Integrand size = 22, antiderivative size = 440 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (1+x^2\right )} \, dx=\frac {\left (A e-B \left (d-\sqrt {d^2+e^2}\right )\right ) \text {arctanh}\left (\frac {\sqrt {d+\sqrt {d^2+e^2}}-\sqrt {2} \sqrt {d+e x}}{\sqrt {d-\sqrt {d^2+e^2}}}\right )}{\sqrt {2} \sqrt {d^2+e^2} \sqrt {d-\sqrt {d^2+e^2}}}-\frac {\left (A e-B \left (d-\sqrt {d^2+e^2}\right )\right ) \text {arctanh}\left (\frac {\sqrt {d+\sqrt {d^2+e^2}}+\sqrt {2} \sqrt {d+e x}}{\sqrt {d-\sqrt {d^2+e^2}}}\right )}{\sqrt {2} \sqrt {d^2+e^2} \sqrt {d-\sqrt {d^2+e^2}}}-\frac {\left (A e-B \left (d+\sqrt {d^2+e^2}\right )\right ) \log \left (d+\sqrt {d^2+e^2}+e x-\sqrt {2} \sqrt {d+\sqrt {d^2+e^2}} \sqrt {d+e x}\right )}{2 \sqrt {2} \sqrt {d^2+e^2} \sqrt {d+\sqrt {d^2+e^2}}}+\frac {\left (A e-B \left (d+\sqrt {d^2+e^2}\right )\right ) \log \left (d+\sqrt {d^2+e^2}+e x+\sqrt {2} \sqrt {d+\sqrt {d^2+e^2}} \sqrt {d+e x}\right )}{2 \sqrt {2} \sqrt {d^2+e^2} \sqrt {d+\sqrt {d^2+e^2}}} \]
1/2*arctanh((-2^(1/2)*(e*x+d)^(1/2)+(d+(d^2+e^2)^(1/2))^(1/2))/(d-(d^2+e^2 )^(1/2))^(1/2))*(A*e-B*(d-(d^2+e^2)^(1/2)))*2^(1/2)/(d^2+e^2)^(1/2)/(d-(d^ 2+e^2)^(1/2))^(1/2)-1/2*arctanh((2^(1/2)*(e*x+d)^(1/2)+(d+(d^2+e^2)^(1/2)) ^(1/2))/(d-(d^2+e^2)^(1/2))^(1/2))*(A*e-B*(d-(d^2+e^2)^(1/2)))*2^(1/2)/(d^ 2+e^2)^(1/2)/(d-(d^2+e^2)^(1/2))^(1/2)-1/4*ln(d+e*x+(d^2+e^2)^(1/2)-2^(1/2 )*(e*x+d)^(1/2)*(d+(d^2+e^2)^(1/2))^(1/2))*(A*e-B*(d+(d^2+e^2)^(1/2)))*2^( 1/2)/(d^2+e^2)^(1/2)/(d+(d^2+e^2)^(1/2))^(1/2)+1/4*ln(d+e*x+(d^2+e^2)^(1/2 )+2^(1/2)*(e*x+d)^(1/2)*(d+(d^2+e^2)^(1/2))^(1/2))*(A*e-B*(d+(d^2+e^2)^(1/ 2)))*2^(1/2)/(d^2+e^2)^(1/2)/(d+(d^2+e^2)^(1/2))^(1/2)
Result contains complex when optimal does not.
Time = 0.21 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.21 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (1+x^2\right )} \, dx=\frac {(-i A+B) \arctan \left (\frac {\sqrt {d+e x}}{\sqrt {-d-i e}}\right )}{\sqrt {-d-i e}}+\frac {(i A+B) \arctan \left (\frac {\sqrt {d+e x}}{\sqrt {-d+i e}}\right )}{\sqrt {-d+i e}} \]
(((-I)*A + B)*ArcTan[Sqrt[d + e*x]/Sqrt[-d - I*e]])/Sqrt[-d - I*e] + ((I*A + B)*ArcTan[Sqrt[d + e*x]/Sqrt[-d + I*e]])/Sqrt[-d + I*e]
Time = 0.85 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {654, 25, 1483, 1142, 25, 27, 1083, 219, 1103}
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.
| \(\displaystyle \int \frac {A+B x}{\left (x^2+1\right ) \sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 654 |
| \(\displaystyle 2 \int -\frac {B d-A e-B (d+e x)}{d^2-2 (d+e x) d+e^2+(d+e x)^2}d\sqrt {d+e x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {B d-A e-B (d+e x)}{d^2-2 (d+e x) d+e^2+(d+e x)^2}d\sqrt {d+e x}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle 2 \left (-\frac {\int \frac {\sqrt {2} (B d-A e) \sqrt {d+\sqrt {d^2+e^2}}-\left (A e-B \left (d+\sqrt {d^2+e^2}\right )\right ) \sqrt {d+e x}}{d+e x+\sqrt {d^2+e^2}+\sqrt {2} \sqrt {d+\sqrt {d^2+e^2}} \sqrt {d+e x}}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {d^2+e^2} \sqrt {\sqrt {d^2+e^2}+d}}-\frac {\int \frac {\sqrt {2} \sqrt {d+\sqrt {d^2+e^2}} (B d-A e)+\left (A e-B \left (d+\sqrt {d^2+e^2}\right )\right ) \sqrt {d+e x}}{d+e x+\sqrt {d^2+e^2}-\sqrt {2} \sqrt {d+\sqrt {d^2+e^2}} \sqrt {d+e x}}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {d^2+e^2} \sqrt {\sqrt {d^2+e^2}+d}}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle 2 \left (-\frac {\frac {1}{2} \left (A e-B \left (\sqrt {d^2+e^2}+d\right )\right ) \int -\frac {\sqrt {2} \left (\sqrt {d+\sqrt {d^2+e^2}}-\sqrt {2} \sqrt {d+e x}\right )}{d+e x+\sqrt {d^2+e^2}-\sqrt {2} \sqrt {d+\sqrt {d^2+e^2}} \sqrt {d+e x}}d\sqrt {d+e x}-\frac {\sqrt {\sqrt {d^2+e^2}+d} \left (A e-B \left (d-\sqrt {d^2+e^2}\right )\right ) \int \frac {1}{d+e x+\sqrt {d^2+e^2}-\sqrt {2} \sqrt {d+\sqrt {d^2+e^2}} \sqrt {d+e x}}d\sqrt {d+e x}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {d^2+e^2} \sqrt {\sqrt {d^2+e^2}+d}}-\frac {-\frac {\sqrt {\sqrt {d^2+e^2}+d} \left (A e-B \left (d-\sqrt {d^2+e^2}\right )\right ) \int \frac {1}{d+e x+\sqrt {d^2+e^2}+\sqrt {2} \sqrt {d+\sqrt {d^2+e^2}} \sqrt {d+e x}}d\sqrt {d+e x}}{\sqrt {2}}-\frac {1}{2} \left (A e-B \left (\sqrt {d^2+e^2}+d\right )\right ) \int \frac {\sqrt {2} \left (\sqrt {d+\sqrt {d^2+e^2}}+\sqrt {2} \sqrt {d+e x}\right )}{d+e x+\sqrt {d^2+e^2}+\sqrt {2} \sqrt {d+\sqrt {d^2+e^2}} \sqrt {d+e x}}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {d^2+e^2} \sqrt {\sqrt {d^2+e^2}+d}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (-\frac {-\frac {\sqrt {\sqrt {d^2+e^2}+d} \left (A e-B \left (d-\sqrt {d^2+e^2}\right )\right ) \int \frac {1}{d+e x+\sqrt {d^2+e^2}-\sqrt {2} \sqrt {d+\sqrt {d^2+e^2}} \sqrt {d+e x}}d\sqrt {d+e x}}{\sqrt {2}}-\frac {1}{2} \left (A e-B \left (\sqrt {d^2+e^2}+d\right )\right ) \int \frac {\sqrt {2} \left (\sqrt {d+\sqrt {d^2+e^2}}-\sqrt {2} \sqrt {d+e x}\right )}{d+e x+\sqrt {d^2+e^2}-\sqrt {2} \sqrt {d+\sqrt {d^2+e^2}} \sqrt {d+e x}}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {d^2+e^2} \sqrt {\sqrt {d^2+e^2}+d}}-\frac {-\frac {\sqrt {\sqrt {d^2+e^2}+d} \left (A e-B \left (d-\sqrt {d^2+e^2}\right )\right ) \int \frac {1}{d+e x+\sqrt {d^2+e^2}+\sqrt {2} \sqrt {d+\sqrt {d^2+e^2}} \sqrt {d+e x}}d\sqrt {d+e x}}{\sqrt {2}}-\frac {1}{2} \left (A e-B \left (\sqrt {d^2+e^2}+d\right )\right ) \int \frac {\sqrt {2} \left (\sqrt {d+\sqrt {d^2+e^2}}+\sqrt {2} \sqrt {d+e x}\right )}{d+e x+\sqrt {d^2+e^2}+\sqrt {2} \sqrt {d+\sqrt {d^2+e^2}} \sqrt {d+e x}}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {d^2+e^2} \sqrt {\sqrt {d^2+e^2}+d}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (-\frac {-\frac {\sqrt {\sqrt {d^2+e^2}+d} \left (A e-B \left (d-\sqrt {d^2+e^2}\right )\right ) \int \frac {1}{d+e x+\sqrt {d^2+e^2}-\sqrt {2} \sqrt {d+\sqrt {d^2+e^2}} \sqrt {d+e x}}d\sqrt {d+e x}}{\sqrt {2}}-\frac {\left (A e-B \left (\sqrt {d^2+e^2}+d\right )\right ) \int \frac {\sqrt {d+\sqrt {d^2+e^2}}-\sqrt {2} \sqrt {d+e x}}{d+e x+\sqrt {d^2+e^2}-\sqrt {2} \sqrt {d+\sqrt {d^2+e^2}} \sqrt {d+e x}}d\sqrt {d+e x}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {d^2+e^2} \sqrt {\sqrt {d^2+e^2}+d}}-\frac {-\frac {\sqrt {\sqrt {d^2+e^2}+d} \left (A e-B \left (d-\sqrt {d^2+e^2}\right )\right ) \int \frac {1}{d+e x+\sqrt {d^2+e^2}+\sqrt {2} \sqrt {d+\sqrt {d^2+e^2}} \sqrt {d+e x}}d\sqrt {d+e x}}{\sqrt {2}}-\frac {\left (A e-B \left (\sqrt {d^2+e^2}+d\right )\right ) \int \frac {\sqrt {d+\sqrt {d^2+e^2}}+\sqrt {2} \sqrt {d+e x}}{d+e x+\sqrt {d^2+e^2}+\sqrt {2} \sqrt {d+\sqrt {d^2+e^2}} \sqrt {d+e x}}d\sqrt {d+e x}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {d^2+e^2} \sqrt {\sqrt {d^2+e^2}+d}}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle 2 \left (-\frac {\sqrt {2} \sqrt {\sqrt {d^2+e^2}+d} \left (A e-B \left (d-\sqrt {d^2+e^2}\right )\right ) \int \frac {1}{-d+2 \left (d-\sqrt {d^2+e^2}\right )-e x}d\left (2 \sqrt {d+e x}-\sqrt {2} \sqrt {d+\sqrt {d^2+e^2}}\right )-\frac {\left (A e-B \left (\sqrt {d^2+e^2}+d\right )\right ) \int \frac {\sqrt {d+\sqrt {d^2+e^2}}-\sqrt {2} \sqrt {d+e x}}{d+e x+\sqrt {d^2+e^2}-\sqrt {2} \sqrt {d+\sqrt {d^2+e^2}} \sqrt {d+e x}}d\sqrt {d+e x}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {d^2+e^2} \sqrt {\sqrt {d^2+e^2}+d}}-\frac {\sqrt {2} \sqrt {\sqrt {d^2+e^2}+d} \left (A e-B \left (d-\sqrt {d^2+e^2}\right )\right ) \int \frac {1}{-d+2 \left (d-\sqrt {d^2+e^2}\right )-e x}d\left (\sqrt {2} \sqrt {d+\sqrt {d^2+e^2}}+2 \sqrt {d+e x}\right )-\frac {\left (A e-B \left (\sqrt {d^2+e^2}+d\right )\right ) \int \frac {\sqrt {d+\sqrt {d^2+e^2}}+\sqrt {2} \sqrt {d+e x}}{d+e x+\sqrt {d^2+e^2}+\sqrt {2} \sqrt {d+\sqrt {d^2+e^2}} \sqrt {d+e x}}d\sqrt {d+e x}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {d^2+e^2} \sqrt {\sqrt {d^2+e^2}+d}}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 2 \left (-\frac {\frac {\sqrt {\sqrt {d^2+e^2}+d} \left (A e-B \left (d-\sqrt {d^2+e^2}\right )\right ) \text {arctanh}\left (\frac {2 \sqrt {d+e x}-\sqrt {2} \sqrt {\sqrt {d^2+e^2}+d}}{\sqrt {2} \sqrt {d-\sqrt {d^2+e^2}}}\right )}{\sqrt {d-\sqrt {d^2+e^2}}}-\frac {\left (A e-B \left (\sqrt {d^2+e^2}+d\right )\right ) \int \frac {\sqrt {d+\sqrt {d^2+e^2}}-\sqrt {2} \sqrt {d+e x}}{d+e x+\sqrt {d^2+e^2}-\sqrt {2} \sqrt {d+\sqrt {d^2+e^2}} \sqrt {d+e x}}d\sqrt {d+e x}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {d^2+e^2} \sqrt {\sqrt {d^2+e^2}+d}}-\frac {\frac {\sqrt {\sqrt {d^2+e^2}+d} \left (A e-B \left (d-\sqrt {d^2+e^2}\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\sqrt {d^2+e^2}+d}+2 \sqrt {d+e x}}{\sqrt {2} \sqrt {d-\sqrt {d^2+e^2}}}\right )}{\sqrt {d-\sqrt {d^2+e^2}}}-\frac {\left (A e-B \left (\sqrt {d^2+e^2}+d\right )\right ) \int \frac {\sqrt {d+\sqrt {d^2+e^2}}+\sqrt {2} \sqrt {d+e x}}{d+e x+\sqrt {d^2+e^2}+\sqrt {2} \sqrt {d+\sqrt {d^2+e^2}} \sqrt {d+e x}}d\sqrt {d+e x}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {d^2+e^2} \sqrt {\sqrt {d^2+e^2}+d}}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 2 \left (-\frac {\frac {\sqrt {\sqrt {d^2+e^2}+d} \left (A e-B \left (d-\sqrt {d^2+e^2}\right )\right ) \text {arctanh}\left (\frac {2 \sqrt {d+e x}-\sqrt {2} \sqrt {\sqrt {d^2+e^2}+d}}{\sqrt {2} \sqrt {d-\sqrt {d^2+e^2}}}\right )}{\sqrt {d-\sqrt {d^2+e^2}}}+\frac {1}{2} \left (A e-B \left (\sqrt {d^2+e^2}+d\right )\right ) \log \left (-\sqrt {2} \sqrt {\sqrt {d^2+e^2}+d} \sqrt {d+e x}+\sqrt {d^2+e^2}+d+e x\right )}{2 \sqrt {2} \sqrt {d^2+e^2} \sqrt {\sqrt {d^2+e^2}+d}}-\frac {\frac {\sqrt {\sqrt {d^2+e^2}+d} \left (A e-B \left (d-\sqrt {d^2+e^2}\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\sqrt {d^2+e^2}+d}+2 \sqrt {d+e x}}{\sqrt {2} \sqrt {d-\sqrt {d^2+e^2}}}\right )}{\sqrt {d-\sqrt {d^2+e^2}}}-\frac {1}{2} \left (A e-B \left (\sqrt {d^2+e^2}+d\right )\right ) \log \left (\sqrt {2} \sqrt {\sqrt {d^2+e^2}+d} \sqrt {d+e x}+\sqrt {d^2+e^2}+d+e x\right )}{2 \sqrt {2} \sqrt {d^2+e^2} \sqrt {\sqrt {d^2+e^2}+d}}\right )\) |
2*(-1/2*((Sqrt[d + Sqrt[d^2 + e^2]]*(A*e - B*(d - Sqrt[d^2 + e^2]))*ArcTan h[(-(Sqrt[2]*Sqrt[d + Sqrt[d^2 + e^2]]) + 2*Sqrt[d + e*x])/(Sqrt[2]*Sqrt[d - Sqrt[d^2 + e^2]])])/Sqrt[d - Sqrt[d^2 + e^2]] + ((A*e - B*(d + Sqrt[d^2 + e^2]))*Log[d + Sqrt[d^2 + e^2] + e*x - Sqrt[2]*Sqrt[d + Sqrt[d^2 + e^2] ]*Sqrt[d + e*x]])/2)/(Sqrt[2]*Sqrt[d^2 + e^2]*Sqrt[d + Sqrt[d^2 + e^2]]) - ((Sqrt[d + Sqrt[d^2 + e^2]]*(A*e - B*(d - Sqrt[d^2 + e^2]))*ArcTanh[(Sqrt [2]*Sqrt[d + Sqrt[d^2 + e^2]] + 2*Sqrt[d + e*x])/(Sqrt[2]*Sqrt[d - Sqrt[d^ 2 + e^2]])])/Sqrt[d - Sqrt[d^2 + e^2]] - ((A*e - B*(d + Sqrt[d^2 + e^2]))* Log[d + Sqrt[d^2 + e^2] + e*x + Sqrt[2]*Sqrt[d + Sqrt[d^2 + e^2]]*Sqrt[d + e*x]])/2)/(2*Sqrt[2]*Sqrt[d^2 + e^2]*Sqrt[d + Sqrt[d^2 + e^2]]))
3.15.66.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d* x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) In t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
Time = 2.17 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.66
| method | result | size |
| pseudoelliptic | \(\frac {-\frac {\sqrt {2 \sqrt {d^{2}+e^{2}}-2 d}\, \sqrt {2 \sqrt {d^{2}+e^{2}}+2 d}\, \left (\ln \left (e x +d +\sqrt {e x +d}\, \sqrt {2 \sqrt {d^{2}+e^{2}}+2 d}+\sqrt {d^{2}+e^{2}}\right )-\ln \left (\sqrt {e x +d}\, \sqrt {2 \sqrt {d^{2}+e^{2}}+2 d}-d -e x -\sqrt {d^{2}+e^{2}}\right )\right ) \left (d A -A \sqrt {d^{2}+e^{2}}+B e \right )}{4}+e \left (A e -B d +B \sqrt {d^{2}+e^{2}}\right ) \left (\arctan \left (\frac {2 \sqrt {e x +d}-\sqrt {2 \sqrt {d^{2}+e^{2}}+2 d}}{\sqrt {2 \sqrt {d^{2}+e^{2}}-2 d}}\right )+\arctan \left (\frac {2 \sqrt {e x +d}+\sqrt {2 \sqrt {d^{2}+e^{2}}+2 d}}{\sqrt {2 \sqrt {d^{2}+e^{2}}-2 d}}\right )\right )}{\sqrt {2 \sqrt {d^{2}+e^{2}}-2 d}\, \sqrt {d^{2}+e^{2}}\, e}\) | \(291\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1365\) |
| default | \(\text {Expression too large to display}\) | \(1365\) |
(-1/4*(2*(d^2+e^2)^(1/2)-2*d)^(1/2)*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*(ln(e*x+ d+(e*x+d)^(1/2)*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)+(d^2+e^2)^(1/2))-ln((e*x+d)^ (1/2)*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)-d-e*x-(d^2+e^2)^(1/2)))*(d*A-A*(d^2+e^ 2)^(1/2)+B*e)+e*(A*e-B*d+B*(d^2+e^2)^(1/2))*(arctan((2*(e*x+d)^(1/2)-(2*(d ^2+e^2)^(1/2)+2*d)^(1/2))/(2*(d^2+e^2)^(1/2)-2*d)^(1/2))+arctan((2*(e*x+d) ^(1/2)+(2*(d^2+e^2)^(1/2)+2*d)^(1/2))/(2*(d^2+e^2)^(1/2)-2*d)^(1/2))))/(2* (d^2+e^2)^(1/2)-2*d)^(1/2)/(d^2+e^2)^(1/2)/e
Leaf count of result is larger than twice the leaf count of optimal. 1545 vs. \(2 (355) = 710\).
Time = 0.39 (sec) , antiderivative size = 1545, normalized size of antiderivative = 3.51 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (1+x^2\right )} \, dx=\text {Too large to display} \]
-1/2*sqrt(-(2*A*B*e + (A^2 - B^2)*d + (d^2 + e^2)*sqrt(-(4*A^2*B^2*d^2 - 4 *(A^3*B - A*B^3)*d*e + (A^4 - 2*A^2*B^2 + B^4)*e^2)/(d^4 + 2*d^2*e^2 + e^4 )))/(d^2 + e^2))*log((2*(A^3*B + A*B^3)*d - (A^4 - B^4)*e)*sqrt(e*x + d) + (2*A*B^2*d^2 - (3*A^2*B - B^3)*d*e + (A^3 - A*B^2)*e^2 + (A*d^3 + B*d^2*e + A*d*e^2 + B*e^3)*sqrt(-(4*A^2*B^2*d^2 - 4*(A^3*B - A*B^3)*d*e + (A^4 - 2*A^2*B^2 + B^4)*e^2)/(d^4 + 2*d^2*e^2 + e^4)))*sqrt(-(2*A*B*e + (A^2 - B^ 2)*d + (d^2 + e^2)*sqrt(-(4*A^2*B^2*d^2 - 4*(A^3*B - A*B^3)*d*e + (A^4 - 2 *A^2*B^2 + B^4)*e^2)/(d^4 + 2*d^2*e^2 + e^4)))/(d^2 + e^2))) + 1/2*sqrt(-( 2*A*B*e + (A^2 - B^2)*d + (d^2 + e^2)*sqrt(-(4*A^2*B^2*d^2 - 4*(A^3*B - A* B^3)*d*e + (A^4 - 2*A^2*B^2 + B^4)*e^2)/(d^4 + 2*d^2*e^2 + e^4)))/(d^2 + e ^2))*log((2*(A^3*B + A*B^3)*d - (A^4 - B^4)*e)*sqrt(e*x + d) - (2*A*B^2*d^ 2 - (3*A^2*B - B^3)*d*e + (A^3 - A*B^2)*e^2 + (A*d^3 + B*d^2*e + A*d*e^2 + B*e^3)*sqrt(-(4*A^2*B^2*d^2 - 4*(A^3*B - A*B^3)*d*e + (A^4 - 2*A^2*B^2 + B^4)*e^2)/(d^4 + 2*d^2*e^2 + e^4)))*sqrt(-(2*A*B*e + (A^2 - B^2)*d + (d^2 + e^2)*sqrt(-(4*A^2*B^2*d^2 - 4*(A^3*B - A*B^3)*d*e + (A^4 - 2*A^2*B^2 + B ^4)*e^2)/(d^4 + 2*d^2*e^2 + e^4)))/(d^2 + e^2))) - 1/2*sqrt(-(2*A*B*e + (A ^2 - B^2)*d - (d^2 + e^2)*sqrt(-(4*A^2*B^2*d^2 - 4*(A^3*B - A*B^3)*d*e + ( A^4 - 2*A^2*B^2 + B^4)*e^2)/(d^4 + 2*d^2*e^2 + e^4)))/(d^2 + e^2))*log((2* (A^3*B + A*B^3)*d - (A^4 - B^4)*e)*sqrt(e*x + d) + (2*A*B^2*d^2 - (3*A^2*B - B^3)*d*e + (A^3 - A*B^2)*e^2 - (A*d^3 + B*d^2*e + A*d*e^2 + B*e^3)*s...
\[ \int \frac {A+B x}{\sqrt {d+e x} \left (1+x^2\right )} \, dx=\int \frac {A + B x}{\sqrt {d + e x} \left (x^{2} + 1\right )}\, dx \]
\[ \int \frac {A+B x}{\sqrt {d+e x} \left (1+x^2\right )} \, dx=\int { \frac {B x + A}{\sqrt {e x + d} {\left (x^{2} + 1\right )}} \,d x } \]
Timed out. \[ \int \frac {A+B x}{\sqrt {d+e x} \left (1+x^2\right )} \, dx=\text {Timed out} \]
Time = 11.00 (sec) , antiderivative size = 1244, normalized size of antiderivative = 2.83 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (1+x^2\right )} \, dx=\text {Too large to display} \]
- atan((((32*B*d*e^2 - 32*A*e^3 + 64*d*e^2*(d + e*x)^(1/2)*((B^2*1i - A^2* 1i + 2*A*B)/(4*(d*1i - e)))^(1/2))*((B^2*1i - A^2*1i + 2*A*B)/(4*(d*1i - e )))^(1/2) + (16*A^2*e^2 - 16*B^2*e^2)*(d + e*x)^(1/2))*((B^2*1i - A^2*1i + 2*A*B)/(4*(d*1i - e)))^(1/2)*1i + ((32*A*e^3 - 32*B*d*e^2 + 64*d*e^2*(d + e*x)^(1/2)*((B^2*1i - A^2*1i + 2*A*B)/(4*(d*1i - e)))^(1/2))*((B^2*1i - A ^2*1i + 2*A*B)/(4*(d*1i - e)))^(1/2) + (16*A^2*e^2 - 16*B^2*e^2)*(d + e*x) ^(1/2))*((B^2*1i - A^2*1i + 2*A*B)/(4*(d*1i - e)))^(1/2)*1i)/(((32*A*e^3 - 32*B*d*e^2 + 64*d*e^2*(d + e*x)^(1/2)*((B^2*1i - A^2*1i + 2*A*B)/(4*(d*1i - e)))^(1/2))*((B^2*1i - A^2*1i + 2*A*B)/(4*(d*1i - e)))^(1/2) + (16*A^2* e^2 - 16*B^2*e^2)*(d + e*x)^(1/2))*((B^2*1i - A^2*1i + 2*A*B)/(4*(d*1i - e )))^(1/2) - ((32*B*d*e^2 - 32*A*e^3 + 64*d*e^2*(d + e*x)^(1/2)*((B^2*1i - A^2*1i + 2*A*B)/(4*(d*1i - e)))^(1/2))*((B^2*1i - A^2*1i + 2*A*B)/(4*(d*1i - e)))^(1/2) + (16*A^2*e^2 - 16*B^2*e^2)*(d + e*x)^(1/2))*((B^2*1i - A^2* 1i + 2*A*B)/(4*(d*1i - e)))^(1/2) + 16*B^3*e^2 + 16*A^2*B*e^2))*((B^2*1i - A^2*1i + 2*A*B)/(4*(d*1i - e)))^(1/2)*2i - atan((((16*A^2*e^2 - 16*B^2*e^ 2)*(d + e*x)^(1/2) + ((B^2 - A^2 + A*B*2i)/(4*(d - e*1i)))^(1/2)*(32*B*d*e ^2 - 32*A*e^3 + 64*d*e^2*(d + e*x)^(1/2)*((B^2 - A^2 + A*B*2i)/(4*(d - e*1 i)))^(1/2)))*((B^2 - A^2 + A*B*2i)/(4*(d - e*1i)))^(1/2)*1i + ((16*A^2*e^2 - 16*B^2*e^2)*(d + e*x)^(1/2) + ((B^2 - A^2 + A*B*2i)/(4*(d - e*1i)))^(1/ 2)*(32*A*e^3 - 32*B*d*e^2 + 64*d*e^2*(d + e*x)^(1/2)*((B^2 - A^2 + A*B*...