\(\int \frac {(a+\frac {b}{c+d x^2})^{3/2}}{x^3} \, dx\) [180]

Optimal result

Integrand size = 21, antiderivative size = 117 \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^3} \, dx=-\frac {b d \sqrt {a+\frac {b}{c+d x^2}}}{c^2}-\frac {(b+a c) \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{2 c^2 x^2}+\frac {3 b \sqrt {b+a c} d \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+\frac {b}{c+d x^2}}}{\sqrt {b+a c}}\right )}{2 c^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^3} \, dx=-\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (a c \left (c+d x^2\right )+b \left (c+3 d x^2\right )\right )}{2 c^2 x^2}+\frac {3 b \sqrt {-b-a c} d \arctan \left (\frac {\sqrt {c} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {-b-a c}}\right )}{2 c^{5/2}} \] Input:

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

Output:

-1/2*(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(a*c*(c + d*x^2) + b*(c + 3*d* 
x^2)))/(c^2*x^2) + (3*b*Sqrt[-b - a*c]*d*ArcTan[(Sqrt[c]*Sqrt[(b + a*c + a 
*d*x^2)/(c + d*x^2)])/Sqrt[-b - a*c]])/(2*c^(5/2))
 

Rubi [A] (warning: unable to verify)

Time = 0.58 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2057, 2053, 2052, 252, 262, 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.

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

\(\Big \downarrow \) 2057

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

\(\Big \downarrow \) 2053

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

\(\Big \downarrow \) 2052

\(\displaystyle -b d \int \frac {x^8}{\left (-c x^4+b+a c\right )^2}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}\)

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

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 252

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x 
)^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* 
(p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c 
}, x] && LtQ[p, -1] && GtQ[m, 1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi 
alQ[a, b, c, 2, m, p, x]
 

rule 262

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

rule 2052

Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)))/((c_) + (d_.)*(x_)))^(p_), x_S 
ymbol] :> With[{q = Denominator[p]}, Simp[q*e*(b*c - a*d)   Subst[Int[x^(q* 
(p + 1) - 1)*(((-a)*e + c*x^q)^m/(b*e - d*x^q)^(m + 2)), x], x, (e*((a + b* 
x)/(c + d*x)))^(1/q)], x]] /; FreeQ[{a, b, c, d, e, m}, x] && FractionQ[p] 
&& IntegerQ[m]
 

rule 2053

Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.) 
))^(p_), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(e*( 
(a + b*x)/(c + d*x)))^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p}, 
 x] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 2057

Int[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u* 
((b + a*c + a*d*x^n)/(c + d*x^n))^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(250\) vs. \(2(99)=198\).

Time = 0.17 (sec) , antiderivative size = 251, normalized size of antiderivative = 2.15

method	result	size
risch	\(-\frac {\left (a c +b \right ) \left (d \,x^{2}+c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{2 c^{2} x^{2}}-\frac {d b \left (-\frac {\left (3 a c +3 b \right ) \ln \left (\frac {2 a \,c^{2}+2 b c +\left (2 a c d +b d \right ) x^{2}+2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,c^{2}+b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}}{x^{2}}\right )}{2 \sqrt {a \,c^{2}+b c}}+\frac {2 a d \,x^{2}+2 a c +2 b}{\sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}}\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}}{2 c^{2} \left (a d \,x^{2}+a c +b \right )}\)	\(251\)
default	\(-\frac {\sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (-2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, a \,d^{3} x^{6}-3 \ln \left (\frac {2 a d \,x^{2} c +b d \,x^{2}+2 a \,c^{2}+2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}+2 b c}{x^{2}}\right ) a b \,c^{2} d^{2} x^{4}-6 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, a c \,d^{2} x^{4}-3 \ln \left (\frac {2 a d \,x^{2} c +b d \,x^{2}+2 a \,c^{2}+2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}+2 b c}{x^{2}}\right ) b^{2} c \,d^{2} x^{4}-2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, b \,d^{2} x^{4}-3 \ln \left (\frac {2 a d \,x^{2} c +b d \,x^{2}+2 a \,c^{2}+2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}+2 b c}{x^{2}}\right ) a b \,c^{3} d \,x^{2}-4 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, a \,c^{2} d \,x^{2}-3 \ln \left (\frac {2 a d \,x^{2} c +b d \,x^{2}+2 a \,c^{2}+2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}+2 b c}{x^{2}}\right ) b^{2} c^{2} d \,x^{2}+4 \sqrt {a \,c^{2}+b c}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, b c d \,x^{2}+2 \sqrt {a \,c^{2}+b c}\, \left (a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c \right )^{\frac {3}{2}} d \,x^{2}-2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, b c d \,x^{2}+2 \sqrt {a \,c^{2}+b c}\, \left (a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c \right )^{\frac {3}{2}} c \right )}{4 \sqrt {a \,c^{2}+b c}\, x^{2} c^{3} \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}}\)	\(820\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 404, normalized size of antiderivative = 3.45 \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^3} \, dx=\left [\frac {3 \, b d x^{2} \sqrt {\frac {a c + b}{c}} \log \left (\frac {{\left (8 \, a^{2} c^{2} + 8 \, a b c + b^{2}\right )} d^{2} x^{4} + 8 \, a^{2} c^{4} + 16 \, a b c^{3} + 8 \, b^{2} c^{2} + 8 \, {\left (2 \, a^{2} c^{3} + 3 \, a b c^{2} + b^{2} c\right )} d x^{2} + 4 \, {\left ({\left (2 \, a c^{2} + b c\right )} d^{2} x^{4} + 2 \, a c^{4} + 2 \, b c^{3} + {\left (4 \, a c^{3} + 3 \, b c^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} \sqrt {\frac {a c + b}{c}}}{x^{4}}\right ) - 4 \, {\left ({\left (a c + 3 \, b\right )} d x^{2} + a c^{2} + b c\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{8 \, c^{2} x^{2}}, -\frac {3 \, b d x^{2} \sqrt {-\frac {a c + b}{c}} \arctan \left (\frac {{\left ({\left (2 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + 2 \, b c\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} \sqrt {-\frac {a c + b}{c}}}{2 \, {\left (a^{2} c^{2} + {\left (a^{2} c + a b\right )} d x^{2} + 2 \, a b c + b^{2}\right )}}\right ) + 2 \, {\left ({\left (a c + 3 \, b\right )} d x^{2} + a c^{2} + b c\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{4 \, c^{2} x^{2}}\right ] \] Input:

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

Output:

[1/8*(3*b*d*x^2*sqrt((a*c + b)/c)*log(((8*a^2*c^2 + 8*a*b*c + b^2)*d^2*x^4 
 + 8*a^2*c^4 + 16*a*b*c^3 + 8*b^2*c^2 + 8*(2*a^2*c^3 + 3*a*b*c^2 + b^2*c)* 
d*x^2 + 4*((2*a*c^2 + b*c)*d^2*x^4 + 2*a*c^4 + 2*b*c^3 + (4*a*c^3 + 3*b*c^ 
2)*d*x^2)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))*sqrt((a*c + b)/c))/x^4) - 
4*((a*c + 3*b)*d*x^2 + a*c^2 + b*c)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))) 
/(c^2*x^2), -1/4*(3*b*d*x^2*sqrt(-(a*c + b)/c)*arctan(1/2*((2*a*c + b)*d*x 
^2 + 2*a*c^2 + 2*b*c)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))*sqrt(-(a*c + b 
)/c)/(a^2*c^2 + (a^2*c + a*b)*d*x^2 + 2*a*b*c + b^2)) + 2*((a*c + 3*b)*d*x 
^2 + a*c^2 + b*c)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/(c^2*x^2)]
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (99) = 198\).

Time = 0.14 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.73 \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^3} \, dx=-\frac {{\left (a b c + b^{2}\right )} d \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{2 \, {\left (a c^{3} + b c^{2} - \frac {{\left (a d x^{2} + a c + b\right )} c^{3}}{d x^{2} + c}\right )}} - \frac {b d \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{c^{2}} - \frac {3 \, {\left (a b c + b^{2}\right )} d \log \left (\frac {c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} - \sqrt {{\left (a c + b\right )} c}}{c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} + \sqrt {{\left (a c + b\right )} c}}\right )}{4 \, \sqrt {{\left (a c + b\right )} c} c^{2}} \] Input:

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

Output:

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

Giac [F]

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

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

Output:

undef
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.50 \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^3} \, dx=\frac {-\sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, a \,c^{3}-\sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, a \,c^{2} d \,x^{2}-\sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, b \,c^{2}-3 \sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, b c d \,x^{2}+3 \sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (\sqrt {a c +b}\, \sqrt {a d \,x^{2}+a c +b}\, c +\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a c +\sqrt {c}\, \sqrt {d \,x^{2}+c}\, b \right ) b c d \,x^{2}+3 \sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (\sqrt {a c +b}\, \sqrt {a d \,x^{2}+a c +b}\, c +\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a c +\sqrt {c}\, \sqrt {d \,x^{2}+c}\, b \right ) b \,d^{2} x^{4}-3 \sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (x \right ) b c d \,x^{2}-3 \sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (x \right ) b \,d^{2} x^{4}}{2 c^{3} x^{2} \left (d \,x^{2}+c \right )} \] Input:

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

Output:

( - sqrt(c + d*x**2)*sqrt(a*c + a*d*x**2 + b)*a*c**3 - sqrt(c + d*x**2)*sq 
rt(a*c + a*d*x**2 + b)*a*c**2*d*x**2 - sqrt(c + d*x**2)*sqrt(a*c + a*d*x** 
2 + b)*b*c**2 - 3*sqrt(c + d*x**2)*sqrt(a*c + a*d*x**2 + b)*b*c*d*x**2 + 3 
*sqrt(c)*sqrt(a*c + b)*log(sqrt(a*c + b)*sqrt(a*c + a*d*x**2 + b)*c + sqrt 
(c)*sqrt(c + d*x**2)*a*c + sqrt(c)*sqrt(c + d*x**2)*b)*b*c*d*x**2 + 3*sqrt 
(c)*sqrt(a*c + b)*log(sqrt(a*c + b)*sqrt(a*c + a*d*x**2 + b)*c + sqrt(c)*s 
qrt(c + d*x**2)*a*c + sqrt(c)*sqrt(c + d*x**2)*b)*b*d**2*x**4 - 3*sqrt(c)* 
sqrt(a*c + b)*log(x)*b*c*d*x**2 - 3*sqrt(c)*sqrt(a*c + b)*log(x)*b*d**2*x* 
*4)/(2*c**3*x**2*(c + d*x**2))