3.515 \(\int \frac{1}{x^2 (1+x)^{3/2} \left (1-x+x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=316 \[ \frac{2}{3 x \sqrt{x+1} \sqrt{x^2-x+1}}+\frac{5 \sqrt{2} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^2-x+1}}-\frac{5 \sqrt{2-\sqrt{3}} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{2\ 3^{3/4} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^2-x+1}}-\frac{5 \left (x^3+1\right )}{3 x \sqrt{x+1} \sqrt{x^2-x+1}}+\frac{5 \left (x^3+1\right )}{3 \sqrt{x+1} \left (x+\sqrt{3}+1\right ) \sqrt{x^2-x+1}} \]

[Out]

2/(3*x*Sqrt[1 + x]*Sqrt[1 - x + x^2]) - (5*(1 + x^3))/(3*x*Sqrt[1 + x]*Sqrt[1 -
x + x^2]) + (5*(1 + x^3))/(3*Sqrt[1 + x]*(1 + Sqrt[3] + x)*Sqrt[1 - x + x^2]) -
(5*Sqrt[2 - Sqrt[3]]*Sqrt[1 + x]*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*Ellipti
cE[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(2*3^(3/4)*Sqrt
[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 - x + x^2]) + (5*Sqrt[2]*Sqrt[1 + x]*Sqrt[(
1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3
] + x)], -7 - 4*Sqrt[3]])/(3*3^(1/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 -
x + x^2])

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Rubi [A]  time = 0.252133, antiderivative size = 316, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261 \[ \frac{2}{3 x \sqrt{x+1} \sqrt{x^2-x+1}}+\frac{5 \sqrt{2} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^2-x+1}}-\frac{5 \sqrt{2-\sqrt{3}} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{2\ 3^{3/4} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^2-x+1}}-\frac{5 \left (x^3+1\right )}{3 x \sqrt{x+1} \sqrt{x^2-x+1}}+\frac{5 \left (x^3+1\right )}{3 \sqrt{x+1} \left (x+\sqrt{3}+1\right ) \sqrt{x^2-x+1}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(1 + x)^(3/2)*(1 - x + x^2)^(3/2)),x]

[Out]

2/(3*x*Sqrt[1 + x]*Sqrt[1 - x + x^2]) - (5*(1 + x^3))/(3*x*Sqrt[1 + x]*Sqrt[1 -
x + x^2]) + (5*(1 + x^3))/(3*Sqrt[1 + x]*(1 + Sqrt[3] + x)*Sqrt[1 - x + x^2]) -
(5*Sqrt[2 - Sqrt[3]]*Sqrt[1 + x]*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*Ellipti
cE[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(2*3^(3/4)*Sqrt
[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 - x + x^2]) + (5*Sqrt[2]*Sqrt[1 + x]*Sqrt[(
1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3
] + x)], -7 - 4*Sqrt[3]])/(3*3^(1/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 -
x + x^2])

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Rubi in Sympy [A]  time = 20.8172, size = 287, normalized size = 0.91 \[ \frac{5 \sqrt{x + 1} \sqrt{x^{2} - x + 1}}{3 \left (x + 1 + \sqrt{3}\right )} - \frac{5 \sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (x + 1\right )^{\frac{3}{2}} \sqrt{x^{2} - x + 1} E\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{6 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x^{3} + 1\right )} + \frac{5 \sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x + 1\right )^{\frac{3}{2}} \sqrt{x^{2} - x + 1} F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{9 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x^{3} + 1\right )} - \frac{5 \sqrt{x + 1} \sqrt{x^{2} - x + 1}}{3 x} + \frac{2 \sqrt{x + 1} \sqrt{x^{2} - x + 1}}{3 x \left (x^{3} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(1+x)**(3/2)/(x**2-x+1)**(3/2),x)

[Out]

5*sqrt(x + 1)*sqrt(x**2 - x + 1)/(3*(x + 1 + sqrt(3))) - 5*3**(1/4)*sqrt((x**2 -
x + 1)/(x + 1 + sqrt(3))**2)*sqrt(-sqrt(3) + 2)*(x + 1)**(3/2)*sqrt(x**2 - x +
1)*elliptic_e(asin((x - sqrt(3) + 1)/(x + 1 + sqrt(3))), -7 - 4*sqrt(3))/(6*sqrt
((x + 1)/(x + 1 + sqrt(3))**2)*(x**3 + 1)) + 5*sqrt(2)*3**(3/4)*sqrt((x**2 - x +
1)/(x + 1 + sqrt(3))**2)*(x + 1)**(3/2)*sqrt(x**2 - x + 1)*elliptic_f(asin((x -
sqrt(3) + 1)/(x + 1 + sqrt(3))), -7 - 4*sqrt(3))/(9*sqrt((x + 1)/(x + 1 + sqrt(
3))**2)*(x**3 + 1)) - 5*sqrt(x + 1)*sqrt(x**2 - x + 1)/(3*x) + 2*sqrt(x + 1)*sqr
t(x**2 - x + 1)/(3*x*(x**3 + 1))

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Mathematica [C]  time = 1.44661, size = 409, normalized size = 1.29 \[ -\frac{5 x^3+3}{3 x \sqrt{x+1} \sqrt{x^2-x+1}}+\frac{5 (x+1)^{3/2} \left (\frac{12 \sqrt{-\frac{i}{\sqrt{3}+3 i}} \left (x^2-x+1\right )}{(x+1)^2}+\frac{i \sqrt{2} \left (\sqrt{3}+3 i\right ) \sqrt{\frac{-\frac{6 i}{x+1}+\sqrt{3}+3 i}{\sqrt{3}+3 i}} \sqrt{\frac{\frac{6 i}{x+1}+\sqrt{3}-3 i}{\sqrt{3}-3 i}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right )|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right )}{\sqrt{x+1}}+\frac{3 \sqrt{2} \left (1-i \sqrt{3}\right ) \sqrt{\frac{-\frac{6 i}{x+1}+\sqrt{3}+3 i}{\sqrt{3}+3 i}} \sqrt{\frac{\frac{6 i}{x+1}+\sqrt{3}-3 i}{\sqrt{3}-3 i}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right )|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right )}{\sqrt{x+1}}\right )}{36 \sqrt{-\frac{i}{\sqrt{3}+3 i}} \sqrt{x^2-x+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(1 + x)^(3/2)*(1 - x + x^2)^(3/2)),x]

[Out]

-(3 + 5*x^3)/(3*x*Sqrt[1 + x]*Sqrt[1 - x + x^2]) + (5*(1 + x)^(3/2)*((12*Sqrt[(-
I)/(3*I + Sqrt[3])]*(1 - x + x^2))/(1 + x)^2 + (3*Sqrt[2]*(1 - I*Sqrt[3])*Sqrt[(
3*I + Sqrt[3] - (6*I)/(1 + x))/(3*I + Sqrt[3])]*Sqrt[(-3*I + Sqrt[3] + (6*I)/(1
+ x))/(-3*I + Sqrt[3])]*EllipticE[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1
+ x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[1 + x] + (I*Sqrt[2]*(3*I + Sqrt[3]
)*Sqrt[(3*I + Sqrt[3] - (6*I)/(1 + x))/(3*I + Sqrt[3])]*Sqrt[(-3*I + Sqrt[3] + (
6*I)/(1 + x))/(-3*I + Sqrt[3])]*EllipticF[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]
/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[1 + x]))/(36*Sqrt[(-I)/(3*
I + Sqrt[3])]*Sqrt[1 - x + x^2])

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Maple [A]  time = 0.046, size = 363, normalized size = 1.2 \[{\frac{1}{ \left ( 6\,{x}^{3}+6 \right ) x}\sqrt{1+x}\sqrt{{x}^{2}-x+1} \left ( 5\,i{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) \sqrt{3}x\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}+15\,\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) x-30\,\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticE} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) x-10\,{x}^{3}-6 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(1+x)^(3/2)/(x^2-x+1)^(3/2),x)

[Out]

1/6*(1+x)^(1/2)*(x^2-x+1)^(1/2)*(5*I*EllipticF((-2*(1+x)/(-3+I*3^(1/2)))^(1/2),(
-(-3+I*3^(1/2))/(I*3^(1/2)+3))^(1/2))*3^(1/2)*x*((I*3^(1/2)-2*x+1)/(I*3^(1/2)+3)
)^(1/2)*((I*3^(1/2)+2*x-1)/(-3+I*3^(1/2)))^(1/2)*(-2*(1+x)/(-3+I*3^(1/2)))^(1/2)
+15*(-2*(1+x)/(-3+I*3^(1/2)))^(1/2)*((I*3^(1/2)-2*x+1)/(I*3^(1/2)+3))^(1/2)*((I*
3^(1/2)+2*x-1)/(-3+I*3^(1/2)))^(1/2)*EllipticF((-2*(1+x)/(-3+I*3^(1/2)))^(1/2),(
-(-3+I*3^(1/2))/(I*3^(1/2)+3))^(1/2))*x-30*(-2*(1+x)/(-3+I*3^(1/2)))^(1/2)*((I*3
^(1/2)-2*x+1)/(I*3^(1/2)+3))^(1/2)*((I*3^(1/2)+2*x-1)/(-3+I*3^(1/2)))^(1/2)*Elli
pticE((-2*(1+x)/(-3+I*3^(1/2)))^(1/2),(-(-3+I*3^(1/2))/(I*3^(1/2)+3))^(1/2))*x-1
0*x^3-6)/(x^3+1)/x

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{2} - x + 1\right )}^{\frac{3}{2}}{\left (x + 1\right )}^{\frac{3}{2}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 - x + 1)^(3/2)*(x + 1)^(3/2)*x^2),x, algorithm="maxima")

[Out]

integrate(1/((x^2 - x + 1)^(3/2)*(x + 1)^(3/2)*x^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (x^{5} + x^{2}\right )} \sqrt{x^{2} - x + 1} \sqrt{x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 - x + 1)^(3/2)*(x + 1)^(3/2)*x^2),x, algorithm="fricas")

[Out]

integral(1/((x^5 + x^2)*sqrt(x^2 - x + 1)*sqrt(x + 1)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \left (x + 1\right )^{\frac{3}{2}} \left (x^{2} - x + 1\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(1+x)**(3/2)/(x**2-x+1)**(3/2),x)

[Out]

Integral(1/(x**2*(x + 1)**(3/2)*(x**2 - x + 1)**(3/2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{2} - x + 1\right )}^{\frac{3}{2}}{\left (x + 1\right )}^{\frac{3}{2}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 - x + 1)^(3/2)*(x + 1)^(3/2)*x^2),x, algorithm="giac")

[Out]

integrate(1/((x^2 - x + 1)^(3/2)*(x + 1)^(3/2)*x^2), x)