\(\int \frac {c+d x^2}{(a+b x^2)^{3/2} (e+f x^2)^3} \, dx\) [342]

Optimal result

Integrand size = 28, antiderivative size = 280 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^3} \, dx=\frac {b \left (8 b^2 c e^2-2 a b e (7 d e-5 c f)-a^2 f (d e+3 c f)\right ) x}{8 a e^2 (b e-a f)^3 \sqrt {a+b x^2}}+\frac {(d e-c f) x}{4 e (b e-a f) \sqrt {a+b x^2} \left (e+f x^2\right )^2}+\frac {(4 b e (d e-2 c f)+a f (d e+3 c f)) x}{8 e^2 (b e-a f)^2 \sqrt {a+b x^2} \left (e+f x^2\right )}+\frac {\left (8 b^2 e^2 (d e-3 c f)-a^2 f^2 (d e+3 c f)+4 a b e f (2 d e+3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{8 e^{5/2} (b e-a f)^{7/2}} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 18.85 (sec) , antiderivative size = 2140, normalized size of antiderivative = 7.64 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^3} \, dx=\text {Result too large to show} \] Input:

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

Output:

(x*((-12*d*(-(b*e) + a*f)*x^2*(1 + (f*x^2)/e)*(-2625*Sqrt[((b*e - a*f)*x^2 
)/(e*(a + b*x^2))] - (5250*f*x^2*Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))])/ 
e - (2310*f^2*x^4*Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))])/e^2 + 70*(((b*e 
 - a*f)*x^2)/(e*(a + b*x^2)))^(3/2) + (560*f*x^2*(((b*e - a*f)*x^2)/(e*(a 
+ b*x^2)))^(3/2))/e + (280*f^2*x^4*(((b*e - a*f)*x^2)/(e*(a + b*x^2)))^(3/ 
2))/e^2 + 2625*ArcTanh[Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))]] + (5250*f* 
x^2*ArcTanh[Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))]])/e + (2310*f^2*x^4*Ar 
cTanh[Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))]])/e^2 - (945*(b*e - a*f)*x^2 
*ArcTanh[Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))]])/(e*(a + b*x^2)) + (2310 
*f*(-(b*e) + a*f)*x^4*ArcTanh[Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))]])/(e 
^2*(a + b*x^2)) + (1050*f^2*(-(b*e) + a*f)*x^6*ArcTanh[Sqrt[((b*e - a*f)*x 
^2)/(e*(a + b*x^2))]])/(e^3*(a + b*x^2)) + 24*(((b*e - a*f)*x^2)/(e*(a + b 
*x^2)))^(7/2)*HypergeometricPFQ[{2, 2, 5/2}, {1, 9/2}, ((b*e - a*f)*x^2)/( 
e*(a + b*x^2))] + (48*f*x^2*(((b*e - a*f)*x^2)/(e*(a + b*x^2)))^(7/2)*Hype 
rgeometricPFQ[{2, 2, 5/2}, {1, 9/2}, ((b*e - a*f)*x^2)/(e*(a + b*x^2))])/e 
 + (24*f^2*x^4*(((b*e - a*f)*x^2)/(e*(a + b*x^2)))^(7/2)*HypergeometricPFQ 
[{2, 2, 5/2}, {1, 9/2}, ((b*e - a*f)*x^2)/(e*(a + b*x^2))])/e^2))/(a + b*x 
^2) + (-(d*e) + c*f)*(-108045*Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))] - (3 
24135*f*x^2*Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))])/e - (324135*f^2*x^4*S 
qrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))])/e^2 - (103320*f^3*x^6*Sqrt[((b*...
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {402, 25, 402, 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.

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 402

\(\displaystyle \frac {\frac {\frac {\int \frac {a \left (8 b^2 (d e-3 c f) e^2+4 a b f (2 d e+3 c f) e-a^2 f^2 (d e+3 c f)\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} \left (a^2 (-f) (3 c f+d e)-2 a b e (7 d e-5 c f)+8 b^2 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} (a c f-5 a d e+4 b c e)}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {a \left (-a^2 f^2 (3 c f+d e)+4 a b e f (3 c f+2 d e)+8 b^2 e^2 (d e-3 c f)\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} \left (a^2 (-f) (3 c f+d e)-2 a b e (7 d e-5 c f)+8 b^2 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} (a c f-5 a d e+4 b c e)}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\frac {\frac {a \left (-a^2 f^2 (3 c f+d e)+4 a b e f (3 c f+2 d e)+8 b^2 e^2 (d e-3 c f)\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} \left (a^2 (-f) (3 c f+d e)-2 a b e (7 d e-5 c f)+8 b^2 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} (a c f-5 a d e+4 b c e)}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (-a^2 f^2 (3 c f+d e)+4 a b e f (3 c f+2 d e)+8 b^2 e^2 (d e-3 c f)\right )}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {f x \sqrt {a+b x^2} \left (a^2 (-f) (3 c f+d e)-2 a b e (7 d e-5 c f)+8 b^2 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} (a c f-5 a d e+4 b c e)}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}+\frac {x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}\)

Input:

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

Output:

((b*c - a*d)*x)/(a*(b*e - a*f)*Sqrt[a + b*x^2]*(e + f*x^2)^2) + ((f*(4*b*c 
*e - 5*a*d*e + a*c*f)*x*Sqrt[a + b*x^2])/(4*e*(b*e - a*f)*(e + f*x^2)^2) + 
 ((f*(8*b^2*c*e^2 - 2*a*b*e*(7*d*e - 5*c*f) - a^2*f*(d*e + 3*c*f))*x*Sqrt[ 
a + b*x^2])/(2*e*(b*e - a*f)*(e + f*x^2)) + (a*(8*b^2*e^2*(d*e - 3*c*f) - 
a^2*f^2*(d*e + 3*c*f) + 4*a*b*e*f*(2*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)))/(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 = 0.98 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.11

method	result	size
pseudoelliptic	\(\frac {-\frac {3 a \sqrt {b \,x^{2}+a}\, \left (f \,x^{2}+e \right )^{2} \left (-\frac {8 b^{2} d \,e^{3}}{3}-\frac {8 b f \left (a d -3 b c \right ) e^{2}}{3}+\frac {a \,f^{2} \left (a d -12 b c \right ) e}{3}+a^{2} c \,f^{3}\right ) \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )}{8}+\frac {5 \left (\frac {8 b^{2} \left (a d -b c \right ) e^{4}}{5}+\frac {8 b f \left (3 a b d \,x^{2}-2 b^{2} c \,x^{2}+d \,a^{2}\right ) e^{3}}{5}-\frac {\left (a^{3} d +12 \left (-\frac {5 x^{2} d}{12}+c \right ) b \,a^{2}+12 b^{2} x^{2} \left (-\frac {7 x^{2} d}{6}+c \right ) a +8 b^{3} c \,x^{4}\right ) f^{2} e^{2}}{5}+a \left (\left (\frac {x^{2} d}{5}+c \right ) a -2 x^{2} b c \right ) \left (b \,x^{2}+a \right ) f^{3} e +\frac {3 a^{2} c \,f^{4} x^{2} \left (b \,x^{2}+a \right )}{5}\right ) \sqrt {\left (a f -b e \right ) e}\, x}{8}}{\left (f \,x^{2}+e \right )^{2} e^{2} \left (a f -b e \right )^{3} \sqrt {\left (a f -b e \right ) e}\, \sqrt {b \,x^{2}+a}\, a}\)	\(311\)
default	\(\text {Expression too large to display}\)	\(4089\)

Input:

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

Output:

5/8*(-3/5*a*(b*x^2+a)^(1/2)*(f*x^2+e)^2*(-8/3*b^2*d*e^3-8/3*b*f*(a*d-3*b*c 
)*e^2+1/3*a*f^2*(a*d-12*b*c)*e+a^2*c*f^3)*arctan(e*(b*x^2+a)^(1/2)/x/((a*f 
-b*e)*e)^(1/2))+(8/5*b^2*(a*d-b*c)*e^4+8/5*b*f*(3*a*b*d*x^2-2*b^2*c*x^2+a^ 
2*d)*e^3-1/5*(a^3*d+12*(-5/12*x^2*d+c)*b*a^2+12*b^2*x^2*(-7/6*x^2*d+c)*a+8 
*b^3*c*x^4)*f^2*e^2+a*((1/5*x^2*d+c)*a-2*x^2*b*c)*(b*x^2+a)*f^3*e+3/5*a^2* 
c*f^4*x^2*(b*x^2+a))*((a*f-b*e)*e)^(1/2)*x)/((a*f-b*e)*e)^(1/2)/(b*x^2+a)^ 
(1/2)/(f*x^2+e)^2/e^2/(a*f-b*e)^3/a
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 996 vs. \(2 (256) = 512\).

Time = 17.04 (sec) , antiderivative size = 2032, normalized size of antiderivative = 7.26 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/32*((8*a^2*b^2*d*e^5 - 3*a^4*c*e^2*f^3 + (8*a*b^3*d*e^3*f^2 - 3*a^3*b*c 
*f^5 - 8*(3*a*b^3*c - a^2*b^2*d)*e^2*f^3 + (12*a^2*b^2*c - a^3*b*d)*e*f^4) 
*x^6 - 8*(3*a^2*b^2*c - a^3*b*d)*e^4*f + (12*a^3*b*c - a^4*d)*e^3*f^2 + (1 
6*a*b^3*d*e^4*f + 6*a^3*b*d*e^2*f^3 - 3*a^4*c*f^5 - 24*(2*a*b^3*c - a^2*b^ 
2*d)*e^3*f^2 + (6*a^3*b*c - a^4*d)*e*f^4)*x^4 + (8*a*b^3*d*e^5 - 6*a^4*c*e 
*f^4 - 24*(a*b^3*c - a^2*b^2*d)*e^4*f - 3*(12*a^2*b^2*c - 5*a^3*b*d)*e^3*f 
^2 + (21*a^3*b*c - 2*a^4*d)*e^2*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)) + 4*((3*a^3*b*c*e*f^5 + 2*(4*b^4*c - 7*a*b^3*d)*e^4 
*f^2 + (2*a*b^3*c + 13*a^2*b^2*d)*e^3*f^3 - (13*a^2*b^2*c - a^3*b*d)*e^2*f 
^4)*x^5 + (3*a^4*c*e*f^5 + 8*(2*b^4*c - 3*a*b^3*d)*e^5*f - (4*a*b^3*c - 19 
*a^2*b^2*d)*e^4*f^2 - (7*a^2*b^2*c - 4*a^3*b*d)*e^3*f^3 - (8*a^3*b*c - a^4 
*d)*e^2*f^4)*x^3 - (8*a*b^3*c*e^5*f - 5*a^4*c*e^2*f^4 - 8*(b^4*c - a*b^3*d 
)*e^6 - 3*(4*a^2*b^2*c + 3*a^3*b*d)*e^4*f^2 + (17*a^3*b*c + a^4*d)*e^3*f^3 
)*x)*sqrt(b*x^2 + a))/(a^2*b^4*e^9 - 4*a^3*b^3*e^8*f + 6*a^4*b^2*e^7*f^2 - 
 4*a^5*b*e^6*f^3 + a^6*e^5*f^4 + (a*b^5*e^7*f^2 - 4*a^2*b^4*e^6*f^3 + 6*a^ 
3*b^3*e^5*f^4 - 4*a^4*b^2*e^4*f^5 + a^5*b*e^3*f^6)*x^6 + (2*a*b^5*e^8*f - 
7*a^2*b^4*e^7*f^2 + 8*a^3*b^3*e^6*f^3 - 2*a^4*b^2*e^5*f^4 - 2*a^5*b*e^4*f^ 
5 + a^6*e^3*f^6)*x^4 + (a*b^5*e^9 - 2*a^2*b^4*e^8*f - 2*a^3*b^3*e^7*f^2...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1012 vs. \(2 (256) = 512\).

Time = 0.69 (sec) , antiderivative size = 1012, normalized size of antiderivative = 3.61 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(b^3*c - a*b^2*d)*x/((a*b^3*e^3 - 3*a^2*b^2*e^2*f + 3*a^3*b*e*f^2 - a^4*f^ 
3)*sqrt(b*x^2 + a)) - 1/8*(8*b^(5/2)*d*e^3 - 24*b^(5/2)*c*e^2*f + 8*a*b^(3 
/2)*d*e^2*f + 12*a*b^(3/2)*c*e*f^2 - a^2*sqrt(b)*d*e*f^2 - 3*a^2*sqrt(b)*c 
*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^3*e^5 - 3*a*b^2*e^4*f + 3*a^2*b*e^3*f^2 - a^3*e^2*f 
^3)*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*e^3*f - 16*(sqrt(b)*x - sqrt(b*x^2 + a))^6*b^(5/2)*c*e^2*f^2 + 12*( 
sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(3/2)*c*e*f^3 - (sqrt(b)*x - sqrt(b*x^2 
 + a))^6*a^2*sqrt(b)*d*e*f^3 - 3*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*sqrt( 
b)*c*f^4 + 48*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*d*e^4 - 80*(sqrt(b)* 
x - sqrt(b*x^2 + a))^4*b^(7/2)*c*e^3*f - 40*(sqrt(b)*x - sqrt(b*x^2 + a))^ 
4*a*b^(5/2)*d*e^3*f + 104*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a*b^(5/2)*c*e^2* 
f^2 + 10*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^(3/2)*d*e^2*f^2 - 54*(sqrt( 
b)*x - sqrt(b*x^2 + a))^4*a^2*b^(3/2)*c*e*f^3 + 3*(sqrt(b)*x - sqrt(b*x^2 
+ a))^4*a^3*sqrt(b)*d*e*f^3 + 9*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^3*sqrt(b 
)*c*f^4 + 40*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*b^(5/2)*d*e^3*f - 64*(sqr 
t(b)*x - sqrt(b*x^2 + a))^2*a^2*b^(5/2)*c*e^2*f^2 - 16*(sqrt(b)*x - sqrt(b 
*x^2 + a))^2*a^3*b^(3/2)*d*e^2*f^2 + 52*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^ 
3*b^(3/2)*c*e*f^3 - 3*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^4*sqrt(b)*d*e*f^3 
- 9*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^4*sqrt(b)*c*f^4 + 6*a^4*b^(3/2)*d...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 6407, normalized size of antiderivative = 22.88 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 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**5*c*e**2*f**4 - 18*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*c*e*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**5*c*f**6*x**4 - 3*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)))*a**5*d*e**3*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*d*e 
**2*f**4*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**5*d*e*f**5*x** 
4 + 60*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*e**3*f**3 + 111*sqr 
t(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sq 
rt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**4*b*c*e**2*f**4*x**2 + 42*sqrt(e)*s 
qrt(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*e*f**5*x**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*b*c*f**6*x**6 + 32*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...