\(\int \frac {\sqrt {a+c x^2}}{x (d+e x+f x^2)} \, dx\) [22]

Optimal result

Integrand size = 27, antiderivative size = 358 \[ \int \frac {\sqrt {a+c x^2}}{x \left (d+e x+f x^2\right )} \, dx=\frac {\left (2 a e f+(c d-a f) \left (e-\sqrt {e^2-4 d f}\right )\right ) \text {arctanh}\left (\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} d \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )}}-\frac {\left (2 a e f+(c d-a f) \left (e+\sqrt {e^2-4 d f}\right )\right ) \text {arctanh}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} d \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )}}-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d} \] Output:

1/2*(2*a*e*f+(-a*f+c*d)*(e-(-4*d*f+e^2)^(1/2)))*arctanh(1/2*(2*a*f-c*(e-(- 
4*d*f+e^2)^(1/2))*x)*2^(1/2)/(2*a*f^2+c*(e^2-2*d*f-e*(-4*d*f+e^2)^(1/2)))^ 
(1/2)/(c*x^2+a)^(1/2))*2^(1/2)/d/(-4*d*f+e^2)^(1/2)/(2*a*f^2+c*(e^2-2*d*f- 
e*(-4*d*f+e^2)^(1/2)))^(1/2)-1/2*(2*a*e*f+(-a*f+c*d)*(e+(-4*d*f+e^2)^(1/2) 
))*arctanh(1/2*(2*a*f-c*(e+(-4*d*f+e^2)^(1/2))*x)*2^(1/2)/(2*a*f^2+c*(e^2- 
2*d*f+e*(-4*d*f+e^2)^(1/2)))^(1/2)/(c*x^2+a)^(1/2))*2^(1/2)/d/(-4*d*f+e^2) 
^(1/2)/(2*a*f^2+c*(e^2-2*d*f+e*(-4*d*f+e^2)^(1/2)))^(1/2)-a^(1/2)*arctanh( 
(c*x^2+a)^(1/2)/a^(1/2))/d
 

Mathematica [C] (verified)

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

Time = 0.53 (sec) , antiderivative size = 310, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {a+c x^2}}{x \left (d+e x+f x^2\right )} \, dx=\frac {\sqrt {a} \left (-\log (x)+\log \left (-\sqrt {a}+\sqrt {a+c x^2}\right )\right )-\text {RootSum}\left [c^2 d+2 \sqrt {a} c e \text {$\#$1}-2 c d \text {$\#$1}^2+4 a f \text {$\#$1}^2-2 \sqrt {a} e \text {$\#$1}^3+d \text {$\#$1}^4\&,\frac {-a c e \log (x)+a c e \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right )+2 \sqrt {a} c d \log (x) \text {$\#$1}-2 a^{3/2} f \log (x) \text {$\#$1}-2 \sqrt {a} c d \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}+2 a^{3/2} f \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}+a e \log (x) \text {$\#$1}^2-a e \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2}{\sqrt {a} c e-2 c d \text {$\#$1}+4 a f \text {$\#$1}-3 \sqrt {a} e \text {$\#$1}^2+2 d \text {$\#$1}^3}\&\right ]}{d} \] Input:

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

Output:

(Sqrt[a]*(-Log[x] + Log[-Sqrt[a] + Sqrt[a + c*x^2]]) - RootSum[c^2*d + 2*S 
qrt[a]*c*e*#1 - 2*c*d*#1^2 + 4*a*f*#1^2 - 2*Sqrt[a]*e*#1^3 + d*#1^4 & , (- 
(a*c*e*Log[x]) + a*c*e*Log[-Sqrt[a] + Sqrt[a + c*x^2] - x*#1] + 2*Sqrt[a]* 
c*d*Log[x]*#1 - 2*a^(3/2)*f*Log[x]*#1 - 2*Sqrt[a]*c*d*Log[-Sqrt[a] + Sqrt[ 
a + c*x^2] - x*#1]*#1 + 2*a^(3/2)*f*Log[-Sqrt[a] + Sqrt[a + c*x^2] - x*#1] 
*#1 + a*e*Log[x]*#1^2 - a*e*Log[-Sqrt[a] + Sqrt[a + c*x^2] - x*#1]*#1^2)/( 
Sqrt[a]*c*e - 2*c*d*#1 + 4*a*f*#1 - 3*Sqrt[a]*e*#1^2 + 2*d*#1^3) & ])/d
 

Rubi [A] (verified)

Time = 1.31 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {7279, 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.

\(\displaystyle \int \frac {\sqrt {a+c x^2}}{x \left (d+e x+f x^2\right )} \, dx\)

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (\frac {\sqrt {a+c x^2} (-e-f x)}{d \left (d+e x+f x^2\right )}+\frac {\sqrt {a+c x^2}}{d x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\left (\left (e-\sqrt {e^2-4 d f}\right ) (c d-a f)+2 a e f\right ) \text {arctanh}\left (\frac {2 a f-c x \left (e-\sqrt {e^2-4 d f}\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} d \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}-\frac {\left (\left (\sqrt {e^2-4 d f}+e\right ) (c d-a f)+2 a e f\right ) \text {arctanh}\left (\frac {2 a f-c x \left (\sqrt {e^2-4 d f}+e\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} d \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d}\)

Input:

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

Output:

((2*a*e*f + (c*d - a*f)*(e - Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e - S 
qrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 
 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c 
*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) - ((2*a*e*f + (c*d - a*f)*(e + Sqrt 
[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqr 
t[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqr 
t[2]*d*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d* 
f])]) - (Sqrt[a]*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/d
 

Defintions of rubi rules used

rule 2009

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

rule 7279

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ 
{v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su 
mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1315\) vs. \(2(315)=630\).

Time = 2.14 (sec) , antiderivative size = 1316, normalized size of antiderivative = 3.68

method	result	size
default	\(\text {Expression too large to display}\)	\(1316\)

Input:

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

Output:

-4*f/(-e+(-4*d*f+e^2)^(1/2))/(e+(-4*d*f+e^2)^(1/2))*((c*x^2+a)^(1/2)-a^(1/ 
2)*ln((2*a+2*a^(1/2)*(c*x^2+a)^(1/2))/x))+2*f/(-e+(-4*d*f+e^2)^(1/2))/(-4* 
d*f+e^2)^(1/2)*(1/2*(4*c*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))^2-4*c*(e-(-4*d* 
f+e^2)^(1/2))/f*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))+2*(-(-4*d*f+e^2)^(1/2)*c 
*e+2*a*f^2-2*d*f*c+c*e^2)/f^2)^(1/2)-1/2*c^(1/2)*(e-(-4*d*f+e^2)^(1/2))/f* 
ln((-1/2*c*(e-(-4*d*f+e^2)^(1/2))/f+c*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2))))/c 
^(1/2)+(c*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))^2-c*(e-(-4*d*f+e^2)^(1/2))/f*( 
x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))+1/2*(-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*d* 
f*c+c*e^2)/f^2)^(1/2))-1/2*(-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*d*f*c+c*e^2) 
/f^2*2^(1/2)/((-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*d*f*c+c*e^2)/f^2)^(1/2)*l 
n(((-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*d*f*c+c*e^2)/f^2-c*(e-(-4*d*f+e^2)^( 
1/2))/f*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))+1/2*2^(1/2)*((-(-4*d*f+e^2)^(1/2 
)*c*e+2*a*f^2-2*d*f*c+c*e^2)/f^2)^(1/2)*(4*c*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/ 
2)))^2-4*c*(e-(-4*d*f+e^2)^(1/2))/f*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))+2*(- 
(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*d*f*c+c*e^2)/f^2)^(1/2))/(x-1/2/f*(-e+(-4 
*d*f+e^2)^(1/2)))))+2*f/(e+(-4*d*f+e^2)^(1/2))/(-4*d*f+e^2)^(1/2)*(1/2*(4* 
c*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)^2-4*c*(e+(-4*d*f+e^2)^(1/2))/f*(x+1/2*( 
e+(-4*d*f+e^2)^(1/2))/f)+2*((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*d*f*c+c*e^2)/ 
f^2)^(1/2)-1/2*c^(1/2)*(e+(-4*d*f+e^2)^(1/2))/f*ln((-1/2*c*(e+(-4*d*f+e^2) 
^(1/2))/f+c*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f))/c^(1/2)+(c*(x+1/2*(e+(-4*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1130 vs. \(2 (313) = 626\).

Time = 21.68 (sec) , antiderivative size = 2269, normalized size of antiderivative = 6.34 \[ \int \frac {\sqrt {a+c x^2}}{x \left (d+e x+f x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/4*(sqrt(2)*d*sqrt((2*c*d^2 + a*e^2 - 2*a*d*f + (d^2*e^2 - 4*d^3*f)*sqr 
t(a^2*e^2/(d^4*e^2 - 4*d^5*f)))/(d^2*e^2 - 4*d^3*f))*log((2*a*c*d*e*x - a^ 
2*e^2 + sqrt(2)*(d^3*e^2 - 4*d^4*f)*sqrt(a^2*e^2/(d^4*e^2 - 4*d^5*f))*sqrt 
(c*x^2 + a)*sqrt((2*c*d^2 + a*e^2 - 2*a*d*f + (d^2*e^2 - 4*d^3*f)*sqrt(a^2 
*e^2/(d^4*e^2 - 4*d^5*f)))/(d^2*e^2 - 4*d^3*f)) - (a*d^2*e^2 - 4*a*d^3*f)* 
sqrt(a^2*e^2/(d^4*e^2 - 4*d^5*f)))/x) - sqrt(2)*d*sqrt((2*c*d^2 + a*e^2 - 
2*a*d*f + (d^2*e^2 - 4*d^3*f)*sqrt(a^2*e^2/(d^4*e^2 - 4*d^5*f)))/(d^2*e^2 
- 4*d^3*f))*log((2*a*c*d*e*x - a^2*e^2 - sqrt(2)*(d^3*e^2 - 4*d^4*f)*sqrt( 
a^2*e^2/(d^4*e^2 - 4*d^5*f))*sqrt(c*x^2 + a)*sqrt((2*c*d^2 + a*e^2 - 2*a*d 
*f + (d^2*e^2 - 4*d^3*f)*sqrt(a^2*e^2/(d^4*e^2 - 4*d^5*f)))/(d^2*e^2 - 4*d 
^3*f)) - (a*d^2*e^2 - 4*a*d^3*f)*sqrt(a^2*e^2/(d^4*e^2 - 4*d^5*f)))/x) - s 
qrt(2)*d*sqrt((2*c*d^2 + a*e^2 - 2*a*d*f - (d^2*e^2 - 4*d^3*f)*sqrt(a^2*e^ 
2/(d^4*e^2 - 4*d^5*f)))/(d^2*e^2 - 4*d^3*f))*log((2*a*c*d*e*x - a^2*e^2 + 
sqrt(2)*(d^3*e^2 - 4*d^4*f)*sqrt(a^2*e^2/(d^4*e^2 - 4*d^5*f))*sqrt(c*x^2 + 
 a)*sqrt((2*c*d^2 + a*e^2 - 2*a*d*f - (d^2*e^2 - 4*d^3*f)*sqrt(a^2*e^2/(d^ 
4*e^2 - 4*d^5*f)))/(d^2*e^2 - 4*d^3*f)) + (a*d^2*e^2 - 4*a*d^3*f)*sqrt(a^2 
*e^2/(d^4*e^2 - 4*d^5*f)))/x) + sqrt(2)*d*sqrt((2*c*d^2 + a*e^2 - 2*a*d*f 
- (d^2*e^2 - 4*d^3*f)*sqrt(a^2*e^2/(d^4*e^2 - 4*d^5*f)))/(d^2*e^2 - 4*d^3* 
f))*log((2*a*c*d*e*x - a^2*e^2 - sqrt(2)*(d^3*e^2 - 4*d^4*f)*sqrt(a^2*e^2/ 
(d^4*e^2 - 4*d^5*f))*sqrt(c*x^2 + a)*sqrt((2*c*d^2 + a*e^2 - 2*a*d*f - ...
 

Sympy [F]

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

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

Output:

Integral(sqrt(a + c*x**2)/(x*(d + e*x + f*x**2)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a+c x^2}}{x \left (d+e x+f x^2\right )} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Degree mismatch inside factorisatio 
n over extensionNot implemented, e.g. for multivariate mod/approx polynomi 
alsError:
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {a+c x^2}}{x \left (d+e x+f x^2\right )} \, dx=\frac {\sqrt {a}\, \mathrm {log}\left (\sqrt {c \,x^{2}+a}-\sqrt {a}\right )-\sqrt {a}\, \mathrm {log}\left (\sqrt {c \,x^{2}+a}+\sqrt {a}\right )-2 \left (\int \frac {\sqrt {c \,x^{2}+a}}{c f \,x^{4}+c e \,x^{3}+a f \,x^{2}+c d \,x^{2}+a e x +a d}d x \right ) a e -2 \left (\int \frac {\sqrt {c \,x^{2}+a}\, x}{c f \,x^{4}+c e \,x^{3}+a f \,x^{2}+c d \,x^{2}+a e x +a d}d x \right ) a f +2 \left (\int \frac {\sqrt {c \,x^{2}+a}\, x}{c f \,x^{4}+c e \,x^{3}+a f \,x^{2}+c d \,x^{2}+a e x +a d}d x \right ) c d}{2 d} \] Input:

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

Output:

(sqrt(a)*log(sqrt(a + c*x**2) - sqrt(a)) - sqrt(a)*log(sqrt(a + c*x**2) + 
sqrt(a)) - 2*int(sqrt(a + c*x**2)/(a*d + a*e*x + a*f*x**2 + c*d*x**2 + c*e 
*x**3 + c*f*x**4),x)*a*e - 2*int((sqrt(a + c*x**2)*x)/(a*d + a*e*x + a*f*x 
**2 + c*d*x**2 + c*e*x**3 + c*f*x**4),x)*a*f + 2*int((sqrt(a + c*x**2)*x)/ 
(a*d + a*e*x + a*f*x**2 + c*d*x**2 + c*e*x**3 + c*f*x**4),x)*c*d)/(2*d)