3.2300 \(\int (1+x)^{3/2} (a+b x) \left (1-x+x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=365 \[ \frac{18 \sqrt{x^2-x+1} \sqrt{x+1} \left (91 a x+55 b x^2\right )}{5005}+\frac{2}{143} \sqrt{x^2-x+1} \left (x^3+1\right ) \sqrt{x+1} \left (13 a x+11 b x^2\right )+\frac{18\ 3^{3/4} \sqrt{2+\sqrt{3}} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} (x+1)^{3/2} \left (91 a-55 \left (1-\sqrt{3}\right ) b\right ) F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{5005 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )}+\frac{54 b \sqrt{x^2-x+1} \sqrt{x+1}}{91 \left (x+\sqrt{3}+1\right )}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} (x+1)^{3/2} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{91 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )} \]

[Out]

(54*b*Sqrt[1 + x]*Sqrt[1 - x + x^2])/(91*(1 + Sqrt[3] + x)) + (18*Sqrt[1 + x]*Sq
rt[1 - x + x^2]*(91*a*x + 55*b*x^2))/5005 + (2*Sqrt[1 + x]*Sqrt[1 - x + x^2]*(13
*a*x + 11*b*x^2)*(1 + x^3))/143 - (27*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]])/(91*Sqrt[(1 + x)/(1 + Sqrt[3]
+ x)^2]*(1 + x^3)) + (18*3^(3/4)*Sqrt[2 + Sqrt[3]]*(91*a - 55*(1 - Sqrt[3])*b)*(
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]])/(5005*Sqrt[(1 + x
)/(1 + Sqrt[3] + x)^2]*(1 + x^3))

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Rubi [A]  time = 0.38049, antiderivative size = 365, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{18 \sqrt{x^2-x+1} \sqrt{x+1} \left (91 a x+55 b x^2\right )}{5005}+\frac{2}{143} \sqrt{x^2-x+1} \left (x^3+1\right ) \sqrt{x+1} \left (13 a x+11 b x^2\right )+\frac{18\ 3^{3/4} \sqrt{2+\sqrt{3}} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} (x+1)^{3/2} \left (91 a-55 \left (1-\sqrt{3}\right ) b\right ) F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{5005 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )}+\frac{54 b \sqrt{x^2-x+1} \sqrt{x+1}}{91 \left (x+\sqrt{3}+1\right )}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} (x+1)^{3/2} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{91 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )} \]

Antiderivative was successfully verified.

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

[Out]

(54*b*Sqrt[1 + x]*Sqrt[1 - x + x^2])/(91*(1 + Sqrt[3] + x)) + (18*Sqrt[1 + x]*Sq
rt[1 - x + x^2]*(91*a*x + 55*b*x^2))/5005 + (2*Sqrt[1 + x]*Sqrt[1 - x + x^2]*(13
*a*x + 11*b*x^2)*(1 + x^3))/143 - (27*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]])/(91*Sqrt[(1 + x)/(1 + Sqrt[3]
+ x)^2]*(1 + x^3)) + (18*3^(3/4)*Sqrt[2 + Sqrt[3]]*(91*a - 55*(1 - Sqrt[3])*b)*(
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]])/(5005*Sqrt[(1 + x
)/(1 + Sqrt[3] + x)^2]*(1 + x^3))

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Rubi in Sympy [A]  time = 34.5167, size = 335, normalized size = 0.92 \[ \frac{54 b \sqrt{x + 1} \sqrt{x^{2} - x + 1}}{91 \left (x + 1 + \sqrt{3}\right )} - \frac{27 \sqrt [4]{3} b \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (x + 1\right )^{\frac{3}{2}} \sqrt{x^{2} - x + 1} E\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{91 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x^{3} + 1\right )} + \sqrt{x + 1} \left (x^{3} + 1\right ) \left (\frac{2 a x}{11} + \frac{2 b x^{2}}{13}\right ) \sqrt{x^{2} - x + 1} + \frac{9 \sqrt{x + 1} \left (\frac{4 a x}{55} + \frac{4 b x^{2}}{91}\right ) \sqrt{x^{2} - x + 1}}{2} + \frac{18 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (x + 1\right )^{\frac{3}{2}} \left (91 a - 55 b + 55 \sqrt{3} b\right ) \sqrt{x^{2} - x + 1} F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{5005 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x^{3} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

54*b*sqrt(x + 1)*sqrt(x**2 - x + 1)/(91*(x + 1 + sqrt(3))) - 27*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))/
(91*sqrt((x + 1)/(x + 1 + sqrt(3))**2)*(x**3 + 1)) + sqrt(x + 1)*(x**3 + 1)*(2*a
*x/11 + 2*b*x**2/13)*sqrt(x**2 - x + 1) + 9*sqrt(x + 1)*(4*a*x/55 + 4*b*x**2/91)
*sqrt(x**2 - x + 1)/2 + 18*3**(3/4)*sqrt((x**2 - x + 1)/(x + 1 + sqrt(3))**2)*sq
rt(sqrt(3) + 2)*(x + 1)**(3/2)*(91*a - 55*b + 55*sqrt(3)*b)*sqrt(x**2 - x + 1)*e
lliptic_f(asin((x - sqrt(3) + 1)/(x + 1 + sqrt(3))), -7 - 4*sqrt(3))/(5005*sqrt(
(x + 1)/(x + 1 + sqrt(3))**2)*(x**3 + 1))

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Mathematica [C]  time = 3.25096, size = 437, normalized size = 1.2 \[ \frac{2 x \sqrt{x+1} \sqrt{x^2-x+1} \left (91 a \left (5 x^3+14\right )+55 b x \left (7 x^3+16\right )\right )}{5005}-\frac{9 (x+1)^{3/2} \left (\frac{\sqrt{2} \sqrt{\frac{-\frac{6 i}{x+1}+\sqrt{3}+3 i}{\sqrt{3}+3 i}} \sqrt{\frac{\frac{6 i}{x+1}+\sqrt{3}-3 i}{\sqrt{3}-3 i}} \left (55 \left (3-i \sqrt{3}\right ) b-182 i \sqrt{3} a\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right )|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right )}{\sqrt{x+1}}-\frac{660 \sqrt{-\frac{i}{\sqrt{3}+3 i}} b \left (x^2-x+1\right )}{(x+1)^2}+\frac{165 i \sqrt{2} \left (\sqrt{3}+i\right ) b \sqrt{\frac{-\frac{6 i}{x+1}+\sqrt{3}+3 i}{\sqrt{3}+3 i}} \sqrt{\frac{\frac{6 i}{x+1}+\sqrt{3}-3 i}{\sqrt{3}-3 i}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right )|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right )}{\sqrt{x+1}}\right )}{10010 \sqrt{-\frac{i}{\sqrt{3}+3 i}} \sqrt{x^2-x+1}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*Sqrt[1 + x]*Sqrt[1 - x + x^2]*(91*a*(14 + 5*x^3) + 55*b*x*(16 + 7*x^3)))/50
05 - (9*(1 + x)^(3/2)*((-660*Sqrt[(-I)/(3*I + Sqrt[3])]*b*(1 - x + x^2))/(1 + x)
^2 + ((165*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]*((-182*I)*Sqrt[3]*a + 55*(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*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 +
x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[1 + x]))/(10010*Sqrt[(-I)/(3*I + Sqr
t[3])]*Sqrt[1 - x + x^2])

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Maple [B]  time = 0.721, size = 608, normalized size = 1.7 \[ -{\frac{1}{5005\,{x}^{3}+5005}\sqrt{1+x}\sqrt{{x}^{2}-x+1} \left ( -770\,b{x}^{8}-910\,a{x}^{7}+2457\,i\sqrt{3}\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) a-1485\,i\sqrt{3}\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) b-2530\,b{x}^{5}-7371\,\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) a-4455\,\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) b+8910\,\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticE} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) b-3458\,a{x}^{4}-1760\,b{x}^{2}-2548\,ax \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5005*(1+x)^(1/2)*(x^2-x+1)^(1/2)*(-770*b*x^8-910*a*x^7+2457*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-1485*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)*Ell
ipticF((-2*(1+x)/(-3+I*3^(1/2)))^(1/2),(-(-3+I*3^(1/2))/(I*3^(1/2)+3))^(1/2))*b-
2530*b*x^5-7371*(-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-4455*(-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+8910*(-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-3458*a*x^4-1760*b*x^2-2548
*a*x)/(x^3+1)

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

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. \[{\rm integral}\left ({\left (b x^{4} + a x^{3} + b x + a\right )} \sqrt{x^{2} - x + 1} \sqrt{x + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^4 + a*x^3 + 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. \[ \int \left (a + b x\right ) \left (x + 1\right )^{\frac{3}{2}} \left (x^{2} - x + 1\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

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)