3.495 \(\int \frac{\sqrt{1+x} \sqrt{1-x+x^2}}{x^3} \, dx\)

Optimal. Leaf size=146 \[ \frac{3^{3/4} \sqrt{2+\sqrt{3}} (x+1)^{3/2} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{2 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )}-\frac{\sqrt{x+1} \sqrt{x^2-x+1}}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.124963, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{3^{3/4} \sqrt{2+\sqrt{3}} (x+1)^{3/2} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{2 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )}-\frac{\sqrt{x+1} \sqrt{x^2-x+1}}{2 x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.0345, size = 126, normalized size = 0.86 \[ \frac{3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (x + 1\right )^{\frac{3}{2}} \sqrt{x^{2} - x + 1} F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{2 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x^{3} + 1\right )} - \frac{\sqrt{x + 1} \sqrt{x^{2} - x + 1}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.675258, size = 185, normalized size = 1.27 \[ \frac{\sqrt{x+1} \left (-\frac{2 \left (x^2-x+1\right )}{x^2}-\frac{3 i \sqrt{2} \sqrt{\frac{-2 i x+\sqrt{3}+i}{\sqrt{3}+3 i}} \sqrt{\frac{2 i x+\sqrt{3}-i}{\sqrt{3}-3 i}} F\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{i (x+1)}{3 i+\sqrt{3}}}\right )|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right )}{\sqrt{-\frac{i (x+1)}{\sqrt{3}+3 i}}}\right )}{4 \sqrt{x^2-x+1}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 + x]*((-2*(1 - x + x^2))/x^2 - ((3*I)*Sqrt[2]*Sqrt[(I + Sqrt[3] - (2*I)*
x)/(3*I + Sqrt[3])]*Sqrt[(-I + Sqrt[3] + (2*I)*x)/(-3*I + Sqrt[3])]*EllipticF[I*
ArcSinh[Sqrt[2]*Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]], (3*I + Sqrt[3])/(3*I - Sq
rt[3])])/Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]))/(4*Sqrt[1 - x + x^2])

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Maple [B]  time = 0.036, size = 259, normalized size = 1.8 \[ -{\frac{1}{ \left ( 4\,{x}^{3}+4 \right ){x}^{2}}\sqrt{1+x}\sqrt{{x}^{2}-x+1} \left ( 3\,i\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) \sqrt{3}{x}^{2}-9\,\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ){x}^{2}+2\,{x}^{3}+2 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(1+x)^(1/2)*(x^2-x+1)^(1/2)*(3*I*(-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))*3^(1/2)*
x^2-9*(-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^2+2*x^3+2)/(x^3+1)/x^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{x^{2} - x + 1} \sqrt{x + 1}}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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