Integrand size = 28, antiderivative size = 291 \[ \int \frac {1}{x^2 \sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx=-\frac {\sqrt {a+b x+c x^2}}{a d x}+\frac {b \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 a^{3/2} d}+\frac {f \text {arctanh}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 d^{3/2} \sqrt {c d-b \sqrt {d} \sqrt {f}+a f}}+\frac {f \text {arctanh}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 d^{3/2} \sqrt {c d+b \sqrt {d} \sqrt {f}+a f}} \] Output:
-(c*x^2+b*x+a)^(1/2)/a/d/x+1/2*b*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+ a)^(1/2))/a^(3/2)/d+1/2*f*arctanh(1/2*(b*d^(1/2)-2*a*f^(1/2)+(2*c*d^(1/2)- b*f^(1/2))*x)/(c*d-b*d^(1/2)*f^(1/2)+a*f)^(1/2)/(c*x^2+b*x+a)^(1/2))/d^(3/ 2)/(c*d-b*d^(1/2)*f^(1/2)+a*f)^(1/2)+1/2*f*arctanh(1/2*(b*d^(1/2)+2*a*f^(1 /2)+(2*c*d^(1/2)+b*f^(1/2))*x)/(c*d+b*d^(1/2)*f^(1/2)+a*f)^(1/2)/(c*x^2+b* x+a)^(1/2))/d^(3/2)/(c*d+b*d^(1/2)*f^(1/2)+a*f)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.72 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^2 \sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx=-\frac {\frac {2 \sqrt {a+x (b+c x)}}{a x}+\frac {2 b \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{a^{3/2}}+f \text {RootSum}\left [b^2 d-a^2 f-4 b \sqrt {c} d \text {$\#$1}+4 c d \text {$\#$1}^2+2 a f \text {$\#$1}^2-f \text {$\#$1}^4\&,\frac {b \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 \sqrt {c} \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}}{b \sqrt {c} d-2 c d \text {$\#$1}-a f \text {$\#$1}+f \text {$\#$1}^3}\&\right ]}{2 d} \] Input:
Integrate[1/(x^2*Sqrt[a + b*x + c*x^2]*(d - f*x^2)),x]
Output:
-1/2*((2*Sqrt[a + x*(b + c*x)])/(a*x) + (2*b*ArcTanh[(Sqrt[c]*x - Sqrt[a + x*(b + c*x)])/Sqrt[a]])/a^(3/2) + f*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*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] - 2*Sqrt[c]*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1)/(b*Sqrt[c]*d - 2*c*d*#1 - a*f*#1 + f*#1^3) & ])/d
Time = 0.84 (sec) , antiderivative size = 291, 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.
\(\displaystyle \int \frac {1}{x^2 \left (d-f x^2\right ) \sqrt {a+b x+c x^2}} \, dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (\frac {f}{d \left (d-f x^2\right ) \sqrt {a+b x+c x^2}}+\frac {1}{d x^2 \sqrt {a+b x+c x^2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 a^{3/2} d}+\frac {f \text {arctanh}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 d^{3/2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}+\frac {f \text {arctanh}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 d^{3/2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}-\frac {\sqrt {a+b x+c x^2}}{a d x}\) |
Input:
Int[1/(x^2*Sqrt[a + b*x + c*x^2]*(d - f*x^2)),x]
Output:
-(Sqrt[a + b*x + c*x^2]/(a*d*x)) + (b*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[ a + b*x + c*x^2])])/(2*a^(3/2)*d) + (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)*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]) + (f*Ar cTanh[(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])
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]
Time = 2.56 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.46
method | result | size |
default | \(\frac {-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}}{d}+\frac {f \ln \left (\frac {\frac {2 b \sqrt {d f}+2 a f +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+f b \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+a f +c d}{f}}\, \sqrt {c \left (x -\frac {\sqrt {d f}}{f}\right )^{2}+\frac {\left (2 c \sqrt {d f}+f b \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+a f +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{2 d \sqrt {d f}\, \sqrt {\frac {b \sqrt {d f}+a f +c d}{f}}}-\frac {f \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 a f +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+f b \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+a f +c d}{f}}\, \sqrt {c \left (x +\frac {\sqrt {d f}}{f}\right )^{2}+\frac {\left (-2 c \sqrt {d f}+f b \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+a f +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{2 d \sqrt {d f}\, \sqrt {\frac {-b \sqrt {d f}+a f +c d}{f}}}\) | \(426\) |
risch | \(-\frac {\sqrt {c \,x^{2}+b x +a}}{a d x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}} d}+\frac {f \ln \left (\frac {\frac {2 b \sqrt {d f}+2 a f +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+f b \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+a f +c d}{f}}\, \sqrt {c \left (x -\frac {\sqrt {d f}}{f}\right )^{2}+\frac {\left (2 c \sqrt {d f}+f b \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+a f +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{2 d \sqrt {d f}\, \sqrt {\frac {b \sqrt {d f}+a f +c d}{f}}}-\frac {f \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 a f +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+f b \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+a f +c d}{f}}\, \sqrt {c \left (x +\frac {\sqrt {d f}}{f}\right )^{2}+\frac {\left (-2 c \sqrt {d f}+f b \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+a f +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{2 d \sqrt {d f}\, \sqrt {\frac {-b \sqrt {d f}+a f +c d}{f}}}\) | \(427\) |
Input:
int(1/x^2/(c*x^2+b*x+a)^(1/2)/(-f*x^2+d),x,method=_RETURNVERBOSE)
Output:
1/d*(-1/a/x*(c*x^2+b*x+a)^(1/2)+1/2*b/a^(3/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2 +b*x+a)^(1/2))/x))+1/2*f/d/(d*f)^(1/2)/((b*(d*f)^(1/2)+a*f+c*d)/f)^(1/2)*l n((2*(b*(d*f)^(1/2)+a*f+c*d)/f+(2*c*(d*f)^(1/2)+f*b)/f*(x-(d*f)^(1/2)/f)+2 *((b*(d*f)^(1/2)+a*f+c*d)/f)^(1/2)*(c*(x-(d*f)^(1/2)/f)^2+(2*c*(d*f)^(1/2) +f*b)/f*(x-(d*f)^(1/2)/f)+(b*(d*f)^(1/2)+a*f+c*d)/f)^(1/2))/(x-(d*f)^(1/2) /f))-1/2*f/d/(d*f)^(1/2)/(1/f*(-b*(d*f)^(1/2)+a*f+c*d))^(1/2)*ln((2/f*(-b* (d*f)^(1/2)+a*f+c*d)+1/f*(-2*c*(d*f)^(1/2)+f*b)*(x+(d*f)^(1/2)/f)+2*(1/f*( -b*(d*f)^(1/2)+a*f+c*d))^(1/2)*(c*(x+(d*f)^(1/2)/f)^2+1/f*(-2*c*(d*f)^(1/2 )+f*b)*(x+(d*f)^(1/2)/f)+1/f*(-b*(d*f)^(1/2)+a*f+c*d))^(1/2))/(x+(d*f)^(1/ 2)/f))
Leaf count of result is larger than twice the leaf count of optimal. 3005 vs. \(2 (223) = 446\).
Time = 170.60 (sec) , antiderivative size = 6018, normalized size of antiderivative = 20.68 \[ \int \frac {1}{x^2 \sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate(1/x^2/(c*x^2+b*x+a)^(1/2)/(-f*x^2+d),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {1}{x^2 \sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx=- \int \frac {1}{- d x^{2} \sqrt {a + b x + c x^{2}} + f x^{4} \sqrt {a + b x + c x^{2}}}\, dx \] Input:
integrate(1/x**2/(c*x**2+b*x+a)**(1/2)/(-f*x**2+d),x)
Output:
-Integral(1/(-d*x**2*sqrt(a + b*x + c*x**2) + f*x**4*sqrt(a + b*x + c*x**2 )), x)
\[ \int \frac {1}{x^2 \sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx=\int { -\frac {1}{\sqrt {c x^{2} + b x + a} {\left (f x^{2} - d\right )} x^{2}} \,d x } \] Input:
integrate(1/x^2/(c*x^2+b*x+a)^(1/2)/(-f*x^2+d),x, algorithm="maxima")
Output:
-integrate(1/(sqrt(c*x^2 + b*x + a)*(f*x^2 - d)*x^2), x)
Timed out. \[ \int \frac {1}{x^2 \sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate(1/x^2/(c*x^2+b*x+a)^(1/2)/(-f*x^2+d),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {1}{x^2 \sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx=\int \frac {1}{x^2\,\left (d-f\,x^2\right )\,\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:
int(1/(x^2*(d - f*x^2)*(a + b*x + c*x^2)^(1/2)),x)
Output:
int(1/(x^2*(d - f*x^2)*(a + b*x + c*x^2)^(1/2)), x)
\[ \int \frac {1}{x^2 \sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx=\int \frac {1}{x^{2} \sqrt {c \,x^{2}+b x +a}\, \left (-f \,x^{2}+d \right )}d x \] Input:
int(1/x^2/(c*x^2+b*x+a)^(1/2)/(-f*x^2+d),x)
Output:
int(1/x^2/(c*x^2+b*x+a)^(1/2)/(-f*x^2+d),x)