3.520 \(\int \frac{1}{(1+x)^{5/2} \left (1-x+x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=168 \[ \frac{14 x}{27 \sqrt{x+1} \sqrt{x^2-x+1}}+\frac{14 \sqrt{2+\sqrt{3}} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{27 \sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^2-x+1}}+\frac{2 x}{9 \sqrt{x+1} \sqrt{x^2-x+1} \left (x^3+1\right )} \]

[Out]

(14*x)/(27*Sqrt[1 + x]*Sqrt[1 - x + x^2]) + (2*x)/(9*Sqrt[1 + x]*Sqrt[1 - x + x^
2]*(1 + x^3)) + (14*Sqrt[2 + Sqrt[3]]*Sqrt[1 + x]*Sqrt[(1 - x + x^2)/(1 + Sqrt[3
] + x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]]
)/(27*3^(1/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 - x + x^2])

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Rubi [A]  time = 0.101846, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{14 x}{27 \sqrt{x+1} \sqrt{x^2-x+1}}+\frac{14 \sqrt{2+\sqrt{3}} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{27 \sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^2-x+1}}+\frac{2 x}{9 \sqrt{x+1} \sqrt{x^2-x+1} \left (x^3+1\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((1 + x)^(5/2)*(1 - x + x^2)^(5/2)),x]

[Out]

(14*x)/(27*Sqrt[1 + x]*Sqrt[1 - x + x^2]) + (2*x)/(9*Sqrt[1 + x]*Sqrt[1 - x + x^
2]*(1 + x^3)) + (14*Sqrt[2 + Sqrt[3]]*Sqrt[1 + x]*Sqrt[(1 - x + x^2)/(1 + Sqrt[3
] + x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]]
)/(27*3^(1/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 - x + x^2])

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Rubi in Sympy [A]  time = 8.15758, size = 162, normalized size = 0.96 \[ \frac{14 x \sqrt{x + 1} \sqrt{x^{2} - x + 1}}{27 \left (x^{3} + 1\right )} + \frac{2 x \sqrt{x + 1} \sqrt{x^{2} - x + 1}}{9 \left (x^{3} + 1\right )^{2}} + \frac{14 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (x + 1\right )^{\frac{3}{2}} \sqrt{x^{2} - x + 1} F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{81 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x^{3} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1+x)**(5/2)/(x**2-x+1)**(5/2),x)

[Out]

14*x*sqrt(x + 1)*sqrt(x**2 - x + 1)/(27*(x**3 + 1)) + 2*x*sqrt(x + 1)*sqrt(x**2
- x + 1)/(9*(x**3 + 1)**2) + 14*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))/(81*sqrt((x + 1)/(x + 1 + sqrt(3))*
*2)*(x**3 + 1))

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Mathematica [C]  time = 0.754859, size = 178, normalized size = 1.06 \[ \frac{\frac{6 x \left (7 x^3+10\right )}{(x+1)^{3/2} \left (x^2-x+1\right )}+\frac{7 i (x+1) \sqrt{1+\frac{6 i}{\left (\sqrt{3}-3 i\right ) (x+1)}} \sqrt{6-\frac{36 i}{\left (\sqrt{3}+3 i\right ) (x+1)}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right )|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right )}{\sqrt{-\frac{i}{\sqrt{3}+3 i}}}}{81 \sqrt{x^2-x+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((1 + x)^(5/2)*(1 - x + x^2)^(5/2)),x]

[Out]

((6*x*(10 + 7*x^3))/((1 + x)^(3/2)*(1 - x + x^2)) + ((7*I)*(1 + x)*Sqrt[1 + (6*I
)/((-3*I + Sqrt[3])*(1 + x))]*Sqrt[6 - (36*I)/((3*I + Sqrt[3])*(1 + x))]*Ellipti
cF[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - S
qrt[3])])/Sqrt[(-I)/(3*I + Sqrt[3])])/(81*Sqrt[1 - x + x^2])

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Maple [B]  time = 0.061, size = 469, normalized size = 2.8 \[ -{\frac{1}{27} \left ( 7\,i\sqrt{3}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ){x}^{3}\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}-21\,{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ){x}^{3}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}+7\,i\sqrt{3}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}-21\,\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) -14\,{x}^{4}-20\,x \right ) \left ( 1+x \right ) ^{-{\frac{3}{2}}} \left ({x}^{2}-x+1 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+x)^(5/2)/(x^2-x+1)^(5/2),x)

[Out]

-1/27*(7*I*3^(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^3*(-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)-21*EllipticF((-2*(1+x)/
(-3+I*3^(1/2)))^(1/2),(-(-3+I*3^(1/2))/(I*3^(1/2)+3))^(1/2))*x^3*((I*3^(1/2)+2*x
-1)/(-3+I*3^(1/2)))^(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)+7*I*3^(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))*(-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)-21*(-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))-14*x^4-20*x)/(x^2-x+1)^(3/2)/(1+x)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{2} - x + 1\right )}^{\frac{5}{2}}{\left (x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 - x + 1)^(5/2)*(x + 1)^(5/2)),x, algorithm="maxima")

[Out]

integrate(1/((x^2 - x + 1)^(5/2)*(x + 1)^(5/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (x^{6} + 2 \, x^{3} + 1\right )} \sqrt{x^{2} - x + 1} \sqrt{x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 - x + 1)^(5/2)*(x + 1)^(5/2)),x, algorithm="fricas")

[Out]

integral(1/((x^6 + 2*x^3 + 1)*sqrt(x^2 - x + 1)*sqrt(x + 1)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (x + 1\right )^{\frac{5}{2}} \left (x^{2} - x + 1\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1+x)**(5/2)/(x**2-x+1)**(5/2),x)

[Out]

Integral(1/((x + 1)**(5/2)*(x**2 - x + 1)**(5/2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{2} - x + 1\right )}^{\frac{5}{2}}{\left (x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 - x + 1)^(5/2)*(x + 1)^(5/2)),x, algorithm="giac")

[Out]

integrate(1/((x^2 - x + 1)^(5/2)*(x + 1)^(5/2)), x)