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

Optimal. Leaf size=170 \[ \frac{6}{55} \sqrt{x+1} \sqrt{x^2-x+1} x+\frac{2}{11} \sqrt{x+1} \sqrt{x^2-x+1} x^4-\frac{4\ 3^{3/4} \sqrt{2+\sqrt{3}} (x+1)^{3/2} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{55 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )} \]

[Out]

(6*x*Sqrt[1 + x]*Sqrt[1 - x + x^2])/55 + (2*x^4*Sqrt[1 + x]*Sqrt[1 - x + x^2])/1
1 - (4*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]])/(55*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*(1 + x^3))

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Rubi [A]  time = 0.157675, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{6}{55} \sqrt{x+1} \sqrt{x^2-x+1} x+\frac{2}{11} \sqrt{x+1} \sqrt{x^2-x+1} x^4-\frac{4\ 3^{3/4} \sqrt{2+\sqrt{3}} (x+1)^{3/2} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{55 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )} \]

Antiderivative was successfully verified.

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

[Out]

(6*x*Sqrt[1 + x]*Sqrt[1 - x + x^2])/55 + (2*x^4*Sqrt[1 + x]*Sqrt[1 - x + x^2])/1
1 - (4*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]])/(55*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*(1 + x^3))

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Rubi in Sympy [A]  time = 12.4889, size = 151, normalized size = 0.89 \[ \frac{2 x^{4} \sqrt{x + 1} \sqrt{x^{2} - x + 1}}{11} + \frac{6 x \sqrt{x + 1} \sqrt{x^{2} - x + 1}}{55} - \frac{4 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (x + 1\right )^{\frac{3}{2}} \sqrt{x^{2} - x + 1} F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{55 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x^{3} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**4*sqrt(x + 1)*sqrt(x**2 - x + 1)/11 + 6*x*sqrt(x + 1)*sqrt(x**2 - x + 1)/55
- 4*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))/(55*sqrt((x + 1)/(x + 1 + sqrt(3))**2)*(x**3 + 1))

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Mathematica [C]  time = 1.43432, size = 221, normalized size = 1.3 \[ \frac{2 \left (x \sqrt{x+1} \left (5 x^5-5 x^4+5 x^3+3 x^2-3 x+3\right )+\sqrt{-\frac{6 i}{\sqrt{3}+3 i}} \left (\sqrt{3}+3 i\right ) (x+1) \sqrt{\frac{\left (\sqrt{3}-3 i\right ) x+\sqrt{3}+3 i}{\left (\sqrt{3}-3 i\right ) (x+1)}} \sqrt{\frac{\left (\sqrt{3}+3 i\right ) x+\sqrt{3}-3 i}{\left (\sqrt{3}+3 i\right ) (x+1)}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right )|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right )\right )}{55 \sqrt{x^2-x+1}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(x*Sqrt[1 + x]*(3 - 3*x + 3*x^2 + 5*x^3 - 5*x^4 + 5*x^5) + Sqrt[(-6*I)/(3*I +
Sqrt[3])]*(3*I + Sqrt[3])*(1 + x)*Sqrt[(3*I + Sqrt[3] + (-3*I + Sqrt[3])*x)/((-
3*I + Sqrt[3])*(1 + x))]*Sqrt[(-3*I + Sqrt[3] + (3*I + Sqrt[3])*x)/((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])]))/(55*Sqrt[1 - x + x^2])

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Maple [A]  time = 0.225, size = 257, normalized size = 1.5 \[{\frac{2}{55\,{x}^{3}+55}\sqrt{1+x}\sqrt{{x}^{2}-x+1} \left ( 5\,{x}^{7}+3\,i\sqrt{3}\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) -9\,\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) +8\,{x}^{4}+3\,x \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/55*(1+x)^(1/2)*(x^2-x+1)^(1/2)*(5*x^7+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)*EllipticF((-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))+8*x^4+3*x)/(x^3+1)

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

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. \[{\rm integral}\left (\sqrt{x^{2} - x + 1} \sqrt{x + 1} x^{3}, x\right ) \]

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. \[ \int x^{3} \sqrt{x + 1} \sqrt{x^{2} - x + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)