+2}}-\frac{11 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}}+\frac{139 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{15 \sqrt{3 x^2+5 x+2}}-\frac{4 \sqrt{3 x^2+5 x+2} (3-10 x)}{15 x^{5/2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.280702, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{139 \sqrt{3 x^2+5 x+2}}{15 \sqrt{x}}-\frac{139 \sqrt{x} (3 x+2)}{15 \sqrt{3 x^2+5 x+2}}-\frac{11 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}}+\frac{139 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{15 \sqrt{3 x^2+5 x+2}}-\frac{4 \sqrt{3 x^2+5 x+2} (3-10 x)}{15 x^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[((2 - 5*x)*Sqrt[2 + 5*x + 3*x^2])/x^(7/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 31.2056, size = 167, normalized size = 0.93 \[ - \frac{139 \sqrt{x} \left (6 x + 4\right )}{30 \sqrt{3 x^{2} + 5 x + 2}} + \frac{139 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{60 \sqrt{3 x^{2} + 5 x + 2}} - \frac{11 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{4 \sqrt{3 x^{2} + 5 x + 2}} + \frac{139 \sqrt{3 x^{2} + 5 x + 2}}{15 \sqrt{x}} - \frac{2 \left (- 20 x + 6\right ) \sqrt{3 x^{2} + 5 x + 2}}{15 x^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2-5*x)*(3*x**2+5*x+2)**(1/2)/x**(7/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.235334, size = 153, normalized size = 0.85 \[ \frac{-26 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{7/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-139 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{7/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+4 \left (30 x^3+41 x^2+5 x-6\right )}{15 x^{5/2} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
[In] Integrate[((2 - 5*x)*Sqrt[2 + 5*x + 3*x^2])/x^(7/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.038, size = 130, normalized size = 0.7 \[{\frac{1}{90} \left ( 87\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{2}-139\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{2}+2502\,{x}^{4}+4890\,{x}^{3}+2652\,{x}^{2}+120\,x-144 \right ){x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2-5*x)*(3*x^2+5*x+2)^(1/2)/x^(7/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (5 \, x - 2\right )}}{x^{\frac{7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(3*x^2 + 5*x + 2)*(5*x - 2)/x^(7/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (5 \, x - 2\right )}}{x^{\frac{7}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(3*x^2 + 5*x + 2)*(5*x - 2)/x^(7/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{2 \sqrt{3 x^{2} + 5 x + 2}}{x^{\frac{7}{2}}}\right )\, dx - \int \frac{5 \sqrt{3 x^{2} + 5 x + 2}}{x^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2-5*x)*(3*x**2+5*x+2)**(1/2)/x**(7/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (5 \, x - 2\right )}}{x^{\frac{7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(3*x^2 + 5*x + 2)*(5*x - 2)/x^(7/2),x, algorithm="giac")
[Out]