\(\int x \sqrt {1+x} \sqrt {1-x+x^2} \, dx\) [474]

Optimal result

Integrand size = 21, antiderivative size = 294 \[ \int x \sqrt {1+x} \sqrt {1-x+x^2} \, dx=\frac {2}{7} x^2 \sqrt {1+x} \sqrt {1-x+x^2}+\frac {6 \sqrt {1+x} \sqrt {1-x+x^2}}{7 \left (1+\sqrt {3}+x\right )}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1+x)^{3/2} \sqrt {1-x+x^2} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} E\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{7 \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \left (1+x^3\right )}+\frac {2 \sqrt {2} 3^{3/4} (1+x)^{3/2} \sqrt {1-x+x^2} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{7 \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \left (1+x^3\right )} \] Output:

2/7*x^2*(1+x)^(1/2)*(x^2-x+1)^(1/2)+6*(1+x)^(1/2)*(x^2-x+1)^(1/2)/(7+7*3^( 
1/2)+7*x)-3/7*3^(1/4)*(1/2*6^(1/2)-1/2*2^(1/2))*(1+x)^(3/2)*(x^2-x+1)^(1/2 
)*((x^2-x+1)/(1+x+3^(1/2))^2)^(1/2)*EllipticE((1+x-3^(1/2))/(1+x+3^(1/2)), 
I*3^(1/2)+2*I)/((1+x)/(1+x+3^(1/2))^2)^(1/2)/(x^3+1)+2/7*2^(1/2)*3^(3/4)*( 
1+x)^(3/2)*(x^2-x+1)^(1/2)*((x^2-x+1)/(1+x+3^(1/2))^2)^(1/2)*EllipticF((1+ 
x-3^(1/2))/(1+x+3^(1/2)),I*3^(1/2)+2*I)/((1+x)/(1+x+3^(1/2))^2)^(1/2)/(x^3 
+1)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 20.64 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.18 \[ \int x \sqrt {1+x} \sqrt {1-x+x^2} \, dx=\frac {\sqrt {1+x} \left (4 x^2 \sqrt {-\frac {i (1+x)}{3 i+\sqrt {3}}} \left (1-x+x^2\right )-3 \sqrt {2} \left (-3 i+\sqrt {3}\right ) \sqrt {\frac {i+\sqrt {3}-2 i x}{3 i+\sqrt {3}}} \sqrt {\frac {-i+\sqrt {3}+2 i x}{-3 i+\sqrt {3}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {-\frac {i (1+x)}{3 i+\sqrt {3}}}\right )|\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )+3 \sqrt {2} \left (-i+\sqrt {3}\right ) \sqrt {\frac {i+\sqrt {3}-2 i x}{3 i+\sqrt {3}}} \sqrt {\frac {-i+\sqrt {3}+2 i x}{-3 i+\sqrt {3}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {-\frac {i (1+x)}{3 i+\sqrt {3}}}\right ),\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )\right )}{14 \sqrt {-\frac {i (1+x)}{3 i+\sqrt {3}}} \sqrt {1-x+x^2}} \] Input:

Integrate[x*Sqrt[1 + x]*Sqrt[1 - x + x^2],x]
 

Output:

(Sqrt[1 + x]*(4*x^2*Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]*(1 - x + x^2) - 3 
*Sqrt[2]*(-3*I + Sqrt[3])*Sqrt[(I + Sqrt[3] - (2*I)*x)/(3*I + Sqrt[3])]*Sq 
rt[(-I + Sqrt[3] + (2*I)*x)/(-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])] + 
3*Sqrt[2]*(-I + Sqrt[3])*Sqrt[(I + Sqrt[3] - (2*I)*x)/(3*I + Sqrt[3])]*Sqr 
t[(-I + Sqrt[3] + (2*I)*x)/(-3*I + Sqrt[3])]*EllipticF[I*ArcSinh[Sqrt[2]*S 
qrt[((-I)*(1 + x))/(3*I + Sqrt[3])]], (3*I + Sqrt[3])/(3*I - Sqrt[3])]))/( 
14*Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]*Sqrt[1 - x + x^2])
 

Rubi [A] (warning: unable to verify)

Time = 0.63 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1210, 811, 832, 759, 2416}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x \sqrt {x+1} \sqrt {x^2-x+1} \, dx\)

\(\Big \downarrow \) 1210

\(\displaystyle \frac {\sqrt {x+1} \sqrt {x^2-x+1} \int x \sqrt {x^3+1}dx}{\sqrt {x^3+1}}\)

\(\Big \downarrow \) 811

\(\displaystyle \frac {\sqrt {x+1} \sqrt {x^2-x+1} \left (\frac {3}{7} \int \frac {x}{\sqrt {x^3+1}}dx+\frac {2}{7} \sqrt {x^3+1} x^2\right )}{\sqrt {x^3+1}}\)

\(\Big \downarrow \) 832

\(\displaystyle \frac {\sqrt {x+1} \sqrt {x^2-x+1} \left (\frac {3}{7} \left (\int \frac {x-\sqrt {3}+1}{\sqrt {x^3+1}}dx-\left (1-\sqrt {3}\right ) \int \frac {1}{\sqrt {x^3+1}}dx\right )+\frac {2}{7} \sqrt {x^3+1} x^2\right )}{\sqrt {x^3+1}}\)

\(\Big \downarrow \) 759

\(\displaystyle \frac {\sqrt {x+1} \sqrt {x^2-x+1} \left (\frac {3}{7} \left (\int \frac {x-\sqrt {3}+1}{\sqrt {x^3+1}}dx-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\right )+\frac {2}{7} \sqrt {x^3+1} x^2\right )}{\sqrt {x^3+1}}\)

\(\Big \downarrow \) 2416

\(\displaystyle \frac {\sqrt {x+1} \sqrt {x^2-x+1} \left (\frac {3}{7} \left (-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}+\frac {2 \sqrt {x^3+1}}{x+\sqrt {3}+1}\right )+\frac {2}{7} \sqrt {x^3+1} x^2\right )}{\sqrt {x^3+1}}\)

Input:

Int[x*Sqrt[1 + x]*Sqrt[1 - x + x^2],x]
 

Output:

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

Defintions of rubi rules used

rule 759

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* 
((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s 
+ r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & 
& PosQ[a]
 

rule 811

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* 
x)^(m + 1)*((a + b*x^n)^p/(c*(m + n*p + 1))), x] + Simp[a*n*(p/(m + n*p + 1 
))   Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && I 
GtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m 
, p, x]
 

rule 832

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] 
], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 - Sqrt[3]))*(s/r)   Int[1/Sqrt[a + b*x 
^3], x], x] + Simp[1/r   Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x 
]] /; FreeQ[{a, b}, x] && PosQ[a]
 

rule 1210

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^FracPart[p]*((a + b*x + 
c*x^2)^FracPart[p]/(a*d + c*e*x^3)^FracPart[p])   Int[(f + g*x)^n*(a*d + c* 
e*x^3)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[b*d + a*e, 
 0] && EqQ[c*d + b*e, 0] && EqQ[m, p]
 

rule 2416

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Simplify[(1 - Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 - Sqrt[3])*(d/c) 
]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x))), x] - S 
imp[3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( 
(1 + Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt 
[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3]) 
*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && Eq 
Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
 

Maple [A] (verified)

Time = 3.20 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.73

method	result	size
elliptic	\(\frac {\sqrt {\left (x +1\right ) \left (x^{2}-x +1\right )}\, \left (\frac {2 x^{2} \sqrt {x^{3}+1}}{7}+\frac {6 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )+\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )\right )}{7 \sqrt {x^{3}+1}}\right )}{\sqrt {x +1}\, \sqrt {x^{2}-x +1}}\)	\(216\)
risch	\(\frac {2 x^{2} \sqrt {x +1}\, \sqrt {x^{2}-x +1}}{7}+\frac {6 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )+\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )\right ) \sqrt {\left (x +1\right ) \left (x^{2}-x +1\right )}}{7 \sqrt {x^{3}+1}\, \sqrt {x +1}\, \sqrt {x^{2}-x +1}}\)	\(223\)
default	\(\frac {\sqrt {x +1}\, \sqrt {x^{2}-x +1}\, \left (3 i \sqrt {3}\, \sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-3}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right )+2 x^{5}-18 \sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-3}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right )+9 \sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-3}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right )+2 x^{2}\right )}{7 x^{3}+7}\)	\(361\)

Input:

int(x*(x+1)^(1/2)*(x^2-x+1)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

((x+1)*(x^2-x+1))^(1/2)/(x+1)^(1/2)/(x^2-x+1)^(1/2)*(2/7*x^2*(x^3+1)^(1/2) 
+6/7*(3/2-1/2*I*3^(1/2))*((x+1)/(3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2-1/2*I*3 
^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2+1/2*I*3^(1/2))/(-3/2+1/2*I*3^( 
1/2)))^(1/2)/(x^3+1)^(1/2)*((-3/2-1/2*I*3^(1/2))*EllipticE(((x+1)/(3/2-1/2 
*I*3^(1/2)))^(1/2),((-3/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2))+(1/2 
+1/2*I*3^(1/2))*EllipticF(((x+1)/(3/2-1/2*I*3^(1/2)))^(1/2),((-3/2+1/2*I*3 
^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.10 \[ \int x \sqrt {1+x} \sqrt {1-x+x^2} \, dx=\frac {2}{7} \, \sqrt {x^{2} - x + 1} \sqrt {x + 1} x^{2} - \frac {6}{7} \, {\rm weierstrassZeta}\left (0, -4, {\rm weierstrassPInverse}\left (0, -4, x\right )\right ) \] Input:

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

Output:

2/7*sqrt(x^2 - x + 1)*sqrt(x + 1)*x^2 - 6/7*weierstrassZeta(0, -4, weierst 
rassPInverse(0, -4, x))
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int x \sqrt {1+x} \sqrt {1-x+x^2} \, dx=\int x \sqrt {x + 1} \sqrt {x^{2} - x + 1}\, dx \] Input:

integrate(x*(1+x)**(1/2)*(x**2-x+1)**(1/2),x)
 

Output:

Integral(x*sqrt(x + 1)*sqrt(x**2 - x + 1), x)
 

Maxima [F]

\[ \int x \sqrt {1+x} \sqrt {1-x+x^2} \, dx=\int { \sqrt {x^{2} - x + 1} \sqrt {x + 1} x \,d x } \] Input:

integrate(x*(1+x)^(1/2)*(x^2-x+1)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int x \sqrt {1+x} \sqrt {1-x+x^2} \, dx=\int { \sqrt {x^{2} - x + 1} \sqrt {x + 1} x \,d x } \] Input:

integrate(x*(1+x)^(1/2)*(x^2-x+1)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int x \sqrt {1+x} \sqrt {1-x+x^2} \, dx=\int x\,\sqrt {x+1}\,\sqrt {x^2-x+1} \,d x \] Input:

int(x*(x + 1)^(1/2)*(x^2 - x + 1)^(1/2),x)
 

Output:

int(x*(x + 1)^(1/2)*(x^2 - x + 1)^(1/2), x)
 

Reduce [F]

\[ \int x \sqrt {1+x} \sqrt {1-x+x^2} \, dx=\frac {2 \sqrt {x +1}\, \sqrt {x^{2}-x +1}\, x^{2}}{7}+\frac {3 \left (\int \frac {\sqrt {x +1}\, \sqrt {x^{2}-x +1}\, x}{x^{3}+1}d x \right )}{7} \] Input:

int(x*(1+x)^(1/2)*(x^2-x+1)^(1/2),x)
 

Output:

(2*sqrt(x + 1)*sqrt(x**2 - x + 1)*x**2 + 3*int((sqrt(x + 1)*sqrt(x**2 - x 
+ 1)*x)/(x**3 + 1),x))/7