\(\int \frac {(c+d x^2)^2}{(a+b x^2)^{3/2} (e+f x^2)^2} \, dx\) [346]

Optimal result

Integrand size = 30, antiderivative size = 218 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx=\frac {d^2 x}{a f^2 \sqrt {a+b x^2}}+\frac {b (d e-c f) (a f (5 d e-c f)-2 b e (d e+c f)) x}{2 a e f^2 (b e-a f)^2 \sqrt {a+b x^2}}-\frac {(d e-c f)^2 x}{2 e f (b e-a f) \sqrt {a+b x^2} \left (e+f x^2\right )}+\frac {(d e-c f) (4 b c e-3 a d e-a c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{2 e^{3/2} (b e-a f)^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.25 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.02 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx=\frac {\frac {\sqrt {e} x \left (2 b^2 c^2 e \left (e+f x^2\right )+a^2 \left (-2 c d e f+c^2 f^2+d^2 e \left (3 e+2 f x^2\right )\right )+a b \left (d^2 e^2 x^2+c^2 f^2 x^2-2 c d e \left (2 e+3 f x^2\right )\right )\right )}{a (b e-a f)^2 \sqrt {a+b x^2} \left (e+f x^2\right )}+\frac {(d e-c f) (-4 b c e+3 a d e+a c f) \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{(-b e+a f)^{5/2}}}{2 e^{3/2}} \] Input:

Integrate[(c + d*x^2)^2/((a + b*x^2)^(3/2)*(e + f*x^2)^2),x]
 

Output:

((Sqrt[e]*x*(2*b^2*c^2*e*(e + f*x^2) + a^2*(-2*c*d*e*f + c^2*f^2 + d^2*e*( 
3*e + 2*f*x^2)) + a*b*(d^2*e^2*x^2 + c^2*f^2*x^2 - 2*c*d*e*(2*e + 3*f*x^2) 
)))/(a*(b*e - a*f)^2*Sqrt[a + b*x^2]*(e + f*x^2)) + ((d*e - c*f)*(-4*b*c*e 
 + 3*a*d*e + a*c*f)*ArcTan[(-(f*x*Sqrt[a + b*x^2]) + Sqrt[b]*(e + f*x^2))/ 
(Sqrt[e]*Sqrt[-(b*e) + a*f])])/(-(b*e) + a*f)^(5/2))/(2*e^(3/2))
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.38, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {425, 402, 25, 27, 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.

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

Input:

Int[(c + d*x^2)^2/((a + b*x^2)^(3/2)*(e + f*x^2)^2),x]
 

Output:

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

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 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 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 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 = 0.99 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.02

method	result	size
pseudoelliptic	\(\frac {-a \sqrt {b \,x^{2}+a}\, \left (\left (3 a d -4 b c \right ) e +a c f \right ) \left (c f -d e \right ) \left (f \,x^{2}+e \right ) \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+\sqrt {\left (a f -b e \right ) e}\, \left (\left (3 a^{2} d^{2}-4 \left (-\frac {x^{2} d}{4}+c \right ) d b a +2 b^{2} c^{2}\right ) e^{2}-2 \left (d \left (-x^{2} d +c \right ) a^{2}+3 a b c d \,x^{2}-b^{2} c^{2} x^{2}\right ) f e +a \,c^{2} f^{2} \left (b \,x^{2}+a \right )\right ) x}{2 \sqrt {\left (a f -b e \right ) e}\, \sqrt {b \,x^{2}+a}\, e \left (f \,x^{2}+e \right ) \left (a f -b e \right )^{2} a}\)	\(222\)
default	\(\text {Expression too large to display}\)	\(2018\)

Input:

int((d*x^2+c)^2/(b*x^2+a)^(3/2)/(f*x^2+e)^2,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 662 vs. \(2 (197) = 394\).

Time = 3.42 (sec) , antiderivative size = 1364, normalized size of antiderivative = 6.26 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate((d*x^2+c)^2/(b*x^2+a)^(3/2)/(f*x^2+e)^2,x, algorithm="fricas")
 

Output:

[1/8*((a^3*c^2*e*f^2 + (a^2*b*c^2*f^3 + (4*a*b^2*c*d - 3*a^2*b*d^2)*e^2*f 
- 2*(2*a*b^2*c^2 - a^2*b*c*d)*e*f^2)*x^4 + (4*a^2*b*c*d - 3*a^3*d^2)*e^3 - 
 2*(2*a^2*b*c^2 - a^3*c*d)*e^2*f + (a^3*c^2*f^3 + (4*a*b^2*c*d - 3*a^2*b*d 
^2)*e^3 - (4*a*b^2*c^2 - 6*a^2*b*c*d + 3*a^3*d^2)*e^2*f - (3*a^2*b*c^2 - 2 
*a^3*c*d)*e*f^2)*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 
)) + 4*((a*b^2*d^2*e^4 - a^2*b*c^2*e*f^3 + (2*b^3*c^2 - 6*a*b^2*c*d + a^2* 
b*d^2)*e^3*f - (a*b^2*c^2 - 6*a^2*b*c*d + 2*a^3*d^2)*e^2*f^2)*x^3 - (a^3*c 
^2*e*f^3 - (2*b^3*c^2 - 4*a*b^2*c*d + 3*a^2*b*d^2)*e^4 + (2*a*b^2*c^2 - 2* 
a^2*b*c*d + 3*a^3*d^2)*e^3*f - (a^2*b*c^2 + 2*a^3*c*d)*e^2*f^2)*x)*sqrt(b* 
x^2 + a))/(a^2*b^3*e^6 - 3*a^3*b^2*e^5*f + 3*a^4*b*e^4*f^2 - a^5*e^3*f^3 + 
 (a*b^4*e^5*f - 3*a^2*b^3*e^4*f^2 + 3*a^3*b^2*e^3*f^3 - a^4*b*e^2*f^4)*x^4 
 + (a*b^4*e^6 - 2*a^2*b^3*e^5*f + 2*a^4*b*e^3*f^3 - a^5*e^2*f^4)*x^2), -1/ 
4*((a^3*c^2*e*f^2 + (a^2*b*c^2*f^3 + (4*a*b^2*c*d - 3*a^2*b*d^2)*e^2*f - 2 
*(2*a*b^2*c^2 - a^2*b*c*d)*e*f^2)*x^4 + (4*a^2*b*c*d - 3*a^3*d^2)*e^3 - 2* 
(2*a^2*b*c^2 - a^3*c*d)*e^2*f + (a^3*c^2*f^3 + (4*a*b^2*c*d - 3*a^2*b*d^2) 
*e^3 - (4*a*b^2*c^2 - 6*a^2*b*c*d + 3*a^3*d^2)*e^2*f - (3*a^2*b*c^2 - 2*a^ 
3*c*d)*e*f^2)*x^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...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate((d*x**2+c)**2/(b*x**2+a)**(3/2)/(f*x**2+e)**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:

integrate((d*x^2+c)^2/(b*x^2+a)^(3/2)/(f*x^2+e)^2,x, algorithm="maxima")
 

Output:

integrate((d*x^2 + c)^2/((b*x^2 + a)^(3/2)*(f*x^2 + e)^2), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 538 vs. \(2 (197) = 394\).

Time = 0.43 (sec) , antiderivative size = 538, normalized size of antiderivative = 2.47 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx=\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{{\left (a b^{2} e^{2} - 2 \, a^{2} b e f + a^{3} f^{2}\right )} \sqrt {b x^{2} + a}} - \frac {{\left (4 \, b^{\frac {3}{2}} c d e^{2} - 3 \, a \sqrt {b} d^{2} e^{2} - 4 \, b^{\frac {3}{2}} c^{2} e f + 2 \, a \sqrt {b} c d e f + a \sqrt {b} c^{2} f^{2}\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} f + 2 \, b e - a f}{2 \, \sqrt {-b^{2} e^{2} + a b e f}}\right )}{2 \, {\left (b^{2} e^{3} - 2 \, a b e^{2} f + a^{2} e f^{2}\right )} \sqrt {-b^{2} e^{2} + a b e f}} + \frac {2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{\frac {3}{2}} d^{2} e^{3} - 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{\frac {3}{2}} c d e^{2} f - {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a \sqrt {b} d^{2} e^{2} f + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{\frac {3}{2}} c^{2} e f^{2} + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a \sqrt {b} c d e f^{2} - {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a \sqrt {b} c^{2} f^{3} + a^{2} \sqrt {b} d^{2} e^{2} f - 2 \, a^{2} \sqrt {b} c d e f^{2} + a^{2} \sqrt {b} c^{2} f^{3}}{{\left (b^{2} e^{3} f - 2 \, a b e^{2} f^{2} + a^{2} e f^{3}\right )} {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} f + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b e - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a f + a^{2} f\right )}} \] Input:

integrate((d*x^2+c)^2/(b*x^2+a)^(3/2)/(f*x^2+e)^2,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx=\int \frac {{\left (d\,x^2+c\right )}^2}{{\left (b\,x^2+a\right )}^{3/2}\,{\left (f\,x^2+e\right )}^2} \,d x \] Input:

int((c + d*x^2)^2/((a + b*x^2)^(3/2)*(e + f*x^2)^2),x)
 

Output:

int((c + d*x^2)^2/((a + b*x^2)^(3/2)*(e + f*x^2)^2), x)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 4917, normalized size of antiderivative = 22.56 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int((d*x^2+c)^2/(b*x^2+a)^(3/2)/(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**4*b*c**2*e*f**3 - 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)))*a**4*b*c**2*f**4*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)))*a**4*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**4*b*c*d*e*f**3*x**2 + 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**4*b*d**2*e**3*f + 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**4*b*d** 
2*e**2*f**2*x**2 + 8*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**3*b**2*c**2 
*e**2*f**2 + 7*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqr 
t(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**3*b**2*c**2*e*f** 
3*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**3*b**2*c**2*f**4*x**4 + 
 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)))*a**3*b**2*c*d*e**3*f + 2*sqrt(e) 
*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqr...