\(\int \frac {\sqrt {a+b x^2} (c+d x^2)}{(e+f x^2)^4} \, dx\) [266]

Optimal result

Integrand size = 28, antiderivative size = 284 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )}{\left (e+f x^2\right )^4} \, dx=-\frac {(d e-c f) x \sqrt {a+b x^2}}{6 e f \left (e+f x^2\right )^3}+\frac {(2 b e (d e+2 c f)-a f (d e+5 c f)) x \sqrt {a+b x^2}}{24 e^2 f (b e-a f) \left (e+f x^2\right )^2}+\frac {\left (4 b^2 e^2 (d e+2 c f)+3 a^2 f^2 (d e+5 c f)-2 a b e f (2 d e+13 c f)\right ) x \sqrt {a+b x^2}}{48 e^3 f (b e-a f)^2 \left (e+f x^2\right )}+\frac {a \left (8 b^2 c e^2+a^2 f (d e+5 c f)-2 a b e (d e+6 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{16 e^{7/2} (b e-a f)^{5/2}} \] Output:

-1/6*(-c*f+d*e)*x*(b*x^2+a)^(1/2)/e/f/(f*x^2+e)^3+1/24*(2*b*e*(2*c*f+d*e)- 
a*f*(5*c*f+d*e))*x*(b*x^2+a)^(1/2)/e^2/f/(-a*f+b*e)/(f*x^2+e)^2+1/48*(4*b^ 
2*e^2*(2*c*f+d*e)+3*a^2*f^2*(5*c*f+d*e)-2*a*b*e*f*(13*c*f+2*d*e))*x*(b*x^2 
+a)^(1/2)/e^3/f/(-a*f+b*e)^2/(f*x^2+e)+1/16*a*(8*b^2*c*e^2+a^2*f*(5*c*f+d* 
e)-2*a*b*e*(6*c*f+d*e))*arctanh((-a*f+b*e)^(1/2)*x/e^(1/2)/(b*x^2+a)^(1/2) 
)/e^(7/2)/(-a*f+b*e)^(5/2)
 

Mathematica [A] (verified)

Time = 11.55 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.33 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )}{\left (e+f x^2\right )^4} \, dx=\frac {x \sqrt {a+b x^2} \left (6 d (b e-a f)^2 \left (e+f x^2\right ) \left (e \left (2 b e \left (2 e+f x^2\right )-a f \left (5 e+3 f x^2\right )\right )+\frac {a (4 b e-3 a f) \left (e+f x^2\right )^2 \text {arctanh}\left (\sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}\right )}{\sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}} \left (a+b x^2\right )}\right )-(d e-c f) \left ((b e-a f) \left (8 b^2 e^2 \left (3 e^2+3 e f x^2+f^2 x^4\right )-2 a b e f \left (30 e^2+35 e f x^2+13 f^2 x^4\right )+a^2 f^2 \left (33 e^2+40 e f x^2+15 f^2 x^4\right )\right )+\frac {3 a \left (8 b^2 e^2-12 a b e f+5 a^2 f^2\right ) \sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}} \left (e+f x^2\right )^3 \text {arctanh}\left (\sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}\right )}{x^2}\right )\right )}{48 e^3 f (b e-a f)^3 \left (e+f x^2\right )^3} \] Input:

Integrate[(Sqrt[a + b*x^2]*(c + d*x^2))/(e + f*x^2)^4,x]
 

Output:

(x*Sqrt[a + b*x^2]*(6*d*(b*e - a*f)^2*(e + f*x^2)*(e*(2*b*e*(2*e + f*x^2) 
- a*f*(5*e + 3*f*x^2)) + (a*(4*b*e - 3*a*f)*(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))] 
*(a + b*x^2))) - (d*e - c*f)*((b*e - a*f)*(8*b^2*e^2*(3*e^2 + 3*e*f*x^2 + 
f^2*x^4) - 2*a*b*e*f*(30*e^2 + 35*e*f*x^2 + 13*f^2*x^4) + a^2*f^2*(33*e^2 
+ 40*e*f*x^2 + 15*f^2*x^4)) + (3*a*(8*b^2*e^2 - 12*a*b*e*f + 5*a^2*f^2)*Sq 
rt[((b*e - a*f)*x^2)/(e*(a + b*x^2))]*(e + f*x^2)^3*ArcTanh[Sqrt[((b*e - a 
*f)*x^2)/(e*(a + b*x^2))]])/x^2)))/(48*e^3*f*(b*e - a*f)^3*(e + f*x^2)^3)
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {401, 25, 402, 25, 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.

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

\(\Big \downarrow \) 401

\(\displaystyle -\frac {\int -\frac {2 b (d e+2 c f) x^2+a (d e+5 c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{6 e f}-\frac {x \sqrt {a+b x^2} (d e-c f)}{6 e f \left (e+f x^2\right )^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {2 b (d e+2 c f) x^2+a (d e+5 c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{6 e f}-\frac {x \sqrt {a+b x^2} (d e-c f)}{6 e f \left (e+f x^2\right )^3}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\frac {\int -\frac {a (3 a f (d e+5 c f)-2 b e (d e+8 c f))-2 b (2 b e (d e+2 c f)-a f (d e+5 c f)) x^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (2 b e (2 c f+d e)-a f (5 c f+d e))}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{6 e f}-\frac {x \sqrt {a+b x^2} (d e-c f)}{6 e f \left (e+f x^2\right )^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {x \sqrt {a+b x^2} (2 b e (2 c f+d e)-a f (5 c f+d e))}{4 e \left (e+f x^2\right )^2 (b e-a f)}-\frac {\int \frac {a (3 a f (d e+5 c f)-2 b e (d e+8 c f))-2 b (2 b e (d e+2 c f)-a f (d e+5 c f)) x^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}}{6 e f}-\frac {x \sqrt {a+b x^2} (d e-c f)}{6 e f \left (e+f x^2\right )^3}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\frac {x \sqrt {a+b x^2} (2 b e (2 c f+d e)-a f (5 c f+d e))}{4 e \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {\int -\frac {3 a f \left (f (d e+5 c f) a^2-2 b e (d e+6 c f) a+8 b^2 c e^2\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {x \sqrt {a+b x^2} \left (3 a^2 f^2 (5 c f+d e)-2 a b e f (13 c f+2 d e)+4 b^2 e^2 (2 c f+d e)\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}}{6 e f}-\frac {x \sqrt {a+b x^2} (d e-c f)}{6 e f \left (e+f x^2\right )^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {x \sqrt {a+b x^2} (2 b e (2 c f+d e)-a f (5 c f+d e))}{4 e \left (e+f x^2\right )^2 (b e-a f)}-\frac {-\frac {3 a f \left (a^2 f (5 c f+d e)-2 a b e (6 c f+d e)+8 b^2 c e^2\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {x \sqrt {a+b x^2} \left (3 a^2 f^2 (5 c f+d e)-2 a b e f (13 c f+2 d e)+4 b^2 e^2 (2 c f+d e)\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}}{6 e f}-\frac {x \sqrt {a+b x^2} (d e-c f)}{6 e f \left (e+f x^2\right )^3}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\frac {x \sqrt {a+b x^2} (2 b e (2 c f+d e)-a f (5 c f+d e))}{4 e \left (e+f x^2\right )^2 (b e-a f)}-\frac {-\frac {3 a f \left (a^2 f (5 c f+d e)-2 a b e (6 c f+d e)+8 b^2 c e^2\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}-\frac {x \sqrt {a+b x^2} \left (3 a^2 f^2 (5 c f+d e)-2 a b e f (13 c f+2 d e)+4 b^2 e^2 (2 c f+d e)\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}}{6 e f}-\frac {x \sqrt {a+b x^2} (d e-c f)}{6 e f \left (e+f x^2\right )^3}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {x \sqrt {a+b x^2} (2 b e (2 c f+d e)-a f (5 c f+d e))}{4 e \left (e+f x^2\right )^2 (b e-a f)}-\frac {-\frac {3 a f \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (a^2 f (5 c f+d e)-2 a b e (6 c f+d e)+8 b^2 c e^2\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} \left (3 a^2 f^2 (5 c f+d e)-2 a b e f (13 c f+2 d e)+4 b^2 e^2 (2 c f+d e)\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}}{6 e f}-\frac {x \sqrt {a+b x^2} (d e-c f)}{6 e f \left (e+f x^2\right )^3}\)

Input:

Int[(Sqrt[a + b*x^2]*(c + d*x^2))/(e + f*x^2)^4,x]
 

Output:

-1/6*((d*e - c*f)*x*Sqrt[a + b*x^2])/(e*f*(e + f*x^2)^3) + (((2*b*e*(d*e + 
 2*c*f) - a*f*(d*e + 5*c*f))*x*Sqrt[a + b*x^2])/(4*e*(b*e - a*f)*(e + f*x^ 
2)^2) - (-1/2*((4*b^2*e^2*(d*e + 2*c*f) + 3*a^2*f^2*(d*e + 5*c*f) - 2*a*b* 
e*f*(2*d*e + 13*c*f))*x*Sqrt[a + b*x^2])/(e*(b*e - a*f)*(e + f*x^2)) - (3* 
a*f*(8*b^2*c*e^2 + a^2*f*(d*e + 5*c*f) - 2*a*b*e*(d*e + 6*c*f))*ArcTanh[(S 
qrt[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)))/(6*e*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 401

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/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1))   Int[(a + b*x^2)^(p + 1)*( 
c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + 
(b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L 
tQ[p, -1] && GtQ[q, 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 = 0.95 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.99

method	result	size
pseudoelliptic	\(-\frac {5 \left (a \left (-\frac {2 b \left (a d -4 b c \right ) e^{2}}{5}+\frac {a f \left (a d -12 b c \right ) e}{5}+a^{2} c \,f^{2}\right ) \left (f \,x^{2}+e \right )^{3} \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )-\frac {11 \left (\frac {2 \left (a d +4 \left (\frac {x^{2} d}{2}+c \right ) b \right ) b \,e^{4}}{11}-\frac {\left (d \,a^{2}+20 \left (\frac {7 x^{2} d}{30}+c \right ) b a -8 \left (\frac {x^{2} d}{6}+c \right ) b^{2} x^{2}\right ) f \,e^{3}}{11}+f^{2} \left (\left (\frac {8 x^{2} d}{33}+c \right ) a^{2}-\frac {70 \left (\frac {2 x^{2} d}{35}+c \right ) b \,x^{2} a}{33}+\frac {8 b^{2} c \,x^{4}}{33}\right ) e^{2}+\frac {40 a \left (\left (\frac {3 x^{2} d}{40}+c \right ) a -\frac {13 x^{2} b c}{20}\right ) x^{2} f^{3} e}{33}+\frac {5 a^{2} c \,f^{4} x^{4}}{11}\right ) \sqrt {\left (a f -b e \right ) e}\, \sqrt {b \,x^{2}+a}\, x}{5}\right )}{16 \sqrt {\left (a f -b e \right ) e}\, \left (f \,x^{2}+e \right )^{3} \left (a f -b e \right )^{2} e^{3}}\)	\(282\)
default	\(\text {Expression too large to display}\)	\(6531\)

Input:

int((b*x^2+a)^(1/2)*(d*x^2+c)/(f*x^2+e)^4,x,method=_RETURNVERBOSE)
 

Output:

-5/16/((a*f-b*e)*e)^(1/2)*(a*(-2/5*b*(a*d-4*b*c)*e^2+1/5*a*f*(a*d-12*b*c)* 
e+a^2*c*f^2)*(f*x^2+e)^3*arctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1/2))-1 
1/5*(2/11*(a*d+4*(1/2*x^2*d+c)*b)*b*e^4-1/11*(d*a^2+20*(7/30*x^2*d+c)*b*a- 
8*(1/6*x^2*d+c)*b^2*x^2)*f*e^3+f^2*((8/33*x^2*d+c)*a^2-70/33*(2/35*x^2*d+c 
)*b*x^2*a+8/33*b^2*c*x^4)*e^2+40/33*a*((3/40*x^2*d+c)*a-13/20*x^2*b*c)*x^2 
*f^3*e+5/11*a^2*c*f^4*x^4)*((a*f-b*e)*e)^(1/2)*(b*x^2+a)^(1/2)*x)/(f*x^2+e 
)^3/(a*f-b*e)^2/e^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 789 vs. \(2 (260) = 520\).

Time = 7.00 (sec) , antiderivative size = 1618, normalized size of antiderivative = 5.70 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )}{\left (e+f x^2\right )^4} \, dx=\text {Too large to display} \] Input:

integrate((b*x^2+a)^(1/2)*(d*x^2+c)/(f*x^2+e)^4,x, algorithm="fricas")
 

Output:

[1/192*(3*(5*a^3*c*e^3*f^2 + (5*a^3*c*f^5 + 2*(4*a*b^2*c - a^2*b*d)*e^2*f^ 
3 - (12*a^2*b*c - a^3*d)*e*f^4)*x^6 + 2*(4*a*b^2*c - a^2*b*d)*e^5 - (12*a^ 
2*b*c - a^3*d)*e^4*f + 3*(5*a^3*c*e*f^4 + 2*(4*a*b^2*c - a^2*b*d)*e^3*f^2 
- (12*a^2*b*c - a^3*d)*e^2*f^3)*x^4 + 3*(5*a^3*c*e^2*f^3 + 2*(4*a*b^2*c - 
a^2*b*d)*e^4*f - (12*a^2*b*c - a^3*d)*e^3*f^2)*x^2)*sqrt(b*e^2 - a*e*f)*lo 
g(((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*((4*b^3*d*e^5*f - 15*a^3*c*e*f^5 + 8* 
(b^3*c - a*b^2*d)*e^4*f^2 - (34*a*b^2*c - 7*a^2*b*d)*e^3*f^3 + (41*a^2*b*c 
 - 3*a^3*d)*e^2*f^4)*x^5 + 2*(6*b^3*d*e^6 - 20*a^3*c*e^2*f^4 + (12*b^3*c - 
 13*a*b^2*d)*e^5*f - (47*a*b^2*c - 11*a^2*b*d)*e^4*f^2 + (55*a^2*b*c - 4*a 
^3*d)*e^3*f^3)*x^3 - 3*(11*a^3*c*e^3*f^3 - 2*(4*b^3*c + a*b^2*d)*e^6 + (28 
*a*b^2*c + 3*a^2*b*d)*e^5*f - (31*a^2*b*c + a^3*d)*e^4*f^2)*x)*sqrt(b*x^2 
+ a))/(b^3*e^10 - 3*a*b^2*e^9*f + 3*a^2*b*e^8*f^2 - a^3*e^7*f^3 + (b^3*e^7 
*f^3 - 3*a*b^2*e^6*f^4 + 3*a^2*b*e^5*f^5 - a^3*e^4*f^6)*x^6 + 3*(b^3*e^8*f 
^2 - 3*a*b^2*e^7*f^3 + 3*a^2*b*e^6*f^4 - a^3*e^5*f^5)*x^4 + 3*(b^3*e^9*f - 
 3*a*b^2*e^8*f^2 + 3*a^2*b*e^7*f^3 - a^3*e^6*f^4)*x^2), -1/96*(3*(5*a^3*c* 
e^3*f^2 + (5*a^3*c*f^5 + 2*(4*a*b^2*c - a^2*b*d)*e^2*f^3 - (12*a^2*b*c - a 
^3*d)*e*f^4)*x^6 + 2*(4*a*b^2*c - a^2*b*d)*e^5 - (12*a^2*b*c - a^3*d)*e^4* 
f + 3*(5*a^3*c*e*f^4 + 2*(4*a*b^2*c - a^2*b*d)*e^3*f^2 - (12*a^2*b*c - ...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )}{\left (e+f x^2\right )^4} \, dx=\text {Timed out} \] Input:

integrate((b*x**2+a)**(1/2)*(d*x**2+c)/(f*x**2+e)**4,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )}{\left (e+f x^2\right )^4} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}}{{\left (f x^{2} + e\right )}^{4}} \,d x } \] Input:

integrate((b*x^2+a)^(1/2)*(d*x^2+c)/(f*x^2+e)^4,x, algorithm="maxima")
 

Output:

integrate(sqrt(b*x^2 + a)*(d*x^2 + c)/(f*x^2 + e)^4, x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1777 vs. \(2 (260) = 520\).

Time = 0.71 (sec) , antiderivative size = 1777, normalized size of antiderivative = 6.26 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )}{\left (e+f x^2\right )^4} \, dx=\text {Too large to display} \] Input:

integrate((b*x^2+a)^(1/2)*(d*x^2+c)/(f*x^2+e)^4,x, algorithm="giac")
 

Output:

-1/16*(8*a*b^(5/2)*c*e^2 - 2*a^2*b^(3/2)*d*e^2 - 12*a^2*b^(3/2)*c*e*f + a^ 
3*sqrt(b)*d*e*f + 5*a^3*sqrt(b)*c*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^5 - 2*a*b*e^4* 
f + a^2*e^3*f^2)*sqrt(-b^2*e^2 + a*b*e*f)) - 1/24*(24*(sqrt(b)*x - sqrt(b* 
x^2 + a))^10*a*b^(5/2)*c*e^2*f^4 - 6*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^2* 
b^(3/2)*d*e^2*f^4 - 36*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^2*b^(3/2)*c*e*f^ 
5 + 3*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^3*sqrt(b)*d*e*f^5 + 15*(sqrt(b)*x 
 - sqrt(b*x^2 + a))^10*a^3*sqrt(b)*c*f^6 - 96*(sqrt(b)*x - sqrt(b*x^2 + a) 
)^8*b^(9/2)*d*e^5*f + 192*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a*b^(7/2)*d*e^4* 
f^2 + 240*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a*b^(7/2)*c*e^3*f^3 - 156*(sqrt( 
b)*x - sqrt(b*x^2 + a))^8*a^2*b^(5/2)*d*e^3*f^3 - 480*(sqrt(b)*x - sqrt(b* 
x^2 + a))^8*a^2*b^(5/2)*c*e^2*f^4 + 60*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^3 
*b^(3/2)*d*e^2*f^4 + 330*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^3*b^(3/2)*c*e*f 
^5 - 15*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^4*sqrt(b)*d*e*f^5 - 75*(sqrt(b)* 
x - sqrt(b*x^2 + a))^8*a^4*sqrt(b)*c*f^6 - 128*(sqrt(b)*x - sqrt(b*x^2 + a 
))^6*b^(11/2)*d*e^6 - 256*(sqrt(b)*x - sqrt(b*x^2 + a))^6*b^(11/2)*c*e^5*f 
 + 320*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(9/2)*d*e^5*f + 1216*(sqrt(b)*x 
 - sqrt(b*x^2 + a))^6*a*b^(9/2)*c*e^4*f^2 - 432*(sqrt(b)*x - sqrt(b*x^2 + 
a))^6*a^2*b^(7/2)*d*e^4*f^2 - 2016*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*b^( 
7/2)*c*e^3*f^3 + 328*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^3*b^(5/2)*d*e^3*...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )}{\left (e+f x^2\right )^4} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\left (d\,x^2+c\right )}{{\left (f\,x^2+e\right )}^4} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.70 (sec) , antiderivative size = 5233, normalized size of antiderivative = 18.43 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )}{\left (e+f x^2\right )^4} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 15*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*c*e**3*f**5 - 45*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*c*e**2*f**6*x**2 - 45*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*c*e*f**7*x**4 - 15*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**4*c*f**8*x**6 - 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*d*e**4*f**4 - 9*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*d*e**3*f* 
*5*x**2 - 9*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*d*e**2*f**6*x**4 - 
 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*d*e*f**7*x**6 + 66*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*c*e**4*f**4 + 198*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*c*e**3*f**5*x**2 + 198*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)/...