Integrand size = 25, antiderivative size = 326 \[ \int \sqrt {1+x} (a+b x) \sqrt {1-x+x^2} \, dx=\frac {6 b \sqrt {1+x} \sqrt {1-x+x^2}}{7 \left (1+\sqrt {3}+x\right )}+\frac {2}{35} \sqrt {1+x} \sqrt {1-x+x^2} \left (7 a x+5 b x^2\right )-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b (1+x)^{3/2} \sqrt {1-x+x^2} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} E\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{7 \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \left (1+x^3\right )}+\frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} \left (7 a-5 \left (1-\sqrt {3}\right ) b\right ) (1+x)^{3/2} \sqrt {1-x+x^2} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{35 \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \left (1+x^3\right )} \]
2/35*(5*b*x^2+7*a*x)*(1+x)^(1/2)*(x^2-x+1)^(1/2)+6/7*b*(1+x)^(1/2)*(x^2-x+ 1)^(1/2)/(1+x+3^(1/2))-3/7*3^(1/4)*b*(1+x)^(3/2)*EllipticE((1+x-3^(1/2))/( 1+x+3^(1/2)),I*3^(1/2)+2*I)*(x^2-x+1)^(1/2)*(1/2*6^(1/2)-1/2*2^(1/2))*((x^ 2-x+1)/(1+x+3^(1/2))^2)^(1/2)/(x^3+1)/((1+x)/(1+x+3^(1/2))^2)^(1/2)+2/35*3 ^(3/4)*(1+x)^(3/2)*EllipticF((1+x-3^(1/2))/(1+x+3^(1/2)),I*3^(1/2)+2*I)*(7 *a-5*b*(1-3^(1/2)))*(x^2-x+1)^(1/2)*(1/2*6^(1/2)+1/2*2^(1/2))*((x^2-x+1)/( 1+x+3^(1/2))^2)^(1/2)/(x^3+1)/((1+x)/(1+x+3^(1/2))^2)^(1/2)
Result contains complex when optimal does not.
Time = 32.13 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.30 \[ \int \sqrt {1+x} (a+b x) \sqrt {1-x+x^2} \, dx=\frac {2}{35} x \sqrt {1+x} (7 a+5 b x) \sqrt {1-x+x^2}-\frac {(1+x)^{3/2} \left (-\frac {60 \sqrt {-\frac {i}{3 i+\sqrt {3}}} b \left (1-x+x^2\right )}{(1+x)^2}+\frac {15 i \sqrt {2} \left (i+\sqrt {3}\right ) b \sqrt {\frac {3 i+\sqrt {3}-\frac {6 i}{1+x}}{3 i+\sqrt {3}}} \sqrt {\frac {-3 i+\sqrt {3}+\frac {6 i}{1+x}}{-3 i+\sqrt {3}}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {6 i}{3 i+\sqrt {3}}}}{\sqrt {1+x}}\right )|\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )}{\sqrt {1+x}}+\frac {\sqrt {2} \left (-14 i \sqrt {3} a+5 \left (3-i \sqrt {3}\right ) b\right ) \sqrt {\frac {3 i+\sqrt {3}-\frac {6 i}{1+x}}{3 i+\sqrt {3}}} \sqrt {\frac {-3 i+\sqrt {3}+\frac {6 i}{1+x}}{-3 i+\sqrt {3}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {6 i}{3 i+\sqrt {3}}}}{\sqrt {1+x}}\right ),\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )}{\sqrt {1+x}}\right )}{70 \sqrt {-\frac {i}{3 i+\sqrt {3}}} \sqrt {1-x+x^2}} \]
(2*x*Sqrt[1 + x]*(7*a + 5*b*x)*Sqrt[1 - x + x^2])/35 - ((1 + x)^(3/2)*((-6 0*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[Sq rt[(-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*ArcSinh[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])
Time = 0.46 (sec) , antiderivative size = 308, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1210, 2392, 27, 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.
| \(\displaystyle \int \sqrt {x+1} \sqrt {x^2-x+1} (a+b x) \, dx\) |
| \(\Big \downarrow \) 1210 |
| \(\displaystyle \frac {\sqrt {x+1} \sqrt {x^2-x+1} \int (a+b x) \sqrt {x^3+1}dx}{\sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 2392 |
| \(\displaystyle \frac {\sqrt {x+1} \sqrt {x^2-x+1} \left (\frac {3}{2} \int \frac {2 (7 a+5 b x)}{35 \sqrt {x^3+1}}dx+\frac {2}{35} \sqrt {x^3+1} \left (7 a x+5 b x^2\right )\right )}{\sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {x+1} \sqrt {x^2-x+1} \left (\frac {3}{35} \int \frac {7 a+5 b x}{\sqrt {x^3+1}}dx+\frac {2}{35} \sqrt {x^3+1} \left (7 a x+5 b x^2\right )\right )}{\sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 2417 |
| \(\displaystyle \frac {\sqrt {x+1} \sqrt {x^2-x+1} \left (\frac {3}{35} \left (\left (7 a-5 \left (1-\sqrt {3}\right ) b\right ) \int \frac {1}{\sqrt {x^3+1}}dx+5 b \int \frac {x-\sqrt {3}+1}{\sqrt {x^3+1}}dx\right )+\frac {2}{35} \sqrt {x^3+1} \left (7 a x+5 b x^2\right )\right )}{\sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {\sqrt {x+1} \sqrt {x^2-x+1} \left (\frac {3}{35} \left (5 b \int \frac {x-\sqrt {3}+1}{\sqrt {x^3+1}}dx+\frac {2 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (7 a-5 \left (1-\sqrt {3}\right ) b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\right )+\frac {2}{35} \sqrt {x^3+1} \left (7 a x+5 b x^2\right )\right )}{\sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle \frac {\sqrt {x+1} \sqrt {x^2-x+1} \left (\frac {3}{35} \left (\frac {2 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (7 a-5 \left (1-\sqrt {3}\right ) b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}+5 b \left (\frac {2 \sqrt {x^3+1}}{x+\sqrt {3}+1}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\right )\right )+\frac {2}{35} \sqrt {x^3+1} \left (7 a x+5 b x^2\right )\right )}{\sqrt {x^3+1}}\) |
(Sqrt[1 + x]*Sqrt[1 - x + x^2]*((2*(7*a*x + 5*b*x^2)*Sqrt[1 + x^3])/35 + ( 3*(5*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]]*(7*a - 5*(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])))/35))/Sqrt[1 + x^3]
3.24.2.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Module[{q = Expon[Pq , x], i}, Simp[(a + b*x^n)^p*Sum[Coeff[Pq, x, i]*(x^(i + 1)/(n*p + i + 1)), {i, 0, q}], x] + Simp[a*n*p Int[(a + b*x^n)^(p - 1)*Sum[Coeff[Pq, x, i]* (x^i/(n*p + i + 1)), {i, 0, q}], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x ] && IGtQ[(n - 1)/2, 0] && GtQ[p, 0]
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 = 0.49 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.06
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (1+x \right ) \left (x^{2}-x +1\right )}\, \left (\frac {2 b \,x^{2} \sqrt {x^{3}+1}}{7}+\frac {2 a x \sqrt {x^{3}+1}}{5}+\frac {6 a \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{5 \sqrt {x^{3}+1}}+\frac {6 b \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) E\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )+\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) F\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )\right )}{7 \sqrt {x^{3}+1}}\right )}{\sqrt {1+x}\, \sqrt {x^{2}-x +1}}\) | \(345\) |
| risch | \(\frac {2 x \left (5 b x +7 a \right ) \sqrt {1+x}\, \sqrt {x^{2}-x +1}}{35}+\frac {\left (\frac {6 a \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{5 \sqrt {x^{3}+1}}+\frac {6 b \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) E\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )+\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) F\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )\right )}{7 \sqrt {x^{3}+1}}\right ) \sqrt {\left (1+x \right ) \left (x^{2}-x +1\right )}}{\sqrt {1+x}\, \sqrt {x^{2}-x +1}}\) | \(348\) |
| default | \(-\frac {\sqrt {1+x}\, \sqrt {x^{2}-x +1}\, \left (21 i \sqrt {3}\, \sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}\, F\left (\sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right ) a -15 i \sqrt {3}\, \sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}\, F\left (\sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right ) b -10 b \,x^{5}-63 \sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}\, F\left (\sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right ) a -45 \sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}\, F\left (\sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right ) b +90 \sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}\, E\left (\sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right ) b -14 a \,x^{4}-10 b \,x^{2}-14 a x \right )}{35 \left (x^{3}+1\right )}\) | \(596\) |
((1+x)*(x^2-x+1))^(1/2)/(1+x)^(1/2)/(x^2-x+1)^(1/2)*(2/7*b*x^2*(x^3+1)^(1/ 2)+2/5*a*x*(x^3+1)^(1/2)+6/5*a*(3/2-1/2*I*3^(1/2))*((1+x)/(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(((1+x)/(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))+ 6/7*b*(3/2-1/2*I*3^(1/2))*((1+x)/(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(((1+x)/(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(((1+x)/(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))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.14 \[ \int \sqrt {1+x} (a+b x) \sqrt {1-x+x^2} \, dx=\frac {2}{35} \, {\left (5 \, b x^{2} + 7 \, a x\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1} + \frac {6}{5} \, a {\rm weierstrassPInverse}\left (0, -4, x\right ) - \frac {6}{7} \, b {\rm weierstrassZeta}\left (0, -4, {\rm weierstrassPInverse}\left (0, -4, x\right )\right ) \]
2/35*(5*b*x^2 + 7*a*x)*sqrt(x^2 - x + 1)*sqrt(x + 1) + 6/5*a*weierstrassPI nverse(0, -4, x) - 6/7*b*weierstrassZeta(0, -4, weierstrassPInverse(0, -4, x))
\[ \int \sqrt {1+x} (a+b x) \sqrt {1-x+x^2} \, dx=\int \left (a + b x\right ) \sqrt {x + 1} \sqrt {x^{2} - x + 1}\, dx \]
\[ \int \sqrt {1+x} (a+b x) \sqrt {1-x+x^2} \, dx=\int { {\left (b x + a\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1} \,d x } \]
\[ \int \sqrt {1+x} (a+b x) \sqrt {1-x+x^2} \, dx=\int { {\left (b x + a\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1} \,d x } \]
Timed out. \[ \int \sqrt {1+x} (a+b x) \sqrt {1-x+x^2} \, dx=\int \sqrt {x+1}\,\left (a+b\,x\right )\,\sqrt {x^2-x+1} \,d x \]