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

Optimal. Leaf size=94 \[ \frac{2}{3} \sqrt{x+1} \sqrt{x^2-x+1}+\frac{2}{9} \sqrt{x+1} \sqrt{x^2-x+1} \left (x^3+1\right )-\frac{2 \sqrt{x+1} \sqrt{x^2-x+1} \tanh ^{-1}\left (\sqrt{x^3+1}\right )}{3 \sqrt{x^3+1}} \]

[Out]

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

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Rubi [A]  time = 0.10512, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{2}{3} \sqrt{x+1} \sqrt{x^2-x+1}+\frac{2}{9} \sqrt{x+1} \sqrt{x^2-x+1} \left (x^3+1\right )-\frac{2 \sqrt{x+1} \sqrt{x^2-x+1} \tanh ^{-1}\left (\sqrt{x^3+1}\right )}{3 \sqrt{x^3+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.7139, size = 83, normalized size = 0.88 \[ \frac{2 \sqrt{x + 1} \left (x^{3} + 1\right ) \sqrt{x^{2} - x + 1}}{9} + \frac{2 \sqrt{x + 1} \sqrt{x^{2} - x + 1}}{3} - \frac{2 \sqrt{x + 1} \sqrt{x^{2} - x + 1} \operatorname{atanh}{\left (\sqrt{x^{3} + 1} \right )}}{3 \sqrt{x^{3} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.467033, size = 201, normalized size = 2.14 \[ \frac{\sqrt{x+1} \left (\frac{2}{9} \left (x^2-x+1\right ) \left (x^3+4\right )+\frac{i \sqrt{2} \sqrt{\frac{-2 i x+\sqrt{3}+i}{\sqrt{3}+3 i}} \sqrt{\frac{2 i x+\sqrt{3}-i}{\sqrt{3}-3 i}} \Pi \left (\frac{3}{2}-\frac{i \sqrt{3}}{2};i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{i (x+1)}{3 i+\sqrt{3}}}\right )|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right )}{\sqrt{-\frac{i (x+1)}{\sqrt{3}+3 i}}}\right )}{\sqrt{x^2-x+1}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 + x]*((2*(1 - x + x^2)*(4 + x^3))/9 + (I*Sqrt[2]*Sqrt[(I + Sqrt[3] - (2*
I)*x)/(3*I + Sqrt[3])]*Sqrt[(-I + Sqrt[3] + (2*I)*x)/(-3*I + Sqrt[3])]*EllipticP
i[3/2 - (I/2)*Sqrt[3], 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])]))/Sqrt[1
- x + x^2]

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Maple [A]  time = 0.012, size = 57, normalized size = 0.6 \[ -{\frac{2}{9}\sqrt{1+x}\sqrt{{x}^{2}-x+1} \left ( -{x}^{3}\sqrt{{x}^{3}+1}+3\,{\it Artanh} \left ( \sqrt{{x}^{3}+1} \right ) -4\,\sqrt{{x}^{3}+1} \right ){\frac{1}{\sqrt{{x}^{3}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9*(1+x)^(1/2)*(x^2-x+1)^(1/2)*(-x^3*(x^3+1)^(1/2)+3*arctanh((x^3+1)^(1/2))-4*
(x^3+1)^(1/2))/(x^3+1)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.289817, size = 88, normalized size = 0.94 \[ \frac{2}{9} \,{\left (x^{3} + 4\right )} \sqrt{x^{2} - x + 1} \sqrt{x + 1} - \frac{1}{3} \, \log \left (\sqrt{x^{2} - x + 1} \sqrt{x + 1} + 1\right ) + \frac{1}{3} \, \log \left (\sqrt{x^{2} - x + 1} \sqrt{x + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*(x^3 + 4)*sqrt(x^2 - x + 1)*sqrt(x + 1) - 1/3*log(sqrt(x^2 - x + 1)*sqrt(x +
1) + 1) + 1/3*log(sqrt(x^2 - x + 1)*sqrt(x + 1) - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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