Integrand size = 25, antiderivative size = 275 \[ \int \frac {a+b x}{\sqrt {1+x} \sqrt {1-x+x^2}} \, dx=\frac {2 b \left (1+x^3\right )}{\sqrt {1+x} \left (1+\sqrt {3}+x\right ) \sqrt {1-x+x^2}}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} b \sqrt {1+x} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} E\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1-x+x^2}}+\frac {2 \sqrt {2+\sqrt {3}} \left (a-\left (1-\sqrt {3}\right ) b\right ) \sqrt {1+x} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1-x+x^2}} \] Output:
2*b*(x^3+1)/(1+x)^(1/2)/(1+x+3^(1/2))/(x^2-x+1)^(1/2)-3^(1/4)*(1/2*6^(1/2) -1/2*2^(1/2))*b*(1+x)^(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^ 2-x+1)^(1/2)+2/3*(1/2*6^(1/2)+1/2*2^(1/2))*(a-(1-3^(1/2))*b)*(1+x)^(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)*3^(3/4)/((1+x)/(1+x+3^(1/2))^2)^(1/2)/(x^2-x+1)^(1/2)
Result contains complex when optimal does not.
Time = 32.35 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.41 \[ \int \frac {a+b x}{\sqrt {1+x} \sqrt {1-x+x^2}} \, dx=-\frac {(1+x)^{3/2} \left (-\frac {12 \sqrt {-\frac {i}{3 i+\sqrt {3}}} b \left (1-x+x^2\right )}{(1+x)^2}+\frac {3 i \sqrt {2} \left (i+\sqrt {3}\right ) b \sqrt {\frac {3 i+\sqrt {3}-\frac {6 i}{1+x}}{3 i+\sqrt {3}}} \sqrt {\frac {-3 i+\sqrt {3}+\frac {6 i}{1+x}}{-3 i+\sqrt {3}}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {6 i}{3 i+\sqrt {3}}}}{\sqrt {1+x}}\right )|\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )}{\sqrt {1+x}}+\frac {\sqrt {2} \left (-2 i \sqrt {3} a+\left (3-i \sqrt {3}\right ) b\right ) \sqrt {\frac {3 i+\sqrt {3}-\frac {6 i}{1+x}}{3 i+\sqrt {3}}} \sqrt {\frac {-3 i+\sqrt {3}+\frac {6 i}{1+x}}{-3 i+\sqrt {3}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {6 i}{3 i+\sqrt {3}}}}{\sqrt {1+x}}\right ),\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )}{\sqrt {1+x}}\right )}{6 \sqrt {-\frac {i}{3 i+\sqrt {3}}} \sqrt {1-x+x^2}} \] Input:
Integrate[(a + b*x)/(Sqrt[1 + x]*Sqrt[1 - x + x^2]),x]
Output:
-1/6*((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[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqr t[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 + Sqrt[3])]*EllipticF[I*ArcSinh[Sqrt[( -6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqr t[1 + x]))/(Sqrt[(-I)/(3*I + Sqrt[3])]*Sqrt[1 - x + x^2])
Time = 0.66 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1210, 2417, 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.
\(\displaystyle \int \frac {a+b x}{\sqrt {x+1} \sqrt {x^2-x+1}} \, dx\) |
\(\Big \downarrow \) 1210 |
\(\displaystyle \frac {\sqrt {x^3+1} \int \frac {a+b x}{\sqrt {x^3+1}}dx}{\sqrt {x+1} \sqrt {x^2-x+1}}\) |
\(\Big \downarrow \) 2417 |
\(\displaystyle \frac {\sqrt {x^3+1} \left (\left (a-\left (1-\sqrt {3}\right ) b\right ) \int \frac {1}{\sqrt {x^3+1}}dx+b \int \frac {x-\sqrt {3}+1}{\sqrt {x^3+1}}dx\right )}{\sqrt {x+1} \sqrt {x^2-x+1}}\) |
\(\Big \downarrow \) 759 |
\(\displaystyle \frac {\sqrt {x^3+1} \left (b \int \frac {x-\sqrt {3}+1}{\sqrt {x^3+1}}dx+\frac {2 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (a-\left (1-\sqrt {3}\right ) b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\right )}{\sqrt {x+1} \sqrt {x^2-x+1}}\) |
\(\Big \downarrow \) 2416 |
\(\displaystyle \frac {\sqrt {x^3+1} \left (\frac {2 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (a-\left (1-\sqrt {3}\right ) b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}+b \left (\frac {2 \sqrt {x^3+1}}{x+\sqrt {3}+1}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\right )\right )}{\sqrt {x+1} \sqrt {x^2-x+1}}\) |
Input:
Int[(a + b*x)/(Sqrt[1 + x]*Sqrt[1 - x + x^2]),x]
Output:
(Sqrt[1 + x^3]*(b*((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*Sqrt[2 + Sqrt[3]]*(a - (1 - Sqrt[3]) *b)*(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])))/(Sqrt[1 + x]*Sqrt[1 - x + x^2])
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_)*((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]
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]
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(c*r - (1 - Sqrt[3])*d*s)/r Int[1/Sqrt[a + b*x^3], x], x] + Simp[d/r Int[((1 - Sqrt[3])*s + r*x)/Sq rt[a + b*x^3], x], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && NeQ[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
Time = 4.88 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.14
method | result | size |
default | \(\frac {\left (-i \operatorname {EllipticF}\left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right ) \sqrt {3}\, a +i \operatorname {EllipticF}\left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right ) \sqrt {3}\, b +3 \operatorname {EllipticF}\left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right ) a +3 \operatorname {EllipticF}\left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right ) b -6 \operatorname {EllipticE}\left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right ) b \right ) \sqrt {x +1}\, \sqrt {x^{2}-x +1}\, \sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-3}}}{x^{3}+1}\) | \(313\) |
elliptic | \(\frac {\sqrt {\left (x +1\right ) \left (x^{2}-x +1\right )}\, \left (\frac {2 a \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+\frac {2 b \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )+\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )\right )}{\sqrt {x^{3}+1}}\right )}{\sqrt {x +1}\, \sqrt {x^{2}-x +1}}\) | \(321\) |
Input:
int((b*x+a)/(x+1)^(1/2)/(x^2-x+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
(-I*EllipticF((-2*(x+1)/(I*3^(1/2)-3))^(1/2),(-(I*3^(1/2)-3)/(I*3^(1/2)+3) )^(1/2))*3^(1/2)*a+I*EllipticF((-2*(x+1)/(I*3^(1/2)-3))^(1/2),(-(I*3^(1/2) -3)/(I*3^(1/2)+3))^(1/2))*3^(1/2)*b+3*EllipticF((-2*(x+1)/(I*3^(1/2)-3))^( 1/2),(-(I*3^(1/2)-3)/(I*3^(1/2)+3))^(1/2))*a+3*EllipticF((-2*(x+1)/(I*3^(1 /2)-3))^(1/2),(-(I*3^(1/2)-3)/(I*3^(1/2)+3))^(1/2))*b-6*EllipticE((-2*(x+1 )/(I*3^(1/2)-3))^(1/2),(-(I*3^(1/2)-3)/(I*3^(1/2)+3))^(1/2))*b)*(x+1)^(1/2 )*(x^2-x+1)^(1/2)*(-2*(x+1)/(I*3^(1/2)-3))^(1/2)*((I*3^(1/2)-2*x+1)/(I*3^( 1/2)+3))^(1/2)*((I*3^(1/2)+2*x-1)/(I*3^(1/2)-3))^(1/2)/(x^3+1)
Time = 0.07 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.07 \[ \int \frac {a+b x}{\sqrt {1+x} \sqrt {1-x+x^2}} \, dx=2 \, a {\rm weierstrassPInverse}\left (0, -4, x\right ) - 2 \, b {\rm weierstrassZeta}\left (0, -4, {\rm weierstrassPInverse}\left (0, -4, x\right )\right ) \] Input:
integrate((b*x+a)/(1+x)^(1/2)/(x^2-x+1)^(1/2),x, algorithm="fricas")
Output:
2*a*weierstrassPInverse(0, -4, x) - 2*b*weierstrassZeta(0, -4, weierstrass PInverse(0, -4, x))
\[ \int \frac {a+b x}{\sqrt {1+x} \sqrt {1-x+x^2}} \, dx=\int \frac {a + b x}{\sqrt {x + 1} \sqrt {x^{2} - x + 1}}\, dx \] Input:
integrate((b*x+a)/(1+x)**(1/2)/(x**2-x+1)**(1/2),x)
Output:
Integral((a + b*x)/(sqrt(x + 1)*sqrt(x**2 - x + 1)), x)
\[ \int \frac {a+b x}{\sqrt {1+x} \sqrt {1-x+x^2}} \, dx=\int { \frac {b x + a}{\sqrt {x^{2} - x + 1} \sqrt {x + 1}} \,d x } \] Input:
integrate((b*x+a)/(1+x)^(1/2)/(x^2-x+1)^(1/2),x, algorithm="maxima")
Output:
integrate((b*x + a)/(sqrt(x^2 - x + 1)*sqrt(x + 1)), x)
\[ \int \frac {a+b x}{\sqrt {1+x} \sqrt {1-x+x^2}} \, dx=\int { \frac {b x + a}{\sqrt {x^{2} - x + 1} \sqrt {x + 1}} \,d x } \] Input:
integrate((b*x+a)/(1+x)^(1/2)/(x^2-x+1)^(1/2),x, algorithm="giac")
Output:
integrate((b*x + a)/(sqrt(x^2 - x + 1)*sqrt(x + 1)), x)
Timed out. \[ \int \frac {a+b x}{\sqrt {1+x} \sqrt {1-x+x^2}} \, dx=\int \frac {a+b\,x}{\sqrt {x+1}\,\sqrt {x^2-x+1}} \,d x \] Input:
int((a + b*x)/((x + 1)^(1/2)*(x^2 - x + 1)^(1/2)),x)
Output:
int((a + b*x)/((x + 1)^(1/2)*(x^2 - x + 1)^(1/2)), x)
\[ \int \frac {a+b x}{\sqrt {1+x} \sqrt {1-x+x^2}} \, dx=\left (\int \frac {\sqrt {x +1}\, \sqrt {x^{2}-x +1}\, x}{x^{3}+1}d x \right ) b +\left (\int \frac {\sqrt {x +1}\, \sqrt {x^{2}-x +1}}{x^{3}+1}d x \right ) a \] Input:
int((b*x+a)/(1+x)^(1/2)/(x^2-x+1)^(1/2),x)
Output:
int((sqrt(x + 1)*sqrt(x**2 - x + 1)*x)/(x**3 + 1),x)*b + int((sqrt(x + 1)* sqrt(x**2 - x + 1))/(x**3 + 1),x)*a