3.2303 \(\int \frac{a+b x}{(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=304 \[ \frac{2 x (a+b x)}{3 \sqrt{x+1} \sqrt{x^2-x+1}}+\frac{2 \sqrt{2+\sqrt{3}} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (a-\sqrt{3} b+b\right ) F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^2-x+1}}+\frac{\sqrt{2-\sqrt{3}} b \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{3^{3/4} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^2-x+1}}-\frac{2 b \left (x^3+1\right )}{3 \sqrt{x+1} \left (x+\sqrt{3}+1\right ) \sqrt{x^2-x+1}} \]

[Out]

(2*x*(a + b*x))/(3*Sqrt[1 + x]*Sqrt[1 - x + x^2]) - (2*b*(1 + x^3))/(3*Sqrt[1 +
x]*(1 + Sqrt[3] + x)*Sqrt[1 - x + x^2]) + (Sqrt[2 - Sqrt[3]]*b*Sqrt[1 + x]*Sqrt[
(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticE[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[
3] + x)], -7 - 4*Sqrt[3]])/(3^(3/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 - x
+ x^2]) + (2*Sqrt[2 + Sqrt[3]]*(a + b - Sqrt[3]*b)*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]])/(3*3^(1/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 - x + x^2])

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Rubi [A]  time = 0.332229, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2 x (a+b x)}{3 \sqrt{x+1} \sqrt{x^2-x+1}}+\frac{2 \sqrt{2+\sqrt{3}} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (a-\sqrt{3} b+b\right ) F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^2-x+1}}+\frac{\sqrt{2-\sqrt{3}} b \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{3^{3/4} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^2-x+1}}-\frac{2 b \left (x^3+1\right )}{3 \sqrt{x+1} \left (x+\sqrt{3}+1\right ) \sqrt{x^2-x+1}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(a + b*x))/(3*Sqrt[1 + x]*Sqrt[1 - x + x^2]) - (2*b*(1 + x^3))/(3*Sqrt[1 +
x]*(1 + Sqrt[3] + x)*Sqrt[1 - x + x^2]) + (Sqrt[2 - Sqrt[3]]*b*Sqrt[1 + x]*Sqrt[
(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticE[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[
3] + x)], -7 - 4*Sqrt[3]])/(3^(3/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 - x
+ x^2]) + (2*Sqrt[2 + Sqrt[3]]*(a + b - Sqrt[3]*b)*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]])/(3*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 = 27.4021, size = 287, normalized size = 0.94 \[ - \frac{2 b \sqrt{x + 1} \sqrt{x^{2} - x + 1}}{3 \left (x + 1 + \sqrt{3}\right )} + \frac{\sqrt [4]{3} b \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (x + 1\right )^{\frac{3}{2}} \sqrt{x^{2} - x + 1} E\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{3 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x^{3} + 1\right )} + \frac{2 x \left (a + b x\right ) \sqrt{x + 1} \sqrt{x^{2} - x + 1}}{3 \left (x^{3} + 1\right )} + \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (x + 1\right )^{\frac{3}{2}} \left (a - \sqrt{3} b + b\right ) \sqrt{x^{2} - x + 1} F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{9 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x^{3} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*b*sqrt(x + 1)*sqrt(x**2 - x + 1)/(3*(x + 1 + sqrt(3))) + 3**(1/4)*b*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))/(3*s
qrt((x + 1)/(x + 1 + sqrt(3))**2)*(x**3 + 1)) + 2*x*(a + b*x)*sqrt(x + 1)*sqrt(x
**2 - x + 1)/(3*(x**3 + 1)) + 2*3**(3/4)*sqrt((x**2 - x + 1)/(x + 1 + sqrt(3))**
2)*sqrt(sqrt(3) + 2)*(x + 1)**(3/2)*(a - sqrt(3)*b + b)*sqrt(x**2 - x + 1)*ellip
tic_f(asin((x - sqrt(3) + 1)/(x + 1 + sqrt(3))), -7 - 4*sqrt(3))/(9*sqrt((x + 1)
/(x + 1 + sqrt(3))**2)*(x**3 + 1))

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Mathematica [C]  time = 1.94125, size = 417, normalized size = 1.37 \[ \frac{12 \sqrt{-\frac{i}{\sqrt{3}+3 i}} x (a+b x)+\sqrt{2} \sqrt{\frac{-\frac{6 i}{x+1}+\sqrt{3}+3 i}{\sqrt{3}+3 i}} \sqrt{-\frac{\frac{6 i}{x+1}+\sqrt{3}-3 i}{-\sqrt{3}+3 i}} (x+1)^{3/2} \left (2 i \sqrt{3} a+\left (3-i \sqrt{3}\right ) b\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right )|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right )-12 \sqrt{-\frac{i}{\sqrt{3}+3 i}} b \left (x^2-x+1\right )+3 i \sqrt{2} \left (\sqrt{3}+i\right ) b \sqrt{\frac{-\frac{6 i}{x+1}+\sqrt{3}+3 i}{\sqrt{3}+3 i}} \sqrt{-\frac{\frac{6 i}{x+1}+\sqrt{3}-3 i}{-\sqrt{3}+3 i}} (x+1)^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right )|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right )}{18 \sqrt{-\frac{i}{\sqrt{3}+3 i}} \sqrt{x+1} \sqrt{x^2-x+1}} \]

Antiderivative was successfully verified.

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

[Out]

(12*Sqrt[(-I)/(3*I + Sqrt[3])]*x*(a + b*x) - 12*Sqrt[(-I)/(3*I + Sqrt[3])]*b*(1
- x + x^2) + (3*I)*Sqrt[2]*(I + Sqrt[3])*b*(1 + x)^(3/2)*Sqrt[(3*I + Sqrt[3] - (
6*I)/(1 + x))/(3*I + Sqrt[3])]*Sqrt[-((-3*I + Sqrt[3] + (6*I)/(1 + x))/(3*I - Sq
rt[3]))]*EllipticE[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + S
qrt[3])/(3*I - Sqrt[3])] + Sqrt[2]*((2*I)*Sqrt[3]*a + (3 - I*Sqrt[3])*b)*(1 + x)
^(3/2)*Sqrt[(3*I + Sqrt[3] - (6*I)/(1 + x))/(3*I + Sqrt[3])]*Sqrt[-((-3*I + Sqrt
[3] + (6*I)/(1 + x))/(3*I - Sqrt[3]))]*EllipticF[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sq
rt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/(18*Sqrt[(-I)/(3*I + Sqr
t[3])]*Sqrt[1 + x]*Sqrt[1 - x + x^2])

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Maple [B]  time = 0.082, size = 584, normalized size = 1.9 \[ -{\frac{1}{3\,{x}^{3}+3}\sqrt{1+x}\sqrt{{x}^{2}-x+1} \left ( i{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) \sqrt{3}a\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}+i{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) \sqrt{3}b\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}-3\,\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) a+3\,\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) b-6\,\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticE} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) b-2\,b{x}^{2}-2\,ax \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(1+x)^(1/2)*(x^2-x+1)^(1/2)*(I*EllipticF((-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)*a*(-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)+
I*EllipticF((-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)*b*(-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)-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)
*EllipticF((-2*(1+x)/(-3+I*3^(1/2)))^(1/2),(-(-3+I*3^(1/2))/(I*3^(1/2)+3))^(1/2)
)*a+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)*EllipticF((-2*(1+x)/(-3+I*3^(1/2)))^(1/2)
,(-(-3+I*3^(1/2))/(I*3^(1/2)+3))^(1/2))*b-6*(-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)*Ell
ipticE((-2*(1+x)/(-3+I*3^(1/2)))^(1/2),(-(-3+I*3^(1/2))/(I*3^(1/2)+3))^(1/2))*b-
2*b*x^2-2*a*x)/(x^3+1)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x + a)/((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. \[ \int \frac{a + b x}{\left (x + 1\right )^{\frac{3}{2}} \left (x^{2} - x + 1\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*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. \[ \int \frac{b x + a}{{\left (x^{2} - x + 1\right )}^{\frac{3}{2}}{\left (x + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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