\(\int \frac {x^3 (c+d x)^2}{(a+b x^2)^3} \, dx\) [232]

Optimal result

Integrand size = 20, antiderivative size = 107 \[ \int \frac {x^3 (c+d x)^2}{\left (a+b x^2\right )^3} \, dx=\frac {a (c+d x)^2}{4 b^2 \left (a+b x^2\right )^2}-\frac {2 b c^2-3 a d^2+5 b c d x}{4 b^3 \left (a+b x^2\right )}+\frac {3 c d \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{4 \sqrt {a} b^{5/2}}+\frac {d^2 \log \left (a+b x^2\right )}{2 b^3} \] Output:

1/4*a*(d*x+c)^2/b^2/(b*x^2+a)^2-1/4*(5*b*c*d*x-3*a*d^2+2*b*c^2)/b^3/(b*x^2 
+a)+3/4*c*d*arctan(b^(1/2)*x/a^(1/2))/a^(1/2)/b^(5/2)+1/2*d^2*ln(b*x^2+a)/ 
b^3
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.97 \[ \int \frac {x^3 (c+d x)^2}{\left (a+b x^2\right )^3} \, dx=\frac {\frac {a \left (-a d^2+b c (c+2 d x)\right )}{\left (a+b x^2\right )^2}+\frac {4 a d^2-b c (2 c+5 d x)}{a+b x^2}+\frac {3 \sqrt {b} c d \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a}}+2 d^2 \log \left (a+b x^2\right )}{4 b^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {531, 27, 2176, 27, 452, 218, 240}

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.

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

\(\Big \downarrow \) 531

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2176

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 452

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 240

\(\displaystyle \frac {a (c+d x)^2}{4 b^2 \left (a+b x^2\right )^2}-\frac {\frac {a (c+d x) (2 c+3 d x)}{2 b \left (a+b x^2\right )}-\frac {a d \left (\frac {3 c \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}+\frac {2 d \log \left (a+b x^2\right )}{b}\right )}{2 b}}{2 a b}\)

Input:

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

Output:

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

Defintions of rubi rules used

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 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 240

Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x 
^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
 

rule 452

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

rule 531

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symb 
ol] :> With[{Qx = PolynomialQuotient[x^m, a + b*x^2, x], e = Coeff[Polynomi 
alRemainder[x^m, a + b*x^2, x], x, 0], f = Coeff[PolynomialRemainder[x^m, a 
 + b*x^2, x], x, 1]}, Simp[(c + d*x)^n*(a*f - b*e*x)*((a + b*x^2)^(p + 1)/( 
2*a*b*(p + 1))), x] + Simp[1/(2*a*b*(p + 1))   Int[(c + d*x)^(n - 1)*(a + b 
*x^2)^(p + 1)*ExpandToSum[2*a*b*(p + 1)*(c + d*x)*Qx - a*d*f*n + b*c*e*(2*p 
 + 3) + b*d*e*(n + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGt 
Q[n, 0] && IGtQ[m, 0] && LtQ[p, -1] && GtQ[n, 1] && IntegerQ[2*p]
 

rule 2176

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{Qx = PolynomialQuotient[Pq, a + b*x^2, x], R = Coeff[PolynomialRema 
inder[Pq, a + b*x^2, x], x, 0], S = Coeff[PolynomialRemainder[Pq, a + b*x^2 
, x], x, 1]}, Simp[(d + e*x)^m*(a + b*x^2)^(p + 1)*((a*S - b*R*x)/(2*a*b*(p 
 + 1))), x] + Simp[1/(2*a*b*(p + 1))   Int[(d + e*x)^(m - 1)*(a + b*x^2)^(p 
 + 1)*ExpandToSum[2*a*b*(p + 1)*(d + e*x)*Qx - a*e*S*m + b*d*R*(2*p + 3) + 
b*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e}, x] && PolyQ[Pq, x 
] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] &&  !(IGtQ[m, 0] && R 
ationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
 

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.03

method	result	size
default	\(\frac {-\frac {5 c d \,x^{3}}{4 b}+\frac {\left (2 a \,d^{2}-b \,c^{2}\right ) x^{2}}{2 b^{2}}-\frac {3 a c d x}{4 b^{2}}+\frac {a \left (3 a \,d^{2}-b \,c^{2}\right )}{4 b^{3}}}{\left (b \,x^{2}+a \right )^{2}}+\frac {d \left (\frac {2 d \ln \left (b \,x^{2}+a \right )}{b}+\frac {3 c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )}{4 b^{2}}\)	\(110\)
risch	\(\frac {-\frac {5 c d \,x^{3}}{4 b}+\frac {\left (2 a \,d^{2}-b \,c^{2}\right ) x^{2}}{2 b^{2}}-\frac {3 a c d x}{4 b^{2}}+\frac {a \left (3 a \,d^{2}-b \,c^{2}\right )}{4 b^{3}}}{\left (b \,x^{2}+a \right )^{2}}+\frac {3 d \ln \left (-\sqrt {-a b}\, x +a \right ) c \sqrt {-a b}}{8 a \,b^{3}}+\frac {d^{2} \ln \left (-\sqrt {-a b}\, x +a \right )}{2 b^{3}}-\frac {3 d \ln \left (\sqrt {-a b}\, x +a \right ) c \sqrt {-a b}}{8 a \,b^{3}}+\frac {d^{2} \ln \left (\sqrt {-a b}\, x +a \right )}{2 b^{3}}\)	\(166\)

Input:

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

Output:

(-5/4*c*d*x^3/b+1/2*(2*a*d^2-b*c^2)/b^2*x^2-3/4*a*c*d*x/b^2+1/4*a*(3*a*d^2 
-b*c^2)/b^3)/(b*x^2+a)^2+1/4*d/b^2*(2*d*ln(b*x^2+a)/b+3*c/(a*b)^(1/2)*arct 
an(b*x/(a*b)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 371, normalized size of antiderivative = 3.47 \[ \int \frac {x^3 (c+d x)^2}{\left (a+b x^2\right )^3} \, dx=\left [-\frac {10 \, a b^{2} c d x^{3} + 6 \, a^{2} b c d x + 2 \, a^{2} b c^{2} - 6 \, a^{3} d^{2} + 4 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b d^{2}\right )} x^{2} + 3 \, {\left (b^{2} c d x^{4} + 2 \, a b c d x^{2} + a^{2} c d\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) - 4 \, {\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \log \left (b x^{2} + a\right )}{8 \, {\left (a b^{5} x^{4} + 2 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}}, -\frac {5 \, a b^{2} c d x^{3} + 3 \, a^{2} b c d x + a^{2} b c^{2} - 3 \, a^{3} d^{2} + 2 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b d^{2}\right )} x^{2} - 3 \, {\left (b^{2} c d x^{4} + 2 \, a b c d x^{2} + a^{2} c d\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) - 2 \, {\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \log \left (b x^{2} + a\right )}{4 \, {\left (a b^{5} x^{4} + 2 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}}\right ] \] Input:

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

Output:

[-1/8*(10*a*b^2*c*d*x^3 + 6*a^2*b*c*d*x + 2*a^2*b*c^2 - 6*a^3*d^2 + 4*(a*b 
^2*c^2 - 2*a^2*b*d^2)*x^2 + 3*(b^2*c*d*x^4 + 2*a*b*c*d*x^2 + a^2*c*d)*sqrt 
(-a*b)*log((b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) - 4*(a*b^2*d^2*x^4 + 
2*a^2*b*d^2*x^2 + a^3*d^2)*log(b*x^2 + a))/(a*b^5*x^4 + 2*a^2*b^4*x^2 + a^ 
3*b^3), -1/4*(5*a*b^2*c*d*x^3 + 3*a^2*b*c*d*x + a^2*b*c^2 - 3*a^3*d^2 + 2* 
(a*b^2*c^2 - 2*a^2*b*d^2)*x^2 - 3*(b^2*c*d*x^4 + 2*a*b*c*d*x^2 + a^2*c*d)* 
sqrt(a*b)*arctan(sqrt(a*b)*x/a) - 2*(a*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3 
*d^2)*log(b*x^2 + a))/(a*b^5*x^4 + 2*a^2*b^4*x^2 + a^3*b^3)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (104) = 208\).

Time = 0.98 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.36 \[ \int \frac {x^3 (c+d x)^2}{\left (a+b x^2\right )^3} \, dx=\left (\frac {d^{2}}{2 b^{3}} - \frac {3 c d \sqrt {- a b^{7}}}{8 a b^{6}}\right ) \log {\left (x + \frac {8 a b^{3} \left (\frac {d^{2}}{2 b^{3}} - \frac {3 c d \sqrt {- a b^{7}}}{8 a b^{6}}\right ) - 4 a d^{2}}{3 b c d} \right )} + \left (\frac {d^{2}}{2 b^{3}} + \frac {3 c d \sqrt {- a b^{7}}}{8 a b^{6}}\right ) \log {\left (x + \frac {8 a b^{3} \left (\frac {d^{2}}{2 b^{3}} + \frac {3 c d \sqrt {- a b^{7}}}{8 a b^{6}}\right ) - 4 a d^{2}}{3 b c d} \right )} + \frac {3 a^{2} d^{2} - a b c^{2} - 3 a b c d x - 5 b^{2} c d x^{3} + x^{2} \cdot \left (4 a b d^{2} - 2 b^{2} c^{2}\right )}{4 a^{2} b^{3} + 8 a b^{4} x^{2} + 4 b^{5} x^{4}} \] Input:

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

Output:

(d**2/(2*b**3) - 3*c*d*sqrt(-a*b**7)/(8*a*b**6))*log(x + (8*a*b**3*(d**2/( 
2*b**3) - 3*c*d*sqrt(-a*b**7)/(8*a*b**6)) - 4*a*d**2)/(3*b*c*d)) + (d**2/( 
2*b**3) + 3*c*d*sqrt(-a*b**7)/(8*a*b**6))*log(x + (8*a*b**3*(d**2/(2*b**3) 
 + 3*c*d*sqrt(-a*b**7)/(8*a*b**6)) - 4*a*d**2)/(3*b*c*d)) + (3*a**2*d**2 - 
 a*b*c**2 - 3*a*b*c*d*x - 5*b**2*c*d*x**3 + x**2*(4*a*b*d**2 - 2*b**2*c**2 
))/(4*a**2*b**3 + 8*a*b**4*x**2 + 4*b**5*x**4)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.10 \[ \int \frac {x^3 (c+d x)^2}{\left (a+b x^2\right )^3} \, dx=\frac {3 \, c d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} b^{2}} - \frac {5 \, b^{2} c d x^{3} + 3 \, a b c d x + a b c^{2} - 3 \, a^{2} d^{2} + 2 \, {\left (b^{2} c^{2} - 2 \, a b d^{2}\right )} x^{2}}{4 \, {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} + \frac {d^{2} \log \left (b x^{2} + a\right )}{2 \, b^{3}} \] Input:

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

Output:

3/4*c*d*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^2) - 1/4*(5*b^2*c*d*x^3 + 3*a*b 
*c*d*x + a*b*c^2 - 3*a^2*d^2 + 2*(b^2*c^2 - 2*a*b*d^2)*x^2)/(b^5*x^4 + 2*a 
*b^4*x^2 + a^2*b^3) + 1/2*d^2*log(b*x^2 + a)/b^3
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.96 \[ \int \frac {x^3 (c+d x)^2}{\left (a+b x^2\right )^3} \, dx=\frac {3 \, c d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} b^{2}} + \frac {d^{2} \log \left (b x^{2} + a\right )}{2 \, b^{3}} - \frac {5 \, b c d x^{3} + 3 \, a c d x + 2 \, {\left (b c^{2} - 2 \, a d^{2}\right )} x^{2} + \frac {a b c^{2} - 3 \, a^{2} d^{2}}{b}}{4 \, {\left (b x^{2} + a\right )}^{2} b^{2}} \] Input:

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

Output:

3/4*c*d*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^2) + 1/2*d^2*log(b*x^2 + a)/b^3 
 - 1/4*(5*b*c*d*x^3 + 3*a*c*d*x + 2*(b*c^2 - 2*a*d^2)*x^2 + (a*b*c^2 - 3*a 
^2*d^2)/b)/((b*x^2 + a)^2*b^2)
 

Mupad [B] (verification not implemented)

Time = 7.30 (sec) , antiderivative size = 241, normalized size of antiderivative = 2.25 \[ \int \frac {x^3 (c+d x)^2}{\left (a+b x^2\right )^3} \, dx=\frac {d^2\,\ln \left (b\,x^2+a\right )}{2\,b^3}-\frac {c^2\,x^2}{2\,\left (a^2\,b+2\,a\,b^2\,x^2+b^3\,x^4\right )}-\frac {a\,c^2}{4\,\left (a^2\,b^2+2\,a\,b^3\,x^2+b^4\,x^4\right )}+\frac {3\,a^2\,d^2}{4\,\left (a^2\,b^3+2\,a\,b^4\,x^2+b^5\,x^4\right )}+\frac {a\,d^2\,x^2}{a^2\,b^2+2\,a\,b^3\,x^2+b^4\,x^4}-\frac {5\,c\,d\,x^3}{4\,\left (a^2\,b+2\,a\,b^2\,x^2+b^3\,x^4\right )}+\frac {3\,c\,d\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{4\,\sqrt {a}\,b^{5/2}}-\frac {3\,a\,c\,d\,x}{4\,\left (a^2\,b^2+2\,a\,b^3\,x^2+b^4\,x^4\right )} \] Input:

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

Output:

(d^2*log(a + b*x^2))/(2*b^3) - (c^2*x^2)/(2*(a^2*b + b^3*x^4 + 2*a*b^2*x^2 
)) - (a*c^2)/(4*(a^2*b^2 + b^4*x^4 + 2*a*b^3*x^2)) + (3*a^2*d^2)/(4*(a^2*b 
^3 + b^5*x^4 + 2*a*b^4*x^2)) + (a*d^2*x^2)/(a^2*b^2 + b^4*x^4 + 2*a*b^3*x^ 
2) - (5*c*d*x^3)/(4*(a^2*b + b^3*x^4 + 2*a*b^2*x^2)) + (3*c*d*atan((b^(1/2 
)*x)/a^(1/2)))/(4*a^(1/2)*b^(5/2)) - (3*a*c*d*x)/(4*(a^2*b^2 + b^4*x^4 + 2 
*a*b^3*x^2))
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.94 \[ \int \frac {x^3 (c+d x)^2}{\left (a+b x^2\right )^3} \, dx=\frac {3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} c d +6 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a b c d \,x^{2}+3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{2} c d \,x^{4}+2 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{3} d^{2}+4 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} b \,d^{2} x^{2}+2 \,\mathrm {log}\left (b \,x^{2}+a \right ) a \,b^{2} d^{2} x^{4}+a^{3} d^{2}-3 a^{2} b c d x -5 a \,b^{2} c d \,x^{3}-2 a \,b^{2} d^{2} x^{4}+b^{3} c^{2} x^{4}}{4 a \,b^{3} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )} \] Input:

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

Output:

(3*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*c*d + 6*sqrt(b)*sqrt 
(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b*c*d*x**2 + 3*sqrt(b)*sqrt(a)*atan((b 
*x)/(sqrt(b)*sqrt(a)))*b**2*c*d*x**4 + 2*log(a + b*x**2)*a**3*d**2 + 4*log 
(a + b*x**2)*a**2*b*d**2*x**2 + 2*log(a + b*x**2)*a*b**2*d**2*x**4 + a**3* 
d**2 - 3*a**2*b*c*d*x - 5*a*b**2*c*d*x**3 - 2*a*b**2*d**2*x**4 + b**3*c**2 
*x**4)/(4*a*b**3*(a**2 + 2*a*b*x**2 + b**2*x**4))