Integrand size = 28, antiderivative size = 286 \[ \int \frac {\sqrt {a+b x+c x^2}}{x^2 \left (d-f x^2\right )} \, dx=-\frac {\sqrt {a+b x+c x^2}}{d x}-\frac {b \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {a} d}+\frac {\sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \text {arctanh}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 d^{3/2}}+\frac {\sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \text {arctanh}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 d^{3/2}} \]
-1/2*b*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))/d/a^(1/2)-(c*x^2 +b*x+a)^(1/2)/d/x+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)/d^(3/2)+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)/d^(3/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.39 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {a+b x+c x^2}}{x^2 \left (d-f x^2\right )} \, dx=-\frac {\frac {2 \sqrt {a+x (b+c x)}}{x}-\frac {2 b \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{\sqrt {a}}+\text {RootSum}\left [b^2 d-a^2 f-4 b \sqrt {c} d \text {$\#$1}+4 c d \text {$\#$1}^2+2 a f \text {$\#$1}^2-f \text {$\#$1}^4\&,\frac {b c d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 c^{3/2} d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 a \sqrt {c} f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+b f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{b \sqrt {c} d-2 c d \text {$\#$1}-a f \text {$\#$1}+f \text {$\#$1}^3}\&\right ]}{2 d} \]
-1/2*((2*Sqrt[a + x*(b + c*x)])/x - (2*b*ArcTanh[(Sqrt[c]*x - Sqrt[a + x*( b + c*x)])/Sqrt[a]])/Sqrt[a] + RootSum[b^2*d - a^2*f - 4*b*Sqrt[c]*d*#1 + 4*c*d*#1^2 + 2*a*f*#1^2 - f*#1^4 & , (b*c*d*Log[-(Sqrt[c]*x) + Sqrt[a + b* x + c*x^2] - #1] - 2*c^(3/2)*d*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1 - 2*a*Sqrt[c]*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1 + b*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1^2)/(b*Sqrt[c]*d - 2 *c*d*#1 - a*f*#1 + f*#1^3) & ])/d
Time = 0.89 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {7276, 2009}
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.
| \(\displaystyle \int \frac {\sqrt {a+b x+c x^2}}{x^2 \left (d-f x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {f \sqrt {a+b x+c x^2}}{d \left (d-f x^2\right )}+\frac {\sqrt {a+b x+c x^2}}{d x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d} \text {arctanh}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 d^{3/2}}+\frac {\sqrt {a f+b \sqrt {d} \sqrt {f}+c d} \text {arctanh}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 d^{3/2}}-\frac {b \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {a} d}-\frac {\sqrt {a+b x+c x^2}}{d x}\) |
-(Sqrt[a + b*x + c*x^2]/(d*x)) - (b*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[a]*d) + (Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Arc Tanh[(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*d^(3/2)) + (Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*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*d^(3/2))
3.1.82.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(463\) vs. \(2(218)=436\).
Time = 0.90 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.62
| method | result | size |
| risch | \(-\frac {\sqrt {c \,x^{2}+b x +a}}{d x}-\frac {\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{\sqrt {a}}-\frac {\left (\sqrt {d f}\, a f +\sqrt {d f}\, c d +b d f \right ) \ln \left (\frac {\frac {2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{d f \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}}-\frac {\left (-\sqrt {d f}\, a f -\sqrt {d f}\, c d +b d f \right ) \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{d f \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}}}{2 d}\) | \(464\) |
| default | \(\frac {-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{a x}+\frac {b \left (\sqrt {c \,x^{2}+b x +a}+\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 \sqrt {c}}-\sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )\right )}{2 a}+\frac {2 c \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{a}}{d}-\frac {f \left (\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}}}\right )}{2 d \sqrt {d f}}+\frac {f \left (\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}}}\right )}{2 d \sqrt {d f}}\) | \(959\) |
-(c*x^2+b*x+a)^(1/2)/d/x-1/2/d*(b/a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x +a)^(1/2))/x)-((d*f)^(1/2)*a*f+(d*f)^(1/2)*c*d+b*d*f)/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))-(-(d*f)^(1/2)*a*f-(d*f)^(1/2)*c*d+b*d*f)/d/f/(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. 543 vs. \(2 (218) = 436\).
Time = 14.92 (sec) , antiderivative size = 1094, normalized size of antiderivative = 3.83 \[ \int \frac {\sqrt {a+b x+c x^2}}{x^2 \left (d-f x^2\right )} \, dx=\left [\frac {a d x \sqrt {\frac {d^{3} \sqrt {\frac {b^{2} f}{d^{5}}} + c d + a f}{d^{3}}} \log \left (\frac {2 \, b c x + 2 \, \sqrt {c x^{2} + b x + a} b d \sqrt {\frac {d^{3} \sqrt {\frac {b^{2} f}{d^{5}}} + c d + a f}{d^{3}}} + b^{2} + {\left (b d^{2} x + 2 \, a d^{2}\right )} \sqrt {\frac {b^{2} f}{d^{5}}}}{x}\right ) - a d x \sqrt {\frac {d^{3} \sqrt {\frac {b^{2} f}{d^{5}}} + c d + a f}{d^{3}}} \log \left (\frac {2 \, b c x - 2 \, \sqrt {c x^{2} + b x + a} b d \sqrt {\frac {d^{3} \sqrt {\frac {b^{2} f}{d^{5}}} + c d + a f}{d^{3}}} + b^{2} + {\left (b d^{2} x + 2 \, a d^{2}\right )} \sqrt {\frac {b^{2} f}{d^{5}}}}{x}\right ) + a d x \sqrt {-\frac {d^{3} \sqrt {\frac {b^{2} f}{d^{5}}} - c d - a f}{d^{3}}} \log \left (\frac {2 \, b c x + 2 \, \sqrt {c x^{2} + b x + a} b d \sqrt {-\frac {d^{3} \sqrt {\frac {b^{2} f}{d^{5}}} - c d - a f}{d^{3}}} + b^{2} - {\left (b d^{2} x + 2 \, a d^{2}\right )} \sqrt {\frac {b^{2} f}{d^{5}}}}{x}\right ) - a d x \sqrt {-\frac {d^{3} \sqrt {\frac {b^{2} f}{d^{5}}} - c d - a f}{d^{3}}} \log \left (\frac {2 \, b c x - 2 \, \sqrt {c x^{2} + b x + a} b d \sqrt {-\frac {d^{3} \sqrt {\frac {b^{2} f}{d^{5}}} - c d - a f}{d^{3}}} + b^{2} - {\left (b d^{2} x + 2 \, a d^{2}\right )} \sqrt {\frac {b^{2} f}{d^{5}}}}{x}\right ) + \sqrt {a} b x \log \left (-\frac {8 \, a b x + {\left (b^{2} + 4 \, a c\right )} x^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right ) - 4 \, \sqrt {c x^{2} + b x + a} a}{4 \, a d x}, \frac {a d x \sqrt {\frac {d^{3} \sqrt {\frac {b^{2} f}{d^{5}}} + c d + a f}{d^{3}}} \log \left (\frac {2 \, b c x + 2 \, \sqrt {c x^{2} + b x + a} b d \sqrt {\frac {d^{3} \sqrt {\frac {b^{2} f}{d^{5}}} + c d + a f}{d^{3}}} + b^{2} + {\left (b d^{2} x + 2 \, a d^{2}\right )} \sqrt {\frac {b^{2} f}{d^{5}}}}{x}\right ) - a d x \sqrt {\frac {d^{3} \sqrt {\frac {b^{2} f}{d^{5}}} + c d + a f}{d^{3}}} \log \left (\frac {2 \, b c x - 2 \, \sqrt {c x^{2} + b x + a} b d \sqrt {\frac {d^{3} \sqrt {\frac {b^{2} f}{d^{5}}} + c d + a f}{d^{3}}} + b^{2} + {\left (b d^{2} x + 2 \, a d^{2}\right )} \sqrt {\frac {b^{2} f}{d^{5}}}}{x}\right ) + a d x \sqrt {-\frac {d^{3} \sqrt {\frac {b^{2} f}{d^{5}}} - c d - a f}{d^{3}}} \log \left (\frac {2 \, b c x + 2 \, \sqrt {c x^{2} + b x + a} b d \sqrt {-\frac {d^{3} \sqrt {\frac {b^{2} f}{d^{5}}} - c d - a f}{d^{3}}} + b^{2} - {\left (b d^{2} x + 2 \, a d^{2}\right )} \sqrt {\frac {b^{2} f}{d^{5}}}}{x}\right ) - a d x \sqrt {-\frac {d^{3} \sqrt {\frac {b^{2} f}{d^{5}}} - c d - a f}{d^{3}}} \log \left (\frac {2 \, b c x - 2 \, \sqrt {c x^{2} + b x + a} b d \sqrt {-\frac {d^{3} \sqrt {\frac {b^{2} f}{d^{5}}} - c d - a f}{d^{3}}} + b^{2} - {\left (b d^{2} x + 2 \, a d^{2}\right )} \sqrt {\frac {b^{2} f}{d^{5}}}}{x}\right ) + 2 \, \sqrt {-a} b x \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{2} + a b x + a^{2}\right )}}\right ) - 4 \, \sqrt {c x^{2} + b x + a} a}{4 \, a d x}\right ] \]
[1/4*(a*d*x*sqrt((d^3*sqrt(b^2*f/d^5) + c*d + a*f)/d^3)*log((2*b*c*x + 2*s qrt(c*x^2 + b*x + a)*b*d*sqrt((d^3*sqrt(b^2*f/d^5) + c*d + a*f)/d^3) + b^2 + (b*d^2*x + 2*a*d^2)*sqrt(b^2*f/d^5))/x) - a*d*x*sqrt((d^3*sqrt(b^2*f/d^ 5) + c*d + a*f)/d^3)*log((2*b*c*x - 2*sqrt(c*x^2 + b*x + a)*b*d*sqrt((d^3* sqrt(b^2*f/d^5) + c*d + a*f)/d^3) + b^2 + (b*d^2*x + 2*a*d^2)*sqrt(b^2*f/d ^5))/x) + a*d*x*sqrt(-(d^3*sqrt(b^2*f/d^5) - c*d - a*f)/d^3)*log((2*b*c*x + 2*sqrt(c*x^2 + b*x + a)*b*d*sqrt(-(d^3*sqrt(b^2*f/d^5) - c*d - a*f)/d^3) + b^2 - (b*d^2*x + 2*a*d^2)*sqrt(b^2*f/d^5))/x) - a*d*x*sqrt(-(d^3*sqrt(b ^2*f/d^5) - c*d - a*f)/d^3)*log((2*b*c*x - 2*sqrt(c*x^2 + b*x + a)*b*d*sqr t(-(d^3*sqrt(b^2*f/d^5) - c*d - a*f)/d^3) + b^2 - (b*d^2*x + 2*a*d^2)*sqrt (b^2*f/d^5))/x) + sqrt(a)*b*x*log(-(8*a*b*x + (b^2 + 4*a*c)*x^2 - 4*sqrt(c *x^2 + b*x + a)*(b*x + 2*a)*sqrt(a) + 8*a^2)/x^2) - 4*sqrt(c*x^2 + b*x + a )*a)/(a*d*x), 1/4*(a*d*x*sqrt((d^3*sqrt(b^2*f/d^5) + c*d + a*f)/d^3)*log(( 2*b*c*x + 2*sqrt(c*x^2 + b*x + a)*b*d*sqrt((d^3*sqrt(b^2*f/d^5) + c*d + a* f)/d^3) + b^2 + (b*d^2*x + 2*a*d^2)*sqrt(b^2*f/d^5))/x) - a*d*x*sqrt((d^3* sqrt(b^2*f/d^5) + c*d + a*f)/d^3)*log((2*b*c*x - 2*sqrt(c*x^2 + b*x + a)*b *d*sqrt((d^3*sqrt(b^2*f/d^5) + c*d + a*f)/d^3) + b^2 + (b*d^2*x + 2*a*d^2) *sqrt(b^2*f/d^5))/x) + a*d*x*sqrt(-(d^3*sqrt(b^2*f/d^5) - c*d - a*f)/d^3)* log((2*b*c*x + 2*sqrt(c*x^2 + b*x + a)*b*d*sqrt(-(d^3*sqrt(b^2*f/d^5) - c* d - a*f)/d^3) + b^2 - (b*d^2*x + 2*a*d^2)*sqrt(b^2*f/d^5))/x) - a*d*x*s...
\[ \int \frac {\sqrt {a+b x+c x^2}}{x^2 \left (d-f x^2\right )} \, dx=- \int \frac {\sqrt {a + b x + c x^{2}}}{- d x^{2} + f x^{4}}\, dx \]
\[ \int \frac {\sqrt {a+b x+c x^2}}{x^2 \left (d-f x^2\right )} \, dx=\int { -\frac {\sqrt {c x^{2} + b x + a}}{{\left (f x^{2} - d\right )} x^{2}} \,d x } \]
Exception generated. \[ \int \frac {\sqrt {a+b x+c x^2}}{x^2 \left (d-f x^2\right )} \, dx=\text {Exception raised: AttributeError} \]
Timed out. \[ \int \frac {\sqrt {a+b x+c x^2}}{x^2 \left (d-f x^2\right )} \, dx=\int \frac {\sqrt {c\,x^2+b\,x+a}}{x^2\,\left (d-f\,x^2\right )} \,d x \]