\(\int \sqrt {a+b x^2} (c+d x^2) (e+f x^2) \, dx\) [262]

Optimal result

Integrand size = 26, antiderivative size = 156 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right ) \, dx=\frac {\left (8 b^2 c e+a^2 d f-2 a b (d e+c f)\right ) x \sqrt {a+b x^2}}{16 b^2}-\frac {(a d f-2 b (d e+c f)) x \left (a+b x^2\right )^{3/2}}{8 b^2}+\frac {d f x^3 \left (a+b x^2\right )^{3/2}}{6 b}+\frac {a \left (8 b^2 c e+a^2 d f-2 a b (d e+c f)\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{5/2}} \] Output:

1/16*(8*b^2*c*e+a^2*d*f-2*a*b*(c*f+d*e))*x*(b*x^2+a)^(1/2)/b^2-1/8*(a*d*f- 
2*b*(c*f+d*e))*x*(b*x^2+a)^(3/2)/b^2+1/6*d*f*x^3*(b*x^2+a)^(3/2)/b+1/16*a* 
(8*b^2*c*e+a^2*d*f-2*a*b*(c*f+d*e))*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^( 
5/2)
 

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.87 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right ) \, dx=\frac {\sqrt {b} x \sqrt {a+b x^2} \left (-3 a^2 d f+2 a b \left (3 d e+3 c f+d f x^2\right )+4 b^2 \left (6 c e+3 d e x^2+3 c f x^2+2 d f x^4\right )\right )-3 a \left (8 b^2 c e+a^2 d f-2 a b (d e+c f)\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{48 b^{5/2}} \] Input:

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

Output:

(Sqrt[b]*x*Sqrt[a + b*x^2]*(-3*a^2*d*f + 2*a*b*(3*d*e + 3*c*f + d*f*x^2) + 
 4*b^2*(6*c*e + 3*d*e*x^2 + 3*c*f*x^2 + 2*d*f*x^4)) - 3*a*(8*b^2*c*e + a^2 
*d*f - 2*a*b*(d*e + c*f))*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(48*b^(5/2) 
)
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {403, 299, 211, 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.

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

\(\Big \downarrow \) 403

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {3 \left (a^2 d f-2 a b (c f+d e)+8 b^2 c e\right ) \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )}{4 b}+\frac {x \left (a+b x^2\right )^{3/2} (-3 a d f+2 b c f+6 b d e)}{4 b}}{6 b}+\frac {f x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {3 \left (a^2 d f-2 a b (c f+d e)+8 b^2 c e\right ) \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )}{4 b}+\frac {x \left (a+b x^2\right )^{3/2} (-3 a d f+2 b c f+6 b d e)}{4 b}}{6 b}+\frac {f x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (a^2 d f-2 a b (c f+d e)+8 b^2 c e\right )}{4 b}+\frac {x \left (a+b x^2\right )^{3/2} (-3 a d f+2 b c f+6 b d e)}{4 b}}{6 b}+\frac {f x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\)

Input:

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

Output:

(f*x*(a + b*x^2)^(3/2)*(c + d*x^2))/(6*b) + (((6*b*d*e + 2*b*c*f - 3*a*d*f 
)*x*(a + b*x^2)^(3/2))/(4*b) + (3*(8*b^2*c*e + a^2*d*f - 2*a*b*(d*e + c*f) 
)*((x*Sqrt[a + b*x^2])/2 + (a*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*Sqr 
t[b])))/(4*b))/(6*b)
 

Defintions of rubi rules used

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

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 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 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]
 

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.79

method	result	size
pseudoelliptic	\(\frac {a \left (8 c e \,b^{2}+a^{2} d f -2 a b \left (c f +d e \right )\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )-\left (\left (-\frac {8 d f \,x^{4}}{3}+\left (-4 c f -4 d e \right ) x^{2}-8 c e \right ) b^{\frac {5}{2}}+a \left (\left (-\frac {2}{3} d f \,x^{2}-2 c f -2 d e \right ) b^{\frac {3}{2}}+a d f \sqrt {b}\right )\right ) \sqrt {b \,x^{2}+a}\, x}{16 b^{\frac {5}{2}}}\)	\(123\)
risch	\(-\frac {x \left (-8 b^{2} d f \,x^{4}-2 a b \,x^{2} d f -12 b^{2} c f \,x^{2}-12 b^{2} d e \,x^{2}+3 a^{2} d f -6 a b c f -6 a b d e -24 c e \,b^{2}\right ) \sqrt {b \,x^{2}+a}}{48 b^{2}}+\frac {a \left (a^{2} d f -2 a b c f -2 a b d e +8 c e \,b^{2}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{16 b^{\frac {5}{2}}}\)	\(131\)
default	\(c e \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )+\left (c f +d e \right ) \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 b}-\frac {a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4 b}\right )+d f \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 b}-\frac {a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4 b}\right )}{2 b}\right )\)	\(189\)

Input:

int((b*x^2+a)^(1/2)*(d*x^2+c)*(f*x^2+e),x,method=_RETURNVERBOSE)
 

Output:

1/16*(a*(8*c*e*b^2+a^2*d*f-2*a*b*(c*f+d*e))*arctanh((b*x^2+a)^(1/2)/x/b^(1 
/2))-((-8/3*d*f*x^4+(-4*c*f-4*d*e)*x^2-8*c*e)*b^(5/2)+a*((-2/3*d*f*x^2-2*c 
*f-2*d*e)*b^(3/2)+a*d*f*b^(1/2)))*(b*x^2+a)^(1/2)*x)/b^(5/2)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.01 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right ) \, dx=\left [\frac {3 \, {\left (2 \, {\left (4 \, a b^{2} c - a^{2} b d\right )} e - {\left (2 \, a^{2} b c - a^{3} d\right )} f\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (8 \, b^{3} d f x^{5} + 2 \, {\left (6 \, b^{3} d e + {\left (6 \, b^{3} c + a b^{2} d\right )} f\right )} x^{3} + 3 \, {\left (2 \, {\left (4 \, b^{3} c + a b^{2} d\right )} e + {\left (2 \, a b^{2} c - a^{2} b d\right )} f\right )} x\right )} \sqrt {b x^{2} + a}}{96 \, b^{3}}, -\frac {3 \, {\left (2 \, {\left (4 \, a b^{2} c - a^{2} b d\right )} e - {\left (2 \, a^{2} b c - a^{3} d\right )} f\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (8 \, b^{3} d f x^{5} + 2 \, {\left (6 \, b^{3} d e + {\left (6 \, b^{3} c + a b^{2} d\right )} f\right )} x^{3} + 3 \, {\left (2 \, {\left (4 \, b^{3} c + a b^{2} d\right )} e + {\left (2 \, a b^{2} c - a^{2} b d\right )} f\right )} x\right )} \sqrt {b x^{2} + a}}{48 \, b^{3}}\right ] \] Input:

integrate((b*x^2+a)^(1/2)*(d*x^2+c)*(f*x^2+e),x, algorithm="fricas")
 

Output:

[1/96*(3*(2*(4*a*b^2*c - a^2*b*d)*e - (2*a^2*b*c - a^3*d)*f)*sqrt(b)*log(- 
2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(8*b^3*d*f*x^5 + 2*(6*b^3*d 
*e + (6*b^3*c + a*b^2*d)*f)*x^3 + 3*(2*(4*b^3*c + a*b^2*d)*e + (2*a*b^2*c 
- a^2*b*d)*f)*x)*sqrt(b*x^2 + a))/b^3, -1/48*(3*(2*(4*a*b^2*c - a^2*b*d)*e 
 - (2*a^2*b*c - a^3*d)*f)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (8 
*b^3*d*f*x^5 + 2*(6*b^3*d*e + (6*b^3*c + a*b^2*d)*f)*x^3 + 3*(2*(4*b^3*c + 
 a*b^2*d)*e + (2*a*b^2*c - a^2*b*d)*f)*x)*sqrt(b*x^2 + a))/b^3]
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.33 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right ) \, dx=\begin {cases} \sqrt {a + b x^{2}} \left (\frac {d f x^{5}}{6} + \frac {x^{3} \left (\frac {a d f}{6} + b c f + b d e\right )}{4 b} + \frac {x \left (a c f + a d e - \frac {3 a \left (\frac {a d f}{6} + b c f + b d e\right )}{4 b} + b c e\right )}{2 b}\right ) + \left (a c e - \frac {a \left (a c f + a d e - \frac {3 a \left (\frac {a d f}{6} + b c f + b d e\right )}{4 b} + b c e\right )}{2 b}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: b \neq 0 \\\sqrt {a} \left (c e x + \frac {d f x^{5}}{5} + \frac {x^{3} \left (c f + d e\right )}{3}\right ) & \text {otherwise} \end {cases} \] Input:

integrate((b*x**2+a)**(1/2)*(d*x**2+c)*(f*x**2+e),x)
 

Output:

Piecewise((sqrt(a + b*x**2)*(d*f*x**5/6 + x**3*(a*d*f/6 + b*c*f + b*d*e)/( 
4*b) + x*(a*c*f + a*d*e - 3*a*(a*d*f/6 + b*c*f + b*d*e)/(4*b) + b*c*e)/(2* 
b)) + (a*c*e - a*(a*c*f + a*d*e - 3*a*(a*d*f/6 + b*c*f + b*d*e)/(4*b) + b* 
c*e)/(2*b))*Piecewise((log(2*sqrt(b)*sqrt(a + b*x**2) + 2*b*x)/sqrt(b), Ne 
(a, 0)), (x*log(x)/sqrt(b*x**2), True)), Ne(b, 0)), (sqrt(a)*(c*e*x + d*f* 
x**5/5 + x**3*(c*f + d*e)/3), True))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.13 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right ) \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} d f x^{3}}{6 \, b} + \frac {1}{2} \, \sqrt {b x^{2} + a} c e x - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} a d f x}{8 \, b^{2}} + \frac {\sqrt {b x^{2} + a} a^{2} d f x}{16 \, b^{2}} + \frac {a c e \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {b}} + \frac {a^{3} d f \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {5}{2}}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d e + c f\right )} x}{4 \, b} - \frac {\sqrt {b x^{2} + a} {\left (d e + c f\right )} a x}{8 \, b} - \frac {{\left (d e + c f\right )} a^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {3}{2}}} \] Input:

integrate((b*x^2+a)^(1/2)*(d*x^2+c)*(f*x^2+e),x, algorithm="maxima")
 

Output:

1/6*(b*x^2 + a)^(3/2)*d*f*x^3/b + 1/2*sqrt(b*x^2 + a)*c*e*x - 1/8*(b*x^2 + 
 a)^(3/2)*a*d*f*x/b^2 + 1/16*sqrt(b*x^2 + a)*a^2*d*f*x/b^2 + 1/2*a*c*e*arc 
sinh(b*x/sqrt(a*b))/sqrt(b) + 1/16*a^3*d*f*arcsinh(b*x/sqrt(a*b))/b^(5/2) 
+ 1/4*(b*x^2 + a)^(3/2)*(d*e + c*f)*x/b - 1/8*sqrt(b*x^2 + a)*(d*e + c*f)* 
a*x/b - 1/8*(d*e + c*f)*a^2*arcsinh(b*x/sqrt(a*b))/b^(3/2)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.94 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right ) \, dx=\frac {1}{48} \, {\left (2 \, {\left (4 \, d f x^{2} + \frac {6 \, b^{4} d e + 6 \, b^{4} c f + a b^{3} d f}{b^{4}}\right )} x^{2} + \frac {3 \, {\left (8 \, b^{4} c e + 2 \, a b^{3} d e + 2 \, a b^{3} c f - a^{2} b^{2} d f\right )}}{b^{4}}\right )} \sqrt {b x^{2} + a} x - \frac {{\left (8 \, a b^{2} c e - 2 \, a^{2} b d e - 2 \, a^{2} b c f + a^{3} d f\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{16 \, b^{\frac {5}{2}}} \] Input:

integrate((b*x^2+a)^(1/2)*(d*x^2+c)*(f*x^2+e),x, algorithm="giac")
 

Output:

1/48*(2*(4*d*f*x^2 + (6*b^4*d*e + 6*b^4*c*f + a*b^3*d*f)/b^4)*x^2 + 3*(8*b 
^4*c*e + 2*a*b^3*d*e + 2*a*b^3*c*f - a^2*b^2*d*f)/b^4)*sqrt(b*x^2 + a)*x - 
 1/16*(8*a*b^2*c*e - 2*a^2*b*d*e - 2*a^2*b*c*f + a^3*d*f)*log(abs(-sqrt(b) 
*x + sqrt(b*x^2 + a)))/b^(5/2)
 

Mupad [F(-1)]

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

int((a + b*x^2)^(1/2)*(c + d*x^2)*(e + f*x^2),x)
 

Output:

int((a + b*x^2)^(1/2)*(c + d*x^2)*(e + f*x^2), x)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.67 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right ) \, dx=\frac {-3 \sqrt {b \,x^{2}+a}\, a^{2} b d f x +6 \sqrt {b \,x^{2}+a}\, a \,b^{2} c f x +6 \sqrt {b \,x^{2}+a}\, a \,b^{2} d e x +2 \sqrt {b \,x^{2}+a}\, a \,b^{2} d f \,x^{3}+24 \sqrt {b \,x^{2}+a}\, b^{3} c e x +12 \sqrt {b \,x^{2}+a}\, b^{3} c f \,x^{3}+12 \sqrt {b \,x^{2}+a}\, b^{3} d e \,x^{3}+8 \sqrt {b \,x^{2}+a}\, b^{3} d f \,x^{5}+3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} d f -6 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b c f -6 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b d e +24 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} c e}{48 b^{3}} \] Input:

int((b*x^2+a)^(1/2)*(d*x^2+c)*(f*x^2+e),x)
 

Output:

( - 3*sqrt(a + b*x**2)*a**2*b*d*f*x + 6*sqrt(a + b*x**2)*a*b**2*c*f*x + 6* 
sqrt(a + b*x**2)*a*b**2*d*e*x + 2*sqrt(a + b*x**2)*a*b**2*d*f*x**3 + 24*sq 
rt(a + b*x**2)*b**3*c*e*x + 12*sqrt(a + b*x**2)*b**3*c*f*x**3 + 12*sqrt(a 
+ b*x**2)*b**3*d*e*x**3 + 8*sqrt(a + b*x**2)*b**3*d*f*x**5 + 3*sqrt(b)*log 
((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**3*d*f - 6*sqrt(b)*log((sqrt(a 
+ b*x**2) + sqrt(b)*x)/sqrt(a))*a**2*b*c*f - 6*sqrt(b)*log((sqrt(a + b*x** 
2) + sqrt(b)*x)/sqrt(a))*a**2*b*d*e + 24*sqrt(b)*log((sqrt(a + b*x**2) + s 
qrt(b)*x)/sqrt(a))*a*b**2*c*e)/(48*b**3)