Integrand size = 20, antiderivative size = 144 \[ \int \sqrt {1+x} \sqrt {1-x+x^2} \, dx=\frac {2}{5} x \sqrt {1+x} \sqrt {1-x+x^2}+\frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} (1+x)^{3/2} \sqrt {1-x+x^2} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{5 \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \left (1+x^3\right )} \] Output:
2/5*x*(1+x)^(1/2)*(x^2-x+1)^(1/2)+2/5*3^(3/4)*(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)*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)
Result contains complex when optimal does not.
Time = 20.56 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.17 \[ \int \sqrt {1+x} \sqrt {1-x+x^2} \, dx=\frac {2 x \sqrt {1+x} \left (1-x+x^2\right )+\frac {i (1+x) \sqrt {1+\frac {6 i}{\left (-3 i+\sqrt {3}\right ) (1+x)}} \sqrt {6-\frac {36 i}{\left (3 i+\sqrt {3}\right ) (1+x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {6 i}{3 i+\sqrt {3}}}}{\sqrt {1+x}}\right ),\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )}{\sqrt {-\frac {i}{3 i+\sqrt {3}}}}}{5 \sqrt {1-x+x^2}} \] Input:
Integrate[Sqrt[1 + x]*Sqrt[1 - x + x^2],x]
Output:
(2*x*Sqrt[1 + x]*(1 - x + x^2) + (I*(1 + x)*Sqrt[1 + (6*I)/((-3*I + Sqrt[3 ])*(1 + x))]*Sqrt[6 - (36*I)/((3*I + Sqrt[3])*(1 + x))]*EllipticF[I*ArcSin h[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3 ])])/Sqrt[(-I)/(3*I + Sqrt[3])])/(5*Sqrt[1 - x + x^2])
Time = 0.39 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1151, 748, 759}
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.
\(\displaystyle \int \sqrt {x+1} \sqrt {x^2-x+1} \, dx\) |
\(\Big \downarrow \) 1151 |
\(\displaystyle \frac {\sqrt {x+1} \sqrt {x^2-x+1} \int \sqrt {x^3+1}dx}{\sqrt {x^3+1}}\) |
\(\Big \downarrow \) 748 |
\(\displaystyle \frac {\sqrt {x+1} \sqrt {x^2-x+1} \left (\frac {3}{5} \int \frac {1}{\sqrt {x^3+1}}dx+\frac {2}{5} \sqrt {x^3+1} x\right )}{\sqrt {x^3+1}}\) |
\(\Big \downarrow \) 759 |
\(\displaystyle \frac {\sqrt {x+1} \sqrt {x^2-x+1} \left (\frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{5 \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}+\frac {2}{5} \sqrt {x^3+1} x\right )}{\sqrt {x^3+1}}\) |
Input:
Int[Sqrt[1 + x]*Sqrt[1 - x + x^2],x]
Output:
(Sqrt[1 + x]*Sqrt[1 - x + x^2]*((2*x*Sqrt[1 + x^3])/5 + (2*3^(3/4)*Sqrt[2 + Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticF[ArcSi n[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(5*Sqrt[(1 + x)/( 1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3])))/Sqrt[1 + x^3]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Simp[a*n*(p/(n*p + 1)) Int[(a + b*x^n)^(p - 1), x], x] /; Fre eQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || LtQ[Denominat or[p + 1/n], Denominator[p]])
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]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Sy mbol] :> Simp[(d + e*x)^FracPart[p]*((a + b*x + c*x^2)^FracPart[p]/(a*d + c *e*x^3)^FracPart[p]) Int[(d + e*x)^(m - p)*(a*d + c*e*x^3)^p, x], x] /; F reeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*d + a*e, 0] && EqQ[c*d + b*e, 0] && IGtQ[m - p + 1, 0] && !IntegerQ[p]
Time = 3.00 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.09
method | result | size |
elliptic | \(\frac {\sqrt {\left (x +1\right ) \left (x^{2}-x +1\right )}\, \left (\frac {2 x \sqrt {x^{3}+1}}{5}+\frac {6 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{5 \sqrt {x^{3}+1}}\right )}{\sqrt {x +1}\, \sqrt {x^{2}-x +1}}\) | \(157\) |
risch | \(\frac {2 x \sqrt {x +1}\, \sqrt {x^{2}-x +1}}{5}+\frac {6 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right ) \sqrt {\left (x +1\right ) \left (x^{2}-x +1\right )}}{5 \sqrt {x^{3}+1}\, \sqrt {x +1}\, \sqrt {x^{2}-x +1}}\) | \(164\) |
default | \(-\frac {\sqrt {x +1}\, \sqrt {x^{2}-x +1}\, \left (3 i \sqrt {3}\, \sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-3}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right )-9 \sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-3}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right )-2 x^{4}-2 x \right )}{5 \left (x^{3}+1\right )}\) | \(252\) |
Input:
int((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)*(2/5*x*(x^3+1)^(1/2)+6 /5*(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)*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)))
Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.17 \[ \int \sqrt {1+x} \sqrt {1-x+x^2} \, dx=\frac {2}{5} \, \sqrt {x^{2} - x + 1} \sqrt {x + 1} x + \frac {6}{5} \, {\rm weierstrassPInverse}\left (0, -4, x\right ) \] Input:
integrate((1+x)^(1/2)*(x^2-x+1)^(1/2),x, algorithm="fricas")
Output:
2/5*sqrt(x^2 - x + 1)*sqrt(x + 1)*x + 6/5*weierstrassPInverse(0, -4, x)
\[ \int \sqrt {1+x} \sqrt {1-x+x^2} \, dx=\int \sqrt {x + 1} \sqrt {x^{2} - x + 1}\, dx \] Input:
integrate((1+x)**(1/2)*(x**2-x+1)**(1/2),x)
Output:
Integral(sqrt(x + 1)*sqrt(x**2 - x + 1), x)
\[ \int \sqrt {1+x} \sqrt {1-x+x^2} \, dx=\int { \sqrt {x^{2} - x + 1} \sqrt {x + 1} \,d x } \] Input:
integrate((1+x)^(1/2)*(x^2-x+1)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(x^2 - x + 1)*sqrt(x + 1), x)
\[ \int \sqrt {1+x} \sqrt {1-x+x^2} \, dx=\int { \sqrt {x^{2} - x + 1} \sqrt {x + 1} \,d x } \] Input:
integrate((1+x)^(1/2)*(x^2-x+1)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(x^2 - x + 1)*sqrt(x + 1), x)
Timed out. \[ \int \sqrt {1+x} \sqrt {1-x+x^2} \, dx=\int \sqrt {x+1}\,\sqrt {x^2-x+1} \,d x \] Input:
int((x + 1)^(1/2)*(x^2 - x + 1)^(1/2),x)
Output:
int((x + 1)^(1/2)*(x^2 - x + 1)^(1/2), x)
\[ \int \sqrt {1+x} \sqrt {1-x+x^2} \, dx=\frac {2 \sqrt {x +1}\, \sqrt {x^{2}-x +1}\, x}{5}+\frac {3 \left (\int \frac {\sqrt {x +1}\, \sqrt {x^{2}-x +1}}{x^{3}+1}d x \right )}{5} \] Input:
int((1+x)^(1/2)*(x^2-x+1)^(1/2),x)
Output:
(2*sqrt(x + 1)*sqrt(x**2 - x + 1)*x + 3*int((sqrt(x + 1)*sqrt(x**2 - x + 1 ))/(x**3 + 1),x))/5