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

Optimal. Leaf size=294 \[ \frac{2}{7} \sqrt{x+1} \sqrt{x^2-x+1} x^2+\frac{6 \sqrt{x+1} \sqrt{x^2-x+1}}{7 \left (x+\sqrt{3}+1\right )}+\frac{2 \sqrt{2} 3^{3/4} (x+1)^{3/2} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{7 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )}-\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (x+1)^{3/2} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{7 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )} \]

[Out]

(2*x^2*Sqrt[1 + x]*Sqrt[1 - x + x^2])/7 + (6*Sqrt[1 + x]*Sqrt[1 - x + x^2])/(7*(
1 + Sqrt[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]])/(7*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*(1 + x^3))
+ (2*Sqrt[2]*3^(3/4)*(1 + x)^(3/2)*Sqrt[1 - x + x^2]*Sqrt[(1 - x + x^2)/(1 + Sq
rt[3] + x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt
[3]])/(7*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*(1 + x^3))

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Rubi [A]  time = 0.200344, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{2}{7} \sqrt{x+1} \sqrt{x^2-x+1} x^2+\frac{6 \sqrt{x+1} \sqrt{x^2-x+1}}{7 \left (x+\sqrt{3}+1\right )}+\frac{2 \sqrt{2} 3^{3/4} (x+1)^{3/2} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{7 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )}-\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (x+1)^{3/2} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{7 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^2*Sqrt[1 + x]*Sqrt[1 - x + x^2])/7 + (6*Sqrt[1 + x]*Sqrt[1 - x + x^2])/(7*(
1 + Sqrt[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]])/(7*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*(1 + x^3))
+ (2*Sqrt[2]*3^(3/4)*(1 + x)^(3/2)*Sqrt[1 - x + x^2]*Sqrt[(1 - x + x^2)/(1 + Sq
rt[3] + x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt
[3]])/(7*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*(1 + x^3))

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Rubi in Sympy [A]  time = 17.1909, size = 262, normalized size = 0.89 \[ \frac{2 x^{2} \sqrt{x + 1} \sqrt{x^{2} - x + 1}}{7} + \frac{6 \sqrt{x + 1} \sqrt{x^{2} - x + 1}}{7 \left (x + 1 + \sqrt{3}\right )} - \frac{3 \sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (x + 1\right )^{\frac{3}{2}} \sqrt{x^{2} - x + 1} E\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{7 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x^{3} + 1\right )} + \frac{2 \sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x + 1\right )^{\frac{3}{2}} \sqrt{x^{2} - x + 1} F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{7 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x^{3} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**2*sqrt(x + 1)*sqrt(x**2 - x + 1)/7 + 6*sqrt(x + 1)*sqrt(x**2 - x + 1)/(7*(x
+ 1 + sqrt(3))) - 3*3**(1/4)*sqrt((x**2 - x + 1)/(x + 1 + sqrt(3))**2)*sqrt(-sq
rt(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))/(7*sqrt((x + 1)/(x + 1 + sqrt(3))**2)*(x**3 +
1)) + 2*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))/(7*sqrt((x + 1)/(x + 1 + sqrt(3))**2)*(x**3 + 1))

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Mathematica [C]  time = 0.843102, size = 347, normalized size = 1.18 \[ \frac{\sqrt{x+1} \left (4 \sqrt{-\frac{i (x+1)}{\sqrt{3}+3 i}} \left (x^2-x+1\right ) x^2+3 \sqrt{2} \left (\sqrt{3}-i\right ) \sqrt{\frac{-2 i x+\sqrt{3}+i}{\sqrt{3}+3 i}} \sqrt{\frac{2 i x+\sqrt{3}-i}{\sqrt{3}-3 i}} F\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{i (x+1)}{3 i+\sqrt{3}}}\right )|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right )-3 \sqrt{2} \left (\sqrt{3}-3 i\right ) \sqrt{\frac{-2 i x+\sqrt{3}+i}{\sqrt{3}+3 i}} \sqrt{\frac{2 i x+\sqrt{3}-i}{\sqrt{3}-3 i}} E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{i (x+1)}{3 i+\sqrt{3}}}\right )|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right )\right )}{14 \sqrt{-\frac{i (x+1)}{\sqrt{3}+3 i}} \sqrt{x^2-x+1}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 + x]*(4*x^2*Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]*(1 - x + x^2) - 3*Sqrt[
2]*(-3*I + Sqrt[3])*Sqrt[(I + Sqrt[3] - (2*I)*x)/(3*I + Sqrt[3])]*Sqrt[(-I + Sqr
t[3] + (2*I)*x)/(-3*I + Sqrt[3])]*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[((-I)*(1 + x)
)/(3*I + Sqrt[3])]], (3*I + Sqrt[3])/(3*I - Sqrt[3])] + 3*Sqrt[2]*(-I + Sqrt[3])
*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 - Sqrt[3])]))/(14*Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]*Sq
rt[1 - x + x^2])

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Maple [A]  time = 0.033, size = 361, normalized size = 1.2 \[{\frac{1}{7\,{x}^{3}+7}\sqrt{1+x}\sqrt{{x}^{2}-x+1} \left ( 3\,i\sqrt{3}\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) +2\,{x}^{5}+9\,\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) -18\,\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticE} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) +2\,{x}^{2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*(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)*El
lipticF((-2*(1+x)/(-3+I*3^(1/2)))^(1/2),(-(-3+I*3^(1/2))/(I*3^(1/2)+3))^(1/2))+2
*x^5+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))-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)*Ell
ipticE((-2*(1+x)/(-3+I*3^(1/2)))^(1/2),(-(-3+I*3^(1/2))/(I*3^(1/2)+3))^(1/2))+2*
x^2)/(x^3+1)

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

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)

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

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]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)