\(\int \frac {(c+d x^2) (e+f x^2)}{(a+b x^2)^{5/2}} \, dx\) [360]

Optimal result

Integrand size = 26, antiderivative size = 113 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {(b c-a d) (b e-a f) x}{3 a b^2 \left (a+b x^2\right )^{3/2}}+\frac {\left (2 b^2 c e-4 a^2 d f+a b (d e+c f)\right ) x}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {d f \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.93 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {-3 a^3 d f x+2 b^3 c e x^3-4 a^2 b d f x^3+a b^2 x \left (3 c e+d e x^2+c f x^2\right )}{3 a^2 b^2 \left (a+b x^2\right )^{3/2}}-\frac {d f \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{b^{5/2}} \] Input:

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

Output:

(-3*a^3*d*f*x + 2*b^3*c*e*x^3 - 4*a^2*b*d*f*x^3 + a*b^2*x*(3*c*e + d*e*x^2 
 + c*f*x^2))/(3*a^2*b^2*(a + b*x^2)^(3/2)) - (d*f*Log[-(Sqrt[b]*x) + Sqrt[ 
a + b*x^2]])/b^(5/2)
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {401, 25, 298, 224, 219}

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.

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 298

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 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 298

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

Maple [A] (verified)

Time = 0.61 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.90

method	result	size
pseudoelliptic	\(\frac {\frac {a^{2} d f \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{b^{\frac {5}{2}}}+\frac {\left (-a^{2} d f +c e \,b^{2}\right ) x}{b^{2} \sqrt {b \,x^{2}+a}}-\frac {\left (a^{2} d f -a b c f -a b d e +c e \,b^{2}\right ) x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}}{a^{2}}\)	\(102\)
default	\(c e \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )+\left (c f +d e \right ) \left (-\frac {x}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {a \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{2 b}\right )+d f \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )\)	\(158\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 322, normalized size of antiderivative = 2.85 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\left [\frac {3 \, {\left (a^{2} b^{2} d f x^{4} + 2 \, a^{3} b d f x^{2} + a^{4} d f\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left ({\left ({\left (2 \, b^{4} c + a b^{3} d\right )} e + {\left (a b^{3} c - 4 \, a^{2} b^{2} d\right )} f\right )} x^{3} + 3 \, {\left (a b^{3} c e - a^{3} b d f\right )} x\right )} \sqrt {b x^{2} + a}}{6 \, {\left (a^{2} b^{5} x^{4} + 2 \, a^{3} b^{4} x^{2} + a^{4} b^{3}\right )}}, -\frac {3 \, {\left (a^{2} b^{2} d f x^{4} + 2 \, a^{3} b d f x^{2} + a^{4} d f\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left ({\left ({\left (2 \, b^{4} c + a b^{3} d\right )} e + {\left (a b^{3} c - 4 \, a^{2} b^{2} d\right )} f\right )} x^{3} + 3 \, {\left (a b^{3} c e - a^{3} b d f\right )} x\right )} \sqrt {b x^{2} + a}}{3 \, {\left (a^{2} b^{5} x^{4} + 2 \, a^{3} b^{4} x^{2} + a^{4} b^{3}\right )}}\right ] \] Input:

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

Output:

[1/6*(3*(a^2*b^2*d*f*x^4 + 2*a^3*b*d*f*x^2 + a^4*d*f)*sqrt(b)*log(-2*b*x^2 
 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(((2*b^4*c + a*b^3*d)*e + (a*b^3*c 
 - 4*a^2*b^2*d)*f)*x^3 + 3*(a*b^3*c*e - a^3*b*d*f)*x)*sqrt(b*x^2 + a))/(a^ 
2*b^5*x^4 + 2*a^3*b^4*x^2 + a^4*b^3), -1/3*(3*(a^2*b^2*d*f*x^4 + 2*a^3*b*d 
*f*x^2 + a^4*d*f)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (((2*b^4*c 
 + a*b^3*d)*e + (a*b^3*c - 4*a^2*b^2*d)*f)*x^3 + 3*(a*b^3*c*e - a^3*b*d*f) 
*x)*sqrt(b*x^2 + a))/(a^2*b^5*x^4 + 2*a^3*b^4*x^2 + a^4*b^3)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 505 vs. \(2 (110) = 220\).

Time = 7.93 (sec) , antiderivative size = 505, normalized size of antiderivative = 4.47 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=c e \left (\frac {3 a x}{3 a^{\frac {7}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {5}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {2 b x^{3}}{3 a^{\frac {7}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {5}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + \frac {c f x^{3}}{3 a^{\frac {5}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {3}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {d e x^{3}}{3 a^{\frac {5}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {3}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + d f \left (\frac {3 a^{\frac {39}{2}} b^{11} \sqrt {1 + \frac {b x^{2}}{a}} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 a^{\frac {37}{2}} b^{12} x^{2} \sqrt {1 + \frac {b x^{2}}{a}} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {3 a^{19} b^{\frac {23}{2}} x}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {4 a^{18} b^{\frac {25}{2}} x^{3}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}}\right ) \] Input:

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

Output:

c*e*(3*a*x/(3*a**(7/2)*sqrt(1 + b*x**2/a) + 3*a**(5/2)*b*x**2*sqrt(1 + b*x 
**2/a)) + 2*b*x**3/(3*a**(7/2)*sqrt(1 + b*x**2/a) + 3*a**(5/2)*b*x**2*sqrt 
(1 + b*x**2/a))) + c*f*x**3/(3*a**(5/2)*sqrt(1 + b*x**2/a) + 3*a**(3/2)*b* 
x**2*sqrt(1 + b*x**2/a)) + d*e*x**3/(3*a**(5/2)*sqrt(1 + b*x**2/a) + 3*a** 
(3/2)*b*x**2*sqrt(1 + b*x**2/a)) + d*f*(3*a**(39/2)*b**11*sqrt(1 + b*x**2/ 
a)*asinh(sqrt(b)*x/sqrt(a))/(3*a**(39/2)*b**(27/2)*sqrt(1 + b*x**2/a) + 3* 
a**(37/2)*b**(29/2)*x**2*sqrt(1 + b*x**2/a)) + 3*a**(37/2)*b**12*x**2*sqrt 
(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(3*a**(39/2)*b**(27/2)*sqrt(1 + b* 
x**2/a) + 3*a**(37/2)*b**(29/2)*x**2*sqrt(1 + b*x**2/a)) - 3*a**19*b**(23/ 
2)*x/(3*a**(39/2)*b**(27/2)*sqrt(1 + b*x**2/a) + 3*a**(37/2)*b**(29/2)*x** 
2*sqrt(1 + b*x**2/a)) - 4*a**18*b**(25/2)*x**3/(3*a**(39/2)*b**(27/2)*sqrt 
(1 + b*x**2/a) + 3*a**(37/2)*b**(29/2)*x**2*sqrt(1 + b*x**2/a)))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.35 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {1}{3} \, d f x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )} + \frac {2 \, c e x}{3 \, \sqrt {b x^{2} + a} a^{2}} + \frac {c e x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {d f x}{3 \, \sqrt {b x^{2} + a} b^{2}} + \frac {d f \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {5}{2}}} - \frac {{\left (d e + c f\right )} x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {{\left (d e + c f\right )} x}{3 \, \sqrt {b x^{2} + a} a b} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.93 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {x {\left (\frac {{\left (2 \, b^{4} c e + a b^{3} d e + a b^{3} c f - 4 \, a^{2} b^{2} d f\right )} x^{2}}{a^{2} b^{3}} + \frac {3 \, {\left (a b^{3} c e - a^{3} b d f\right )}}{a^{2} b^{3}}\right )}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} - \frac {d f \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{b^{\frac {5}{2}}} \] Input:

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

Output:

1/3*x*((2*b^4*c*e + a*b^3*d*e + a*b^3*c*f - 4*a^2*b^2*d*f)*x^2/(a^2*b^3) + 
 3*(a*b^3*c*e - a^3*b*d*f)/(a^2*b^3))/(b*x^2 + a)^(3/2) - d*f*log(abs(-sqr 
t(b)*x + sqrt(b*x^2 + a)))/b^(5/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 341, normalized size of antiderivative = 3.02 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {-3 \sqrt {b \,x^{2}+a}\, a^{3} b d f x -4 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d f \,x^{3}+3 \sqrt {b \,x^{2}+a}\, a \,b^{3} c e x +\sqrt {b \,x^{2}+a}\, a \,b^{3} c f \,x^{3}+\sqrt {b \,x^{2}+a}\, a \,b^{3} d e \,x^{3}+2 \sqrt {b \,x^{2}+a}\, b^{4} c e \,x^{3}+3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} d f +6 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b d f \,x^{2}+3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b^{2} d f \,x^{4}+\sqrt {b}\, a^{3} b c f +\sqrt {b}\, a^{3} b d e -2 \sqrt {b}\, a^{2} b^{2} c e +2 \sqrt {b}\, a^{2} b^{2} c f \,x^{2}+2 \sqrt {b}\, a^{2} b^{2} d e \,x^{2}-4 \sqrt {b}\, a \,b^{3} c e \,x^{2}+\sqrt {b}\, a \,b^{3} c f \,x^{4}+\sqrt {b}\, a \,b^{3} d e \,x^{4}-2 \sqrt {b}\, b^{4} c e \,x^{4}}{3 a^{2} b^{3} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )} \] Input:

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

Output:

( - 3*sqrt(a + b*x**2)*a**3*b*d*f*x - 4*sqrt(a + b*x**2)*a**2*b**2*d*f*x** 
3 + 3*sqrt(a + b*x**2)*a*b**3*c*e*x + sqrt(a + b*x**2)*a*b**3*c*f*x**3 + s 
qrt(a + b*x**2)*a*b**3*d*e*x**3 + 2*sqrt(a + b*x**2)*b**4*c*e*x**3 + 3*sqr 
t(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**4*d*f + 6*sqrt(b)*log( 
(sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**3*b*d*f*x**2 + 3*sqrt(b)*log((s 
qrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**2*b**2*d*f*x**4 + sqrt(b)*a**3*b* 
c*f + sqrt(b)*a**3*b*d*e - 2*sqrt(b)*a**2*b**2*c*e + 2*sqrt(b)*a**2*b**2*c 
*f*x**2 + 2*sqrt(b)*a**2*b**2*d*e*x**2 - 4*sqrt(b)*a*b**3*c*e*x**2 + sqrt( 
b)*a*b**3*c*f*x**4 + sqrt(b)*a*b**3*d*e*x**4 - 2*sqrt(b)*b**4*c*e*x**4)/(3 
*a**2*b**3*(a**2 + 2*a*b*x**2 + b**2*x**4))