\(\int \frac {(c+d x^2)^2}{\sqrt {a+b x^2} (e+f x^2)^3} \, dx\) [327]

Optimal result

Integrand size = 30, antiderivative size = 261 \[ \int \frac {\left (c+d x^2\right )^2}{\sqrt {a+b x^2} \left (e+f x^2\right )^3} \, dx=\frac {d^2 x \sqrt {a+b x^2}}{4 e f (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (c+d x^2\right )^2}{4 e (b e-a f) \left (e+f x^2\right )^2}-\frac {(d e-c f) (2 b e (d e-3 c f)+a f (d e+3 c f)) x \sqrt {a+b x^2}}{8 e^2 f (b e-a f)^2 \left (e+f x^2\right )}+\frac {\left (8 b^2 c^2 e^2-8 a b c e (d e+c f)+a^2 \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{8 e^{5/2} (b e-a f)^{5/2}} \] Output:

1/4*d^2*x*(b*x^2+a)^(1/2)/e/f/(-a*f+b*e)-1/4*f*x*(b*x^2+a)^(1/2)*(d*x^2+c) 
^2/e/(-a*f+b*e)/(f*x^2+e)^2-1/8*(-c*f+d*e)*(2*b*e*(-3*c*f+d*e)+a*f*(3*c*f+ 
d*e))*x*(b*x^2+a)^(1/2)/e^2/f/(-a*f+b*e)^2/(f*x^2+e)+1/8*(8*b^2*c^2*e^2-8* 
a*b*c*e*(c*f+d*e)+a^2*(3*c^2*f^2+2*c*d*e*f+3*d^2*e^2))*arctanh((-a*f+b*e)^ 
(1/2)*x/e^(1/2)/(b*x^2+a)^(1/2))/e^(5/2)/(-a*f+b*e)^(5/2)
 

Mathematica [A] (verified)

Time = 11.30 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.43 \[ \int \frac {\left (c+d x^2\right )^2}{\sqrt {a+b x^2} \left (e+f x^2\right )^3} \, dx=\frac {d (d e-c f) x \left (f \left (a+b x^2\right )-\frac {(2 b e-a f) \left (e+f x^2\right ) \text {arctanh}\left (\sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}\right )}{e \sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}}\right )}{e f^2 (b e-a f) \sqrt {a+b x^2} \left (e+f x^2\right )}-\frac {(d e-c f)^2 x \left (e f \left (a+b x^2\right ) \left (2 b e \left (4 e+3 f x^2\right )-a f \left (5 e+3 f x^2\right )\right )-\frac {\left (8 b^2 e^2-8 a b e f+3 a^2 f^2\right ) \left (e+f x^2\right )^2 \text {arctanh}\left (\sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}\right )}{\sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}}\right )}{8 e^3 f^2 (b e-a f)^2 \sqrt {a+b x^2} \left (e+f x^2\right )^2}+\frac {d^2 \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f^2 \sqrt {b e-a f}} \] Input:

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

Output:

(d*(d*e - c*f)*x*(f*(a + b*x^2) - ((2*b*e - a*f)*(e + f*x^2)*ArcTanh[Sqrt[ 
((b*e - a*f)*x^2)/(e*(a + b*x^2))]])/(e*Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x 
^2))])))/(e*f^2*(b*e - a*f)*Sqrt[a + b*x^2]*(e + f*x^2)) - ((d*e - c*f)^2* 
x*(e*f*(a + b*x^2)*(2*b*e*(4*e + 3*f*x^2) - a*f*(5*e + 3*f*x^2)) - ((8*b^2 
*e^2 - 8*a*b*e*f + 3*a^2*f^2)*(e + f*x^2)^2*ArcTanh[Sqrt[((b*e - a*f)*x^2) 
/(e*(a + b*x^2))]])/Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))]))/(8*e^3*f^2*( 
b*e - a*f)^2*Sqrt[a + b*x^2]*(e + f*x^2)^2) + (d^2*ArcTanh[(Sqrt[b*e - a*f 
]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(Sqrt[e]*f^2*Sqrt[b*e - a*f])
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.34, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {425, 402, 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.

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 402

\(\displaystyle \frac {d \left (\frac {\int \frac {2 b c e-a (d e+c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d \left (\frac {(2 b c e-a (c f+d e)) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {d \left (\frac {(2 b c e-a (c f+d e)) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {d \left (\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (2 b c e-a (c f+d e))}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {d \left (\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (2 b c e-a (c f+d e))}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {\int \frac {f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (a f (3 c f+d e)+2 b e (d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d \left (\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (2 b c e-a (c f+d e))}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {\left (a^2 f (3 c f+d e)-4 a b e (2 c f+d e)+8 b^2 c e^2\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (a f (3 c f+d e)+2 b e (d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {d \left (\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (2 b c e-a (c f+d e))}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {\left (a^2 f (3 c f+d e)-4 a b e (2 c f+d e)+8 b^2 c e^2\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (a f (3 c f+d e)+2 b e (d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {d \left (\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (2 b c e-a (c f+d e))}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (a^2 f (3 c f+d e)-4 a b e (2 c f+d e)+8 b^2 c e^2\right )}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {x \sqrt {a+b x^2} (a f (3 c f+d e)+2 b e (d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\)

Input:

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

Output:

(d*(((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 - ((d*e - c*f)*(((d*e - c*f)*x*Sqrt[a 
 + b*x^2])/(4*e*(b*e - a*f)*(e + f*x^2)^2) + (((2*b*e*(d*e - 3*c*f) + a*f* 
(d*e + 3*c*f))*x*Sqrt[a + b*x^2])/(2*e*(b*e - a*f)*(e + f*x^2)) + ((8*b^2* 
c*e^2 - 4*a*b*e*(d*e + 2*c*f) + a^2*f*(d*e + 3*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)))/(4*e*(b* 
e - a*f))))/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 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 = 1.01 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.81

method	result	size
pseudoelliptic	\(\frac {5 \sqrt {\left (a f -b e \right ) e}\, \left (\frac {\left (3 a d -8 \left (\frac {x^{2} d}{4}+c \right ) b \right ) e^{2}}{5}+f \left (\left (x^{2} d +c \right ) a -\frac {6 x^{2} b c}{5}\right ) e +\frac {3 a c \,f^{2} x^{2}}{5}\right ) \left (c f -d e \right ) \sqrt {b \,x^{2}+a}\, x -3 \left (\left (a^{2} d^{2}-\frac {8}{3} a b c d +\frac {8}{3} b^{2} c^{2}\right ) e^{2}+\frac {2 a c f \left (a d -4 b c \right ) e}{3}+a^{2} c^{2} f^{2}\right ) \left (f \,x^{2}+e \right )^{2} \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )}{8 \sqrt {\left (a f -b e \right ) e}\, \left (a f -b e \right )^{2} e^{2} \left (f \,x^{2}+e \right )^{2}}\)	\(211\)
default	\(\text {Expression too large to display}\)	\(1983\)

Input:

int((d*x^2+c)^2/(b*x^2+a)^(1/2)/(f*x^2+e)^3,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 639 vs. \(2 (237) = 474\).

Time = 2.88 (sec) , antiderivative size = 1318, normalized size of antiderivative = 5.05 \[ \int \frac {\left (c+d x^2\right )^2}{\sqrt {a+b x^2} \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((d*x^2+c)^2/(b*x^2+a)^(1/2)/(f*x^2+e)^3,x, algorithm="fricas")
 

Output:

[1/32*((3*a^2*c^2*e^2*f^2 + (8*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*e^4 - 2*(4 
*a*b*c^2 - a^2*c*d)*e^3*f + (3*a^2*c^2*f^4 + (8*b^2*c^2 - 8*a*b*c*d + 3*a^ 
2*d^2)*e^2*f^2 - 2*(4*a*b*c^2 - a^2*c*d)*e*f^3)*x^4 + 2*(3*a^2*c^2*e*f^3 + 
 (8*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*e^3*f - 2*(4*a*b*c^2 - a^2*c*d)*e^2*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)*sq 
rt(b*e^2 - a*e*f)*sqrt(b*x^2 + a))/(f^2*x^4 + 2*e*f*x^2 + e^2)) + 4*((2*b^ 
2*d^2*e^5 - 3*a^2*c^2*e*f^4 + (4*b^2*c*d - 7*a*b*d^2)*e^4*f - (6*b^2*c^2 + 
 2*a*b*c*d - 5*a^2*d^2)*e^3*f^2 + (9*a*b*c^2 - 2*a^2*c*d)*e^2*f^3)*x^3 - ( 
5*a^2*c^2*e^2*f^3 - (8*b^2*c*d - 3*a*b*d^2)*e^5 + (8*b^2*c^2 + 10*a*b*c*d 
- 3*a^2*d^2)*e^4*f - (13*a*b*c^2 + 2*a^2*c*d)*e^3*f^2)*x)*sqrt(b*x^2 + a)) 
/(b^3*e^8 - 3*a*b^2*e^7*f + 3*a^2*b*e^6*f^2 - a^3*e^5*f^3 + (b^3*e^6*f^2 - 
 3*a*b^2*e^5*f^3 + 3*a^2*b*e^4*f^4 - a^3*e^3*f^5)*x^4 + 2*(b^3*e^7*f - 3*a 
*b^2*e^6*f^2 + 3*a^2*b*e^5*f^3 - a^3*e^4*f^4)*x^2), -1/16*((3*a^2*c^2*e^2* 
f^2 + (8*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*e^4 - 2*(4*a*b*c^2 - a^2*c*d)*e^ 
3*f + (3*a^2*c^2*f^4 + (8*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*e^2*f^2 - 2*(4* 
a*b*c^2 - a^2*c*d)*e*f^3)*x^4 + 2*(3*a^2*c^2*e*f^3 + (8*b^2*c^2 - 8*a*b*c* 
d + 3*a^2*d^2)*e^3*f - 2*(4*a*b*c^2 - a^2*c*d)*e^2*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*e^2 - a^2*e*f)*x)) - 2*((2*b^2*...
 

Sympy [F]

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

integrate((d*x**2+c)**2/(b*x**2+a)**(1/2)/(f*x**2+e)**3,x)
 

Output:

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

Maxima [F]

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

integrate((d*x^2+c)^2/(b*x^2+a)^(1/2)/(f*x^2+e)^3,x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1350 vs. \(2 (237) = 474\).

Time = 0.36 (sec) , antiderivative size = 1350, normalized size of antiderivative = 5.17 \[ \int \frac {\left (c+d x^2\right )^2}{\sqrt {a+b x^2} \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((d*x^2+c)^2/(b*x^2+a)^(1/2)/(f*x^2+e)^3,x, algorithm="giac")
 

Output:

-1/8*(8*b^(5/2)*c^2*e^2 - 8*a*b^(3/2)*c*d*e^2 + 3*a^2*sqrt(b)*d^2*e^2 - 8* 
a*b^(3/2)*c^2*e*f + 2*a^2*sqrt(b)*c*d*e*f + 3*a^2*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^4 - 2*a*b*e^3*f + a^2*e^2*f^2)*sqrt(-b^2*e^2 + a*b*e*f)) + 1 
/4*(8*(sqrt(b)*x - sqrt(b*x^2 + a))^6*b^(5/2)*d^2*e^4*f - 16*(sqrt(b)*x - 
sqrt(b*x^2 + a))^6*a*b^(3/2)*d^2*e^3*f^2 - 8*(sqrt(b)*x - sqrt(b*x^2 + a)) 
^6*b^(5/2)*c^2*e^2*f^3 + 8*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(3/2)*c*d*e 
^2*f^3 + 5*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*sqrt(b)*d^2*e^2*f^3 + 8*(sq 
rt(b)*x - sqrt(b*x^2 + a))^6*a*b^(3/2)*c^2*e*f^4 - 2*(sqrt(b)*x - sqrt(b*x 
^2 + a))^6*a^2*sqrt(b)*c*d*e*f^4 - 3*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*s 
qrt(b)*c^2*f^5 + 16*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*d^2*e^5 + 32*( 
sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*c*d*e^4*f - 56*(sqrt(b)*x - sqrt(b* 
x^2 + a))^4*a*b^(5/2)*d^2*e^4*f - 48*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/ 
2)*c^2*e^3*f^2 - 16*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a*b^(5/2)*c*d*e^3*f^2 
+ 46*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^(3/2)*d^2*e^3*f^2 + 72*(sqrt(b) 
*x - sqrt(b*x^2 + a))^4*a*b^(5/2)*c^2*e^2*f^3 - 4*(sqrt(b)*x - sqrt(b*x^2 
+ a))^4*a^2*b^(3/2)*c*d*e^2*f^3 - 15*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^3*s 
qrt(b)*d^2*e^2*f^3 - 42*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^(3/2)*c^2*e* 
f^4 + 6*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^3*sqrt(b)*c*d*e*f^4 + 9*(sqrt(b) 
*x - sqrt(b*x^2 + a))^4*a^3*sqrt(b)*c^2*f^5 + 8*(sqrt(b)*x - sqrt(b*x^2...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 4829, normalized size of antiderivative = 18.50 \[ \int \frac {\left (c+d x^2\right )^2}{\sqrt {a+b x^2} \left (e+f x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 6*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*c**2*e**2*f**4 - 12*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)))*a**3*c**2*e*f**5*x**2 - 6*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*c**2*f**6*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)/(sqr 
t(e)*sqrt(b)))*a**3*c*d*e**3*f**3 - 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*c*d*e**2*f**4*x**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)))*a**3 
*c*d*e*f**5*x**4 - 6*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*d**2*e**4 
*f**2 - 12*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*d**2*e**3*f**3*x**2 
 - 6*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*d**2*e**2*f**4*x**4 + 28* 
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**2*b*c**2*e**3*f**3 + 56*sqrt(e)* 
sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt...