\(\int \frac {\sqrt {a+b x+c x^2}}{x^2 (d-f x^2)} \, dx\) [49]

Optimal result

Integrand size = 28, antiderivative size = 286 \[ \int \frac {\sqrt {a+b x+c x^2}}{x^2 \left (d-f x^2\right )} \, dx=-\frac {\sqrt {a+b x+c x^2}}{d x}-\frac {b \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {a} d}+\frac {\sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \text {arctanh}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 d^{3/2}}+\frac {\sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \text {arctanh}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 d^{3/2}} \] Output:

-(c*x^2+b*x+a)^(1/2)/d/x-1/2*b*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a) 
^(1/2))/a^(1/2)/d+1/2*(c*d-b*d^(1/2)*f^(1/2)+a*f)^(1/2)*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)+1/2*(c*d+b*d^(1/2)*f^(1/2)+a*f)^(1/2)*arc 
tanh(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)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.70 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {a+b x+c x^2}}{x^2 \left (d-f x^2\right )} \, dx=-\frac {\frac {2 \sqrt {a+x (b+c x)}}{x}-\frac {2 b \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{\sqrt {a}}+\text {RootSum}\left [b^2 d-a^2 f-4 b \sqrt {c} d \text {$\#$1}+4 c d \text {$\#$1}^2+2 a f \text {$\#$1}^2-f \text {$\#$1}^4\&,\frac {b c d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 c^{3/2} d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 a \sqrt {c} f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+b f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{b \sqrt {c} d-2 c d \text {$\#$1}-a f \text {$\#$1}+f \text {$\#$1}^3}\&\right ]}{2 d} \] Input:

Integrate[Sqrt[a + b*x + c*x^2]/(x^2*(d - f*x^2)),x]
 

Output:

-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
 

Rubi [A] (verified)

Time = 0.88 (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

\(\displaystyle \frac {\sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d} \text {arctanh}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 d^{3/2}}+\frac {\sqrt {a f+b \sqrt {d} \sqrt {f}+c d} \text {arctanh}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 d^{3/2}}-\frac {b \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {a} d}-\frac {\sqrt {a+b x+c x^2}}{d x}\)

Input:

Int[Sqrt[a + b*x + c*x^2]/(x^2*(d - f*x^2)),x]
 

Output:

-(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))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7276

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(461\) vs. \(2(218)=436\).

Time = 2.46 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.62

method	result	size
risch	\(-\frac {\sqrt {c \,x^{2}+b x +a}}{d x}-\frac {\frac {\left (\sqrt {d f}\, a f +\sqrt {d f}\, c d -b d f \right ) \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 a f +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+f b \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+a f +c d}{f}}\, \sqrt {c \left (x +\frac {\sqrt {d f}}{f}\right )^{2}+\frac {\left (-2 c \sqrt {d f}+f b \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+a f +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{d f \sqrt {\frac {-b \sqrt {d f}+a f +c d}{f}}}-\frac {\left (\sqrt {d f}\, a f +\sqrt {d f}\, c d +b d f \right ) \ln \left (\frac {\frac {2 b \sqrt {d f}+2 a f +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+f b \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+a f +c d}{f}}\, \sqrt {c \left (x -\frac {\sqrt {d f}}{f}\right )^{2}+\frac {\left (2 c \sqrt {d f}+f b \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+a f +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{d f \sqrt {\frac {b \sqrt {d f}+a f +c d}{f}}}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{\sqrt {a}}}{2 d}\)	\(462\)
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 {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}}+\frac {\left (2 c \sqrt {d f}+f b \right ) \ln \left (\frac {\frac {2 c \sqrt {d f}+f b}{2 f}+c \left (x -\frac {\sqrt {d f}}{f}\right )}{\sqrt {c}}+\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}}\right )}{2 f \sqrt {c}}-\frac {\left (b \sqrt {d f}+a f +c d \right ) \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 )}{f \sqrt {\frac {b \sqrt {d f}+a f +c d}{f}}}\right )}{2 d \sqrt {d f}}+\frac {f \left (\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}}+\frac {\left (-2 c \sqrt {d f}+f b \right ) \ln \left (\frac {\frac {-2 c \sqrt {d f}+f b}{2 f}+c \left (x +\frac {\sqrt {d f}}{f}\right )}{\sqrt {c}}+\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}}\right )}{2 f \sqrt {c}}-\frac {\left (-b \sqrt {d f}+a f +c d \right ) \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 )}{f \sqrt {\frac {-b \sqrt {d f}+a f +c d}{f}}}\right )}{2 d \sqrt {d f}}\)	\(959\)

Input:

int((c*x^2+b*x+a)^(1/2)/x^2/(-f*x^2+d),x,method=_RETURNVERBOSE)
 

Output:

-(c*x^2+b*x+a)^(1/2)/d/x-1/2/d*(((d*f)^(1/2)*a*f+(d*f)^(1/2)*c*d-b*d*f)/d/ 
f/(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))-((d*f)^(1/2)*a*f 
+(d*f)^(1/2)*c*d+b*d*f)/d/f/((b*(d*f)^(1/2)+a*f+c*d)/f)^(1/2)*ln((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))+b/a^(1 
/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 543 vs. \(2 (218) = 436\).

Time = 13.70 (sec) , antiderivative size = 1094, normalized size of antiderivative = 3.83 \[ \int \frac {\sqrt {a+b x+c x^2}}{x^2 \left (d-f x^2\right )} \, dx =\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^(1/2)/x^2/(-f*x^2+d),x, algorithm="fricas")
 

Output:

[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...
 

Sympy [F]

\[ \int \frac {\sqrt {a+b x+c x^2}}{x^2 \left (d-f x^2\right )} \, dx=- \int \frac {\sqrt {a + b x + c x^{2}}}{- d x^{2} + f x^{4}}\, dx \] Input:

integrate((c*x**2+b*x+a)**(1/2)/x**2/(-f*x**2+d),x)
                                                                                    
                                                                                    
 

Output:

-Integral(sqrt(a + b*x + c*x**2)/(-d*x**2 + f*x**4), x)
 

Maxima [F]

\[ \int \frac {\sqrt {a+b x+c x^2}}{x^2 \left (d-f x^2\right )} \, dx=\int { -\frac {\sqrt {c x^{2} + b x + a}}{{\left (f x^{2} - d\right )} x^{2}} \,d x } \] Input:

integrate((c*x^2+b*x+a)^(1/2)/x^2/(-f*x^2+d),x, algorithm="maxima")
 

Output:

-integrate(sqrt(c*x^2 + b*x + a)/((f*x^2 - d)*x^2), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x+c x^2}}{x^2 \left (d-f x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((c*x^2+b*x+a)^(1/2)/x^2/(-f*x^2+d),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x+c x^2}}{x^2 \left (d-f x^2\right )} \, dx=\int \frac {\sqrt {c\,x^2+b\,x+a}}{x^2\,\left (d-f\,x^2\right )} \,d x \] Input:

int((a + b*x + c*x^2)^(1/2)/(x^2*(d - f*x^2)),x)
 

Output:

int((a + b*x + c*x^2)^(1/2)/(x^2*(d - f*x^2)), x)
 

Reduce [F]

\[ \int \frac {\sqrt {a+b x+c x^2}}{x^2 \left (d-f x^2\right )} \, dx=\int \frac {\sqrt {c \,x^{2}+b x +a}}{x^{2} \left (-f \,x^{2}+d \right )}d x \] Input:

int((c*x^2+b*x+a)^(1/2)/x^2/(-f*x^2+d),x)
 

Output:

int((c*x^2+b*x+a)^(1/2)/x^2/(-f*x^2+d),x)