Integrand size = 29, antiderivative size = 780 \[ \int \frac {A+B x}{\left (a+c x^2\right ) \sqrt {d+e x+f x^2}} \, dx=\frac {\sqrt {a B e+A \left (c d-a f-\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )} \sqrt {-A c e+B \left (c d-a f+\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )} \text {arctanh}\left (\frac {\sqrt {e} \left (a \left (A c e-B \left (c d-a f+\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right )-c \left (a B e+A \left (c d-a f-\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right ) x\right )}{\sqrt {2} \sqrt {a} \sqrt {c} \sqrt {a B e+A \left (c d-a f-\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )} \sqrt {-A c e+B \left (c d-a f+\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )} \sqrt {d+e x+f x^2}}\right )}{\sqrt {2} \sqrt {a} \sqrt {c} \sqrt {e} \sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}}-\frac {\sqrt {-A c e+B \left (c d-a f-\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )} \sqrt {a B e+A \left (c d-a f+\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )} \text {arctanh}\left (\frac {\sqrt {e} \left (a \left (A c e-B \left (c d-a f-\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right )-c \left (a B e+A \left (c d-a f+\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right ) x\right )}{\sqrt {2} \sqrt {a} \sqrt {c} \sqrt {-A c e+B \left (c d-a f-\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )} \sqrt {a B e+A \left (c d-a f+\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )} \sqrt {d+e x+f x^2}}\right )}{\sqrt {2} \sqrt {a} \sqrt {c} \sqrt {e} \sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}} \]
-1/2*arctanh(1/2*e^(1/2)*(a*(A*c*e-B*(c*d-a*f-(c^2*d^2+a^2*f^2+a*c*(-2*d*f +e^2))^(1/2)))-c*x*(B*a*e+A*(c*d-a*f+(c^2*d^2+a^2*f^2+a*c*(-2*d*f+e^2))^(1 /2))))*2^(1/2)/a^(1/2)/c^(1/2)/(f*x^2+e*x+d)^(1/2)/(-A*c*e+B*(c*d-a*f-(c^2 *d^2+a^2*f^2+a*c*(-2*d*f+e^2))^(1/2)))^(1/2)/(B*a*e+A*(c*d-a*f+(c^2*d^2+a^ 2*f^2+a*c*(-2*d*f+e^2))^(1/2)))^(1/2))*(-A*c*e+B*(c*d-a*f-(c^2*d^2+a^2*f^2 +a*c*(-2*d*f+e^2))^(1/2)))^(1/2)*(B*a*e+A*(c*d-a*f+(c^2*d^2+a^2*f^2+a*c*(- 2*d*f+e^2))^(1/2)))^(1/2)*2^(1/2)/a^(1/2)/c^(1/2)/e^(1/2)/(c^2*d^2+a^2*f^2 +a*c*(-2*d*f+e^2))^(1/2)+1/2*arctanh(1/2*e^(1/2)*(-c*x*(B*a*e+A*(c*d-a*f-( c^2*d^2+a^2*f^2+a*c*(-2*d*f+e^2))^(1/2)))+a*(A*c*e-B*(c*d-a*f+(c^2*d^2+a^2 *f^2+a*c*(-2*d*f+e^2))^(1/2))))*2^(1/2)/a^(1/2)/c^(1/2)/(f*x^2+e*x+d)^(1/2 )/(B*a*e+A*(c*d-a*f-(c^2*d^2+a^2*f^2+a*c*(-2*d*f+e^2))^(1/2)))^(1/2)/(-A*c *e+B*(c*d-a*f+(c^2*d^2+a^2*f^2+a*c*(-2*d*f+e^2))^(1/2)))^(1/2))*(B*a*e+A*( c*d-a*f-(c^2*d^2+a^2*f^2+a*c*(-2*d*f+e^2))^(1/2)))^(1/2)*(-A*c*e+B*(c*d-a* f+(c^2*d^2+a^2*f^2+a*c*(-2*d*f+e^2))^(1/2)))^(1/2)*2^(1/2)/a^(1/2)/c^(1/2) /e^(1/2)/(c^2*d^2+a^2*f^2+a*c*(-2*d*f+e^2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.37 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.28 \[ \int \frac {A+B x}{\left (a+c x^2\right ) \sqrt {d+e x+f x^2}} \, dx=\frac {1}{2} \text {RootSum}\left [c d^2+a e^2-4 a e \sqrt {f} \text {$\#$1}-2 c d \text {$\#$1}^2+4 a f \text {$\#$1}^2+c \text {$\#$1}^4\&,\frac {B d \log \left (-\sqrt {f} x+\sqrt {d+e x+f x^2}-\text {$\#$1}\right )-A e \log \left (-\sqrt {f} x+\sqrt {d+e x+f x^2}-\text {$\#$1}\right )+2 A \sqrt {f} \log \left (-\sqrt {f} x+\sqrt {d+e x+f x^2}-\text {$\#$1}\right ) \text {$\#$1}-B \log \left (-\sqrt {f} x+\sqrt {d+e x+f x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{a e \sqrt {f}+c d \text {$\#$1}-2 a f \text {$\#$1}-c \text {$\#$1}^3}\&\right ] \]
RootSum[c*d^2 + a*e^2 - 4*a*e*Sqrt[f]*#1 - 2*c*d*#1^2 + 4*a*f*#1^2 + c*#1^ 4 & , (B*d*Log[-(Sqrt[f]*x) + Sqrt[d + e*x + f*x^2] - #1] - A*e*Log[-(Sqrt [f]*x) + Sqrt[d + e*x + f*x^2] - #1] + 2*A*Sqrt[f]*Log[-(Sqrt[f]*x) + Sqrt [d + e*x + f*x^2] - #1]*#1 - B*Log[-(Sqrt[f]*x) + Sqrt[d + e*x + f*x^2] - #1]*#1^2)/(a*e*Sqrt[f] + c*d*#1 - 2*a*f*#1 - c*#1^3) & ]/2
Time = 2.36 (sec) , antiderivative size = 874, normalized size of antiderivative = 1.12, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1369, 25, 1363, 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.
| \(\displaystyle \int \frac {A+B x}{\left (a+c x^2\right ) \sqrt {d+e x+f x^2}} \, dx\) |
| \(\Big \downarrow \) 1369 |
| \(\displaystyle \frac {\int -\frac {a B e+A \left (c d-a f-\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )+\left (A c e-B \left (c d-a f+\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right ) x}{\left (c x^2+a\right ) \sqrt {f x^2+e x+d}}dx}{2 \sqrt {a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}}-\frac {\int -\frac {a B e+A \left (c d-a f+\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )+\left (A c e-B \left (c d-a f-\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right ) x}{\left (c x^2+a\right ) \sqrt {f x^2+e x+d}}dx}{2 \sqrt {a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {a B e+A \left (c d-a f+\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )+\left (A c e-B \left (c d-a f-\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right ) x}{\left (c x^2+a\right ) \sqrt {f x^2+e x+d}}dx}{2 \sqrt {a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}}-\frac {\int \frac {a B e+A \left (c d-a f-\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )+\left (A c e-B \left (c d-a f+\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right ) x}{\left (c x^2+a\right ) \sqrt {f x^2+e x+d}}dx}{2 \sqrt {a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}}\) |
| \(\Big \downarrow \) 1363 |
| \(\displaystyle \frac {a \left (a B e+A \left (c d-a f-\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right ) \left (A c e-B \left (c d-a f+\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right ) \int \frac {1}{2 c \left (a B e+A \left (c d-a f-\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right ) \left (A c e-B \left (c d-a f+\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right ) a^2+\frac {e \left (a \left (A c e-B \left (c d-a f+\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right )-c \left (a B e+A \left (c d-a f-\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right ) x\right )^2 a}{f x^2+e x+d}}d\frac {a \left (A c e-B \left (c d-a f+\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right )-c \left (a B e+A \left (c d-a f-\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right ) x}{\sqrt {f x^2+e x+d}}}{\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}}-\frac {a \left (A c e-B \left (c d-a f-\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right ) \left (a B e+A \left (c d-a f+\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right ) \int \frac {1}{2 c \left (A c e-B \left (c d-a f-\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right ) \left (a B e+A \left (c d-a f+\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right ) a^2+\frac {e \left (a \left (A c e-B \left (c d-a f-\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right )-c \left (a B e+A \left (c d-a f+\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right ) x\right )^2 a}{f x^2+e x+d}}d\frac {a \left (A c e-B \left (c d-a f-\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right )-c \left (a B e+A \left (c d-a f+\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right ) x}{\sqrt {f x^2+e x+d}}}{\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\left (A c e-B \left (c d-a f-\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right ) \sqrt {a B e+A \left (c d-a f+\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )} \text {arctanh}\left (\frac {\sqrt {e} \left (a \left (A c e-B \left (c d-a f-\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right )-c \left (a B e+A \left (c d-a f+\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right ) x\right )}{\sqrt {2} \sqrt {a} \sqrt {c} \sqrt {B \left (c d-a f-\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )-A c e} \sqrt {a B e+A \left (c d-a f+\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )} \sqrt {f x^2+e x+d}}\right )}{\sqrt {2} \sqrt {a} \sqrt {c} \sqrt {e} \sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )} \sqrt {B \left (c d-a f-\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )-A c e}}-\frac {\sqrt {a B e+A \left (c d-a f-\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )} \left (A c e-B \left (c d-a f+\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right ) \text {arctanh}\left (\frac {\sqrt {e} \left (a \left (A c e-B \left (c d-a f+\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right )-c \left (a B e+A \left (c d-a f-\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right ) x\right )}{\sqrt {2} \sqrt {a} \sqrt {c} \sqrt {a B e+A \left (c d-a f-\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )} \sqrt {B \left (c d-a f+\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )-A c e} \sqrt {f x^2+e x+d}}\right )}{\sqrt {2} \sqrt {a} \sqrt {c} \sqrt {e} \sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )} \sqrt {B \left (c d-a f+\sqrt {c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )-A c e}}\) |
-((Sqrt[a*B*e + A*(c*d - a*f - Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)] )]*(A*c*e - B*(c*d - a*f + Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)]))*A rcTanh[(Sqrt[e]*(a*(A*c*e - B*(c*d - a*f + Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e ^2 - 2*d*f)])) - c*(a*B*e + A*(c*d - a*f - Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e ^2 - 2*d*f)]))*x))/(Sqrt[2]*Sqrt[a]*Sqrt[c]*Sqrt[a*B*e + A*(c*d - a*f - Sq rt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*Sqrt[-(A*c*e) + B*(c*d - a*f + Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*Sqrt[d + e*x + f*x^2])])/(S qrt[2]*Sqrt[a]*Sqrt[c]*Sqrt[e]*Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)] *Sqrt[-(A*c*e) + B*(c*d - a*f + Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f) ])])) + ((A*c*e - B*(c*d - a*f - Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f )]))*Sqrt[a*B*e + A*(c*d - a*f + Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f )])]*ArcTanh[(Sqrt[e]*(a*(A*c*e - B*(c*d - a*f - Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])) - c*(a*B*e + A*(c*d - a*f + Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)]))*x))/(Sqrt[2]*Sqrt[a]*Sqrt[c]*Sqrt[-(A*c*e) + B*(c*d - a*f - Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*Sqrt[a*B*e + A*(c*d - a*f + Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*Sqrt[d + e*x + f*x^2] )])/(Sqrt[2]*Sqrt[a]*Sqrt[c]*Sqrt[e]*Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2 *d*f)]*Sqrt[-(A*c*e) + B*(c*d - a*f - Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])])
3.1.22.3.1 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[((g_) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f _.)*(x_)^2]), x_Symbol] :> Simp[-2*a*g*h Subst[Int[1/Simp[2*a^2*g*h*c + a *e*x^2, x], x], x, Simp[a*h - g*c*x, x]/Sqrt[d + e*x + f*x^2]], x] /; FreeQ [{a, c, d, e, f, g, h}, x] && EqQ[a*h^2*e + 2*g*h*(c*d - a*f) - g^2*c*e, 0]
Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + ( f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[(c*d - a*f)^2 + a*c*e^2, 2]}, Simp [1/(2*q) Int[Simp[(-a)*h*e - g*(c*d - a*f - q) + (h*(c*d - a*f + q) - g*c *e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - Simp[1/(2*q) Int[ Simp[(-a)*h*e - g*(c*d - a*f + q) + (h*(c*d - a*f - q) - g*c*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f, g, h}, x] && NeQ[e^2 - 4*d*f, 0] && NegQ[(-a)*c]
Time = 0.77 (sec) , antiderivative size = 425, normalized size of antiderivative = 0.54
| method | result | size |
| default | \(-\frac {\left (-A c +B \sqrt {-a c}\right ) \ln \left (\frac {-\frac {2 \left (e \sqrt {-a c}+f a -c d \right )}{c}+\frac {\left (-2 \sqrt {-a c}\, f +c e \right ) \left (x +\frac {\sqrt {-a c}}{c}\right )}{c}+2 \sqrt {-\frac {e \sqrt {-a c}+f a -c d}{c}}\, \sqrt {f \left (x +\frac {\sqrt {-a c}}{c}\right )^{2}+\frac {\left (-2 \sqrt {-a c}\, f +c e \right ) \left (x +\frac {\sqrt {-a c}}{c}\right )}{c}-\frac {e \sqrt {-a c}+f a -c d}{c}}}{x +\frac {\sqrt {-a c}}{c}}\right )}{2 \sqrt {-a c}\, c \sqrt {-\frac {e \sqrt {-a c}+f a -c d}{c}}}-\frac {\left (A c +B \sqrt {-a c}\right ) \ln \left (\frac {-\frac {2 \left (-e \sqrt {-a c}+f a -c d \right )}{c}+\frac {\left (2 \sqrt {-a c}\, f +c e \right ) \left (x -\frac {\sqrt {-a c}}{c}\right )}{c}+2 \sqrt {-\frac {-e \sqrt {-a c}+f a -c d}{c}}\, \sqrt {f \left (x -\frac {\sqrt {-a c}}{c}\right )^{2}+\frac {\left (2 \sqrt {-a c}\, f +c e \right ) \left (x -\frac {\sqrt {-a c}}{c}\right )}{c}-\frac {-e \sqrt {-a c}+f a -c d}{c}}}{x -\frac {\sqrt {-a c}}{c}}\right )}{2 \sqrt {-a c}\, c \sqrt {-\frac {-e \sqrt {-a c}+f a -c d}{c}}}\) | \(425\) |
-1/2*(-A*c+B*(-a*c)^(1/2))/(-a*c)^(1/2)/c/(-(e*(-a*c)^(1/2)+f*a-c*d)/c)^(1 /2)*ln((-2*(e*(-a*c)^(1/2)+f*a-c*d)/c+1/c*(-2*(-a*c)^(1/2)*f+c*e)*(x+(-a*c )^(1/2)/c)+2*(-(e*(-a*c)^(1/2)+f*a-c*d)/c)^(1/2)*(f*(x+(-a*c)^(1/2)/c)^2+1 /c*(-2*(-a*c)^(1/2)*f+c*e)*(x+(-a*c)^(1/2)/c)-(e*(-a*c)^(1/2)+f*a-c*d)/c)^ (1/2))/(x+(-a*c)^(1/2)/c))-1/2*(A*c+B*(-a*c)^(1/2))/(-a*c)^(1/2)/c/(-(-e*( -a*c)^(1/2)+f*a-c*d)/c)^(1/2)*ln((-2*(-e*(-a*c)^(1/2)+f*a-c*d)/c+(2*(-a*c) ^(1/2)*f+c*e)/c*(x-(-a*c)^(1/2)/c)+2*(-(-e*(-a*c)^(1/2)+f*a-c*d)/c)^(1/2)* (f*(x-(-a*c)^(1/2)/c)^2+(2*(-a*c)^(1/2)*f+c*e)/c*(x-(-a*c)^(1/2)/c)-(-e*(- a*c)^(1/2)+f*a-c*d)/c)^(1/2))/(x-(-a*c)^(1/2)/c))
Leaf count of result is larger than twice the leaf count of optimal. 6861 vs. \(2 (703) = 1406\).
Time = 28.21 (sec) , antiderivative size = 6861, normalized size of antiderivative = 8.80 \[ \int \frac {A+B x}{\left (a+c x^2\right ) \sqrt {d+e x+f x^2}} \, dx=\text {Too large to display} \]
\[ \int \frac {A+B x}{\left (a+c x^2\right ) \sqrt {d+e x+f x^2}} \, dx=\int \frac {A + B x}{\left (a + c x^{2}\right ) \sqrt {d + e x + f x^{2}}}\, dx \]
\[ \int \frac {A+B x}{\left (a+c x^2\right ) \sqrt {d+e x+f x^2}} \, dx=\int { \frac {B x + A}{{\left (c x^{2} + a\right )} \sqrt {f x^{2} + e x + d}} \,d x } \]
Exception generated. \[ \int \frac {A+B x}{\left (a+c x^2\right ) \sqrt {d+e x+f x^2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(con st gen &
Timed out. \[ \int \frac {A+B x}{\left (a+c x^2\right ) \sqrt {d+e x+f x^2}} \, dx=\int \frac {A+B\,x}{\left (c\,x^2+a\right )\,\sqrt {f\,x^2+e\,x+d}} \,d x \]