3.2301 \(\int \sqrt{1+x} (a+b x) \sqrt{1-x+x^2} \, dx\)

Optimal. Leaf size=326 \[ \frac{2}{35} \sqrt{x^2-x+1} \sqrt{x+1} \left (7 a x+5 b x^2\right )+\frac{2\ 3^{3/4} \sqrt{2+\sqrt{3}} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} (x+1)^{3/2} \left (7 a-5 \left (1-\sqrt{3}\right ) b\right ) F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{35 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )}+\frac{6 b \sqrt{x^2-x+1} \sqrt{x+1}}{7 \left (x+\sqrt{3}+1\right )}-\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} (x+1)^{3/2} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{7 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )} \]

[Out]

(6*b*Sqrt[1 + x]*Sqrt[1 - x + x^2])/(7*(1 + Sqrt[3] + x)) + (2*Sqrt[1 + x]*Sqrt[
1 - x + x^2]*(7*a*x + 5*b*x^2))/35 - (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*b*(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]])/(7*Sqrt[(1 + x)/(1 + Sqrt[3]
+ x)^2]*(1 + x^3)) + (2*3^(3/4)*Sqrt[2 + Sqrt[3]]*(7*a - 5*(1 - Sqrt[3])*b)*(1
+ x)^(3/2)*Sqrt[1 - x + x^2]*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticF[A
rcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(35*Sqrt[(1 + x)/(1
+ Sqrt[3] + x)^2]*(1 + x^3))

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Rubi [A]  time = 0.297787, antiderivative size = 326, 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 \[ \frac{2}{35} \sqrt{x^2-x+1} \sqrt{x+1} \left (7 a x+5 b x^2\right )+\frac{2\ 3^{3/4} \sqrt{2+\sqrt{3}} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} (x+1)^{3/2} \left (7 a-5 \left (1-\sqrt{3}\right ) b\right ) F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{35 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )}+\frac{6 b \sqrt{x^2-x+1} \sqrt{x+1}}{7 \left (x+\sqrt{3}+1\right )}-\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} (x+1)^{3/2} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{7 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )} \]

Antiderivative was successfully verified.

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

[Out]

(6*b*Sqrt[1 + x]*Sqrt[1 - x + x^2])/(7*(1 + Sqrt[3] + x)) + (2*Sqrt[1 + x]*Sqrt[
1 - x + x^2]*(7*a*x + 5*b*x^2))/35 - (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*b*(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]])/(7*Sqrt[(1 + x)/(1 + Sqrt[3]
+ x)^2]*(1 + x^3)) + (2*3^(3/4)*Sqrt[2 + Sqrt[3]]*(7*a - 5*(1 - Sqrt[3])*b)*(1
+ x)^(3/2)*Sqrt[1 - x + x^2]*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticF[A
rcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(35*Sqrt[(1 + x)/(1
+ Sqrt[3] + x)^2]*(1 + x^3))

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Rubi in Sympy [A]  time = 27.5581, size = 294, normalized size = 0.9 \[ \frac{6 b \sqrt{x + 1} \sqrt{x^{2} - x + 1}}{7 \left (x + 1 + \sqrt{3}\right )} - \frac{3 \sqrt [4]{3} b \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (x + 1\right )^{\frac{3}{2}} \sqrt{x^{2} - x + 1} E\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{7 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x^{3} + 1\right )} + \sqrt{x + 1} \left (\frac{2 a x}{5} + \frac{2 b x^{2}}{7}\right ) \sqrt{x^{2} - x + 1} + \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (x + 1\right )^{\frac{3}{2}} \left (7 a - 5 b + 5 \sqrt{3} b\right ) \sqrt{x^{2} - x + 1} F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{35 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x^{3} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*b*sqrt(x + 1)*sqrt(x**2 - x + 1)/(7*(x + 1 + sqrt(3))) - 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))/(7*
sqrt((x + 1)/(x + 1 + sqrt(3))**2)*(x**3 + 1)) + sqrt(x + 1)*(2*a*x/5 + 2*b*x**2
/7)*sqrt(x**2 - x + 1) + 2*3**(3/4)*sqrt((x**2 - x + 1)/(x + 1 + sqrt(3))**2)*sq
rt(sqrt(3) + 2)*(x + 1)**(3/2)*(7*a - 5*b + 5*sqrt(3)*b)*sqrt(x**2 - x + 1)*elli
ptic_f(asin((x - sqrt(3) + 1)/(x + 1 + sqrt(3))), -7 - 4*sqrt(3))/(35*sqrt((x +
1)/(x + 1 + sqrt(3))**2)*(x**3 + 1))

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Mathematica [C]  time = 2.95429, size = 423, normalized size = 1.3 \[ \frac{2}{35} x \sqrt{x+1} \sqrt{x^2-x+1} (7 a+5 b x)-\frac{(x+1)^{3/2} \left (\frac{\sqrt{2} \sqrt{\frac{-\frac{6 i}{x+1}+\sqrt{3}+3 i}{\sqrt{3}+3 i}} \sqrt{\frac{\frac{6 i}{x+1}+\sqrt{3}-3 i}{\sqrt{3}-3 i}} \left (5 \left (3-i \sqrt{3}\right ) b-14 i \sqrt{3} a\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right )|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right )}{\sqrt{x+1}}-\frac{60 \sqrt{-\frac{i}{\sqrt{3}+3 i}} b \left (x^2-x+1\right )}{(x+1)^2}+\frac{15 i \sqrt{2} \left (\sqrt{3}+i\right ) b \sqrt{\frac{-\frac{6 i}{x+1}+\sqrt{3}+3 i}{\sqrt{3}+3 i}} \sqrt{\frac{\frac{6 i}{x+1}+\sqrt{3}-3 i}{\sqrt{3}-3 i}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right )|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right )}{\sqrt{x+1}}\right )}{70 \sqrt{-\frac{i}{\sqrt{3}+3 i}} \sqrt{x^2-x+1}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*Sqrt[1 + x]*(7*a + 5*b*x)*Sqrt[1 - x + x^2])/35 - ((1 + x)^(3/2)*((-60*Sqrt
[(-I)/(3*I + Sqrt[3])]*b*(1 - x + x^2))/(1 + x)^2 + ((15*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[Sqrt[(-6*I)/(3*I + Sqrt[3]
)]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[1 + x] + (Sqrt[2]*((-14*
I)*Sqrt[3]*a + 5*(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 + Sqrt[3])]*EllipticF[I*Ar
cSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])
])/Sqrt[1 + x]))/(70*Sqrt[(-I)/(3*I + Sqrt[3])]*Sqrt[1 - x + x^2])

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Maple [B]  time = 0.019, size = 596, normalized size = 1.8 \[ -{\frac{1}{35\,{x}^{3}+35}\sqrt{1+x}\sqrt{{x}^{2}-x+1} \left ( 21\,i\sqrt{3}\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) a-15\,i\sqrt{3}\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) b-10\,b{x}^{5}-63\,\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) a-45\,\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) b+90\,\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticE} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) b-14\,a{x}^{4}-10\,b{x}^{2}-14\,ax \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/35*(1+x)^(1/2)*(x^2-x+1)^(1/2)*(21*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)
)*a-15*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))*b-10*b*x^5-63*(-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-45*(-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+90*(-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))*b-14*a*x^4-10*b*x^2-14*a*x)/(x^3+1)

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

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. \[{\rm integral}\left ({\left (b x + a\right )} \sqrt{x^{2} - x + 1} \sqrt{x + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*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. \[ \int \left (a + b x\right ) \sqrt{x + 1} \sqrt{x^{2} - x + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+x)**(1/2)*(b*x+a)*(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. \[ \int{\left (b x + a\right )} \sqrt{x^{2} - x + 1} \sqrt{x + 1}\,{d x} \]

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)