\(\int \frac {c+d x^2}{(a+b x^2)^{5/2} (e+f x^2)^2} \, dx\) [362]

Optimal result

Integrand size = 28, antiderivative size = 243 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx=\frac {b (2 b c e-5 a d e+3 a c f) x}{6 a e (b e-a f)^2 \left (a+b x^2\right )^{3/2}}+\frac {b \left (4 b^2 c e^2+2 a b e (d e-8 c f)+a^2 f (13 d e-3 c f)\right ) x}{6 a^2 e (b e-a f)^3 \sqrt {a+b x^2}}+\frac {(d e-c f) x}{2 e (b e-a f) \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )}-\frac {f (2 b e (2 d e-3 c f)+a f (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{2 e^{3/2} (b e-a f)^{7/2}} \] Output:

1/6*b*(3*a*c*f-5*a*d*e+2*b*c*e)*x/a/e/(-a*f+b*e)^2/(b*x^2+a)^(3/2)+1/6*b*( 
4*b^2*c*e^2+2*a*b*e*(-8*c*f+d*e)+a^2*f*(-3*c*f+13*d*e))*x/a^2/e/(-a*f+b*e) 
^3/(b*x^2+a)^(1/2)+1/2*(-c*f+d*e)*x/e/(-a*f+b*e)/(b*x^2+a)^(3/2)/(f*x^2+e) 
-1/2*f*(2*b*e*(-3*c*f+2*d*e)+a*f*(c*f+d*e))*arctanh((-a*f+b*e)^(1/2)*x/e^( 
1/2)/(b*x^2+a)^(1/2))/e^(3/2)/(-a*f+b*e)^(7/2)
 

Mathematica [A] (verified)

Time = 15.59 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.88 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx=\frac {1}{6} \left (x \sqrt {a+b x^2} \left (\frac {2 b (b c-a d)}{a (b e-a f)^2 \left (a+b x^2\right )^2}-\frac {2 b \left (2 b^2 c e+5 a^2 d f+a b (d e-8 c f)\right )}{a^2 (-b e+a f)^3 \left (a+b x^2\right )}+\frac {3 f^2 (d e-c f)}{e (b e-a f)^3 \left (e+f x^2\right )}\right )+\frac {3 f (2 b e (2 d e-3 c f)+a f (d e+c f)) \arctan \left (\frac {\sqrt {-b e+a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{e^{3/2} (-b e+a f)^{7/2}}\right ) \] Input:

Integrate[(c + d*x^2)/((a + b*x^2)^(5/2)*(e + f*x^2)^2),x]
 

Output:

(x*Sqrt[a + b*x^2]*((2*b*(b*c - a*d))/(a*(b*e - a*f)^2*(a + b*x^2)^2) - (2 
*b*(2*b^2*c*e + 5*a^2*d*f + a*b*(d*e - 8*c*f)))/(a^2*(-(b*e) + a*f)^3*(a + 
 b*x^2)) + (3*f^2*(d*e - c*f))/(e*(b*e - a*f)^3*(e + f*x^2))) + (3*f*(2*b* 
e*(2*d*e - 3*c*f) + a*f*(d*e + c*f))*ArcTan[(Sqrt[-(b*e) + a*f]*x)/(Sqrt[e 
]*Sqrt[a + b*x^2])])/(e^(3/2)*(-(b*e) + a*f)^(7/2)))/6
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.22, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {402, 25, 402, 25, 27, 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.

\(\displaystyle \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx\)

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 402

\(\displaystyle \frac {\frac {x \left (4 a^2 d f+a b (d e-7 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}-\frac {\int -\frac {f \left (2 \left (4 d f a^2+b (d e-7 c f) a+2 b^2 c e\right ) x^2+a (2 b c e-5 a d e+3 a c f)\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{a (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {f \left (2 \left (4 d f a^2+b (d e-7 c f) a+2 b^2 c e\right ) x^2+a (2 b c e-5 a d e+3 a c f)\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{a (b e-a f)}+\frac {x \left (4 a^2 d f+a b (d e-7 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {f \int \frac {2 \left (4 d f a^2+b (d e-7 c f) a+2 b^2 c e\right ) x^2+a (2 b c e-5 a d e+3 a c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{a (b e-a f)}+\frac {x \left (4 a^2 d f+a b (d e-7 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\frac {f \left (\frac {\int -\frac {3 a^2 (2 b e (2 d e-3 c f)+a f (d e+c f))}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (a^2 f (13 d e-3 c f)+2 a b e (d e-8 c f)+4 b^2 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{a (b e-a f)}+\frac {x \left (4 a^2 d f+a b (d e-7 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {f \left (\frac {x \sqrt {a+b x^2} \left (a^2 f (13 d e-3 c f)+2 a b e (d e-8 c f)+4 b^2 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {3 a^2 (a f (c f+d e)+2 b e (2 d e-3 c f)) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}\right )}{a (b e-a f)}+\frac {x \left (4 a^2 d f+a b (d e-7 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\frac {f \left (\frac {x \sqrt {a+b x^2} \left (a^2 f (13 d e-3 c f)+2 a b e (d e-8 c f)+4 b^2 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {3 a^2 (a f (c f+d e)+2 b e (2 d e-3 c f)) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}\right )}{a (b e-a f)}+\frac {x \left (4 a^2 d f+a b (d e-7 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {f \left (\frac {x \sqrt {a+b x^2} \left (a^2 f (13 d e-3 c f)+2 a b e (d e-8 c f)+4 b^2 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {3 a^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (c f+d e)+2 b e (2 d e-3 c f))}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{a (b e-a f)}+\frac {x \left (4 a^2 d f+a b (d e-7 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\)

Input:

Int[(c + d*x^2)/((a + b*x^2)^(5/2)*(e + f*x^2)^2),x]
 

Output:

((b*c - a*d)*x)/(3*a*(b*e - a*f)*(a + b*x^2)^(3/2)*(e + f*x^2)) + (((2*b^2 
*c*e + 4*a^2*d*f + a*b*(d*e - 7*c*f))*x)/(a*(b*e - a*f)*Sqrt[a + b*x^2]*(e 
 + f*x^2)) + (f*(((4*b^2*c*e^2 + 2*a*b*e*(d*e - 8*c*f) + a^2*f*(13*d*e - 3 
*c*f))*x*Sqrt[a + b*x^2])/(2*e*(b*e - a*f)*(e + f*x^2)) - (3*a^2*(2*b*e*(2 
*d*e - 3*c*f) + a*f*(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))))/(a*(b*e - a*f)))/(3*a*(b*e 
- a*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]
 

Maple [A] (verified)

Time = 1.14 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.16

method	result	size
pseudoelliptic	\(\frac {-a^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (\left (c \,f^{2}+d e f \right ) a -6 b c e f +4 b d \,e^{2}\right ) \left (f \,x^{2}+e \right ) f \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+\left (\left (c \,f^{3}-e \,f^{2} d \right ) a^{4}+2 b f \left (c \,f^{2} x^{2}-3 d e f \,x^{2}-2 d \,e^{2}\right ) a^{3}+6 \left (\left (-\frac {5 x^{2} d}{9}+c \right ) e^{2}+f \,x^{2} \left (-\frac {13 x^{2} d}{18}+c \right ) e +\frac {c \,f^{2} x^{4}}{6}\right ) b^{2} f \,a^{2}-2 \left (e \left (\frac {x^{2} d}{3}+c \right )-\frac {8 c f \,x^{2}}{3}\right ) b^{3} \left (f \,x^{2}+e \right ) e a -\frac {4 b^{4} c \,e^{2} x^{2} \left (f \,x^{2}+e \right )}{3}\right ) \sqrt {\left (a f -b e \right ) e}\, x}{2 \sqrt {\left (a f -b e \right ) e}\, \left (b \,x^{2}+a \right )^{\frac {3}{2}} e \left (f \,x^{2}+e \right ) \left (a f -b e \right )^{3} a^{2}}\)	\(283\)
default	\(\text {Expression too large to display}\)	\(3485\)

Input:

int((d*x^2+c)/(b*x^2+a)^(5/2)/(f*x^2+e)^2,x,method=_RETURNVERBOSE)
 

Output:

1/2*(-a^2*(b*x^2+a)^(3/2)*((c*f^2+d*e*f)*a-6*b*c*e*f+4*b*d*e^2)*(f*x^2+e)* 
f*arctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1/2))+((c*f^3-d*e*f^2)*a^4+2*b 
*f*(c*f^2*x^2-3*d*e*f*x^2-2*d*e^2)*a^3+6*((-5/9*x^2*d+c)*e^2+f*x^2*(-13/18 
*x^2*d+c)*e+1/6*c*f^2*x^4)*b^2*f*a^2-2*(e*(1/3*x^2*d+c)-8/3*c*f*x^2)*b^3*( 
f*x^2+e)*e*a-4/3*b^4*c*e^2*x^2*(f*x^2+e))*((a*f-b*e)*e)^(1/2)*x)/((a*f-b*e 
)*e)^(1/2)/(b*x^2+a)^(3/2)/e/(f*x^2+e)/(a*f-b*e)^3/a^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 937 vs. \(2 (219) = 438\).

Time = 7.25 (sec) , antiderivative size = 1914, normalized size of antiderivative = 7.88 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate((d*x^2+c)/(b*x^2+a)^(5/2)/(f*x^2+e)^2,x, algorithm="fricas")
 

Output:

[-1/24*(3*(4*a^4*b*d*e^3*f + a^5*c*e*f^3 + (4*a^2*b^3*d*e^2*f^2 + a^3*b^2* 
c*f^4 - (6*a^2*b^3*c - a^3*b^2*d)*e*f^3)*x^6 - (6*a^4*b*c - a^5*d)*e^2*f^2 
 + (4*a^2*b^3*d*e^3*f + 2*a^4*b*c*f^4 - 3*(2*a^2*b^3*c - 3*a^3*b^2*d)*e^2* 
f^2 - (11*a^3*b^2*c - 2*a^4*b*d)*e*f^3)*x^4 + (8*a^3*b^2*d*e^3*f + a^5*c*f 
^4 - 6*(2*a^3*b^2*c - a^4*b*d)*e^2*f^2 - (4*a^4*b*c - a^5*d)*e*f^3)*x^2)*s 
qrt(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*((3*a^3*b^2*c*e* 
f^4 + 2*(2*b^5*c + a*b^4*d)*e^4*f - (20*a*b^4*c - 11*a^2*b^3*d)*e^3*f^2 + 
13*(a^2*b^3*c - a^3*b^2*d)*e^2*f^3)*x^5 + 2*(3*a^4*b*c*e*f^4 + (2*b^5*c + 
a*b^4*d)*e^5 - (7*a*b^4*c - 4*a^2*b^3*d)*e^4*f - 4*(a^2*b^3*c - a^3*b^2*d) 
*e^3*f^2 + 3*(2*a^3*b^2*c - 3*a^4*b*d)*e^2*f^3)*x^3 + 3*(2*a*b^4*c*e^5 + a 
^5*c*e*f^4 - 4*(2*a^2*b^3*c - a^3*b^2*d)*e^4*f + 3*(2*a^3*b^2*c - a^4*b*d) 
*e^3*f^2 - (a^4*b*c + a^5*d)*e^2*f^3)*x)*sqrt(b*x^2 + a))/(a^4*b^4*e^7 - 4 
*a^5*b^3*e^6*f + 6*a^6*b^2*e^5*f^2 - 4*a^7*b*e^4*f^3 + a^8*e^3*f^4 + (a^2* 
b^6*e^6*f - 4*a^3*b^5*e^5*f^2 + 6*a^4*b^4*e^4*f^3 - 4*a^5*b^3*e^3*f^4 + a^ 
6*b^2*e^2*f^5)*x^6 + (a^2*b^6*e^7 - 2*a^3*b^5*e^6*f - 2*a^4*b^4*e^5*f^2 + 
8*a^5*b^3*e^4*f^3 - 7*a^6*b^2*e^3*f^4 + 2*a^7*b*e^2*f^5)*x^4 + (2*a^3*b^5* 
e^7 - 7*a^4*b^4*e^6*f + 8*a^5*b^3*e^5*f^2 - 2*a^6*b^2*e^4*f^3 - 2*a^7*b*e^ 
3*f^4 + a^8*e^2*f^5)*x^2), 1/12*(3*(4*a^4*b*d*e^3*f + a^5*c*e*f^3 + (4*...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate((d*x**2+c)/(b*x**2+a)**(5/2)/(f*x**2+e)**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:

integrate((d*x^2+c)/(b*x^2+a)^(5/2)/(f*x^2+e)^2,x, algorithm="maxima")
 

Output:

integrate((d*x^2 + c)/((b*x^2 + a)^(5/2)*(f*x^2 + e)^2), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 839 vs. \(2 (219) = 438\).

Time = 0.42 (sec) , antiderivative size = 839, normalized size of antiderivative = 3.45 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

integrate((d*x^2+c)/(b*x^2+a)^(5/2)/(f*x^2+e)^2,x, algorithm="giac")
 

Output:

1/3*((2*b^8*c*e^4 + a*b^7*d*e^4 - 14*a*b^7*c*e^3*f + 2*a^2*b^6*d*e^3*f + 3 
0*a^2*b^6*c*e^2*f^2 - 12*a^3*b^5*d*e^2*f^2 - 26*a^3*b^5*c*e*f^3 + 14*a^4*b 
^4*d*e*f^3 + 8*a^4*b^4*c*f^4 - 5*a^5*b^3*d*f^4)*x^2/(a^2*b^7*e^6 - 6*a^3*b 
^6*e^5*f + 15*a^4*b^5*e^4*f^2 - 20*a^5*b^4*e^3*f^3 + 15*a^6*b^3*e^2*f^4 - 
6*a^7*b^2*e*f^5 + a^8*b*f^6) + 3*(a*b^7*c*e^4 - 6*a^2*b^6*c*e^3*f + 2*a^3* 
b^5*d*e^3*f + 12*a^3*b^5*c*e^2*f^2 - 6*a^4*b^4*d*e^2*f^2 - 10*a^4*b^4*c*e* 
f^3 + 6*a^5*b^3*d*e*f^3 + 3*a^5*b^3*c*f^4 - 2*a^6*b^2*d*f^4)/(a^2*b^7*e^6 
- 6*a^3*b^6*e^5*f + 15*a^4*b^5*e^4*f^2 - 20*a^5*b^4*e^3*f^3 + 15*a^6*b^3*e 
^2*f^4 - 6*a^7*b^2*e*f^5 + a^8*b*f^6))*x/(b*x^2 + a)^(3/2) + 1/2*(4*b^(3/2 
)*d*e^2*f - 6*b^(3/2)*c*e*f^2 + a*sqrt(b)*d*e*f^2 + a*sqrt(b)*c*f^3)*arcta 
n(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^3*e^4 - 3*a*b^2*e^3*f + 3*a^2*b*e^2*f^2 - a^3*e*f^3)*sqrt(-b^2 
*e^2 + a*b*e*f)) + (2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b^(3/2)*d*e^2*f - 2* 
(sqrt(b)*x - sqrt(b*x^2 + a))^2*b^(3/2)*c*e*f^2 - (sqrt(b)*x - sqrt(b*x^2 
+ a))^2*a*sqrt(b)*d*e*f^2 + (sqrt(b)*x - sqrt(b*x^2 + a))^2*a*sqrt(b)*c*f^ 
3 + a^2*sqrt(b)*d*e*f^2 - a^2*sqrt(b)*c*f^3)/((b^3*e^4 - 3*a*b^2*e^3*f + 3 
*a^2*b*e^2*f^2 - a^3*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. \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx=\int \frac {d\,x^2+c}{{\left (b\,x^2+a\right )}^{5/2}\,{\left (f\,x^2+e\right )}^2} \,d x \] Input:

int((c + d*x^2)/((a + b*x^2)^(5/2)*(e + f*x^2)^2),x)
 

Output:

int((c + d*x^2)/((a + b*x^2)^(5/2)*(e + f*x^2)^2), x)
 

Reduce [B] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 5210, normalized size of antiderivative = 21.44 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int((d*x^2+c)/(b*x^2+a)^(5/2)/(f*x^2+e)^2,x)
 

Output:

( - 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**6*c*e*f**4 - 3*sqrt(e)*sqr 
t(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sq 
rt(b)*x)/(sqrt(e)*sqrt(b)))*a**6*c*f**5*x**2 - 3*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**6*d*e**2*f**3 - 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**6*d*e*f**4*x**2 + 6*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqr 
t(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**5*b*c*e** 
2*f**3 - 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**5*b*c*f**5*x**4 - 24* 
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**5*b*d*e**3*f**2 - 30*sqrt(e)*sqr 
t(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sq 
rt(b)*x)/(sqrt(e)*sqrt(b)))*a**5*b*d*e**2*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**5*b*d*e*f**4*x**4 + 72*sqrt(e)*sqrt(a*f - b*e)*ata 
n((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e 
)*sqrt(b)))*a**4*b**2*c*e**3*f**2 + 84*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)*sqr...