\(\int \frac {(c+d x^2)^{3/2}}{x^2 (a+b x^2)} \, dx\) [937]

Optimal result

Integrand size = 24, antiderivative size = 102 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )} \, dx=-\frac {c \sqrt {c+d x^2}}{a x}-\frac {(b c-a d)^{3/2} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{3/2} b}+\frac {d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b} \] Output:

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

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.26 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )} \, dx=\frac {(b c-a d)^{3/2} x \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )-\sqrt {a} \left (b c \sqrt {c+d x^2}+a d^{3/2} x \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right )\right )}{a^{3/2} b x} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {376, 25, 398, 224, 219, 291, 218}

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.

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

\(\Big \downarrow \) 376

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 398

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 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 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 376

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) 
, x_Symbol] :> Simp[c*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1 
)/(a*e*(m + 1))), x] - Simp[1/(a*e^2*(m + 1))   Int[(e*x)^(m + 2)*(a + b*x^ 
2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b*c - a*d)*(m + 1) + 2*c*(b*c*(p + 1) + a* 
d*(q - 1)) + d*((b*c - a*d)*(m + 1) + 2*b*c*(p + q))*x^2, x], x], x] /; Fre 
eQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && LtQ[m, -1] & 
& IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
 

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]
 

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.12

method	result	size
pseudoelliptic	\(\frac {-x \left (a d -b c \right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {x^{2} d +c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )+\left (\operatorname {arctanh}\left (\frac {\sqrt {x^{2} d +c}}{x \sqrt {d}}\right ) d^{\frac {3}{2}} a x -\sqrt {x^{2} d +c}\, b c \right ) \sqrt {\left (a d -b c \right ) a}}{\sqrt {\left (a d -b c \right ) a}\, a x b}\)	\(114\)
risch	\(-\frac {c \sqrt {x^{2} d +c}}{a x}+\frac {\frac {d^{\frac {3}{2}} a \ln \left (\sqrt {d}\, x +\sqrt {x^{2} d +c}\right )}{b}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}}{a}\)	\(401\)
default	\(\text {Expression too large to display}\)	\(1330\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 718, normalized size of antiderivative = 7.04 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )} \, dx=\left [\frac {2 \, a d^{\frac {3}{2}} x \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) - {\left (b c - a d\right )} x \sqrt {-\frac {b c - a d}{a}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \, {\left (a^{2} c x - {\left (a b c - 2 \, a^{2} d\right )} x^{3}\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{a}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, \sqrt {d x^{2} + c} b c}{4 \, a b x}, -\frac {4 \, a \sqrt {-d} d x \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (b c - a d\right )} x \sqrt {-\frac {b c - a d}{a}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \, {\left (a^{2} c x - {\left (a b c - 2 \, a^{2} d\right )} x^{3}\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{a}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, \sqrt {d x^{2} + c} b c}{4 \, a b x}, \frac {a d^{\frac {3}{2}} x \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) - {\left (b c - a d\right )} x \sqrt {\frac {b c - a d}{a}} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{a}}}{2 \, {\left ({\left (b c d - a d^{2}\right )} x^{3} + {\left (b c^{2} - a c d\right )} x\right )}}\right ) - 2 \, \sqrt {d x^{2} + c} b c}{2 \, a b x}, -\frac {2 \, a \sqrt {-d} d x \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (b c - a d\right )} x \sqrt {\frac {b c - a d}{a}} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{a}}}{2 \, {\left ({\left (b c d - a d^{2}\right )} x^{3} + {\left (b c^{2} - a c d\right )} x\right )}}\right ) + 2 \, \sqrt {d x^{2} + c} b c}{2 \, a b x}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

sage0*x
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 414, normalized size of antiderivative = 4.06 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )} \, dx=\frac {-\sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{2}+c}+\sqrt {d}\, \sqrt {b}\, x \right ) a d x +\sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{2}+c}+\sqrt {d}\, \sqrt {b}\, x \right ) b c x -\sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{2}+c}+\sqrt {d}\, \sqrt {b}\, x \right ) a d x +\sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{2}+c}+\sqrt {d}\, \sqrt {b}\, x \right ) b c x +\sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}+2 \sqrt {d}\, \sqrt {d \,x^{2}+c}\, b x +2 a d +2 b d \,x^{2}\right ) a d x -\sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}+2 \sqrt {d}\, \sqrt {d \,x^{2}+c}\, b x +2 a d +2 b d \,x^{2}\right ) b c x -2 \sqrt {d \,x^{2}+c}\, a b c +2 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d \,x^{2}+c}+\sqrt {d}\, x}{\sqrt {c}}\right ) a^{2} d x -2 \sqrt {d}\, a b c x}{2 a^{2} b x} \] Input:

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

Output:

( - sqrt(a)*sqrt(a*d - b*c)*log( - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) 
- 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*a*d*x + sqr 
t(a)*sqrt(a*d - b*c)*log( - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d 
 + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*b*c*x - sqrt(a)*sq 
rt(a*d - b*c)*log(sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + 
sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*a*d*x + sqrt(a)*sqrt(a*d - b 
*c)*log(sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(b)*sq 
rt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*b*c*x + sqrt(a)*sqrt(a*d - b*c)*log(2* 
sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*sqrt(d)*sqrt(c + d*x**2)*b*x + 2*a*d + 
 2*b*d*x**2)*a*d*x - sqrt(a)*sqrt(a*d - b*c)*log(2*sqrt(d)*sqrt(a)*sqrt(a* 
d - b*c) + 2*sqrt(d)*sqrt(c + d*x**2)*b*x + 2*a*d + 2*b*d*x**2)*b*c*x - 2* 
sqrt(c + d*x**2)*a*b*c + 2*sqrt(d)*log((sqrt(c + d*x**2) + sqrt(d)*x)/sqrt 
(c))*a**2*d*x - 2*sqrt(d)*a*b*c*x)/(2*a**2*b*x)