3.502 \(\int \frac{(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}}{x^3} \, dx\)

Optimal. Leaf size=175 \[ \frac{9}{10} x \sqrt{x^2-x+1} \sqrt{x+1}-\frac{\sqrt{x^2-x+1} \left (x^3+1\right ) \sqrt{x+1}}{2 x^2}+\frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} (x+1)^{3/2} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{10 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )} \]

[Out]

(9*x*Sqrt[1 + x]*Sqrt[1 - x + x^2])/10 - (Sqrt[1 + x]*Sqrt[1 - x + x^2]*(1 + x^3
))/(2*x^2) + (9*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]])/(10*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*(1 + x^3))

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Rubi [A]  time = 0.144037, antiderivative size = 175, 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 \[ \frac{9}{10} x \sqrt{x^2-x+1} \sqrt{x+1}-\frac{\sqrt{x^2-x+1} \left (x^3+1\right ) \sqrt{x+1}}{2 x^2}+\frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} (x+1)^{3/2} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{10 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )} \]

Antiderivative was successfully verified.

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

[Out]

(9*x*Sqrt[1 + x]*Sqrt[1 - x + x^2])/10 - (Sqrt[1 + x]*Sqrt[1 - x + x^2]*(1 + x^3
))/(2*x^2) + (9*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]])/(10*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*(1 + x^3))

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Rubi in Sympy [A]  time = 10.9406, size = 155, normalized size = 0.89 \[ \frac{9 x \sqrt{x + 1} \sqrt{x^{2} - x + 1}}{10} + \frac{9 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (x + 1\right )^{\frac{3}{2}} \sqrt{x^{2} - x + 1} F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{10 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x^{3} + 1\right )} - \frac{\sqrt{x + 1} \left (x^{3} + 1\right ) \sqrt{x^{2} - x + 1}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9*x*sqrt(x + 1)*sqrt(x**2 - x + 1)/10 + 9*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))/(10*sqrt((x + 1)/(x + 1 +
sqrt(3))**2)*(x**3 + 1)) - sqrt(x + 1)*(x**3 + 1)*sqrt(x**2 - x + 1)/(2*x**2)

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Mathematica [C]  time = 0.646539, size = 192, normalized size = 1.1 \[ \frac{\sqrt{x+1} \left (\frac{2 \left (x^2-x+1\right ) \left (4 x^3-5\right )}{x^2}-\frac{27 i \sqrt{2} \sqrt{\frac{-2 i x+\sqrt{3}+i}{\sqrt{3}+3 i}} \sqrt{\frac{2 i x+\sqrt{3}-i}{\sqrt{3}-3 i}} F\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{i (x+1)}{3 i+\sqrt{3}}}\right )|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right )}{\sqrt{-\frac{i (x+1)}{\sqrt{3}+3 i}}}\right )}{20 \sqrt{x^2-x+1}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 + x]*((2*(1 - x + x^2)*(-5 + 4*x^3))/x^2 - ((27*I)*Sqrt[2]*Sqrt[(I + Sqr
t[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])])/Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]))/(20*Sqrt[1 - x + x^
2])

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Maple [A]  time = 0.019, size = 264, normalized size = 1.5 \[ -{\frac{1}{ \left ( 20\,{x}^{3}+20 \right ){x}^{2}}\sqrt{1+x}\sqrt{{x}^{2}-x+1} \left ( 27\,i\sqrt{3}\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ){x}^{2}-81\,\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ){x}^{2}-8\,{x}^{6}+2\,{x}^{3}+10 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*(1+x)^(1/2)*(x^2-x+1)^(1/2)*(27*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)
)*x^2-81*(-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^2-8*x^6+2*x^3+10)/(x^3+1)/x^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x^{2} - x + 1\right )}^{\frac{3}{2}}{\left (x + 1\right )}^{\frac{3}{2}}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 - x + 1)^(3/2)*(x + 1)^(3/2)/x^3,x, algorithm="maxima")

[Out]

integrate((x^2 - x + 1)^(3/2)*(x + 1)^(3/2)/x^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (x^{3} + 1\right )} \sqrt{x^{2} - x + 1} \sqrt{x + 1}}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 - x + 1)^(3/2)*(x + 1)^(3/2)/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x + 1\right )^{\frac{3}{2}} \left (x^{2} - x + 1\right )^{\frac{3}{2}}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 - x + 1)^(3/2)*(x + 1)^(3/2)/x^3,x, algorithm="giac")

[Out]

integrate((x^2 - x + 1)^(3/2)*(x + 1)^(3/2)/x^3, x)