Integrand size = 25, antiderivative size = 266 \[ \int \frac {\sqrt {a+b x+c x^2}}{d-f x^2} \, dx=-\frac {\sqrt {c} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{f}+\frac {\sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \text {arctanh}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {d} f}+\frac {\sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \text {arctanh}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {d} f} \]
-arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))*c^(1/2)/f+1/2*arctanh( 1/2*(b*d^(1/2)-2*a*f^(1/2)+x*(2*c*d^(1/2)-b*f^(1/2)))/(c*x^2+b*x+a)^(1/2)/ (c*d+a*f-b*d^(1/2)*f^(1/2))^(1/2))*(c*d+a*f-b*d^(1/2)*f^(1/2))^(1/2)/f/d^( 1/2)+1/2*arctanh(1/2*(b*d^(1/2)+2*a*f^(1/2)+x*(2*c*d^(1/2)+b*f^(1/2)))/(c* x^2+b*x+a)^(1/2)/(c*d+a*f+b*d^(1/2)*f^(1/2))^(1/2))*(c*d+a*f+b*d^(1/2)*f^( 1/2))^(1/2)/f/d^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.36 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.46 \[ \int \frac {\sqrt {a+b x+c x^2}}{d-f x^2} \, dx=-\frac {-4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a}-\sqrt {a+x (b+c x)}}\right )+\text {RootSum}\left [c^2 d-b^2 f+4 \sqrt {a} b f \text {$\#$1}-2 c d \text {$\#$1}^2-4 a f \text {$\#$1}^2+d \text {$\#$1}^4\&,\frac {-c^2 d \log (x)+b^2 f \log (x)-a c f \log (x)+c^2 d \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right )-b^2 f \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right )+a c f \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right )-2 \sqrt {a} b f \log (x) \text {$\#$1}+2 \sqrt {a} b f \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}+c d \log (x) \text {$\#$1}^2+a f \log (x) \text {$\#$1}^2-c d \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-a f \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-\sqrt {a} b f+c d \text {$\#$1}+2 a f \text {$\#$1}-d \text {$\#$1}^3}\&\right ]}{2 f} \]
-1/2*(-4*Sqrt[c]*ArcTanh[(Sqrt[c]*x)/(Sqrt[a] - Sqrt[a + x*(b + c*x)])] + RootSum[c^2*d - b^2*f + 4*Sqrt[a]*b*f*#1 - 2*c*d*#1^2 - 4*a*f*#1^2 + d*#1^ 4 & , (-(c^2*d*Log[x]) + b^2*f*Log[x] - a*c*f*Log[x] + c^2*d*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] - b^2*f*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^ 2] - x*#1] + a*c*f*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] - 2*Sqrt[a ]*b*f*Log[x]*#1 + 2*Sqrt[a]*b*f*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*# 1]*#1 + c*d*Log[x]*#1^2 + a*f*Log[x]*#1^2 - c*d*Log[-Sqrt[a] + Sqrt[a + b* x + c*x^2] - x*#1]*#1^2 - a*f*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] *#1^2)/(-(Sqrt[a]*b*f) + c*d*#1 + 2*a*f*#1 - d*#1^3) & ])/f
Time = 0.45 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {1321, 1092, 219, 1366, 25, 27, 1154, 219}
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.
| \(\displaystyle \int \frac {\sqrt {a+b x+c x^2}}{d-f x^2} \, dx\) |
| \(\Big \downarrow \) 1321 |
| \(\displaystyle \frac {\int \frac {c d+a f+b f x}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{f}-\frac {c \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {\int \frac {c d+a f+b f x}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{f}-\frac {2 c \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\int \frac {c d+a f+b f x}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{f}-\frac {\sqrt {c} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{f}\) |
| \(\Big \downarrow \) 1366 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt {f} \left (\frac {a f+c d}{\sqrt {d}}+b \sqrt {f}\right ) \int \frac {1}{\sqrt {f} \left (\sqrt {d}-\sqrt {f} x\right ) \sqrt {c x^2+b x+a}}dx+\frac {1}{2} \sqrt {f} \left (b \sqrt {f}-\frac {a f+c d}{\sqrt {d}}\right ) \int -\frac {1}{\sqrt {f} \left (\sqrt {f} x+\sqrt {d}\right ) \sqrt {c x^2+b x+a}}dx}{f}-\frac {\sqrt {c} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt {f} \left (\frac {a f+c d}{\sqrt {d}}+b \sqrt {f}\right ) \int \frac {1}{\sqrt {f} \left (\sqrt {d}-\sqrt {f} x\right ) \sqrt {c x^2+b x+a}}dx-\frac {1}{2} \sqrt {f} \left (b \sqrt {f}-\frac {a f+c d}{\sqrt {d}}\right ) \int \frac {1}{\sqrt {f} \left (\sqrt {f} x+\sqrt {d}\right ) \sqrt {c x^2+b x+a}}dx}{f}-\frac {\sqrt {c} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {a f+c d}{\sqrt {d}}+b \sqrt {f}\right ) \int \frac {1}{\left (\sqrt {d}-\sqrt {f} x\right ) \sqrt {c x^2+b x+a}}dx-\frac {1}{2} \left (b \sqrt {f}-\frac {a f+c d}{\sqrt {d}}\right ) \int \frac {1}{\left (\sqrt {f} x+\sqrt {d}\right ) \sqrt {c x^2+b x+a}}dx}{f}-\frac {\sqrt {c} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{f}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\left (b \sqrt {f}-\frac {a f+c d}{\sqrt {d}}\right ) \int \frac {1}{4 \left (-\sqrt {d} \sqrt {f} b+c d+a f\right )-\frac {\left (-2 \sqrt {f} a+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x+b \sqrt {d}\right )^2}{c x^2+b x+a}}d\left (-\frac {-2 \sqrt {f} a+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x+b \sqrt {d}}{\sqrt {c x^2+b x+a}}\right )-\left (\frac {a f+c d}{\sqrt {d}}+b \sqrt {f}\right ) \int \frac {1}{4 \left (\sqrt {d} \sqrt {f} b+c d+a f\right )-\frac {\left (2 \sqrt {f} a+\left (\sqrt {f} b+2 c \sqrt {d}\right ) x+b \sqrt {d}\right )^2}{c x^2+b x+a}}d\left (-\frac {2 \sqrt {f} a+\left (\sqrt {f} b+2 c \sqrt {d}\right ) x+b \sqrt {d}}{\sqrt {c x^2+b x+a}}\right )}{f}-\frac {\sqrt {c} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\left (\frac {a f+c d}{\sqrt {d}}+b \sqrt {f}\right ) \text {arctanh}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}-\frac {\left (b \sqrt {f}-\frac {a f+c d}{\sqrt {d}}\right ) \text {arctanh}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}}{f}-\frac {\sqrt {c} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{f}\) |
-((Sqrt[c]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/f) + (- 1/2*((b*Sqrt[f] - (c*d + a*f)/Sqrt[d])*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f] + ((b*Sqrt[f] + (c* d + a*f)/Sqrt[d])*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt [f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2 *Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]))/f
3.1.80.3.1 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[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]/((d_) + (f_.)*(x_)^2), x_Symbol] :> Simp[c/f Int[1/Sqrt[a + b*x + c*x^2], x], x] - Simp[1/f Int[(c*d - a*f - b*f*x)/(Sqrt[a + b*x + c*x^2]*(d + f*x^2)), x], x] /; FreeQ[{a, b, c, d, f}, x] && NeQ[b^2 - 4*a*c, 0]
Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + ( f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Simp[(h/2 + c*(g/(2*q ))) Int[1/((-q + c*x)*Sqrt[d + e*x + f*x^2]), x], x] + Simp[(h/2 - c*(g/( 2*q))) Int[1/((q + c*x)*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] && PosQ[(-a)*c]
Leaf count of result is larger than twice the leaf count of optimal. \(771\) vs. \(2(202)=404\).
Time = 0.68 (sec) , antiderivative size = 772, normalized size of antiderivative = 2.90
| method | result | size |
| default | \(-\frac {\sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}+\frac {\left (2 c \sqrt {d f}+b f \right ) \ln \left (\frac {\frac {2 c \sqrt {d f}+b f}{2 f}+c \left (x -\frac {\sqrt {d f}}{f}\right )}{\sqrt {c}}+\sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}\right )}{2 f \sqrt {c}}-\frac {\left (b \sqrt {d f}+f a +c d \right ) \ln \left (\frac {\frac {2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{f \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}}}{2 \sqrt {d f}}+\frac {\sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}+\frac {\left (-2 c \sqrt {d f}+b f \right ) \ln \left (\frac {\frac {-2 c \sqrt {d f}+b f}{2 f}+c \left (x +\frac {\sqrt {d f}}{f}\right )}{\sqrt {c}}+\sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}\right )}{2 f \sqrt {c}}-\frac {\left (-b \sqrt {d f}+f a +c d \right ) \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{f \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}}}{2 \sqrt {d f}}\) | \(772\) |
-1/2/(d*f)^(1/2)*(((x-(d*f)^(1/2)/f)^2*c+(2*c*(d*f)^(1/2)+b*f)/f*(x-(d*f)^ (1/2)/f)+(b*(d*f)^(1/2)+f*a+c*d)/f)^(1/2)+1/2*(2*c*(d*f)^(1/2)+b*f)/f*ln(( 1/2*(2*c*(d*f)^(1/2)+b*f)/f+c*(x-(d*f)^(1/2)/f))/c^(1/2)+((x-(d*f)^(1/2)/f )^2*c+(2*c*(d*f)^(1/2)+b*f)/f*(x-(d*f)^(1/2)/f)+(b*(d*f)^(1/2)+f*a+c*d)/f) ^(1/2))/c^(1/2)-(b*(d*f)^(1/2)+f*a+c*d)/f/((b*(d*f)^(1/2)+f*a+c*d)/f)^(1/2 )*ln((2*(b*(d*f)^(1/2)+f*a+c*d)/f+(2*c*(d*f)^(1/2)+b*f)/f*(x-(d*f)^(1/2)/f )+2*((b*(d*f)^(1/2)+f*a+c*d)/f)^(1/2)*((x-(d*f)^(1/2)/f)^2*c+(2*c*(d*f)^(1 /2)+b*f)/f*(x-(d*f)^(1/2)/f)+(b*(d*f)^(1/2)+f*a+c*d)/f)^(1/2))/(x-(d*f)^(1 /2)/f)))+1/2/(d*f)^(1/2)*(((x+(d*f)^(1/2)/f)^2*c+1/f*(-2*c*(d*f)^(1/2)+b*f )*(x+(d*f)^(1/2)/f)+1/f*(-b*(d*f)^(1/2)+f*a+c*d))^(1/2)+1/2/f*(-2*c*(d*f)^ (1/2)+b*f)*ln((1/2/f*(-2*c*(d*f)^(1/2)+b*f)+c*(x+(d*f)^(1/2)/f))/c^(1/2)+( (x+(d*f)^(1/2)/f)^2*c+1/f*(-2*c*(d*f)^(1/2)+b*f)*(x+(d*f)^(1/2)/f)+1/f*(-b *(d*f)^(1/2)+f*a+c*d))^(1/2))/c^(1/2)-1/f*(-b*(d*f)^(1/2)+f*a+c*d)/(1/f*(- b*(d*f)^(1/2)+f*a+c*d))^(1/2)*ln((2/f*(-b*(d*f)^(1/2)+f*a+c*d)+1/f*(-2*c*( d*f)^(1/2)+b*f)*(x+(d*f)^(1/2)/f)+2*(1/f*(-b*(d*f)^(1/2)+f*a+c*d))^(1/2)*( (x+(d*f)^(1/2)/f)^2*c+1/f*(-2*c*(d*f)^(1/2)+b*f)*(x+(d*f)^(1/2)/f)+1/f*(-b *(d*f)^(1/2)+f*a+c*d))^(1/2))/(x+(d*f)^(1/2)/f)))
Leaf count of result is larger than twice the leaf count of optimal. 568 vs. \(2 (202) = 404\).
Time = 65.90 (sec) , antiderivative size = 1139, normalized size of antiderivative = 4.28 \[ \int \frac {\sqrt {a+b x+c x^2}}{d-f x^2} \, dx=\left [\frac {f \sqrt {\frac {d f^{2} \sqrt {\frac {b^{2}}{d f^{3}}} + c d + a f}{d f^{2}}} \log \left (\frac {2 \, b c x + 2 \, \sqrt {c x^{2} + b x + a} b f \sqrt {\frac {d f^{2} \sqrt {\frac {b^{2}}{d f^{3}}} + c d + a f}{d f^{2}}} + b^{2} + {\left (b f^{2} x + 2 \, a f^{2}\right )} \sqrt {\frac {b^{2}}{d f^{3}}}}{x}\right ) - f \sqrt {\frac {d f^{2} \sqrt {\frac {b^{2}}{d f^{3}}} + c d + a f}{d f^{2}}} \log \left (\frac {2 \, b c x - 2 \, \sqrt {c x^{2} + b x + a} b f \sqrt {\frac {d f^{2} \sqrt {\frac {b^{2}}{d f^{3}}} + c d + a f}{d f^{2}}} + b^{2} + {\left (b f^{2} x + 2 \, a f^{2}\right )} \sqrt {\frac {b^{2}}{d f^{3}}}}{x}\right ) + f \sqrt {-\frac {d f^{2} \sqrt {\frac {b^{2}}{d f^{3}}} - c d - a f}{d f^{2}}} \log \left (\frac {2 \, b c x + 2 \, \sqrt {c x^{2} + b x + a} b f \sqrt {-\frac {d f^{2} \sqrt {\frac {b^{2}}{d f^{3}}} - c d - a f}{d f^{2}}} + b^{2} - {\left (b f^{2} x + 2 \, a f^{2}\right )} \sqrt {\frac {b^{2}}{d f^{3}}}}{x}\right ) - f \sqrt {-\frac {d f^{2} \sqrt {\frac {b^{2}}{d f^{3}}} - c d - a f}{d f^{2}}} \log \left (\frac {2 \, b c x - 2 \, \sqrt {c x^{2} + b x + a} b f \sqrt {-\frac {d f^{2} \sqrt {\frac {b^{2}}{d f^{3}}} - c d - a f}{d f^{2}}} + b^{2} - {\left (b f^{2} x + 2 \, a f^{2}\right )} \sqrt {\frac {b^{2}}{d f^{3}}}}{x}\right ) + 2 \, \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right )}{4 \, f}, \frac {f \sqrt {\frac {d f^{2} \sqrt {\frac {b^{2}}{d f^{3}}} + c d + a f}{d f^{2}}} \log \left (\frac {2 \, b c x + 2 \, \sqrt {c x^{2} + b x + a} b f \sqrt {\frac {d f^{2} \sqrt {\frac {b^{2}}{d f^{3}}} + c d + a f}{d f^{2}}} + b^{2} + {\left (b f^{2} x + 2 \, a f^{2}\right )} \sqrt {\frac {b^{2}}{d f^{3}}}}{x}\right ) - f \sqrt {\frac {d f^{2} \sqrt {\frac {b^{2}}{d f^{3}}} + c d + a f}{d f^{2}}} \log \left (\frac {2 \, b c x - 2 \, \sqrt {c x^{2} + b x + a} b f \sqrt {\frac {d f^{2} \sqrt {\frac {b^{2}}{d f^{3}}} + c d + a f}{d f^{2}}} + b^{2} + {\left (b f^{2} x + 2 \, a f^{2}\right )} \sqrt {\frac {b^{2}}{d f^{3}}}}{x}\right ) + f \sqrt {-\frac {d f^{2} \sqrt {\frac {b^{2}}{d f^{3}}} - c d - a f}{d f^{2}}} \log \left (\frac {2 \, b c x + 2 \, \sqrt {c x^{2} + b x + a} b f \sqrt {-\frac {d f^{2} \sqrt {\frac {b^{2}}{d f^{3}}} - c d - a f}{d f^{2}}} + b^{2} - {\left (b f^{2} x + 2 \, a f^{2}\right )} \sqrt {\frac {b^{2}}{d f^{3}}}}{x}\right ) - f \sqrt {-\frac {d f^{2} \sqrt {\frac {b^{2}}{d f^{3}}} - c d - a f}{d f^{2}}} \log \left (\frac {2 \, b c x - 2 \, \sqrt {c x^{2} + b x + a} b f \sqrt {-\frac {d f^{2} \sqrt {\frac {b^{2}}{d f^{3}}} - c d - a f}{d f^{2}}} + b^{2} - {\left (b f^{2} x + 2 \, a f^{2}\right )} \sqrt {\frac {b^{2}}{d f^{3}}}}{x}\right ) + 4 \, \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right )}{4 \, f}\right ] \]
[1/4*(f*sqrt((d*f^2*sqrt(b^2/(d*f^3)) + c*d + a*f)/(d*f^2))*log((2*b*c*x + 2*sqrt(c*x^2 + b*x + a)*b*f*sqrt((d*f^2*sqrt(b^2/(d*f^3)) + c*d + a*f)/(d *f^2)) + b^2 + (b*f^2*x + 2*a*f^2)*sqrt(b^2/(d*f^3)))/x) - f*sqrt((d*f^2*s qrt(b^2/(d*f^3)) + c*d + a*f)/(d*f^2))*log((2*b*c*x - 2*sqrt(c*x^2 + b*x + a)*b*f*sqrt((d*f^2*sqrt(b^2/(d*f^3)) + c*d + a*f)/(d*f^2)) + b^2 + (b*f^2 *x + 2*a*f^2)*sqrt(b^2/(d*f^3)))/x) + f*sqrt(-(d*f^2*sqrt(b^2/(d*f^3)) - c *d - a*f)/(d*f^2))*log((2*b*c*x + 2*sqrt(c*x^2 + b*x + a)*b*f*sqrt(-(d*f^2 *sqrt(b^2/(d*f^3)) - c*d - a*f)/(d*f^2)) + b^2 - (b*f^2*x + 2*a*f^2)*sqrt( b^2/(d*f^3)))/x) - f*sqrt(-(d*f^2*sqrt(b^2/(d*f^3)) - c*d - a*f)/(d*f^2))* log((2*b*c*x - 2*sqrt(c*x^2 + b*x + a)*b*f*sqrt(-(d*f^2*sqrt(b^2/(d*f^3)) - c*d - a*f)/(d*f^2)) + b^2 - (b*f^2*x + 2*a*f^2)*sqrt(b^2/(d*f^3)))/x) + 2*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 + 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c))/f, 1/4*(f*sqrt((d*f^2*sqrt(b^2/(d*f^3)) + c*d + a*f )/(d*f^2))*log((2*b*c*x + 2*sqrt(c*x^2 + b*x + a)*b*f*sqrt((d*f^2*sqrt(b^2 /(d*f^3)) + c*d + a*f)/(d*f^2)) + b^2 + (b*f^2*x + 2*a*f^2)*sqrt(b^2/(d*f^ 3)))/x) - f*sqrt((d*f^2*sqrt(b^2/(d*f^3)) + c*d + a*f)/(d*f^2))*log((2*b*c *x - 2*sqrt(c*x^2 + b*x + a)*b*f*sqrt((d*f^2*sqrt(b^2/(d*f^3)) + c*d + a*f )/(d*f^2)) + b^2 + (b*f^2*x + 2*a*f^2)*sqrt(b^2/(d*f^3)))/x) + f*sqrt(-(d* f^2*sqrt(b^2/(d*f^3)) - c*d - a*f)/(d*f^2))*log((2*b*c*x + 2*sqrt(c*x^2 + b*x + a)*b*f*sqrt(-(d*f^2*sqrt(b^2/(d*f^3)) - c*d - a*f)/(d*f^2)) + b^2...
\[ \int \frac {\sqrt {a+b x+c x^2}}{d-f x^2} \, dx=- \int \frac {\sqrt {a + b x + c x^{2}}}{- d + f x^{2}}\, dx \]
Exception generated. \[ \int \frac {\sqrt {a+b x+c x^2}}{d-f x^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(((c*sqrt(4*d*f))/(2*f^2)>0)', se e `assume?
Exception generated. \[ \int \frac {\sqrt {a+b x+c x^2}}{d-f x^2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {\sqrt {a+b x+c x^2}}{d-f x^2} \, dx=\int \frac {\sqrt {c\,x^2+b\,x+a}}{d-f\,x^2} \,d x \]