\(\int \frac {\sqrt {a+b x^2}}{(c+d x^2)^2 (e+f x^2)} \, dx\) [284]

Optimal result

Integrand size = 30, antiderivative size = 182 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^2 \left (e+f x^2\right )} \, dx=\frac {d x \sqrt {a+b x^2}}{2 c (d e-c f) \left (c+d x^2\right )}+\frac {\left (2 b c^2 f+a d (d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \sqrt {b e-a f} \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} (d e-c f)^2} \] Output:

1/2*d*x*(b*x^2+a)^(1/2)/c/(-c*f+d*e)/(d*x^2+c)+1/2*(2*b*c^2*f+a*d*(-3*c*f+ 
d*e))*arctanh((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a)^(1/2))/c^(3/2)/(-a*d+b* 
c)^(1/2)/(-c*f+d*e)^2-f*(-a*f+b*e)^(1/2)*arctanh((-a*f+b*e)^(1/2)*x/e^(1/2 
)/(b*x^2+a)^(1/2))/e^(1/2)/(-c*f+d*e)^2
 

Mathematica [A] (verified)

Time = 1.42 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^2 \left (e+f x^2\right )} \, dx=\frac {1}{2} \left (-\frac {d x \sqrt {a+b x^2}}{c (-d e+c f) \left (c+d x^2\right )}-\frac {\left (2 b c^2 f+a d (d e-3 c f)\right ) \arctan \left (\frac {-d x \sqrt {a+b x^2}+\sqrt {b} \left (c+d x^2\right )}{\sqrt {c} \sqrt {-b c+a d}}\right )}{c^{3/2} \sqrt {-b c+a d} (d e-c f)^2}-\frac {2 f \sqrt {-b e+a f} \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{\sqrt {e} (d e-c f)^2}\right ) \] Input:

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

Output:

(-((d*x*Sqrt[a + b*x^2])/(c*(-(d*e) + c*f)*(c + d*x^2))) - ((2*b*c^2*f + a 
*d*(d*e - 3*c*f))*ArcTan[(-(d*x*Sqrt[a + b*x^2]) + Sqrt[b]*(c + d*x^2))/(S 
qrt[c]*Sqrt[-(b*c) + a*d])])/(c^(3/2)*Sqrt[-(b*c) + a*d]*(d*e - c*f)^2) - 
(2*f*Sqrt[-(b*e) + a*f]*ArcTan[(-(f*x*Sqrt[a + b*x^2]) + Sqrt[b]*(e + f*x^ 
2))/(Sqrt[e]*Sqrt[-(b*e) + a*f])])/(Sqrt[e]*(d*e - c*f)^2))/2
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.41, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {421, 25, 301, 224, 219, 291, 221, 401, 25, 27, 398, 224, 219, 291, 221}

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.

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

\(\Big \downarrow \) 421

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 301

\(\displaystyle \frac {f^2 \left (\frac {b \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{(d e-c f)^2}+\frac {d \int \frac {\sqrt {b x^2+a} \left (-d f x^2+d e-2 c f\right )}{\left (d x^2+c\right )^2}dx}{(d e-c f)^2}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {f^2 \left (\frac {b \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{(d e-c f)^2}+\frac {d \int \frac {\sqrt {b x^2+a} \left (-d f x^2+d e-2 c f\right )}{\left (d x^2+c\right )^2}dx}{(d e-c f)^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {f^2 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{(d e-c f)^2}+\frac {d \int \frac {\sqrt {b x^2+a} \left (-d f x^2+d e-2 c f\right )}{\left (d x^2+c\right )^2}dx}{(d e-c f)^2}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {f^2 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {(b e-a f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}\right )}{(d e-c f)^2}+\frac {d \int \frac {\sqrt {b x^2+a} \left (-d f x^2+d e-2 c f\right )}{\left (d x^2+c\right )^2}dx}{(d e-c f)^2}\)

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 398

\(\displaystyle \frac {d \left (\frac {\frac {\left (a d (d e-3 c f)+2 b c^2 f\right ) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}-\frac {2 b c f \int \frac {1}{\sqrt {b x^2+a}}dx}{d}}{2 c}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 c \left (c+d x^2\right )}\right )}{(d e-c f)^2}+\frac {f^2 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}\right )}{(d e-c f)^2}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {d \left (\frac {\frac {\left (a d (d e-3 c f)+2 b c^2 f\right ) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}-\frac {2 b c f \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}}{2 c}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 c \left (c+d x^2\right )}\right )}{(d e-c f)^2}+\frac {f^2 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}\right )}{(d e-c f)^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {d \left (\frac {\frac {\left (a d (d e-3 c f)+2 b c^2 f\right ) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}-\frac {2 \sqrt {b} c f \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}}{2 c}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 c \left (c+d x^2\right )}\right )}{(d e-c f)^2}+\frac {f^2 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}\right )}{(d e-c f)^2}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {d \left (\frac {\frac {\left (a d (d e-3 c f)+2 b c^2 f\right ) \int \frac {1}{c-\frac {(b c-a d) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}-\frac {2 \sqrt {b} c f \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}}{2 c}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 c \left (c+d x^2\right )}\right )}{(d e-c f)^2}+\frac {f^2 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}\right )}{(d e-c f)^2}\)

\(\Big \downarrow \) 221

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

Input:

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

Output:

(d*(((d*e - c*f)*x*Sqrt[a + b*x^2])/(2*c*(c + d*x^2)) + ((-2*Sqrt[b]*c*f*A 
rcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/d + ((2*b*c^2*f + a*d*(d*e - 3*c*f))* 
ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(Sqrt[c]*d*Sqrt[b* 
c - a*d]))/(2*c)))/(d*e - c*f)^2 + (f^2*((Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt 
[a + b*x^2]])/f - (Sqrt[b*e - a*f]*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sq 
rt[a + b*x^2])])/(Sqrt[e]*f)))/(d*e - c*f)^2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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])
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 291

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst 
[Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, 
d}, x] && NeQ[b*c - a*d, 0]
 

rule 301

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[b/ 
d   Int[(a + b*x^2)^(p - 1), x], x] - Simp[(b*c - a*d)/d   Int[(a + b*x^2)^ 
(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] 
&& GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4] || (EqQ[p, 2/3] && E 
qQ[b*c + 3*a*d, 0]))
 

rule 398

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) 
, x_Symbol] :> Simp[f/b   Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ 
b   Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} 
, x]
 

rule 401

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x 
_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ 
q/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1))   Int[(a + b*x^2)^(p + 1)*( 
c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + 
(b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L 
tQ[p, -1] && GtQ[q, 0]
 

rule 421

Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( 
x_)^2), x_Symbol] :> Simp[b^2/(b*c - a*d)^2   Int[(c + d*x^2)^(q + 2)*((e + 
 f*x^2)^r/(a + b*x^2)), x], x] - Simp[d/(b*c - a*d)^2   Int[(c + d*x^2)^q*( 
e + f*x^2)^r*(2*b*c - a*d + b*d*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, r} 
, x] && LtQ[q, -1]
 

Maple [A] (verified)

Time = 1.43 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.15

method	result	size
pseudoelliptic	\(-\frac {-3 \left (x^{2} d +c \right ) \sqrt {\left (a f -b e \right ) e}\, \left (a c d f -\frac {1}{3} a \,d^{2} e -\frac {2}{3} b \,c^{2} f \right ) \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )+\left (2 c f \left (x^{2} d +c \right ) \left (a f -b e \right ) \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+d \left (c f -d e \right ) \sqrt {b \,x^{2}+a}\, x \sqrt {\left (a f -b e \right ) e}\right ) \sqrt {\left (a d -b c \right ) c}}{2 \sqrt {\left (a d -b c \right ) c}\, \sqrt {\left (a f -b e \right ) e}\, \left (x^{2} d +c \right ) \left (c f -d e \right )^{2} c}\)	\(210\)
default	\(\text {Expression too large to display}\)	\(2862\)

Input:

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

Output:

-1/2/((a*d-b*c)*c)^(1/2)*(-3*(d*x^2+c)*((a*f-b*e)*e)^(1/2)*(a*c*d*f-1/3*a* 
d^2*e-2/3*b*c^2*f)*arctan(c*(b*x^2+a)^(1/2)/x/((a*d-b*c)*c)^(1/2))+(2*c*f* 
(d*x^2+c)*(a*f-b*e)*arctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1/2))+d*(c*f 
-d*e)*(b*x^2+a)^(1/2)*x*((a*f-b*e)*e)^(1/2))*((a*d-b*c)*c)^(1/2))/((a*f-b* 
e)*e)^(1/2)/(d*x^2+c)/(c*f-d*e)^2/c
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 376 vs. \(2 (156) = 312\).

Time = 0.87 (sec) , antiderivative size = 376, normalized size of antiderivative = 2.07 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^2 \left (e+f x^2\right )} \, dx=-\frac {{\left (a \sqrt {b} d^{2} e + 2 \, b^{\frac {3}{2}} c^{2} f - 3 \, a \sqrt {b} c d f\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt {-b^{2} c^{2} + a b c d}}\right )}{2 \, {\left (c d^{2} e^{2} - 2 \, c^{2} d e f + c^{3} f^{2}\right )} \sqrt {-b^{2} c^{2} + a b c d}} + \frac {{\left (b^{\frac {3}{2}} e f - a \sqrt {b} f^{2}\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} f + 2 \, b e - a f}{2 \, \sqrt {-b^{2} e^{2} + a b e f}}\right )}{\sqrt {-b^{2} e^{2} + a b e f} {\left (d^{2} e^{2} - 2 \, c d e f + c^{2} f^{2}\right )}} + \frac {2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{\frac {3}{2}} c - {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a \sqrt {b} d + a^{2} \sqrt {b} d}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} d + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b c - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a d + a^{2} d\right )} {\left (c d e - c^{2} f\right )}} \] Input:

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

Output:

-1/2*(a*sqrt(b)*d^2*e + 2*b^(3/2)*c^2*f - 3*a*sqrt(b)*c*d*f)*arctan(1/2*(( 
sqrt(b)*x - sqrt(b*x^2 + a))^2*d + 2*b*c - a*d)/sqrt(-b^2*c^2 + a*b*c*d))/ 
((c*d^2*e^2 - 2*c^2*d*e*f + c^3*f^2)*sqrt(-b^2*c^2 + a*b*c*d)) + (b^(3/2)* 
e*f - a*sqrt(b)*f^2)*arctan(1/2*((sqrt(b)*x - sqrt(b*x^2 + a))^2*f + 2*b*e 
 - a*f)/sqrt(-b^2*e^2 + a*b*e*f))/(sqrt(-b^2*e^2 + a*b*e*f)*(d^2*e^2 - 2*c 
*d*e*f + c^2*f^2)) + (2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b^(3/2)*c - (sqrt( 
b)*x - sqrt(b*x^2 + a))^2*a*sqrt(b)*d + a^2*sqrt(b)*d)/(((sqrt(b)*x - sqrt 
(b*x^2 + a))^4*d + 4*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b*c - 2*(sqrt(b)*x - 
sqrt(b*x^2 + a))^2*a*d + a^2*d)*(c*d*e - c^2*f))
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 1.27 (sec) , antiderivative size = 1408, normalized size of antiderivative = 7.74 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^2 \left (e+f x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

(3*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2 
) - sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*a*c**2*d*e*f - sqrt(c)*sqrt(a*d 
- b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)* 
x)/(sqrt(c)*sqrt(b)))*a*c*d**2*e**2 + 3*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt 
(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt( 
b)))*a*c*d**2*e*f*x**2 - sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - s 
qrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*a*d**3*e** 
2*x**2 - 2*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a 
+ b*x**2) - sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*b*c**3*e*f - 2*sqrt(c)*s 
qrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)* 
sqrt(b)*x)/(sqrt(c)*sqrt(b)))*b*c**2*d*e*f*x**2 + 3*sqrt(c)*sqrt(a*d - b*c 
)*atan((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x**2) + sqrt(d)*sqrt(b)*x)/(s 
qrt(c)*sqrt(b)))*a*c**2*d*e*f - sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b 
*c) + sqrt(d)*sqrt(a + b*x**2) + sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*a*c 
*d**2*e**2 + 3*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) + sqrt(d)*sqr 
t(a + b*x**2) + sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*a*c*d**2*e*f*x**2 - 
sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x**2) + 
 sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*a*d**3*e**2*x**2 - 2*sqrt(c)*sqrt(a 
*d - b*c)*atan((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x**2) + sqrt(d)*sqrt( 
b)*x)/(sqrt(c)*sqrt(b)))*b*c**3*e*f - 2*sqrt(c)*sqrt(a*d - b*c)*atan((s...