\(\int \frac {c+d x^2}{(a+b x^2)^{5/2} (e+f x^2)^3} \, dx\) [363]

Optimal result

Integrand size = 28, antiderivative size = 375 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\frac {b \left (8 b^2 c e^2-4 a b e (8 d e-9 c f)-3 a^2 f (d e+3 c f)\right ) x}{24 a e^2 (b e-a f)^3 \left (a+b x^2\right )^{3/2}}+\frac {b \left (16 b^3 c e^3+2 a^2 b e f (47 d e-21 c f)+8 a b^2 e^2 (d e-11 c f)+3 a^3 f^2 (d e+3 c f)\right ) x}{24 a^2 e^2 (b e-a f)^4 \sqrt {a+b x^2}}+\frac {(d e-c f) x}{4 e (b e-a f) \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}+\frac {(2 b e (3 d e-5 c f)+a f (d e+3 c f)) x}{8 e^2 (b e-a f)^2 \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )}-\frac {f \left (24 b^2 e^2 (d e-2 c f)-a^2 f^2 (d e+3 c f)+4 a b e f (3 d e+4 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{8 e^{5/2} (b e-a f)^{9/2}} \] Output:

1/24*b*(8*b^2*c*e^2-4*a*b*e*(-9*c*f+8*d*e)-3*a^2*f*(3*c*f+d*e))*x/a/e^2/(- 
a*f+b*e)^3/(b*x^2+a)^(3/2)+1/24*b*(16*b^3*c*e^3+2*a^2*b*e*f*(-21*c*f+47*d* 
e)+8*a*b^2*e^2*(-11*c*f+d*e)+3*a^3*f^2*(3*c*f+d*e))*x/a^2/e^2/(-a*f+b*e)^4 
/(b*x^2+a)^(1/2)+1/4*(-c*f+d*e)*x/e/(-a*f+b*e)/(b*x^2+a)^(3/2)/(f*x^2+e)^2 
+1/8*(2*b*e*(-5*c*f+3*d*e)+a*f*(3*c*f+d*e))*x/e^2/(-a*f+b*e)^2/(b*x^2+a)^( 
3/2)/(f*x^2+e)-1/8*f*(24*b^2*e^2*(-2*c*f+d*e)-a^2*f^2*(3*c*f+d*e)+4*a*b*e* 
f*(4*c*f+3*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)^(9/2)
 

Mathematica [A] (verified)

Time = 15.98 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.78 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\frac {1}{24} \left (x \sqrt {a+b x^2} \left (\frac {8 b^2 (-b c+a d)}{a (-b e+a f)^3 \left (a+b x^2\right )^2}+\frac {8 b^2 \left (2 b^2 c e+8 a^2 d f+a b (d e-11 c f)\right )}{a^2 (b e-a f)^4 \left (a+b x^2\right )}+\frac {6 f^2 (d e-c f)}{e (b e-a f)^3 \left (e+f x^2\right )^2}+\frac {3 f^2 (2 b e (5 d e-7 c f)+a f (d e+3 c f))}{e^2 (b e-a f)^4 \left (e+f x^2\right )}\right )+\frac {3 f \left (-24 b^2 e^2 (d e-2 c f)+a^2 f^2 (d e+3 c f)-4 a b e f (3 d e+4 c f)\right ) \arctan \left (\frac {\sqrt {-b e+a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{e^{5/2} (-b e+a f)^{9/2}}\right ) \] Input:

Integrate[(c + d*x^2)/((a + b*x^2)^(5/2)*(e + f*x^2)^3),x]
 

Output:

(x*Sqrt[a + b*x^2]*((8*b^2*(-(b*c) + a*d))/(a*(-(b*e) + a*f)^3*(a + b*x^2) 
^2) + (8*b^2*(2*b^2*c*e + 8*a^2*d*f + a*b*(d*e - 11*c*f)))/(a^2*(b*e - a*f 
)^4*(a + b*x^2)) + (6*f^2*(d*e - c*f))/(e*(b*e - a*f)^3*(e + f*x^2)^2) + ( 
3*f^2*(2*b*e*(5*d*e - 7*c*f) + a*f*(d*e + 3*c*f)))/(e^2*(b*e - a*f)^4*(e + 
 f*x^2))) + (3*f*(-24*b^2*e^2*(d*e - 2*c*f) + a^2*f^2*(d*e + 3*c*f) - 4*a* 
b*e*f*(3*d*e + 4*c*f))*ArcTan[(Sqrt[-(b*e) + a*f]*x)/(Sqrt[e]*Sqrt[a + b*x 
^2])])/(e^(5/2)*(-(b*e) + a*f)^(9/2)))/24
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.16, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {402, 25, 402, 25, 27, 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.

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 402

\(\displaystyle \frac {\frac {x \left (6 a^2 d f+a b (d e-9 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\int -\frac {f \left (4 \left (6 d f a^2+b (d e-9 c f) a+2 b^2 c e\right ) x^2+a (4 b c e-7 a d e+3 a c f)\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{a (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {f \left (4 \left (6 d f a^2+b (d e-9 c f) a+2 b^2 c e\right ) x^2+a (4 b c e-7 a d e+3 a c f)\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{a (b e-a f)}+\frac {x \left (6 a^2 d f+a b (d e-9 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {f \int \frac {4 \left (6 d f a^2+b (d e-9 c f) a+2 b^2 c e\right ) x^2+a (4 b c e-7 a d e+3 a c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{a (b e-a f)}+\frac {x \left (6 a^2 d f+a b (d e-9 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\frac {f \left (\frac {\int \frac {2 b \left (f (31 d e-3 c f) a^2+4 b e (d e-10 c f) a+8 b^2 c e^2\right ) x^2+a \left (-3 f (d e+3 c f) a^2-4 b e (8 d e-9 c f) a+8 b^2 c e^2\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (a^2 f (31 d e-3 c f)+4 a b e (d e-10 c f)+8 b^2 c e^2\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{a (b e-a f)}+\frac {x \left (6 a^2 d f+a b (d e-9 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\frac {f \left (\frac {\frac {\int -\frac {3 a^2 \left (24 b^2 (d e-2 c f) e^2+4 a b f (3 d e+4 c f) e-a^2 f^2 (d e+3 c f)\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (3 a^3 f^2 (3 c f+d e)+2 a^2 b e f (47 d e-21 c f)+8 a b^2 e^2 (d e-11 c f)+16 b^3 c e^3\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (a^2 f (31 d e-3 c f)+4 a b e (d e-10 c f)+8 b^2 c e^2\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{a (b e-a f)}+\frac {x \left (6 a^2 d f+a b (d e-9 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {f \left (\frac {\frac {x \sqrt {a+b x^2} \left (3 a^3 f^2 (3 c f+d e)+2 a^2 b e f (47 d e-21 c f)+8 a b^2 e^2 (d e-11 c f)+16 b^3 c e^3\right )}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {3 a^2 \left (-a^2 f^2 (3 c f+d e)+4 a b e f (4 c f+3 d e)+24 b^2 e^2 (d e-2 c f)\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (a^2 f (31 d e-3 c f)+4 a b e (d e-10 c f)+8 b^2 c e^2\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{a (b e-a f)}+\frac {x \left (6 a^2 d f+a b (d e-9 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\frac {f \left (\frac {\frac {x \sqrt {a+b x^2} \left (3 a^3 f^2 (3 c f+d e)+2 a^2 b e f (47 d e-21 c f)+8 a b^2 e^2 (d e-11 c f)+16 b^3 c e^3\right )}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {3 a^2 \left (-a^2 f^2 (3 c f+d e)+4 a b e f (4 c f+3 d e)+24 b^2 e^2 (d e-2 c f)\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (a^2 f (31 d e-3 c f)+4 a b e (d e-10 c f)+8 b^2 c e^2\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{a (b e-a f)}+\frac {x \left (6 a^2 d f+a b (d e-9 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {x \left (6 a^2 d f+a b (d e-9 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}+\frac {f \left (\frac {x \sqrt {a+b x^2} \left (a^2 f (31 d e-3 c f)+4 a b e (d e-10 c f)+8 b^2 c e^2\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}+\frac {\frac {x \sqrt {a+b x^2} \left (3 a^3 f^2 (3 c f+d e)+2 a^2 b e f (47 d e-21 c f)+8 a b^2 e^2 (d e-11 c f)+16 b^3 c e^3\right )}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {3 a^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (-a^2 f^2 (3 c f+d e)+4 a b e f (4 c f+3 d e)+24 b^2 e^2 (d e-2 c f)\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\)

Input:

Int[(c + d*x^2)/((a + b*x^2)^(5/2)*(e + f*x^2)^3),x]
 

Output:

((b*c - a*d)*x)/(3*a*(b*e - a*f)*(a + b*x^2)^(3/2)*(e + f*x^2)^2) + (((2*b 
^2*c*e + 6*a^2*d*f + a*b*(d*e - 9*c*f))*x)/(a*(b*e - a*f)*Sqrt[a + b*x^2]* 
(e + f*x^2)^2) + (f*(((8*b^2*c*e^2 + 4*a*b*e*(d*e - 10*c*f) + a^2*f*(31*d* 
e - 3*c*f))*x*Sqrt[a + b*x^2])/(4*e*(b*e - a*f)*(e + f*x^2)^2) + (((16*b^3 
*c*e^3 + 2*a^2*b*e*f*(47*d*e - 21*c*f) + 8*a*b^2*e^2*(d*e - 11*c*f) + 3*a^ 
3*f^2*(d*e + 3*c*f))*x*Sqrt[a + b*x^2])/(2*e*(b*e - a*f)*(e + f*x^2)) - (3 
*a^2*(24*b^2*e^2*(d*e - 2*c*f) - a^2*f^2*(d*e + 3*c*f) + 4*a*b*e*f*(3*d*e 
+ 4*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)))/(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 = 1.05 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.19

method	result	size
pseudoelliptic	\(\frac {-\frac {3 a^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (f^{2} \left (c f +\frac {d e}{3}\right ) a^{2}-\frac {16 b f e \left (c f +\frac {3 d e}{4}\right ) a}{3}+16 b^{2} c \,e^{2} f -8 b^{2} d \,e^{3}\right ) \left (f \,x^{2}+e \right )^{2} f \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )}{8}+\frac {5 \left (\left (\frac {3 c \,f^{2} x^{2}}{5}+e \left (\frac {x^{2} d}{5}+c \right ) f -\frac {d \,e^{2}}{5}\right ) f^{3} a^{5}-\frac {16 \left (-\frac {3 c \,f^{3} x^{4}}{8}+\frac {\left (-\frac {x^{2} d}{2}+c \right ) x^{2} e \,f^{2}}{4}+e^{2} \left (-\frac {x^{2} d}{2}+c \right ) f -\frac {3 d \,e^{3}}{4}\right ) b \,f^{2} a^{4}}{5}-\frac {32 \left (-\frac {3 c \,f^{4} x^{6}}{32}+\frac {23 x^{4} e \left (-\frac {x^{2} d}{23}+c \right ) f^{3}}{32}+e^{2} x^{2} \left (-\frac {43 x^{2} d}{32}+c \right ) f^{2}-\frac {9 d \,e^{3} f \,x^{2}}{4}-\frac {3 d \,e^{4}}{4}\right ) b^{2} f \,a^{3}}{5}-\frac {32 b^{3} \left (\frac {7 c \,f^{3} x^{6}}{16}+\frac {3 \left (-\frac {47 x^{2} d}{72}+c \right ) x^{4} e \,f^{2}}{2}+2 \left (-\frac {41 x^{2} d}{48}+c \right ) x^{2} e^{2} f +e^{3} \left (-\frac {2 x^{2} d}{3}+c \right )\right ) f e \,a^{2}}{5}+\frac {8 \left (-\frac {11 c f \,x^{2}}{3}+e \left (\frac {x^{2} d}{3}+c \right )\right ) b^{4} \left (f \,x^{2}+e \right )^{2} e^{2} a}{5}+\frac {16 b^{5} c \,e^{3} x^{2} \left (f \,x^{2}+e \right )^{2}}{15}\right ) \sqrt {\left (a f -b e \right ) e}\, x}{8}}{\left (f \,x^{2}+e \right )^{2} e^{2} \left (a f -b e \right )^{4} \sqrt {\left (a f -b e \right ) e}\, \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2}}\)	\(447\)
default	\(\text {Expression too large to display}\)	\(7175\)

Input:

int((d*x^2+c)/(b*x^2+a)^(5/2)/(f*x^2+e)^3,x,method=_RETURNVERBOSE)
 

Output:

5/8/((a*f-b*e)*e)^(1/2)/(b*x^2+a)^(3/2)*(-3/5*a^2*(b*x^2+a)^(3/2)*(f^2*(c* 
f+1/3*d*e)*a^2-16/3*b*f*e*(c*f+3/4*d*e)*a+16*b^2*c*e^2*f-8*b^2*d*e^3)*(f*x 
^2+e)^2*f*arctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1/2))+((3/5*c*f^2*x^2+ 
e*(1/5*x^2*d+c)*f-1/5*d*e^2)*f^3*a^5-16/5*(-3/8*c*f^3*x^4+1/4*(-1/2*x^2*d+ 
c)*x^2*e*f^2+e^2*(-1/2*x^2*d+c)*f-3/4*d*e^3)*b*f^2*a^4-32/5*(-3/32*c*f^4*x 
^6+23/32*x^4*e*(-1/23*x^2*d+c)*f^3+e^2*x^2*(-43/32*x^2*d+c)*f^2-9/4*d*e^3* 
f*x^2-3/4*d*e^4)*b^2*f*a^3-32/5*b^3*(7/16*c*f^3*x^6+3/2*(-47/72*x^2*d+c)*x 
^4*e*f^2+2*(-41/48*x^2*d+c)*x^2*e^2*f+e^3*(-2/3*x^2*d+c))*f*e*a^2+8/5*(-11 
/3*c*f*x^2+e*(1/3*x^2*d+c))*b^4*(f*x^2+e)^2*e^2*a+16/15*b^5*c*e^3*x^2*(f*x 
^2+e)^2)*((a*f-b*e)*e)^(1/2)*x)/(f*x^2+e)^2/e^2/(a*f-b*e)^4/a^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1564 vs. \(2 (347) = 694\).

Time = 31.98 (sec) , antiderivative size = 3168, normalized size of antiderivative = 8.45 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((d*x^2+c)/(b*x^2+a)^(5/2)/(f*x^2+e)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate((d*x**2+c)/(b*x**2+a)**(5/2)/(f*x**2+e)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}^{3}} \,d x } \] Input:

integrate((d*x^2+c)/(b*x^2+a)^(5/2)/(f*x^2+e)^3,x, algorithm="maxima")
 

Output:

integrate((d*x^2 + c)/((b*x^2 + a)^(5/2)*(f*x^2 + e)^3), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1567 vs. \(2 (347) = 694\).

Time = 0.44 (sec) , antiderivative size = 1567, normalized size of antiderivative = 4.18 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((d*x^2+c)/(b*x^2+a)^(5/2)/(f*x^2+e)^3,x, algorithm="giac")
 

Output:

1/3*((2*b^10*c*e^5 + a*b^9*d*e^5 - 19*a*b^9*c*e^4*f + 4*a^2*b^8*d*e^4*f + 
56*a^2*b^8*c*e^3*f^2 - 26*a^3*b^7*d*e^3*f^2 - 74*a^3*b^7*c*e^2*f^3 + 44*a^ 
4*b^6*d*e^2*f^3 + 46*a^4*b^6*c*e*f^4 - 31*a^5*b^5*d*e*f^4 - 11*a^5*b^5*c*f 
^5 + 8*a^6*b^4*d*f^5)*x^2/(a^2*b^9*e^8 - 8*a^3*b^8*e^7*f + 28*a^4*b^7*e^6* 
f^2 - 56*a^5*b^6*e^5*f^3 + 70*a^6*b^5*e^4*f^4 - 56*a^7*b^4*e^3*f^5 + 28*a^ 
8*b^3*e^2*f^6 - 8*a^9*b^2*e*f^7 + a^10*b*f^8) + 3*(a*b^9*c*e^5 - 8*a^2*b^8 
*c*e^4*f + 3*a^3*b^7*d*e^4*f + 22*a^3*b^7*c*e^3*f^2 - 12*a^4*b^6*d*e^3*f^2 
 - 28*a^4*b^6*c*e^2*f^3 + 18*a^5*b^5*d*e^2*f^3 + 17*a^5*b^5*c*e*f^4 - 12*a 
^6*b^4*d*e*f^4 - 4*a^6*b^4*c*f^5 + 3*a^7*b^3*d*f^5)/(a^2*b^9*e^8 - 8*a^3*b 
^8*e^7*f + 28*a^4*b^7*e^6*f^2 - 56*a^5*b^6*e^5*f^3 + 70*a^6*b^5*e^4*f^4 - 
56*a^7*b^4*e^3*f^5 + 28*a^8*b^3*e^2*f^6 - 8*a^9*b^2*e*f^7 + a^10*b*f^8))*x 
/(b*x^2 + a)^(3/2) + 1/8*(24*b^(5/2)*d*e^3*f - 48*b^(5/2)*c*e^2*f^2 + 12*a 
*b^(3/2)*d*e^2*f^2 + 16*a*b^(3/2)*c*e*f^3 - a^2*sqrt(b)*d*e*f^3 - 3*a^2*sq 
rt(b)*c*f^4)*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^4*e^6 - 4*a*b^3*e^5*f + 6*a^2*b^2*e^4*f^2 - 
4*a^3*b*e^3*f^3 + a^4*e^2*f^4)*sqrt(-b^2*e^2 + a*b*e*f)) + 1/4*(16*(sqrt(b 
)*x - sqrt(b*x^2 + a))^6*b^(5/2)*d*e^3*f^2 - 24*(sqrt(b)*x - sqrt(b*x^2 + 
a))^6*b^(5/2)*c*e^2*f^3 - 4*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(3/2)*d*e^ 
2*f^3 + 16*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(3/2)*c*e*f^4 - (sqrt(b)*x 
- sqrt(b*x^2 + a))^6*a^2*sqrt(b)*d*e*f^4 - 3*(sqrt(b)*x - sqrt(b*x^2 + ...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\int \frac {d\,x^2+c}{{\left (b\,x^2+a\right )}^{5/2}\,{\left (f\,x^2+e\right )}^3} \,d x \] Input:

int((c + d*x^2)/((a + b*x^2)^(5/2)*(e + f*x^2)^3),x)
 

Output:

int((c + d*x^2)/((a + b*x^2)^(5/2)*(e + f*x^2)^3), x)
 

Reduce [B] (verification not implemented)

Time = 1.21 (sec) , antiderivative size = 9199, normalized size of antiderivative = 24.53 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int((d*x^2+c)/(b*x^2+a)^(5/2)/(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**7*c*e**2*f**5 - 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**7*c*e*f**6*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**7*c*f**7*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**7*d*e**3*f**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**7*d*e 
**2*f**5*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**7*d*e*f**6*x** 
4 + 120*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*b*c*e**3*f**4 + 222*sq 
rt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - s 
qrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**6*b*c*e**2*f**5*x**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)*sqrt(b)))*a**6*b*c*e*f**6*x**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**6*b*c*f**7*x**6 + 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...