\(\int \frac {(a+b x^2)^{3/2}}{(c+d x^2) (e+f x^2)^3} \, dx\) [310]

Optimal result

Integrand size = 30, antiderivative size = 286 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=\frac {(b e-a f) x \sqrt {a+b x^2}}{4 e (d e-c f) \left (e+f x^2\right )^2}-\frac {(a f (7 d e-3 c f)-2 b e (d e+c f)) x \sqrt {a+b x^2}}{8 e^2 (d e-c f)^2 \left (e+f x^2\right )}+\frac {d (b c-a d)^{3/2} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} (d e-c f)^3}-\frac {\left (8 b^2 c d e^3-4 a b d e^2 (3 d e+c f)+a^2 f \left (15 d^2 e^2-10 c d e f+3 c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{8 e^{5/2} \sqrt {b e-a f} (d e-c f)^3} \] Output:

1/4*(-a*f+b*e)*x*(b*x^2+a)^(1/2)/e/(-c*f+d*e)/(f*x^2+e)^2-1/8*(a*f*(-3*c*f 
+7*d*e)-2*b*e*(c*f+d*e))*x*(b*x^2+a)^(1/2)/e^2/(-c*f+d*e)^2/(f*x^2+e)+d*(- 
a*d+b*c)^(3/2)*arctanh((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a)^(1/2))/c^(1/2) 
/(-c*f+d*e)^3-1/8*(8*b^2*c*d*e^3-4*a*b*d*e^2*(c*f+3*d*e)+a^2*f*(3*c^2*f^2- 
10*c*d*e*f+15*d^2*e^2))*arctanh((-a*f+b*e)^(1/2)*x/e^(1/2)/(b*x^2+a)^(1/2) 
)/e^(5/2)/(-a*f+b*e)^(1/2)/(-c*f+d*e)^3
 

Mathematica [A] (verified)

Time = 10.58 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=-\frac {x \sqrt {a+b x^2} \left (-2 b e \left (c f^2 x^2+d e \left (2 e+f x^2\right )\right )+a f \left (-c f \left (5 e+3 f x^2\right )+d e \left (9 e+7 f x^2\right )\right )\right )}{8 e^2 (d e-c f)^2 \left (e+f x^2\right )^2}-\frac {d (-b c+a d)^{3/2} \arctan \left (\frac {\sqrt {-b c+a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} (-d e+c f)^3}-\frac {\left (8 b^2 c d e^3-4 a b d e^2 (3 d e+c f)+a^2 f \left (15 d^2 e^2-10 c d e f+3 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {-b e+a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{8 e^{5/2} \sqrt {-b e+a f} (d e-c f)^3} \] Input:

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

Output:

-1/8*(x*Sqrt[a + b*x^2]*(-2*b*e*(c*f^2*x^2 + d*e*(2*e + f*x^2)) + a*f*(-(c 
*f*(5*e + 3*f*x^2)) + d*e*(9*e + 7*f*x^2))))/(e^2*(d*e - c*f)^2*(e + f*x^2 
)^2) - (d*(-(b*c) + a*d)^(3/2)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt 
[a + b*x^2])])/(Sqrt[c]*(-(d*e) + c*f)^3) - ((8*b^2*c*d*e^3 - 4*a*b*d*e^2* 
(3*d*e + c*f) + a^2*f*(15*d^2*e^2 - 10*c*d*e*f + 3*c^2*f^2))*ArcTan[(Sqrt[ 
-(b*e) + a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(8*e^(5/2)*Sqrt[-(b*e) + a*f] 
*(d*e - c*f)^3)
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.41, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {419, 25, 401, 27, 402, 25, 27, 291, 221, 422, 301, 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.

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

\(\Big \downarrow \) 419

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 422

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

\(\Big \downarrow \) 301

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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]
 

rule 422

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

Maple [A] (verified)

Time = 1.67 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.08

method	result	size
pseudoelliptic	\(-\frac {3 \left (\left (\left (-4 a b \,d^{2}+\frac {8}{3} b^{2} c d \right ) e^{3}+5 a d \left (a d -\frac {4 b c}{15}\right ) f \,e^{2}-\frac {10 a^{2} c d e \,f^{2}}{3}+a^{2} c^{2} f^{3}\right ) \sqrt {\left (a d -b c \right ) c}\, \left (f \,x^{2}+e \right )^{2} \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )-\frac {8 \sqrt {\left (a f -b e \right ) e}\, \left (d \,e^{2} \left (f \,x^{2}+e \right )^{2} \left (a d -b c \right )^{2} \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )+\frac {5 \left (c f -d e \right ) \sqrt {\left (a d -b c \right ) c}\, \sqrt {b \,x^{2}+a}\, x \left (\frac {4 b d \,e^{3}}{5}-\frac {9 \left (-\frac {2 b \,x^{2}}{9}+a \right ) d f \,e^{2}}{5}+f^{2} \left (-\frac {7 a d \,x^{2}}{5}+c \left (\frac {2 b \,x^{2}}{5}+a \right )\right ) e +\frac {3 a c \,f^{3} x^{2}}{5}\right )}{8}\right )}{3}\right )}{8 \sqrt {\left (a d -b c \right ) c}\, \sqrt {\left (a f -b e \right ) e}\, \left (c f -d e \right )^{3} \left (f \,x^{2}+e \right )^{2} e^{2}}\)	\(310\)
default	\(\text {Expression too large to display}\)	\(8550\)

Input:

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

Output:

-3/8/((a*d-b*c)*c)^(1/2)/((a*f-b*e)*e)^(1/2)*(((-4*a*b*d^2+8/3*b^2*c*d)*e^ 
3+5*a*d*(a*d-4/15*b*c)*f*e^2-10/3*a^2*c*d*e*f^2+a^2*c^2*f^3)*((a*d-b*c)*c) 
^(1/2)*(f*x^2+e)^2*arctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1/2))-8/3*((a 
*f-b*e)*e)^(1/2)*(d*e^2*(f*x^2+e)^2*(a*d-b*c)^2*arctan(c*(b*x^2+a)^(1/2)/x 
/((a*d-b*c)*c)^(1/2))+5/8*(c*f-d*e)*((a*d-b*c)*c)^(1/2)*(b*x^2+a)^(1/2)*x* 
(4/5*b*d*e^3-9/5*(-2/9*b*x^2+a)*d*f*e^2+f^2*(-7/5*a*d*x^2+c*(2/5*b*x^2+a)) 
*e+3/5*a*c*f^3*x^2)))/(c*f-d*e)^3/(f*x^2+e)^2/e^2
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1054 vs. \(2 (257) = 514\).

Time = 1.92 (sec) , antiderivative size = 1054, normalized size of antiderivative = 3.69 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-(b^(5/2)*c^2*d - 2*a*b^(3/2)*c*d^2 + a^2*sqrt(b)*d^3)*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))/((d^3* 
e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 - c^3*f^3)*sqrt(-b^2*c^2 + a*b*c*d)) + 
 1/8*(8*b^(5/2)*c*d*e^3 - 12*a*b^(3/2)*d^2*e^3 - 4*a*b^(3/2)*c*d*e^2*f + 1 
5*a^2*sqrt(b)*d^2*e^2*f - 10*a^2*sqrt(b)*c*d*e*f^2 + 3*a^2*sqrt(b)*c^2*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))/((d^3*e^5 - 3*c*d^2*e^4*f + 3*c^2*d*e^3*f^2 - c^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) 
*c*e^2*f^2 - 12*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(3/2)*d*e^2*f^2 + 7*(s 
qrt(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 + 16*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2) 
*d*e^4 + 16*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*c*e^3*f - 72*(sqrt(b)* 
x - sqrt(b*x^2 + a))^4*a*b^(5/2)*d*e^3*f + 8*(sqrt(b)*x - sqrt(b*x^2 + a)) 
^4*a*b^(5/2)*c*e^2*f^2 + 62*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^(3/2)*d* 
e^2*f^2 - 18*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^(3/2)*c*e*f^3 - 21*(sqr 
t(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 + 16*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*b^(5/ 
2)*d*e^3*f + 8*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*b^(5/2)*c*e^2*f^2 - 52* 
(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^3*b^(3/2)*d*e^2*f^2 + 16*(sqrt(b)*x - sq 
rt(b*x^2 + a))^2*a^3*b^(3/2)*c*e*f^3 + 21*(sqrt(b)*x - sqrt(b*x^2 + a))...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 5.57 (sec) , antiderivative size = 7621, normalized size of antiderivative = 26.65 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(16*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**3*d**2*e**5*f**3 + 32*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**3*d**2*e**4*f**4*x**2 + 16*sqrt(c)*sqr 
t(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sq 
rt(b)*x)/(sqrt(c)*sqrt(b)))*a**3*d**2*e**3*f**5*x**4 - 16*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**2*b*c*d*e**5*f**3 - 32*sqrt(c)*sqrt(a*d - b*c)*a 
tan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(sqrt 
(c)*sqrt(b)))*a**2*b*c*d*e**4*f**4*x**2 - 16*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**2*b*c*d*e**3*f**5*x**4 - 48*sqrt(c)*sqrt(a*d - b*c)*atan((sqr 
t(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt 
(b)))*a**2*b*d**2*e**6*f**2 - 96*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* 
*2*b*d**2*e**5*f**3*x**2 - 48*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**2* 
b*d**2*e**4*f**4*x**4 + 48*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*b**2*c 
*d*e**6*f**2 + 96*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(...