Optimal. Leaf size=109 \[ \frac{\left (7 (x-1)^2+26\right ) (x-1)}{432 \sqrt{-(x-1)^4-2 (x-1)^2+3}}+\frac{\left ((x-1)^2+5\right ) (x-1)}{72 \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}-\frac{11 F\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{144 \sqrt{3}}+\frac{7 E\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{144 \sqrt{3}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.20915, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304 \[ -\frac{\left ((x-1)^2+5\right ) (1-x)}{72 \left (-(1-x)^4-2 (1-x)^2+3\right )^{3/2}}-\frac{\left (7 (1-x)^2+26\right ) (1-x)}{432 \sqrt{-(1-x)^4-2 (1-x)^2+3}}-\frac{11 F\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{144 \sqrt{3}}+\frac{7 E\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{144 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[(8*x - 8*x^2 + 4*x^3 - x^4)^(-5/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 22.7036, size = 97, normalized size = 0.89 \[ \frac{\left (x - 1\right ) \left (2 \left (x - 1\right )^{2} + 10\right )}{144 \left (- \left (x - 1\right )^{4} - 2 \left (x - 1\right )^{2} + 3\right )^{\frac{3}{2}}} + \frac{\left (x - 1\right ) \left (112 \left (x - 1\right )^{2} + 416\right )}{6912 \sqrt{- \left (x - 1\right )^{4} - 2 \left (x - 1\right )^{2} + 3}} - \frac{7 \sqrt{3} E\left (\operatorname{asin}{\left (x - 1 \right )}\middle | - \frac{1}{3}\right )}{432} + \frac{11 \sqrt{3} F\left (\operatorname{asin}{\left (x - 1 \right )}\middle | - \frac{1}{3}\right )}{432} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(-x**4+4*x**3-8*x**2+8*x)**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 1.7223, size = 298, normalized size = 2.73 \[ \frac{\frac{7 i \sqrt{2} (x-2) \sqrt{\frac{x^2-2 x+4}{x^2}} x^2 E\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt{3}+i-\frac{4 i}{x}}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-i+\sqrt{3}}\right )}{\sqrt{-\frac{i (x-2)}{\left (\sqrt{3}-i\right ) x}}}+\frac{7 x^6-37 x^5+115 x^4-226 x^3+274 x^2-19 i \sqrt{2} \sqrt{-\frac{i (x-2)}{\left (\sqrt{3}-i\right ) x}} \sqrt{\frac{x^2-2 x+4}{x^2}} \left (x^3-4 x^2+8 x-8\right ) x^3 F\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt{3}+i-\frac{4 i}{x}}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-i+\sqrt{3}}\right )-232 x+36}{x^3-4 x^2+8 x-8}}{432 x \sqrt{-x \left (x^3-4 x^2+8 x-8\right )}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(8*x - 8*x^2 + 4*x^3 - x^4)^(-5/2),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.052, size = 1039, normalized size = 9.5 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(-x^4+4*x^3-8*x^2+8*x)^(5/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x^4 + 4*x^3 - 8*x^2 + 8*x)^(-5/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (x^{8} - 8 \, x^{7} + 32 \, x^{6} - 80 \, x^{5} + 128 \, x^{4} - 128 \, x^{3} + 64 \, x^{2}\right )} \sqrt{-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x^4 + 4*x^3 - 8*x^2 + 8*x)^(-5/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (- x^{4} + 4 x^{3} - 8 x^{2} + 8 x\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(-x**4+4*x**3-8*x**2+8*x)**(5/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x^4 + 4*x^3 - 8*x^2 + 8*x)^(-5/2),x, algorithm="giac")
[Out]