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

Optimal. Leaf size=325 \[ \frac{18}{91} \sqrt{x+1} \sqrt{x^2-x+1} x^2+\frac{54 \sqrt{x+1} \sqrt{x^2-x+1}}{91 \left (x+\sqrt{3}+1\right )}+\frac{2}{13} \sqrt{x+1} \sqrt{x^2-x+1} \left (x^3+1\right ) x^2+\frac{18 \sqrt{2} 3^{3/4} (x+1)^{3/2} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{91 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (x+1)^{3/2} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{91 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )} \]

[Out]

(18*x^2*Sqrt[1 + x]*Sqrt[1 - x + x^2])/91 + (54*Sqrt[1 + x]*Sqrt[1 - x + x^2])/(
91*(1 + Sqrt[3] + x)) + (2*x^2*Sqrt[1 + x]*Sqrt[1 - x + x^2]*(1 + x^3))/13 - (27
*3^(1/4)*Sqrt[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]])/(91*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*(1 + x^3)) + (18*Sqrt[2]*3^(3/4
)*(1 + x)^(3/2)*Sqrt[1 - x + x^2]*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*Ellipt
icF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(91*Sqrt[(1 +
x)/(1 + Sqrt[3] + x)^2]*(1 + x^3))

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Rubi [A]  time = 0.230249, antiderivative size = 325, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{18}{91} \sqrt{x+1} \sqrt{x^2-x+1} x^2+\frac{54 \sqrt{x+1} \sqrt{x^2-x+1}}{91 \left (x+\sqrt{3}+1\right )}+\frac{2}{13} \sqrt{x+1} \sqrt{x^2-x+1} \left (x^3+1\right ) x^2+\frac{18 \sqrt{2} 3^{3/4} (x+1)^{3/2} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{91 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (x+1)^{3/2} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{91 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )} \]

Antiderivative was successfully verified.

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

[Out]

(18*x^2*Sqrt[1 + x]*Sqrt[1 - x + x^2])/91 + (54*Sqrt[1 + x]*Sqrt[1 - x + x^2])/(
91*(1 + Sqrt[3] + x)) + (2*x^2*Sqrt[1 + x]*Sqrt[1 - x + x^2]*(1 + x^3))/13 - (27
*3^(1/4)*Sqrt[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]])/(91*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*(1 + x^3)) + (18*Sqrt[2]*3^(3/4
)*(1 + x)^(3/2)*Sqrt[1 - x + x^2]*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*Ellipt
icF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(91*Sqrt[(1 +
x)/(1 + Sqrt[3] + x)^2]*(1 + x^3))

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Rubi in Sympy [A]  time = 19.7817, size = 291, normalized size = 0.9 \[ \frac{2 x^{2} \sqrt{x + 1} \left (x^{3} + 1\right ) \sqrt{x^{2} - x + 1}}{13} + \frac{18 x^{2} \sqrt{x + 1} \sqrt{x^{2} - x + 1}}{91} + \frac{54 \sqrt{x + 1} \sqrt{x^{2} - x + 1}}{91 \left (x + 1 + \sqrt{3}\right )} - \frac{27 \sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (x + 1\right )^{\frac{3}{2}} \sqrt{x^{2} - x + 1} E\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{91 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x^{3} + 1\right )} + \frac{18 \sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x + 1\right )^{\frac{3}{2}} \sqrt{x^{2} - x + 1} F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{91 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x^{3} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**2*sqrt(x + 1)*(x**3 + 1)*sqrt(x**2 - x + 1)/13 + 18*x**2*sqrt(x + 1)*sqrt(x
**2 - x + 1)/91 + 54*sqrt(x + 1)*sqrt(x**2 - x + 1)/(91*(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))/(91*sqrt((x + 1)/(x + 1 + sqrt(3))**2)*(x**3 + 1)) + 18*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))/(91*sqr
t((x + 1)/(x + 1 + sqrt(3))**2)*(x**3 + 1))

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Mathematica [C]  time = 0.832645, size = 244, normalized size = 0.75 \[ \frac{\sqrt{x+1} \left (4 x^2 \left (x^2-x+1\right ) \left (7 x^3+16\right )-\frac{27 \sqrt{2} \sqrt{\frac{2 i x+\sqrt{3}-i}{\sqrt{3}-3 i}} \left (\left (\sqrt{3}-3 i\right ) E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{i (x+1)}{3 i+\sqrt{3}}}\right )|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right )-\left (\sqrt{3}-i\right ) F\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{i (x+1)}{3 i+\sqrt{3}}}\right )|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right )\right )}{\sqrt{-\frac{i (x+1)}{-2 i x+\sqrt{3}+i}}}\right )}{182 \sqrt{x^2-x+1}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(Sqrt[1 + x]*(4*x^2*(1 - x + x^2)*(16 + 7*x^3) - (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)]))/(182*Sq
rt[1 - x + x^2])

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Maple [A]  time = 0.034, size = 366, normalized size = 1.1 \[{\frac{1}{91\,{x}^{3}+91}\sqrt{1+x}\sqrt{{x}^{2}-x+1} \left ( 14\,{x}^{8}+27\,i\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) \sqrt{3}+46\,{x}^{5}+81\,\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) -162\,\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticE} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) +32\,{x}^{2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/91*(1+x)^(1/2)*(x^2-x+1)^(1/2)*(14*x^8+27*I*(-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)*E
llipticF((-2*(1+x)/(-3+I*3^(1/2)))^(1/2),(-(-3+I*3^(1/2))/(I*3^(1/2)+3))^(1/2))*
3^(1/2)+46*x^5+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))-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))+32*x^2)/(x^3+1)

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

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 [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (x^{4} + x\right )} \sqrt{x^{2} - x + 1} \sqrt{x + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((x^4 + x)*sqrt(x^2 - x + 1)*sqrt(x + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)