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

Optimal. Leaf size=323 \[ \frac{9}{7} \sqrt{x+1} \sqrt{x^2-x+1} x^2+\frac{27 \sqrt{x+1} \sqrt{x^2-x+1}}{7 \left (x+\sqrt{3}+1\right )}-\frac{\sqrt{x+1} \sqrt{x^2-x+1} \left (x^3+1\right )}{x}+\frac{9 \sqrt{2} 3^{3/4} (x+1)^{3/2} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{7 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (x+1)^{3/2} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{14 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )} \]

[Out]

(9*x^2*Sqrt[1 + x]*Sqrt[1 - x + x^2])/7 + (27*Sqrt[1 + x]*Sqrt[1 - x + x^2])/(7*
(1 + Sqrt[3] + x)) - (Sqrt[1 + x]*Sqrt[1 - x + x^2]*(1 + x^3))/x - (27*3^(1/4)*S
qrt[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]])
/(14*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*(1 + x^3)) + (9*Sqrt[2]*3^(3/4)*(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]])/(7*Sqrt[(1 + x)/(1 + Sqrt
[3] + x)^2]*(1 + x^3))

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Rubi [A]  time = 0.248497, antiderivative size = 323, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261 \[ \frac{9}{7} \sqrt{x+1} \sqrt{x^2-x+1} x^2+\frac{27 \sqrt{x+1} \sqrt{x^2-x+1}}{7 \left (x+\sqrt{3}+1\right )}-\frac{\sqrt{x+1} \sqrt{x^2-x+1} \left (x^3+1\right )}{x}+\frac{9 \sqrt{2} 3^{3/4} (x+1)^{3/2} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{7 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (x+1)^{3/2} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{14 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )} \]

Antiderivative was successfully verified.

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

[Out]

(9*x^2*Sqrt[1 + x]*Sqrt[1 - x + x^2])/7 + (27*Sqrt[1 + x]*Sqrt[1 - x + x^2])/(7*
(1 + Sqrt[3] + x)) - (Sqrt[1 + x]*Sqrt[1 - x + x^2]*(1 + x^3))/x - (27*3^(1/4)*S
qrt[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]])
/(14*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*(1 + x^3)) + (9*Sqrt[2]*3^(3/4)*(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]])/(7*Sqrt[(1 + x)/(1 + Sqrt
[3] + x)^2]*(1 + x^3))

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Rubi in Sympy [A]  time = 20.9339, size = 286, normalized size = 0.89 \[ \frac{9 x^{2} \sqrt{x + 1} \sqrt{x^{2} - x + 1}}{7} + \frac{27 \sqrt{x + 1} \sqrt{x^{2} - x + 1}}{7 \left (x + 1 + \sqrt{3}\right )} - \frac{27 \sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (x + 1\right )^{\frac{3}{2}} \sqrt{x^{2} - x + 1} E\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{14 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x^{3} + 1\right )} + \frac{9 \sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x + 1\right )^{\frac{3}{2}} \sqrt{x^{2} - x + 1} F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{7 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x^{3} + 1\right )} - \frac{\sqrt{x + 1} \left (x^{3} + 1\right ) \sqrt{x^{2} - x + 1}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9*x**2*sqrt(x + 1)*sqrt(x**2 - x + 1)/7 + 27*sqrt(x + 1)*sqrt(x**2 - x + 1)/(7*(
x + 1 + sqrt(3))) - 27*3**(1/4)*sqrt((x**2 - x + 1)/(x + 1 + sqrt(3))**2)*sqrt(-
sqrt(3) + 2)*(x + 1)**(3/2)*sqrt(x**2 - x + 1)*elliptic_e(asin((x - sqrt(3) + 1)
/(x + 1 + sqrt(3))), -7 - 4*sqrt(3))/(14*sqrt((x + 1)/(x + 1 + sqrt(3))**2)*(x**
3 + 1)) + 9*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)) - sqrt(x + 1)*(
x**3 + 1)*sqrt(x**2 - x + 1)/x

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Mathematica [C]  time = 0.845893, size = 244, normalized size = 0.76 \[ \frac{\sqrt{x+1} \left (\frac{4 \left (x^2-x+1\right ) \left (2 x^3-7\right )}{x}-\frac{27 \sqrt{2} \sqrt{\frac{2 i x+\sqrt{3}-i}{\sqrt{3}-3 i}} \left (\left (\sqrt{3}-3 i\right ) E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{i (x+1)}{3 i+\sqrt{3}}}\right )|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right )-\left (\sqrt{3}-i\right ) F\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{i (x+1)}{3 i+\sqrt{3}}}\right )|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right )\right )}{\sqrt{-\frac{i (x+1)}{-2 i x+\sqrt{3}+i}}}\right )}{28 \sqrt{x^2-x+1}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(Sqrt[1 + x]*((4*(1 - x + x^2)*(-7 + 2*x^3))/x - (27*Sqrt[2]*Sqrt[(-I + Sqrt[3]
+ (2*I)*x)/(-3*I + Sqrt[3])]*((-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])] - (-I + Sqrt[
3])*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]], (3*I + Sq
rt[3])/(3*I - Sqrt[3])]))/Sqrt[((-I)*(1 + x))/(I + Sqrt[3] - (2*I)*x)]))/(28*Sqr
t[1 - x + x^2])

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Maple [A]  time = 0.022, size = 368, normalized size = 1.1 \[{\frac{1}{14\,x \left ({x}^{3}+1 \right ) }\sqrt{1+x}\sqrt{{x}^{2}-x+1} \left ( 27\,i\sqrt{3}\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) x+4\,{x}^{6}-162\,\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticE} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) x+81\,\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) x-10\,{x}^{3}-14 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/14*(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+4*x^6-162*(-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)*EllipticE((-2*(1+x)/(-3+I*3^(1/2))
)^(1/2),(-(-3+I*3^(1/2))/(I*3^(1/2)+3))^(1/2))*x+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-10*x^3-14)/x/(x^3+1)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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