\(\int \frac {(c+d x^2)^2}{(a+b x^2)^{3/2} (e+f x^2)} \, dx\) [345]

Optimal result

Integrand size = 30, antiderivative size = 134 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {(b c-a d)^2 x}{a b (b e-a f) \sqrt {a+b x^2}}+\frac {d^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2} f}-\frac {(d e-c f)^2 \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f (b e-a f)^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.16 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=-\frac {(b c-a d)^2 x}{a b (-b e+a f) \sqrt {a+b x^2}}-\frac {(d e-c f)^2 \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{\sqrt {e} f (-b e+a f)^{3/2}}-\frac {d^2 \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{b^{3/2} f} \] Input:

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

Output:

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(326\) vs. \(2(134)=268\).

Time = 0.65 (sec) , antiderivative size = 326, normalized size of antiderivative = 2.43, 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, 25, 27, 299, 224, 219, 403, 25, 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.

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

\(\Big \downarrow \) 419

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 403

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 398

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(((b*c - a*d)*(b*e - a*f)*x*(c + d*x^2))/(a*Sqrt[a + b*x^2]) + (-1/2*(d*(2 
*b^2*c*e + 2*a^2*d*f - a*b*(3*d*e + c*f))*x*Sqrt[a + b*x^2])/b + (a*(2*a^2 
*d^2*f + 2*b^2*c*(2*d*e - c*f) - a*b*d*(3*d*e + c*f))*ArcTanh[(Sqrt[b]*x)/ 
Sqrt[a + b*x^2]])/(2*b^(3/2)))/a)/(b*e - a*f)^2 - (f*(d*e - c*f)*((d*x*Sqr 
t[a + b*x^2])/(2*f) - (((2*b*d*e - 2*b*c*f - a*d*f)*ArcTanh[(Sqrt[b]*x)/Sq 
rt[a + b*x^2]])/(Sqrt[b]*f) - (2*Sqrt[b*e - a*f]*(d*e - c*f)*ArcTanh[(Sqrt 
[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(Sqrt[e]*f))/(2*f)))/(b*e - a*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 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 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 403

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + 
 q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1))   Int[(a + b*x^2)^p*(c 
+ d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + 
f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, 
 d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 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 = 0.89 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.24

method	result	size
pseudoelliptic	\(-\frac {a \,b^{\frac {5}{2}} \left (c f -d e \right )^{2} \sqrt {b \,x^{2}+a}\, \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+\sqrt {\left (a f -b e \right ) e}\, \left (-a \,d^{2} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right ) \left (a f -b e \right ) b \sqrt {b \,x^{2}+a}+\left (-a d +b c \right )^{2} b^{\frac {3}{2}} x f \right )}{\sqrt {\left (a f -b e \right ) e}\, \sqrt {b \,x^{2}+a}\, \left (a f -b e \right ) b^{\frac {5}{2}} f a}\)	\(166\)
default	\(\frac {d \left (d f \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )+\frac {2 c f x}{a \sqrt {b \,x^{2}+a}}-\frac {d e x}{a \sqrt {b \,x^{2}+a}}\right )}{f^{2}}-\frac {\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \left (\frac {f}{\left (a f -b e \right ) \sqrt {\left (x +\frac {\sqrt {-e f}}{f}\right )^{2} b -\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}+\frac {2 b \sqrt {-e f}\, \left (2 b \left (x +\frac {\sqrt {-e f}}{f}\right )-\frac {2 b \sqrt {-e f}}{f}\right )}{\left (a f -b e \right ) \left (\frac {4 b \left (a f -b e \right )}{f}+\frac {4 b^{2} e}{f}\right ) \sqrt {\left (x +\frac {\sqrt {-e f}}{f}\right )^{2} b -\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}-\frac {f \ln \left (\frac {\frac {2 a f -2 b e}{f}-\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {\left (x +\frac {\sqrt {-e f}}{f}\right )^{2} b -\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}{x +\frac {\sqrt {-e f}}{f}}\right )}{\left (a f -b e \right ) \sqrt {\frac {a f -b e}{f}}}\right )}{2 f^{2} \sqrt {-e f}}+\frac {\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \left (\frac {f}{\left (a f -b e \right ) \sqrt {\left (x -\frac {\sqrt {-e f}}{f}\right )^{2} b +\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}-\frac {2 b \sqrt {-e f}\, \left (2 b \left (x -\frac {\sqrt {-e f}}{f}\right )+\frac {2 b \sqrt {-e f}}{f}\right )}{\left (a f -b e \right ) \left (\frac {4 b \left (a f -b e \right )}{f}+\frac {4 b^{2} e}{f}\right ) \sqrt {\left (x -\frac {\sqrt {-e f}}{f}\right )^{2} b +\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}-\frac {f \ln \left (\frac {\frac {2 a f -2 b e}{f}+\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {\left (x -\frac {\sqrt {-e f}}{f}\right )^{2} b +\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}{x -\frac {\sqrt {-e f}}{f}}\right )}{\left (a f -b e \right ) \sqrt {\frac {a f -b e}{f}}}\right )}{2 f^{2} \sqrt {-e f}}\)	\(850\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 432 vs. \(2 (116) = 232\).

Time = 2.14 (sec) , antiderivative size = 1822, normalized size of antiderivative = 13.60 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/4*(4*((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*e^2*f - (a*b^3*c^2 - 2*a^2* 
b^2*c*d + a^3*b*d^2)*e*f^2)*sqrt(b*x^2 + a)*x + 2*(a^2*b^2*d^2*e^3 - 2*a^3 
*b*d^2*e^2*f + a^4*d^2*e*f^2 + (a*b^3*d^2*e^3 - 2*a^2*b^2*d^2*e^2*f + a^3* 
b*d^2*e*f^2)*x^2)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) 
- (a^2*b^2*d^2*e^2 - 2*a^2*b^2*c*d*e*f + a^2*b^2*c^2*f^2 + (a*b^3*d^2*e^2 
- 2*a*b^3*c*d*e*f + a*b^3*c^2*f^2)*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)))/(a^2*b^4*e^3*f - 2*a^3*b^3*e^2*f^2 + a^4*b^2*e*f^3 + 
 (a*b^5*e^3*f - 2*a^2*b^4*e^2*f^2 + a^3*b^3*e*f^3)*x^2), 1/4*(4*((b^4*c^2 
- 2*a*b^3*c*d + a^2*b^2*d^2)*e^2*f - (a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^ 
2)*e*f^2)*sqrt(b*x^2 + a)*x - 4*(a^2*b^2*d^2*e^3 - 2*a^3*b*d^2*e^2*f + a^4 
*d^2*e*f^2 + (a*b^3*d^2*e^3 - 2*a^2*b^2*d^2*e^2*f + a^3*b*d^2*e*f^2)*x^2)* 
sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (a^2*b^2*d^2*e^2 - 2*a^2*b^2 
*c*d*e*f + a^2*b^2*c^2*f^2 + (a*b^3*d^2*e^2 - 2*a*b^3*c*d*e*f + a*b^3*c^2* 
f^2)*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)*s 
qrt(b*e^2 - a*e*f)*sqrt(b*x^2 + a))/(f^2*x^4 + 2*e*f*x^2 + e^2)))/(a^2*b^4 
*e^3*f - 2*a^3*b^3*e^2*f^2 + a^4*b^2*e*f^3 + (a*b^5*e^3*f - 2*a^2*b^4*e^2* 
f^2 + a^3*b^3*e*f^3)*x^2), 1/2*(2*((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 1363, normalized size of antiderivative = 10.17 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

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