\(\int \frac {(a+b x^2)^{3/2}}{(c+d x^2) (e+f x^2)^2} \, dx\) [309]

Optimal result

Integrand size = 30, antiderivative size = 183 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx=\frac {(b e-a f) x \sqrt {a+b x^2}}{2 e (d e-c f) \left (e+f x^2\right )}+\frac {(b c-a d)^{3/2} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} (d e-c f)^2}-\frac {\sqrt {b e-a f} (2 b c e-3 a d e+a c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{2 e^{3/2} (d e-c f)^2} \] Output:

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

Mathematica [A] (verified)

Time = 1.17 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx=\frac {\frac {(b e-a f) (d e-c f) x \sqrt {a+b x^2}}{e \left (e+f x^2\right )}-\frac {2 (-b c+a d)^{3/2} \arctan \left (\frac {-d x \sqrt {a+b x^2}+\sqrt {b} \left (c+d x^2\right )}{\sqrt {c} \sqrt {-b c+a d}}\right )}{\sqrt {c}}-\frac {\sqrt {-b e+a f} (2 b c e-3 a d e+a c f) \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{e^{3/2}}}{2 (d e-c f)^2} \] Input:

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

Output:

(((b*e - a*f)*(d*e - c*f)*x*Sqrt[a + b*x^2])/(e*(e + f*x^2)) - (2*(-(b*c) 
+ a*d)^(3/2)*ArcTan[(-(d*x*Sqrt[a + b*x^2]) + Sqrt[b]*(c + d*x^2))/(Sqrt[c 
]*Sqrt[-(b*c) + a*d])])/Sqrt[c] - (Sqrt[-(b*e) + a*f]*(2*b*c*e - 3*a*d*e + 
 a*c*f)*ArcTan[(-(f*x*Sqrt[a + b*x^2]) + Sqrt[b]*(e + f*x^2))/(Sqrt[e]*Sqr 
t[-(b*e) + a*f])])/e^(3/2))/(2*(d*e - c*f)^2)
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.46, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {419, 25, 301, 224, 219, 291, 221, 401, 27, 398, 224, 219, 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.

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

\(\Big \downarrow \) 419

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 301

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 398

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

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 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

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 301

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[b/ 
d   Int[(a + b*x^2)^(p - 1), x], x] - Simp[(b*c - a*d)/d   Int[(a + b*x^2)^ 
(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] 
&& GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4] || (EqQ[p, 2/3] && E 
qQ[b*c + 3*a*d, 0]))
 

rule 398

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) 
, x_Symbol] :> Simp[f/b   Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ 
b   Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} 
, x]
 

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 419

Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( 
x_)^2), x_Symbol] :> Simp[b*((b*e - a*f)/(b*c - a*d)^2)   Int[(c + d*x^2)^( 
q + 2)*((e + f*x^2)^(r - 1)/(a + b*x^2)), x], x] - Simp[1/(b*c - a*d)^2   I 
nt[(c + d*x^2)^q*(e + f*x^2)^(r - 1)*(2*b*c*d*e - a*d^2*e - b*c^2*f + d^2*( 
b*e - a*f)*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[q, -1] && Gt 
Q[r, 1]
 

Maple [A] (verified)

Time = 1.46 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.21

method	result	size
pseudoelliptic	\(\frac {-\left (\left (a f +2 b e \right ) c -3 a d e \right ) \sqrt {\left (a d -b c \right ) c}\, \left (a f -b e \right ) \left (f \,x^{2}+e \right ) \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+\sqrt {\left (a f -b e \right ) e}\, \left (-2 e \left (a d -b c \right )^{2} \left (f \,x^{2}+e \right ) \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )+\sqrt {\left (a d -b c \right ) c}\, \left (a f -b e \right ) \left (c f -d e \right ) \sqrt {b \,x^{2}+a}\, x \right )}{2 \sqrt {\left (a d -b c \right ) c}\, \sqrt {\left (a f -b e \right ) e}\, \left (c f -d e \right )^{2} e \left (f \,x^{2}+e \right )}\)	\(222\)
default	\(\text {Expression too large to display}\)	\(4846\)

Input:

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

Output:

1/2*(-((a*f+2*b*e)*c-3*a*d*e)*((a*d-b*c)*c)^(1/2)*(a*f-b*e)*(f*x^2+e)*arct 
an(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1/2))+((a*f-b*e)*e)^(1/2)*(-2*e*(a*d 
-b*c)^2*(f*x^2+e)*arctan(c*(b*x^2+a)^(1/2)/x/((a*d-b*c)*c)^(1/2))+((a*d-b* 
c)*c)^(1/2)*(a*f-b*e)*(c*f-d*e)*(b*x^2+a)^(1/2)*x))/((a*d-b*c)*c)^(1/2)/(( 
a*f-b*e)*e)^(1/2)/(c*f-d*e)^2/e/(f*x^2+e)
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 458 vs. \(2 (157) = 314\).

Time = 0.87 (sec) , antiderivative size = 458, normalized size of antiderivative = 2.50 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx=-\frac {{\left (b^{\frac {5}{2}} c^{2} - 2 \, a b^{\frac {3}{2}} c d + a^{2} \sqrt {b} d^{2}\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt {-b^{2} c^{2} + a b c d}}\right )}{\sqrt {-b^{2} c^{2} + a b c d} {\left (d^{2} e^{2} - 2 \, c d e f + c^{2} f^{2}\right )}} + \frac {{\left (2 \, b^{\frac {5}{2}} c e^{2} - 3 \, a b^{\frac {3}{2}} d e^{2} - a b^{\frac {3}{2}} c e f + 3 \, a^{2} \sqrt {b} d e f - a^{2} \sqrt {b} c f^{2}\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} f + 2 \, b e - a f}{2 \, \sqrt {-b^{2} e^{2} + a b e f}}\right )}{2 \, {\left (d^{2} e^{3} - 2 \, c d e^{2} f + c^{2} e f^{2}\right )} \sqrt {-b^{2} e^{2} + a b e f}} + \frac {2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{\frac {5}{2}} e^{2} - 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a b^{\frac {3}{2}} e f + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} \sqrt {b} f^{2} + a^{2} b^{\frac {3}{2}} e f - a^{3} \sqrt {b} f^{2}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} f + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b e - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a f + a^{2} f\right )} {\left (d e^{2} f - c e f^{2}\right )}} \] Input:

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

Output:

-(b^(5/2)*c^2 - 2*a*b^(3/2)*c*d + a^2*sqrt(b)*d^2)*arctan(1/2*((sqrt(b)*x 
- sqrt(b*x^2 + a))^2*d + 2*b*c - a*d)/sqrt(-b^2*c^2 + a*b*c*d))/(sqrt(-b^2 
*c^2 + a*b*c*d)*(d^2*e^2 - 2*c*d*e*f + c^2*f^2)) + 1/2*(2*b^(5/2)*c*e^2 - 
3*a*b^(3/2)*d*e^2 - a*b^(3/2)*c*e*f + 3*a^2*sqrt(b)*d*e*f - a^2*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))/((d^2*e^3 - 2*c*d*e^2*f + c^2*e*f^2)*sqrt(-b^2*e^2 + a*b 
*e*f)) + (2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b^(5/2)*e^2 - 3*(sqrt(b)*x - s 
qrt(b*x^2 + a))^2*a*b^(3/2)*e*f + (sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*sqrt 
(b)*f^2 + a^2*b^(3/2)*e*f - a^3*sqrt(b)*f^2)/(((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)*(d*e^2*f - c*e*f^2))
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 1.29 (sec) , antiderivative size = 1306, normalized size of antiderivative = 7.14 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 2*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x 
**2) - sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*a*d*e**3 - 2*sqrt(c)*sqrt(a*d 
 - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b) 
*x)/(sqrt(c)*sqrt(b)))*a*d*e**2*f*x**2 + 2*sqrt(c)*sqrt(a*d - b*c)*atan((s 
qrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(sqrt(c)*sq 
rt(b)))*b*c*e**3 + 2*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt( 
d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*b*c*e**2*f*x** 
2 - 2*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x 
**2) + sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*a*d*e**3 - 2*sqrt(c)*sqrt(a*d 
 - b*c)*atan((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x**2) + sqrt(d)*sqrt(b) 
*x)/(sqrt(c)*sqrt(b)))*a*d*e**2*f*x**2 + 2*sqrt(c)*sqrt(a*d - b*c)*atan((s 
qrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x**2) + sqrt(d)*sqrt(b)*x)/(sqrt(c)*sq 
rt(b)))*b*c*e**3 + 2*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) + sqrt( 
d)*sqrt(a + b*x**2) + sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*b*c*e**2*f*x** 
2 - 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*c**2*e*f - 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*c**2*f**2*x**2 + 3*sqrt(e)*sqrt(a*f - b*e)*atan((sq 
rt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqr 
t(b)))*a*c*d*e**2 + 3*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - s...