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

Optimal. Leaf size=144 \[ \frac{2}{5} x \sqrt{x^2-x+1} \sqrt{x+1}+\frac{2\ 3^{3/4} \sqrt{2+\sqrt{3}} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} (x+1)^{3/2} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{5 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )} \]

[Out]

(2*x*Sqrt[1 + x]*Sqrt[1 - x + x^2])/5 + (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]])/(5*Sqrt[(1 + x)/(1 + Sqrt[3
] + x)^2]*(1 + x^3))

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Rubi [A]  time = 0.0822216, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{2}{5} x \sqrt{x^2-x+1} \sqrt{x+1}+\frac{2\ 3^{3/4} \sqrt{2+\sqrt{3}} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} (x+1)^{3/2} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{5 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*Sqrt[1 + x]*Sqrt[1 - x + x^2])/5 + (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]])/(5*Sqrt[(1 + x)/(1 + Sqrt[3
] + x)^2]*(1 + x^3))

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Rubi in Sympy [A]  time = 7.17304, size = 128, normalized size = 0.89 \[ \frac{2 x \sqrt{x + 1} \sqrt{x^{2} - x + 1}}{5} + \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (x + 1\right )^{\frac{3}{2}} \sqrt{x^{2} - x + 1} F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{5 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x^{3} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x*sqrt(x + 1)*sqrt(x**2 - x + 1)/5 + 2*3**(3/4)*sqrt((x**2 - x + 1)/(x + 1 + s
qrt(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))/(5*sqrt((x + 1)/(x + 1 + s
qrt(3))**2)*(x**3 + 1))

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Mathematica [C]  time = 0.839915, size = 169, normalized size = 1.17 \[ \frac{2 x \sqrt{x+1} \left (x^2-x+1\right )+\frac{i (x+1) \sqrt{1+\frac{6 i}{\left (\sqrt{3}-3 i\right ) (x+1)}} \sqrt{6-\frac{36 i}{\left (\sqrt{3}+3 i\right ) (x+1)}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right )|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right )}{\sqrt{-\frac{i}{\sqrt{3}+3 i}}}}{5 \sqrt{x^2-x+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.031, size = 252, normalized size = 1.8 \[ -{\frac{1}{5\,{x}^{3}+5}\sqrt{1+x}\sqrt{{x}^{2}-x+1} \left ( 3\,i\sqrt{3}\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) -9\,\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) -2\,{x}^{4}-2\,x \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*(1+x)^(1/2)*(x^2-x+1)^(1/2)*(3*I*3^(1/2)*(-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)*E
llipticF((-2*(1+x)/(-3+I*3^(1/2)))^(1/2),(-(-3+I*3^(1/2))/(I*3^(1/2)+3))^(1/2))-
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))-2*x^4-2*x)/(x^3+1)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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