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

Optimal. Leaf size=287 \[ -\frac{\sqrt{x^2-x+1} \sqrt{x+1}}{x}+\frac{3 \sqrt{x^2-x+1} \sqrt{x+1}}{x+\sqrt{3}+1}+\frac{\sqrt{2} 3^{3/4} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} (x+1)^{3/2} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )}-\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} (x+1)^{3/2} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{2 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )} \]

[Out]

-((Sqrt[1 + x]*Sqrt[1 - x + x^2])/x) + (3*Sqrt[1 + x]*Sqrt[1 - x + x^2])/(1 + Sq
rt[3] + x) - (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*(1 + x)^(3/2)*Sqrt[1 - x + x^2]*Sqrt[(
1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticE[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)) + (Sqr
t[2]*3^(3/4)*(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]])/(Sq
rt[(1 + x)/(1 + Sqrt[3] + x)^2]*(1 + x^3))

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Rubi [A]  time = 0.21022, antiderivative size = 287, 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 \[ -\frac{\sqrt{x^2-x+1} \sqrt{x+1}}{x}+\frac{3 \sqrt{x^2-x+1} \sqrt{x+1}}{x+\sqrt{3}+1}+\frac{\sqrt{2} 3^{3/4} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} (x+1)^{3/2} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )}-\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} (x+1)^{3/2} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{2 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[1 + x]*Sqrt[1 - x + x^2])/x) + (3*Sqrt[1 + x]*Sqrt[1 - x + x^2])/(1 + Sq
rt[3] + x) - (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*(1 + x)^(3/2)*Sqrt[1 - x + x^2]*Sqrt[(
1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticE[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)) + (Sqr
t[2]*3^(3/4)*(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]])/(Sq
rt[(1 + x)/(1 + Sqrt[3] + x)^2]*(1 + x^3))

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Rubi in Sympy [A]  time = 18.3855, size = 250, normalized size = 0.87 \[ \frac{3 \sqrt{x + 1} \sqrt{x^{2} - x + 1}}{x + 1 + \sqrt{3}} - \frac{3 \sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (x + 1\right )^{\frac{3}{2}} \sqrt{x^{2} - x + 1} E\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{2 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x^{3} + 1\right )} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x + 1\right )^{\frac{3}{2}} \sqrt{x^{2} - x + 1} F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{\sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x^{3} + 1\right )} - \frac{\sqrt{x + 1} \sqrt{x^{2} - x + 1}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*sqrt(x + 1)*sqrt(x**2 - x + 1)/(x + 1 + sqrt(3)) - 3*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)*e
lliptic_e(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(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))/(sqrt((x + 1)/(x + 1 + sqrt(3))**2)*
(x**3 + 1)) - sqrt(x + 1)*sqrt(x**2 - x + 1)/x

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Mathematica [C]  time = 0.684009, size = 349, normalized size = 1.22 \[ -\frac{\sqrt{x+1} \sqrt{x^2-x+1}}{x}+\frac{3 \sqrt{1+\frac{2 i (x+1)}{\sqrt{3}-3 i}} \sqrt{1-\frac{2 i (x+1)}{\sqrt{3}+3 i}} \left (\frac{\left (\sqrt{3}-i\right ) \sqrt{-\frac{i}{\sqrt{3}+3 i}} \sqrt{x+1} F\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{i (x+1)}{3 i+\sqrt{3}}}\right )|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right )}{\sqrt{-\frac{i (x+1)}{\sqrt{3}+3 i}}}-\frac{\left (\sqrt{3}-3 i\right ) \sqrt{-\frac{i}{\sqrt{3}+3 i}} \sqrt{x+1} E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{i (x+1)}{3 i+\sqrt{3}}}\right )|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right )}{\sqrt{-\frac{i (x+1)}{\sqrt{3}+3 i}}}\right )}{2 \sqrt{2} \sqrt{-\frac{i}{\sqrt{3}+3 i}} \sqrt{(x+1)^2-3 (x+1)+3}} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[1 + x]*Sqrt[1 - x + x^2])/x) + (3*Sqrt[1 + ((2*I)*(1 + x))/(-3*I + Sqrt[
3])]*Sqrt[1 - ((2*I)*(1 + x))/(3*I + Sqrt[3])]*(-(((-3*I + Sqrt[3])*Sqrt[(-I)/(3
*I + Sqrt[3])]*Sqrt[1 + x]*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[((-I)*(1 + x))/(3*I
+ Sqrt[3])]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[((-I)*(1 + x))/(3*I + Sqrt[
3])]) + ((-I + Sqrt[3])*Sqrt[(-I)/(3*I + Sqrt[3])]*Sqrt[1 + x]*EllipticF[I*ArcSi
nh[Sqrt[2]*Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]], (3*I + Sqrt[3])/(3*I - Sqrt[3]
)])/Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]))/(2*Sqrt[2]*Sqrt[(-I)/(3*I + Sqrt[3])]
*Sqrt[3 - 3*(1 + x) + (1 + x)^2])

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Maple [A]  time = 0.036, size = 363, normalized size = 1.3 \[{\frac{1}{2\,x \left ({x}^{3}+1 \right ) }\sqrt{1+x}\sqrt{{x}^{2}-x+1} \left ( 3\,i\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) \sqrt{3}x+9\,\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) x-18\,\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticE} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) x-2\,{x}^{3}-2 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(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
+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-18*(-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)*Ellip
ticE((-2*(1+x)/(-3+I*3^(1/2)))^(1/2),(-(-3+I*3^(1/2))/(I*3^(1/2)+3))^(1/2))*x-2*
x^3-2)/x/(x^3+1)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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