\(\int \frac {(c+d x^2)^2}{\sqrt {a+b x^2} (e+f x^2)^2} \, dx\) [326]

Optimal result

Integrand size = 30, antiderivative size = 170 \[ \int \frac {\left (c+d x^2\right )^2}{\sqrt {a+b x^2} \left (e+f x^2\right )^2} \, dx=-\frac {(d e-c f)^2 x \sqrt {a+b x^2}}{2 e f (b e-a f) \left (e+f x^2\right )}+\frac {d^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f^2}-\frac {(d e-c f) (2 b e (d e+c f)-a f (3 d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{2 e^{3/2} f^2 (b e-a f)^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.11 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.08 \[ \int \frac {\left (c+d x^2\right )^2}{\sqrt {a+b x^2} \left (e+f x^2\right )^2} \, dx=-\frac {\frac {f (d e-c f)^2 x \sqrt {a+b x^2}}{e (b e-a f) \left (e+f x^2\right )}+\frac {(d e-c f) (2 b e (d e+c f)-a f (3 d e+c f)) \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{e^{3/2} (-b e+a f)^{3/2}}+\frac {2 d^2 \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{\sqrt {b}}}{2 f^2} \] Input:

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

Output:

-1/2*((f*(d*e - c*f)^2*x*Sqrt[a + b*x^2])/(e*(b*e - a*f)*(e + f*x^2)) + (( 
d*e - c*f)*(2*b*e*(d*e + c*f) - a*f*(3*d*e + c*f))*ArcTan[(-(f*x*Sqrt[a + 
b*x^2]) + Sqrt[b]*(e + f*x^2))/(Sqrt[e]*Sqrt[-(b*e) + a*f])])/(e^(3/2)*(-( 
b*e) + a*f)^(3/2)) + (2*d^2*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/Sqrt[b])/ 
f^2
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.32, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {425, 398, 224, 219, 291, 221, 402, 27, 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.

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 398

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

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 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 402

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 + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) 
 Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) 
*(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, q}, x] && LtQ[p, -1]
 

rule 425

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

Maple [A] (verified)

Time = 1.08 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.14

method	result	size
pseudoelliptic	\(\frac {2 \left (b d \,e^{2}+f \left (b c -\frac {3 a d}{2}\right ) e -\frac {a c \,f^{2}}{2}\right ) \left (c f -d e \right ) \sqrt {b}\, \left (f \,x^{2}+e \right ) \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+\sqrt {\left (a f -b e \right ) e}\, \left (2 d^{2} e \left (f \,x^{2}+e \right ) \left (a f -b e \right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )+\sqrt {b \,x^{2}+a}\, \left (c f -d e \right )^{2} \sqrt {b}\, x f \right )}{2 \sqrt {\left (a f -b e \right ) e}\, \sqrt {b}\, f^{2} \left (a f -b e \right ) e \left (f \,x^{2}+e \right )}\)	\(193\)
default	\(\frac {d^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{f^{2} \sqrt {b}}-\frac {\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \left (-\frac {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}}}{\left (a f -b e \right ) \left (x -\frac {\sqrt {-e f}}{f}\right )}+\frac {b \sqrt {-e f}\, \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 )}{\left (a f -b e \right ) \sqrt {\frac {a f -b e}{f}}}\right )}{4 f^{3} e}-\frac {\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \left (-\frac {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}}}{\left (a f -b e \right ) \left (x +\frac {\sqrt {-e f}}{f}\right )}-\frac {b \sqrt {-e f}\, \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 )}{\left (a f -b e \right ) \sqrt {\frac {a f -b e}{f}}}\right )}{4 f^{3} e}+\frac {\left (c^{2} f^{2}+2 c d e f -3 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 )}{4 f^{2} \sqrt {-e f}\, e \sqrt {\frac {a f -b e}{f}}}-\frac {\left (c^{2} f^{2}+2 c d e f -3 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 )}{4 f^{2} \sqrt {-e f}\, e \sqrt {\frac {a f -b e}{f}}}\)	\(928\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 464 vs. \(2 (148) = 296\).

Time = 4.40 (sec) , antiderivative size = 1947, normalized size of antiderivative = 11.45 \[ \int \frac {\left (c+d x^2\right )^2}{\sqrt {a+b x^2} \left (e+f x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/8*(4*(b^2*d^2*e^4*f - a*b*c^2*e*f^4 - (2*b^2*c*d + a*b*d^2)*e^3*f^2 + 
(b^2*c^2 + 2*a*b*c*d)*e^2*f^3)*sqrt(b*x^2 + a)*x - 4*(b^2*d^2*e^5 - 2*a*b* 
d^2*e^4*f + a^2*d^2*e^3*f^2 + (b^2*d^2*e^4*f - 2*a*b*d^2*e^3*f^2 + a^2*d^2 
*e^2*f^3)*x^2)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + ( 
2*b^2*d^2*e^4 - 3*a*b*d^2*e^3*f + a*b*c^2*e*f^3 - 2*(b^2*c^2 - a*b*c*d)*e^ 
2*f^2 + (2*b^2*d^2*e^3*f - 3*a*b*d^2*e^2*f^2 + a*b*c^2*f^4 - 2*(b^2*c^2 - 
a*b*c*d)*e*f^3)*x^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^5*f^2 - 2*a*b^2*e^4*f^3 + a^2*b*e^3*f^4 + (b^3*e^4*f^3 - 2*a*b^2 
*e^3*f^4 + a^2*b*e^2*f^5)*x^2), -1/8*(4*(b^2*d^2*e^4*f - a*b*c^2*e*f^4 - ( 
2*b^2*c*d + a*b*d^2)*e^3*f^2 + (b^2*c^2 + 2*a*b*c*d)*e^2*f^3)*sqrt(b*x^2 + 
 a)*x + 8*(b^2*d^2*e^5 - 2*a*b*d^2*e^4*f + a^2*d^2*e^3*f^2 + (b^2*d^2*e^4* 
f - 2*a*b*d^2*e^3*f^2 + a^2*d^2*e^2*f^3)*x^2)*sqrt(-b)*arctan(sqrt(-b)*x/s 
qrt(b*x^2 + a)) + (2*b^2*d^2*e^4 - 3*a*b*d^2*e^3*f + a*b*c^2*e*f^3 - 2*(b^ 
2*c^2 - a*b*c*d)*e^2*f^2 + (2*b^2*d^2*e^3*f - 3*a*b*d^2*e^2*f^2 + a*b*c^2* 
f^4 - 2*(b^2*c^2 - a*b*c*d)*e*f^3)*x^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^5*f^2 - 2*a*b^2*e^4*f^3 + a^2*b*e^3*f^4 + ...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 499 vs. \(2 (148) = 296\).

Time = 0.16 (sec) , antiderivative size = 499, normalized size of antiderivative = 2.94 \[ \int \frac {\left (c+d x^2\right )^2}{\sqrt {a+b x^2} \left (e+f x^2\right )^2} \, dx=\frac {{\left (2 \, b^{\frac {3}{2}} d^{2} e^{3} - 3 \, a \sqrt {b} d^{2} e^{2} f - 2 \, b^{\frac {3}{2}} c^{2} e f^{2} + 2 \, a \sqrt {b} c d e f^{2} + a \sqrt {b} c^{2} f^{3}\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} f + 2 \, b e - a f}{2 \, \sqrt {-b^{2} e^{2} + a b e f}}\right )}{2 \, {\left (b e^{2} f^{2} - a e f^{3}\right )} \sqrt {-b^{2} e^{2} + a b e f}} - \frac {d^{2} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right )}{2 \, \sqrt {b} f^{2}} - \frac {2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{\frac {3}{2}} d^{2} e^{3} - 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{\frac {3}{2}} c d e^{2} f - {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a \sqrt {b} d^{2} e^{2} f + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{\frac {3}{2}} c^{2} e f^{2} + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a \sqrt {b} c d e f^{2} - {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a \sqrt {b} c^{2} f^{3} + a^{2} \sqrt {b} d^{2} e^{2} f - 2 \, a^{2} \sqrt {b} c d e f^{2} + a^{2} \sqrt {b} c^{2} f^{3}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} f + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b e - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a f + a^{2} f\right )} {\left (b e^{2} f^{2} - a e f^{3}\right )}} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 1649, normalized size of antiderivative = 9.70 \[ \int \frac {\left (c+d x^2\right )^2}{\sqrt {a+b x^2} \left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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