3.2302 \(\int \frac{a+b x}{\sqrt{1+x} \sqrt{1-x+x^2}} \, dx\)

Optimal. Leaf size=275 \[ \frac{2 \sqrt{2+\sqrt{3}} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (a-\left (1-\sqrt{3}\right ) b\right ) F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^2-x+1}}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} b \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^2-x+1}}+\frac{2 b \left (x^3+1\right )}{\sqrt{x+1} \left (x+\sqrt{3}+1\right ) \sqrt{x^2-x+1}} \]

[Out]

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

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Rubi [A]  time = 0.238455, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{2 \sqrt{2+\sqrt{3}} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (a-\left (1-\sqrt{3}\right ) b\right ) F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^2-x+1}}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} b \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^2-x+1}}+\frac{2 b \left (x^3+1\right )}{\sqrt{x+1} \left (x+\sqrt{3}+1\right ) \sqrt{x^2-x+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.1098, size = 250, normalized size = 0.91 \[ \frac{2 b \sqrt{x + 1} \sqrt{x^{2} - x + 1}}{x + 1 + \sqrt{3}} - \frac{\sqrt [4]{3} b \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (x + 1\right )^{\frac{3}{2}} \sqrt{x^{2} - x + 1} E\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{\sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x^{3} + 1\right )} + \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (x + 1\right )^{\frac{3}{2}} \left (a - b + \sqrt{3} b\right ) \sqrt{x^{2} - x + 1} F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{3 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x^{3} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b*sqrt(x + 1)*sqrt(x**2 - x + 1)/(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))/(sqrt((x
+ 1)/(x + 1 + sqrt(3))**2)*(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 - b + sqrt(3)*b)*sqrt(x**2 - x
+ 1)*elliptic_f(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))

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Mathematica [C]  time = 2.12906, size = 389, normalized size = 1.41 \[ -\frac{(x+1)^{3/2} \left (\frac{\sqrt{2} \sqrt{\frac{-\frac{6 i}{x+1}+\sqrt{3}+3 i}{\sqrt{3}+3 i}} \sqrt{\frac{\frac{6 i}{x+1}+\sqrt{3}-3 i}{\sqrt{3}-3 i}} \left (\left (3-i \sqrt{3}\right ) b-2 i \sqrt{3} a\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right )|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right )}{\sqrt{x+1}}-\frac{12 \sqrt{-\frac{i}{\sqrt{3}+3 i}} b \left (x^2-x+1\right )}{(x+1)^2}+\frac{3 i \sqrt{2} \left (\sqrt{3}+i\right ) b \sqrt{\frac{-\frac{6 i}{x+1}+\sqrt{3}+3 i}{\sqrt{3}+3 i}} \sqrt{\frac{\frac{6 i}{x+1}+\sqrt{3}-3 i}{\sqrt{3}-3 i}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right )|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right )}{\sqrt{x+1}}\right )}{6 \sqrt{-\frac{i}{\sqrt{3}+3 i}} \sqrt{x^2-x+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.05, size = 313, normalized size = 1.1 \[{\frac{1}{{x}^{3}+1} \left ( -i{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) \sqrt{3}a+i{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) \sqrt{3}b+3\,{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) a+3\,{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) b-6\,{\it EllipticE} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) b \right ) \sqrt{1+x}\sqrt{{x}^{2}-x+1}\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-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+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+3*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*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*EllipticE((-2*(1+x)/(-3+I*3^(1/2)))^(1/2
),(-(-3+I*3^(1/2))/(I*3^(1/2)+3))^(1/2))*b)*(1+x)^(1/2)*(x^2-x+1)^(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)/(x^3+1)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{a + b x}{\sqrt{x + 1} \sqrt{x^{2} - x + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*x)/(sqrt(x + 1)*sqrt(x**2 - x + 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{b x + a}{\sqrt{x^{2} - x + 1} \sqrt{x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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