Integrand size = 31, antiderivative size = 82 \[ \int \frac {1}{\sqrt {d+e x+f x^2} \left (a+b x+\frac {b f x^2}{e}\right )} \, dx=-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {b d-a e} (e+2 f x)}{\sqrt {e} \sqrt {b e-4 a f} \sqrt {d+e x+f x^2}}\right )}{\sqrt {b d-a e} \sqrt {b e-4 a f}} \]
-2*arctanh((2*f*x+e)*(-a*e+b*d)^(1/2)/e^(1/2)/(-4*a*f+b*e)^(1/2)/(f*x^2+e* x+d)^(1/2))*e^(1/2)/(-a*e+b*d)^(1/2)/(-4*a*f+b*e)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.41 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.49 \[ \int \frac {1}{\sqrt {d+e x+f x^2} \left (a+b x+\frac {b f x^2}{e}\right )} \, dx=e \text {RootSum}\left [a e f^2-2 b \sqrt {d} e f \text {$\#$1}+b e^2 \text {$\#$1}^2+4 b d f \text {$\#$1}^2-2 a e f \text {$\#$1}^2-2 b \sqrt {d} e \text {$\#$1}^3+a e \text {$\#$1}^4\&,\frac {-f \log (x)+f \log \left (-\sqrt {d}+\sqrt {d+e x+f x^2}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^2-\log \left (-\sqrt {d}+\sqrt {d+e x+f x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-b \sqrt {d} e f+b e^2 \text {$\#$1}+4 b d f \text {$\#$1}-2 a e f \text {$\#$1}-3 b \sqrt {d} e \text {$\#$1}^2+2 a e \text {$\#$1}^3}\&\right ] \]
e*RootSum[a*e*f^2 - 2*b*Sqrt[d]*e*f*#1 + b*e^2*#1^2 + 4*b*d*f*#1^2 - 2*a*e *f*#1^2 - 2*b*Sqrt[d]*e*#1^3 + a*e*#1^4 & , (-(f*Log[x]) + f*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1] + Log[x]*#1^2 - Log[-Sqrt[d] + Sqrt[d + e* x + f*x^2] - x*#1]*#1^2)/(-(b*Sqrt[d]*e*f) + b*e^2*#1 + 4*b*d*f*#1 - 2*a*e *f*#1 - 3*b*Sqrt[d]*e*#1^2 + 2*a*e*#1^3) & ]
Time = 0.24 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {1313, 221}
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.
| \(\displaystyle \int \frac {1}{\sqrt {d+e x+f x^2} \left (a+\frac {b f x^2}{e}+b x\right )} \, dx\) |
| \(\Big \downarrow \) 1313 |
| \(\displaystyle -2 e \int \frac {1}{e (b e-4 a f)-\frac {(b d-a e) (e+2 f x)^2}{f x^2+e x+d}}d\frac {e+2 f x}{\sqrt {f x^2+e x+d}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {2 \sqrt {e} \text {arctanh}\left (\frac {(e+2 f x) \sqrt {b d-a e}}{\sqrt {e} \sqrt {b e-4 a f} \sqrt {d+e x+f x^2}}\right )}{\sqrt {b d-a e} \sqrt {b e-4 a f}}\) |
(-2*Sqrt[e]*ArcTanh[(Sqrt[b*d - a*e]*(e + 2*f*x))/(Sqrt[e]*Sqrt[b*e - 4*a* f]*Sqrt[d + e*x + f*x^2])])/(Sqrt[b*d - a*e]*Sqrt[b*e - 4*a*f])
3.1.2.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[1/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*( x_)^2]), x_Symbol] :> Simp[-2*e Subst[Int[1/(e*(b*e - 4*a*f) - (b*d - a*e )*x^2), x], x, (e + 2*f*x)/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[c*e - b*f, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(490\) vs. \(2(68)=136\).
Time = 1.00 (sec) , antiderivative size = 491, normalized size of antiderivative = 5.99
| method | result | size |
| default | \(e \left (-\frac {\ln \left (\frac {-\frac {2 \left (a e -b d \right )}{b}+\frac {\sqrt {-b e \left (4 f a -b e \right )}\, \left (x -\frac {-b e +\sqrt {-b e \left (4 f a -b e \right )}}{2 b f}\right )}{b}+2 \sqrt {-\frac {a e -b d}{b}}\, \sqrt {\left (x -\frac {-b e +\sqrt {-b e \left (4 f a -b e \right )}}{2 b f}\right )^{2} f +\frac {\sqrt {-b e \left (4 f a -b e \right )}\, \left (x -\frac {-b e +\sqrt {-b e \left (4 f a -b e \right )}}{2 b f}\right )}{b}-\frac {a e -b d}{b}}}{x -\frac {-b e +\sqrt {-b e \left (4 f a -b e \right )}}{2 b f}}\right )}{\sqrt {-b e \left (4 f a -b e \right )}\, \sqrt {-\frac {a e -b d}{b}}}+\frac {\ln \left (\frac {-\frac {2 \left (a e -b d \right )}{b}-\frac {\sqrt {-b e \left (4 f a -b e \right )}\, \left (x +\frac {b e +\sqrt {-b e \left (4 f a -b e \right )}}{2 b f}\right )}{b}+2 \sqrt {-\frac {a e -b d}{b}}\, \sqrt {\left (x +\frac {b e +\sqrt {-b e \left (4 f a -b e \right )}}{2 b f}\right )^{2} f -\frac {\sqrt {-b e \left (4 f a -b e \right )}\, \left (x +\frac {b e +\sqrt {-b e \left (4 f a -b e \right )}}{2 b f}\right )}{b}-\frac {a e -b d}{b}}}{x +\frac {b e +\sqrt {-b e \left (4 f a -b e \right )}}{2 b f}}\right )}{\sqrt {-b e \left (4 f a -b e \right )}\, \sqrt {-\frac {a e -b d}{b}}}\right )\) | \(491\) |
e*(-1/(-b*e*(4*a*f-b*e))^(1/2)/(-(a*e-b*d)/b)^(1/2)*ln((-2*(a*e-b*d)/b+(-b *e*(4*a*f-b*e))^(1/2)/b*(x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)+2*(-(a *e-b*d)/b)^(1/2)*((x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)^2*f+(-b*e*(4 *a*f-b*e))^(1/2)/b*(x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)-(a*e-b*d)/b )^(1/2))/(x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f))+1/(-b*e*(4*a*f-b*e)) ^(1/2)/(-(a*e-b*d)/b)^(1/2)*ln((-2*(a*e-b*d)/b-(-b*e*(4*a*f-b*e))^(1/2)/b* (x+1/2*(b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)+2*(-(a*e-b*d)/b)^(1/2)*((x+1/2* (b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)^2*f-(-b*e*(4*a*f-b*e))^(1/2)/b*(x+1/2* (b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)-(a*e-b*d)/b)^(1/2))/(x+1/2*(b*e+(-b*e* (4*a*f-b*e))^(1/2))/b/f)))
Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (68) = 136\).
Time = 0.48 (sec) , antiderivative size = 1079, normalized size of antiderivative = 13.16 \[ \int \frac {1}{\sqrt {d+e x+f x^2} \left (a+b x+\frac {b f x^2}{e}\right )} \, dx=\left [\frac {1}{2} \, \sqrt {\frac {e}{b^{2} d e - a b e^{2} - 4 \, {\left (a b d - a^{2} e\right )} f}} \log \left (\frac {8 \, b^{2} d^{2} e^{4} - 8 \, a b d e^{5} + a^{2} e^{6} + 16 \, a^{2} d^{2} e^{2} f^{2} + {\left (b^{2} e^{4} f^{2} + 16 \, {\left (b^{2} d^{2} - 8 \, a b d e + 8 \, a^{2} e^{2}\right )} f^{4} + 8 \, {\left (3 \, b^{2} d e^{2} - 4 \, a b e^{3}\right )} f^{3}\right )} x^{4} + 2 \, {\left (b^{2} e^{5} f + 16 \, {\left (b^{2} d^{2} e - 8 \, a b d e^{2} + 8 \, a^{2} e^{3}\right )} f^{3} + 8 \, {\left (3 \, b^{2} d e^{3} - 4 \, a b e^{4}\right )} f^{2}\right )} x^{3} + {\left (b^{2} e^{6} - 32 \, {\left (3 \, a b d^{2} e - 4 \, a^{2} d e^{2}\right )} f^{3} + 16 \, {\left (3 \, b^{2} d^{2} e^{2} - 13 \, a b d e^{3} + 10 \, a^{2} e^{4}\right )} f^{2} + 2 \, {\left (16 \, b^{2} d e^{4} - 19 \, a b e^{5}\right )} f\right )} x^{2} - 8 \, {\left (4 \, a b d^{2} e^{3} - 3 \, a^{2} d e^{4}\right )} f + 2 \, {\left (4 \, b^{2} d e^{5} - 3 \, a b e^{6} - 16 \, {\left (3 \, a b d^{2} e^{2} - 4 \, a^{2} d e^{3}\right )} f^{2} + 8 \, {\left (2 \, b^{2} d^{2} e^{3} - 5 \, a b d e^{4} + 2 \, a^{2} e^{5}\right )} f\right )} x - 4 \, {\left (2 \, b^{3} d^{2} e^{4} - 3 \, a b^{2} d e^{5} + a^{2} b e^{6} - 2 \, {\left (16 \, {\left (a b^{2} d^{2} - 3 \, a^{2} b d e + 2 \, a^{3} e^{2}\right )} f^{4} - 4 \, {\left (b^{3} d^{2} e - 4 \, a b^{2} d e^{2} + 3 \, a^{2} b e^{3}\right )} f^{3} - {\left (b^{3} d e^{3} - a b^{2} e^{4}\right )} f^{2}\right )} x^{3} + 16 \, {\left (a^{2} b d^{2} e^{2} - a^{3} d e^{3}\right )} f^{2} - 3 \, {\left (16 \, {\left (a b^{2} d^{2} e - 3 \, a^{2} b d e^{2} + 2 \, a^{3} e^{3}\right )} f^{3} - 4 \, {\left (b^{3} d^{2} e^{2} - 4 \, a b^{2} d e^{3} + 3 \, a^{2} b e^{4}\right )} f^{2} - {\left (b^{3} d e^{4} - a b^{2} e^{5}\right )} f\right )} x^{2} - 4 \, {\left (3 \, a b^{2} d^{2} e^{3} - 4 \, a^{2} b d e^{4} + a^{3} e^{5}\right )} f + {\left (b^{3} d e^{5} - a b^{2} e^{6} + 32 \, {\left (a^{2} b d^{2} e - a^{3} d e^{2}\right )} f^{3} - 40 \, {\left (a b^{2} d^{2} e^{2} - 2 \, a^{2} b d e^{3} + a^{3} e^{4}\right )} f^{2} + 2 \, {\left (4 \, b^{3} d^{2} e^{3} - 11 \, a b^{2} d e^{4} + 7 \, a^{2} b e^{5}\right )} f\right )} x\right )} \sqrt {f x^{2} + e x + d} \sqrt {\frac {e}{b^{2} d e - a b e^{2} - 4 \, {\left (a b d - a^{2} e\right )} f}}}{b^{2} f^{2} x^{4} + 2 \, b^{2} e f x^{3} + 2 \, a b e^{2} x + a^{2} e^{2} + {\left (b^{2} e^{2} + 2 \, a b e f\right )} x^{2}}\right ), -\sqrt {-\frac {e}{b^{2} d e - a b e^{2} - 4 \, {\left (a b d - a^{2} e\right )} f}} \arctan \left (-\frac {{\left (2 \, b d e^{2} - a e^{3} - 4 \, a d e f + {\left (b e^{2} f + 4 \, {\left (b d - 2 \, a e\right )} f^{2}\right )} x^{2} + {\left (b e^{3} + 4 \, {\left (b d e - 2 \, a e^{2}\right )} f\right )} x\right )} \sqrt {f x^{2} + e x + d} \sqrt {-\frac {e}{b^{2} d e - a b e^{2} - 4 \, {\left (a b d - a^{2} e\right )} f}}}{2 \, {\left (2 \, e f^{2} x^{3} + 3 \, e^{2} f x^{2} + d e^{2} + {\left (e^{3} + 2 \, d e f\right )} x\right )}}\right )\right ] \]
[1/2*sqrt(e/(b^2*d*e - a*b*e^2 - 4*(a*b*d - a^2*e)*f))*log((8*b^2*d^2*e^4 - 8*a*b*d*e^5 + a^2*e^6 + 16*a^2*d^2*e^2*f^2 + (b^2*e^4*f^2 + 16*(b^2*d^2 - 8*a*b*d*e + 8*a^2*e^2)*f^4 + 8*(3*b^2*d*e^2 - 4*a*b*e^3)*f^3)*x^4 + 2*(b ^2*e^5*f + 16*(b^2*d^2*e - 8*a*b*d*e^2 + 8*a^2*e^3)*f^3 + 8*(3*b^2*d*e^3 - 4*a*b*e^4)*f^2)*x^3 + (b^2*e^6 - 32*(3*a*b*d^2*e - 4*a^2*d*e^2)*f^3 + 16* (3*b^2*d^2*e^2 - 13*a*b*d*e^3 + 10*a^2*e^4)*f^2 + 2*(16*b^2*d*e^4 - 19*a*b *e^5)*f)*x^2 - 8*(4*a*b*d^2*e^3 - 3*a^2*d*e^4)*f + 2*(4*b^2*d*e^5 - 3*a*b* e^6 - 16*(3*a*b*d^2*e^2 - 4*a^2*d*e^3)*f^2 + 8*(2*b^2*d^2*e^3 - 5*a*b*d*e^ 4 + 2*a^2*e^5)*f)*x - 4*(2*b^3*d^2*e^4 - 3*a*b^2*d*e^5 + a^2*b*e^6 - 2*(16 *(a*b^2*d^2 - 3*a^2*b*d*e + 2*a^3*e^2)*f^4 - 4*(b^3*d^2*e - 4*a*b^2*d*e^2 + 3*a^2*b*e^3)*f^3 - (b^3*d*e^3 - a*b^2*e^4)*f^2)*x^3 + 16*(a^2*b*d^2*e^2 - a^3*d*e^3)*f^2 - 3*(16*(a*b^2*d^2*e - 3*a^2*b*d*e^2 + 2*a^3*e^3)*f^3 - 4 *(b^3*d^2*e^2 - 4*a*b^2*d*e^3 + 3*a^2*b*e^4)*f^2 - (b^3*d*e^4 - a*b^2*e^5) *f)*x^2 - 4*(3*a*b^2*d^2*e^3 - 4*a^2*b*d*e^4 + a^3*e^5)*f + (b^3*d*e^5 - a *b^2*e^6 + 32*(a^2*b*d^2*e - a^3*d*e^2)*f^3 - 40*(a*b^2*d^2*e^2 - 2*a^2*b* d*e^3 + a^3*e^4)*f^2 + 2*(4*b^3*d^2*e^3 - 11*a*b^2*d*e^4 + 7*a^2*b*e^5)*f) *x)*sqrt(f*x^2 + e*x + d)*sqrt(e/(b^2*d*e - a*b*e^2 - 4*(a*b*d - a^2*e)*f) ))/(b^2*f^2*x^4 + 2*b^2*e*f*x^3 + 2*a*b*e^2*x + a^2*e^2 + (b^2*e^2 + 2*a*b *e*f)*x^2)), -sqrt(-e/(b^2*d*e - a*b*e^2 - 4*(a*b*d - a^2*e)*f))*arctan(-1 /2*(2*b*d*e^2 - a*e^3 - 4*a*d*e*f + (b*e^2*f + 4*(b*d - 2*a*e)*f^2)*x^2...
\[ \int \frac {1}{\sqrt {d+e x+f x^2} \left (a+b x+\frac {b f x^2}{e}\right )} \, dx=e \int \frac {1}{a e \sqrt {d + e x + f x^{2}} + b e x \sqrt {d + e x + f x^{2}} + b f x^{2} \sqrt {d + e x + f x^{2}}}\, dx \]
e*Integral(1/(a*e*sqrt(d + e*x + f*x**2) + b*e*x*sqrt(d + e*x + f*x**2) + b*f*x**2*sqrt(d + e*x + f*x**2)), x)
Exception generated. \[ \int \frac {1}{\sqrt {d+e x+f x^2} \left (a+b x+\frac {b f x^2}{e}\right )} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e*(4*a*f-b*e)>0)', see `assume?` for more
Leaf count of result is larger than twice the leaf count of optimal. 851 vs. \(2 (68) = 136\).
Time = 0.96 (sec) , antiderivative size = 851, normalized size of antiderivative = 10.38 \[ \int \frac {1}{\sqrt {d+e x+f x^2} \left (a+b x+\frac {b f x^2}{e}\right )} \, dx=-\frac {\sqrt {b^{2} d e^{2} - a b e^{3} - 4 \, a b d e f + 4 \, a^{2} e^{2} f} \log \left ({\left | -{\left (\sqrt {f} x - \sqrt {f x^{2} + e x + d}\right )}^{2} b e^{2} f - 4 \, {\left (\sqrt {f} x - \sqrt {f x^{2} + e x + d}\right )}^{2} b d f^{2} + 8 \, {\left (\sqrt {f} x - \sqrt {f x^{2} + e x + d}\right )}^{2} a e f^{2} - {\left (\sqrt {f} x - \sqrt {f x^{2} + e x + d}\right )} b e^{3} \sqrt {f} - 4 \, {\left (\sqrt {f} x - \sqrt {f x^{2} + e x + d}\right )} b d e f^{\frac {3}{2}} + 8 \, {\left (\sqrt {f} x - \sqrt {f x^{2} + e x + d}\right )} a e^{2} f^{\frac {3}{2}} - 3 \, b d e^{2} f + 2 \, a e^{3} f + 4 \, b d^{2} f^{2} + 4 \, \sqrt {b^{2} d e^{2} - a b e^{3} - 4 \, a b d e f + 4 \, a^{2} e^{2} f} {\left (\sqrt {f} x - \sqrt {f x^{2} + e x + d}\right )}^{2} f^{\frac {3}{2}} + 4 \, \sqrt {b^{2} d e^{2} - a b e^{3} - 4 \, a b d e f + 4 \, a^{2} e^{2} f} {\left (\sqrt {f} x - \sqrt {f x^{2} + e x + d}\right )} e f + \sqrt {b^{2} d e^{2} - a b e^{3} - 4 \, a b d e f + 4 \, a^{2} e^{2} f} e^{2} \sqrt {f} \right |}\right )}{b^{2} d e - a b e^{2} - 4 \, a b d f + 4 \, a^{2} e f} + \frac {\sqrt {b^{2} d e^{2} - a b e^{3} - 4 \, a b d e f + 4 \, a^{2} e^{2} f} \log \left ({\left | -{\left (\sqrt {f} x - \sqrt {f x^{2} + e x + d}\right )}^{2} b e^{2} f - 4 \, {\left (\sqrt {f} x - \sqrt {f x^{2} + e x + d}\right )}^{2} b d f^{2} + 8 \, {\left (\sqrt {f} x - \sqrt {f x^{2} + e x + d}\right )}^{2} a e f^{2} - {\left (\sqrt {f} x - \sqrt {f x^{2} + e x + d}\right )} b e^{3} \sqrt {f} - 4 \, {\left (\sqrt {f} x - \sqrt {f x^{2} + e x + d}\right )} b d e f^{\frac {3}{2}} + 8 \, {\left (\sqrt {f} x - \sqrt {f x^{2} + e x + d}\right )} a e^{2} f^{\frac {3}{2}} - 3 \, b d e^{2} f + 2 \, a e^{3} f + 4 \, b d^{2} f^{2} - 4 \, \sqrt {b^{2} d e^{2} - a b e^{3} - 4 \, a b d e f + 4 \, a^{2} e^{2} f} {\left (\sqrt {f} x - \sqrt {f x^{2} + e x + d}\right )}^{2} f^{\frac {3}{2}} - 4 \, \sqrt {b^{2} d e^{2} - a b e^{3} - 4 \, a b d e f + 4 \, a^{2} e^{2} f} {\left (\sqrt {f} x - \sqrt {f x^{2} + e x + d}\right )} e f - \sqrt {b^{2} d e^{2} - a b e^{3} - 4 \, a b d e f + 4 \, a^{2} e^{2} f} e^{2} \sqrt {f} \right |}\right )}{b^{2} d e - a b e^{2} - 4 \, a b d f + 4 \, a^{2} e f} \]
-sqrt(b^2*d*e^2 - a*b*e^3 - 4*a*b*d*e*f + 4*a^2*e^2*f)*log(abs(-(sqrt(f)*x - sqrt(f*x^2 + e*x + d))^2*b*e^2*f - 4*(sqrt(f)*x - sqrt(f*x^2 + e*x + d) )^2*b*d*f^2 + 8*(sqrt(f)*x - sqrt(f*x^2 + e*x + d))^2*a*e*f^2 - (sqrt(f)*x - sqrt(f*x^2 + e*x + d))*b*e^3*sqrt(f) - 4*(sqrt(f)*x - sqrt(f*x^2 + e*x + d))*b*d*e*f^(3/2) + 8*(sqrt(f)*x - sqrt(f*x^2 + e*x + d))*a*e^2*f^(3/2) - 3*b*d*e^2*f + 2*a*e^3*f + 4*b*d^2*f^2 + 4*sqrt(b^2*d*e^2 - a*b*e^3 - 4*a *b*d*e*f + 4*a^2*e^2*f)*(sqrt(f)*x - sqrt(f*x^2 + e*x + d))^2*f^(3/2) + 4* sqrt(b^2*d*e^2 - a*b*e^3 - 4*a*b*d*e*f + 4*a^2*e^2*f)*(sqrt(f)*x - sqrt(f* x^2 + e*x + d))*e*f + sqrt(b^2*d*e^2 - a*b*e^3 - 4*a*b*d*e*f + 4*a^2*e^2*f )*e^2*sqrt(f)))/(b^2*d*e - a*b*e^2 - 4*a*b*d*f + 4*a^2*e*f) + sqrt(b^2*d*e ^2 - a*b*e^3 - 4*a*b*d*e*f + 4*a^2*e^2*f)*log(abs(-(sqrt(f)*x - sqrt(f*x^2 + e*x + d))^2*b*e^2*f - 4*(sqrt(f)*x - sqrt(f*x^2 + e*x + d))^2*b*d*f^2 + 8*(sqrt(f)*x - sqrt(f*x^2 + e*x + d))^2*a*e*f^2 - (sqrt(f)*x - sqrt(f*x^2 + e*x + d))*b*e^3*sqrt(f) - 4*(sqrt(f)*x - sqrt(f*x^2 + e*x + d))*b*d*e*f ^(3/2) + 8*(sqrt(f)*x - sqrt(f*x^2 + e*x + d))*a*e^2*f^(3/2) - 3*b*d*e^2*f + 2*a*e^3*f + 4*b*d^2*f^2 - 4*sqrt(b^2*d*e^2 - a*b*e^3 - 4*a*b*d*e*f + 4* a^2*e^2*f)*(sqrt(f)*x - sqrt(f*x^2 + e*x + d))^2*f^(3/2) - 4*sqrt(b^2*d*e^ 2 - a*b*e^3 - 4*a*b*d*e*f + 4*a^2*e^2*f)*(sqrt(f)*x - sqrt(f*x^2 + e*x + d ))*e*f - sqrt(b^2*d*e^2 - a*b*e^3 - 4*a*b*d*e*f + 4*a^2*e^2*f)*e^2*sqrt(f) ))/(b^2*d*e - a*b*e^2 - 4*a*b*d*f + 4*a^2*e*f)
Timed out. \[ \int \frac {1}{\sqrt {d+e x+f x^2} \left (a+b x+\frac {b f x^2}{e}\right )} \, dx=\int \frac {1}{\sqrt {f\,x^2+e\,x+d}\,\left (a+b\,x+\frac {b\,f\,x^2}{e}\right )} \,d x \]