\(\int \frac {(c+d x^2)^2}{\sqrt {a+b x^2} (e+f x^2)} \, dx\) [325]

Optimal result

Integrand size = 30, antiderivative size = 135 \[ \int \frac {\left (c+d x^2\right )^2}{\sqrt {a+b x^2} \left (e+f x^2\right )} \, dx=\frac {d^2 x \sqrt {a+b x^2}}{2 b f}-\frac {d (2 b d e-4 b c f+a d f) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2} f^2}+\frac {(d e-c f)^2 \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f^2 \sqrt {b e-a f}} \] Output:

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

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.10 \[ \int \frac {\left (c+d x^2\right )^2}{\sqrt {a+b x^2} \left (e+f x^2\right )} \, dx=\frac {\frac {d^2 f x \sqrt {a+b x^2}}{b}-\frac {2 (d e-c f)^2 \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{\sqrt {e} \sqrt {-b e+a f}}+\frac {d (2 b d e-4 b c f+a d f) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{b^{3/2}}}{2 f^2} \] Input:

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

Output:

((d^2*f*x*Sqrt[a + b*x^2])/b - (2*(d*e - c*f)^2*ArcTan[(-(f*x*Sqrt[a + b*x 
^2]) + Sqrt[b]*(e + f*x^2))/(Sqrt[e]*Sqrt[-(b*e) + a*f])])/(Sqrt[e]*Sqrt[- 
(b*e) + a*f]) + (d*(2*b*d*e - 4*b*c*f + a*d*f)*Log[-(Sqrt[b]*x) + Sqrt[a + 
 b*x^2]])/b^(3/2))/(2*f^2)
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.24, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {420, 299, 224, 219, 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.

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

\(\Big \downarrow \) 420

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 398

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

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 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 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 420

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

Maple [A] (verified)

Time = 0.92 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.99

method	result	size
pseudoelliptic	\(-\frac {b^{\frac {5}{2}} \left (c f -d e \right )^{2} \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )-\frac {d \left (\left (\left (4 c f -2 d e \right ) b^{2}-d f a b \right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )+d f \sqrt {b \,x^{2}+a}\, x \,b^{\frac {3}{2}}\right ) \sqrt {\left (a f -b e \right ) e}}{2}}{\sqrt {\left (a f -b e \right ) e}\, b^{\frac {5}{2}} f^{2}}\)	\(133\)
risch	\(\frac {d^{2} x \sqrt {b \,x^{2}+a}}{2 b f}-\frac {\frac {d \left (a d f -4 b c f +2 b d e \right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{f \sqrt {b}}-\frac {b \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \ln \left (\frac {\frac {2 a f -2 b e}{f}-\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {\left (x +\frac {\sqrt {-e f}}{f}\right )^{2} b -\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}{x +\frac {\sqrt {-e f}}{f}}\right )}{\sqrt {-e f}\, f \sqrt {\frac {a f -b e}{f}}}+\frac {b \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \ln \left (\frac {\frac {2 a f -2 b e}{f}+\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {\left (x -\frac {\sqrt {-e f}}{f}\right )^{2} b +\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}{x -\frac {\sqrt {-e f}}{f}}\right )}{\sqrt {-e f}\, f \sqrt {\frac {a f -b e}{f}}}}{2 b f}\)	\(418\)
default	\(\frac {d \left (d f \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )+\frac {2 c f \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}-\frac {d e \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}\right )}{f^{2}}+\frac {\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \ln \left (\frac {\frac {2 a f -2 b e}{f}-\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {\left (x +\frac {\sqrt {-e f}}{f}\right )^{2} b -\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}{x +\frac {\sqrt {-e f}}{f}}\right )}{2 f^{2} \sqrt {-e f}\, \sqrt {\frac {a f -b e}{f}}}-\frac {\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \ln \left (\frac {\frac {2 a f -2 b e}{f}+\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {\left (x -\frac {\sqrt {-e f}}{f}\right )^{2} b +\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}{x -\frac {\sqrt {-e f}}{f}}\right )}{2 f^{2} \sqrt {-e f}\, \sqrt {\frac {a f -b e}{f}}}\)	\(441\)

Input:

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

Output:

-(b^(5/2)*(c*f-d*e)^2*arctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1/2))-1/2* 
d*(((4*c*f-2*d*e)*b^2-d*f*a*b)*arctanh((b*x^2+a)^(1/2)/x/b^(1/2))+d*f*(b*x 
^2+a)^(1/2)*x*b^(3/2))*((a*f-b*e)*e)^(1/2))/((a*f-b*e)*e)^(1/2)/b^(5/2)/f^ 
2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 259 vs. \(2 (113) = 226\).

Time = 1.04 (sec) , antiderivative size = 1131, normalized size of antiderivative = 8.38 \[ \int \frac {\left (c+d x^2\right )^2}{\sqrt {a+b x^2} \left (e+f x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/4*(2*(b^2*d^2*e^2*f - a*b*d^2*e*f^2)*sqrt(b*x^2 + a)*x + (2*b^2*d^2*e^3 
 - (4*b^2*c*d + a*b*d^2)*e^2*f + (4*a*b*c*d - a^2*d^2)*e*f^2)*sqrt(b)*log( 
-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + (b^2*d^2*e^2 - 2*b^2*c*d*e*f 
 + b^2*c^2*f^2)*sqrt(b*e^2 - a*e*f)*log(((8*b^2*e^2 - 8*a*b*e*f + a^2*f^2) 
*x^4 + a^2*e^2 + 2*(4*a*b*e^2 - 3*a^2*e*f)*x^2 + 4*((2*b*e - a*f)*x^3 + a* 
e*x)*sqrt(b*e^2 - a*e*f)*sqrt(b*x^2 + a))/(f^2*x^4 + 2*e*f*x^2 + e^2)))/(b 
^3*e^2*f^2 - a*b^2*e*f^3), 1/4*(2*(b^2*d^2*e^2*f - a*b*d^2*e*f^2)*sqrt(b*x 
^2 + a)*x + 2*(2*b^2*d^2*e^3 - (4*b^2*c*d + a*b*d^2)*e^2*f + (4*a*b*c*d - 
a^2*d^2)*e*f^2)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (b^2*d^2*e^2 
 - 2*b^2*c*d*e*f + b^2*c^2*f^2)*sqrt(b*e^2 - a*e*f)*log(((8*b^2*e^2 - 8*a* 
b*e*f + a^2*f^2)*x^4 + a^2*e^2 + 2*(4*a*b*e^2 - 3*a^2*e*f)*x^2 + 4*((2*b*e 
 - a*f)*x^3 + a*e*x)*sqrt(b*e^2 - a*e*f)*sqrt(b*x^2 + a))/(f^2*x^4 + 2*e*f 
*x^2 + e^2)))/(b^3*e^2*f^2 - a*b^2*e*f^3), 1/4*(2*(b^2*d^2*e^2*f - a*b*d^2 
*e*f^2)*sqrt(b*x^2 + a)*x - 2*(b^2*d^2*e^2 - 2*b^2*c*d*e*f + b^2*c^2*f^2)* 
sqrt(-b*e^2 + a*e*f)*arctan(1/2*sqrt(-b*e^2 + a*e*f)*((2*b*e - a*f)*x^2 + 
a*e)*sqrt(b*x^2 + a)/((b^2*e^2 - a*b*e*f)*x^3 + (a*b*e^2 - a^2*e*f)*x)) + 
(2*b^2*d^2*e^3 - (4*b^2*c*d + a*b*d^2)*e^2*f + (4*a*b*c*d - a^2*d^2)*e*f^2 
)*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a))/(b^3*e^2*f^2 - 
a*b^2*e*f^3), 1/2*((b^2*d^2*e^2*f - a*b*d^2*e*f^2)*sqrt(b*x^2 + a)*x - (b^ 
2*d^2*e^2 - 2*b^2*c*d*e*f + b^2*c^2*f^2)*sqrt(-b*e^2 + a*e*f)*arctan(1/...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 578, normalized size of antiderivative = 4.28 \[ \int \frac {\left (c+d x^2\right )^2}{\sqrt {a+b x^2} \left (e+f x^2\right )} \, dx=\frac {-2 \sqrt {e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {a f -b e}-\sqrt {f}\, \sqrt {b \,x^{2}+a}-\sqrt {f}\, \sqrt {b}\, x}{\sqrt {e}\, \sqrt {b}}\right ) b^{2} c^{2} f^{2}+4 \sqrt {e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {a f -b e}-\sqrt {f}\, \sqrt {b \,x^{2}+a}-\sqrt {f}\, \sqrt {b}\, x}{\sqrt {e}\, \sqrt {b}}\right ) b^{2} c d e f -2 \sqrt {e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {a f -b e}-\sqrt {f}\, \sqrt {b \,x^{2}+a}-\sqrt {f}\, \sqrt {b}\, x}{\sqrt {e}\, \sqrt {b}}\right ) b^{2} d^{2} e^{2}-2 \sqrt {e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {a f -b e}+\sqrt {f}\, \sqrt {b \,x^{2}+a}+\sqrt {f}\, \sqrt {b}\, x}{\sqrt {e}\, \sqrt {b}}\right ) b^{2} c^{2} f^{2}+4 \sqrt {e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {a f -b e}+\sqrt {f}\, \sqrt {b \,x^{2}+a}+\sqrt {f}\, \sqrt {b}\, x}{\sqrt {e}\, \sqrt {b}}\right ) b^{2} c d e f -2 \sqrt {e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {a f -b e}+\sqrt {f}\, \sqrt {b \,x^{2}+a}+\sqrt {f}\, \sqrt {b}\, x}{\sqrt {e}\, \sqrt {b}}\right ) b^{2} d^{2} e^{2}+\sqrt {b \,x^{2}+a}\, a b \,d^{2} e \,f^{2} x -\sqrt {b \,x^{2}+a}\, b^{2} d^{2} e^{2} f x -\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} d^{2} e \,f^{2}+4 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b c d e \,f^{2}-\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b \,d^{2} e^{2} f -4 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c d \,e^{2} f +2 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} d^{2} e^{3}}{2 b^{2} e \,f^{2} \left (a f -b e \right )} \] Input:

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

Output:

( - 2*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x 
**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*b**2*c**2*f**2 + 4*sqrt(e)*sq 
rt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*s 
qrt(b)*x)/(sqrt(e)*sqrt(b)))*b**2*c*d*e*f - 2*sqrt(e)*sqrt(a*f - b*e)*atan 
((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e) 
*sqrt(b)))*b**2*d**2*e**2 - 2*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e 
) + sqrt(f)*sqrt(a + b*x**2) + sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*b**2* 
c**2*f**2 + 4*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) + sqrt(f)*sqrt 
(a + b*x**2) + sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*b**2*c*d*e*f - 2*sqrt 
(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) + sqrt(f)*sqrt(a + b*x**2) + sqr 
t(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*b**2*d**2*e**2 + sqrt(a + b*x**2)*a*b*d 
**2*e*f**2*x - sqrt(a + b*x**2)*b**2*d**2*e**2*f*x - sqrt(b)*log((sqrt(a + 
 b*x**2) + sqrt(b)*x)/sqrt(a))*a**2*d**2*e*f**2 + 4*sqrt(b)*log((sqrt(a + 
b*x**2) + sqrt(b)*x)/sqrt(a))*a*b*c*d*e*f**2 - sqrt(b)*log((sqrt(a + b*x** 
2) + sqrt(b)*x)/sqrt(a))*a*b*d**2*e**2*f - 4*sqrt(b)*log((sqrt(a + b*x**2) 
 + sqrt(b)*x)/sqrt(a))*b**2*c*d*e**2*f + 2*sqrt(b)*log((sqrt(a + b*x**2) + 
 sqrt(b)*x)/sqrt(a))*b**2*d**2*e**3)/(2*b**2*e*f**2*(a*f - b*e))