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

Optimal. Leaf size=66 \[ \frac{2}{3} \sqrt{x+1} \sqrt{x^2-x+1}-\frac{2 \sqrt{x+1} \sqrt{x^2-x+1} \tanh ^{-1}\left (\sqrt{x^3+1}\right )}{3 \sqrt{x^3+1}} \]

[Out]

(2*Sqrt[1 + x]*Sqrt[1 - x + x^2])/3 - (2*Sqrt[1 + x]*Sqrt[1 - x + x^2]*ArcTanh[S
qrt[1 + x^3]])/(3*Sqrt[1 + x^3])

_______________________________________________________________________________________

Rubi [A]  time = 0.0891741, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{2}{3} \sqrt{x+1} \sqrt{x^2-x+1}-\frac{2 \sqrt{x+1} \sqrt{x^2-x+1} \tanh ^{-1}\left (\sqrt{x^3+1}\right )}{3 \sqrt{x^3+1}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 + x]*Sqrt[1 - x + x^2])/3 - (2*Sqrt[1 + x]*Sqrt[1 - x + x^2]*ArcTanh[S
qrt[1 + x^3]])/(3*Sqrt[1 + x^3])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.78869, size = 58, normalized size = 0.88 \[ \frac{2 \sqrt{x + 1} \sqrt{x^{2} - x + 1}}{3} - \frac{2 \sqrt{x + 1} \sqrt{x^{2} - x + 1} \operatorname{atanh}{\left (\sqrt{x^{3} + 1} \right )}}{3 \sqrt{x^{3} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x + 1)*sqrt(x**2 - x + 1)/3 - 2*sqrt(x + 1)*sqrt(x**2 - x + 1)*atanh(sqrt
(x**3 + 1))/(3*sqrt(x**3 + 1))

_______________________________________________________________________________________

Mathematica [C]  time = 0.625561, size = 197, normalized size = 2.98 \[ \frac{\sqrt{x+1} \left (2 \left (x^2-x+1\right )+\frac{3 i \sqrt{2} \sqrt{\frac{-2 i x+\sqrt{3}+i}{\sqrt{3}+3 i}} \sqrt{\frac{2 i x+\sqrt{3}-i}{\sqrt{3}-3 i}} \Pi \left (\frac{3}{2}-\frac{i \sqrt{3}}{2};i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{i (x+1)}{3 i+\sqrt{3}}}\right )|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right )}{\sqrt{-\frac{i (x+1)}{\sqrt{3}+3 i}}}\right )}{3 \sqrt{x^2-x+1}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 + x]*(2*(1 - x + 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])]*EllipticPi[3/2 - (I
/2)*Sqrt[3], I*ArcSinh[Sqrt[2]*Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]], (3*I + Sqr
t[3])/(3*I - Sqrt[3])])/Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]))/(3*Sqrt[1 - x + x
^2])

_______________________________________________________________________________________

Maple [A]  time = 0.031, size = 43, normalized size = 0.7 \[ -{\frac{2}{3}\sqrt{1+x}\sqrt{{x}^{2}-x+1} \left ( -\sqrt{{x}^{3}+1}+{\it Artanh} \left ( \sqrt{{x}^{3}+1} \right ) \right ){\frac{1}{\sqrt{{x}^{3}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(1+x)^(1/2)*(x^2-x+1)^(1/2)*(-(x^3+1)^(1/2)+arctanh((x^3+1)^(1/2)))/(x^3+1)
^(1/2)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.286542, size = 81, normalized size = 1.23 \[ \frac{2}{3} \, \sqrt{x^{2} - x + 1} \sqrt{x + 1} - \frac{1}{3} \, \log \left (\sqrt{x^{2} - x + 1} \sqrt{x + 1} + 1\right ) + \frac{1}{3} \, \log \left (\sqrt{x^{2} - x + 1} \sqrt{x + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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