\(\int \frac {x}{(a+\frac {b}{c+d x^2})^{3/2}} \, dx\) [204]

Optimal result

Integrand size = 19, antiderivative size = 97 \[ \int \frac {x}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\frac {b}{a^2 d \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{2 a^2 d}-\frac {3 b \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{c+d x^2}}}{\sqrt {a}}\right )}{2 a^{5/2} d} \] Output:

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.18 \[ \int \frac {x}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\frac {\left (c+d x^2\right ) \sqrt {\frac {b+a \left (c+d x^2\right )}{c+d x^2}} \left (3 b+a \left (c+d x^2\right )\right )}{2 a^2 d \left (b+a \left (c+d x^2\right )\right )}-\frac {3 b \text {arctanh}\left (\frac {\sqrt {\frac {b+a \left (c+d x^2\right )}{c+d x^2}}}{\sqrt {a}}\right )}{2 a^{5/2} d} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2024, 773, 52, 61, 73, 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.

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

\(\Big \downarrow \) 2024

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

\(\Big \downarrow \) 773

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

\(\Big \downarrow \) 52

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

\(\Big \downarrow \) 61

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
 

rule 61

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0 
] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d 
, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

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 773

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^ 
2, x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] &&  !IntegerQ[p]
 

rule 2024

Int[((a_.) + (b_.)*(Pq_)^(n_.))^(p_.)*(Qr_), x_Symbol] :> With[{q = Expon[P 
q, x], r = Expon[Qr, x]}, Simp[Coeff[Qr, x, r]/(q*Coeff[Pq, x, q])   Subst[ 
Int[(a + b*x^n)^p, x], x, Pq], x] /; EqQ[r, q - 1] && EqQ[Coeff[Qr, x, r]*D 
[Pq, x], q*Coeff[Pq, x, q]*Qr]] /; FreeQ[{a, b, n, p}, x] && PolyQ[Pq, x] & 
& PolyQ[Qr, x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(216\) vs. \(2(83)=166\).

Time = 0.46 (sec) , antiderivative size = 217, normalized size of antiderivative = 2.24

method	result	size
risch	\(\frac {a d \,x^{2}+a c +b}{2 d \,a^{2} \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}-\frac {b \left (\frac {3 \ln \left (\frac {a c d +\frac {1}{2} b d +a \,d^{2} x^{2}}{\sqrt {a \,d^{2}}}+\sqrt {a \,c^{2}+b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}\right )}{2 \sqrt {a \,d^{2}}}-\frac {2 \left (d \,x^{2}+c \right )}{d \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}}{2 a^{2} \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\)	\(217\)
derivativedivides	\(-\frac {\sqrt {\frac {\left (d \,x^{2}+c \right ) a +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (-6 \sqrt {\left (d \,x^{2}+c \right ) \left (\left (d \,x^{2}+c \right ) a +b \right )}\, a^{\frac {5}{2}} \left (d \,x^{2}+c \right )^{2}+3 \ln \left (\frac {2 \sqrt {\left (d \,x^{2}+c \right ) \left (\left (d \,x^{2}+c \right ) a +b \right )}\, \sqrt {a}+2 \left (d \,x^{2}+c \right ) a +b}{2 \sqrt {a}}\right ) a^{2} b \left (d \,x^{2}+c \right )^{2}+4 a^{\frac {3}{2}} {\left (\left (d \,x^{2}+c \right ) \left (\left (d \,x^{2}+c \right ) a +b \right )\right )}^{\frac {3}{2}}-12 \sqrt {\left (d \,x^{2}+c \right ) \left (\left (d \,x^{2}+c \right ) a +b \right )}\, a^{\frac {3}{2}} b \left (d \,x^{2}+c \right )+6 \ln \left (\frac {2 \sqrt {\left (d \,x^{2}+c \right ) \left (\left (d \,x^{2}+c \right ) a +b \right )}\, \sqrt {a}+2 \left (d \,x^{2}+c \right ) a +b}{2 \sqrt {a}}\right ) a \,b^{2} \left (d \,x^{2}+c \right )-6 \sqrt {\left (d \,x^{2}+c \right ) \left (\left (d \,x^{2}+c \right ) a +b \right )}\, \sqrt {a}\, b^{2}+3 \ln \left (\frac {2 \sqrt {\left (d \,x^{2}+c \right ) \left (\left (d \,x^{2}+c \right ) a +b \right )}\, \sqrt {a}+2 \left (d \,x^{2}+c \right ) a +b}{2 \sqrt {a}}\right ) b^{3}\right )}{4 d \,a^{\frac {5}{2}} \sqrt {\left (d \,x^{2}+c \right ) \left (\left (d \,x^{2}+c \right ) a +b \right )}\, {\left (\left (d \,x^{2}+c \right ) a +b \right )}^{2}}\)	\(363\)
default	\(\frac {\sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (-3 \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +2 \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) a b \,d^{2} x^{2}+2 \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, a d \,x^{2}-3 \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +2 \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) a b c d +2 \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, a c -3 \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +2 \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) b^{2} d +2 \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, b +4 \sqrt {a \,d^{2}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, b \right )}{4 a^{2} d \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, \sqrt {a \,d^{2}}\, \left (a d \,x^{2}+a c +b \right )}\)	\(478\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (83) = 166\).

Time = 0.16 (sec) , antiderivative size = 395, normalized size of antiderivative = 4.07 \[ \int \frac {x}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\left [\frac {3 \, {\left (a b d x^{2} + a b c + b^{2}\right )} \sqrt {a} \log \left (8 \, a^{2} d^{2} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (2 \, a^{2} c + a b\right )} d x^{2} + 8 \, a b c + b^{2} - 4 \, {\left (2 \, a d^{2} x^{4} + {\left (4 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + b c\right )} \sqrt {a} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}\right ) + 4 \, {\left (a^{2} d^{2} x^{4} + a^{2} c^{2} + {\left (2 \, a^{2} c + 3 \, a b\right )} d x^{2} + 3 \, a b c\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{8 \, {\left (a^{4} d^{2} x^{2} + {\left (a^{4} c + a^{3} b\right )} d\right )}}, \frac {3 \, {\left (a b d x^{2} + a b c + b^{2}\right )} \sqrt {-a} \arctan \left (\frac {{\left (2 \, a d x^{2} + 2 \, a c + b\right )} \sqrt {-a} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{2 \, {\left (a^{2} d x^{2} + a^{2} c + a b\right )}}\right ) + 2 \, {\left (a^{2} d^{2} x^{4} + a^{2} c^{2} + {\left (2 \, a^{2} c + 3 \, a b\right )} d x^{2} + 3 \, a b c\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{4 \, {\left (a^{4} d^{2} x^{2} + {\left (a^{4} c + a^{3} b\right )} d\right )}}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.66 \[ \int \frac {x}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\frac {2 \, a b - \frac {3 \, {\left (a d x^{2} + a c + b\right )} b}{d x^{2} + c}}{2 \, {\left (a^{3} d \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} - a^{2} d \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {3}{2}}\right )}} + \frac {3 \, b \log \left (-\frac {\sqrt {a} - \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{\sqrt {a} + \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}\right )}{4 \, a^{\frac {5}{2}} d} \] Input:

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [B] (verification not implemented)

Time = 11.16 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.63 \[ \int \frac {x}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\frac {{\left (\frac {a\,\left (d\,x^2+c\right )}{b}+1\right )}^{3/2}\,\left (d\,x^2+c\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {3}{2},\frac {5}{2};\ \frac {7}{2};\ -\frac {a\,\left (d\,x^2+c\right )}{b}\right )}{5\,d\,{\left (a+\frac {b}{d\,x^2+c}\right )}^{3/2}} \] Input:

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

Output:

(((a*(c + d*x^2))/b + 1)^(3/2)*(c + d*x^2)*hypergeom([3/2, 5/2], 7/2, -(a* 
(c + d*x^2))/b))/(5*d*(a + b/(c + d*x^2))^(3/2))
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.13 \[ \int \frac {x}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\frac {\sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, a^{2} c +\sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, a^{2} d \,x^{2}+3 \sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, a b +3 \sqrt {a}\, \mathrm {log}\left (-\sqrt {a}\, \sqrt {a d \,x^{2}+a c +b}+\sqrt {d \,x^{2}+c}\, a \right ) a b c +3 \sqrt {a}\, \mathrm {log}\left (-\sqrt {a}\, \sqrt {a d \,x^{2}+a c +b}+\sqrt {d \,x^{2}+c}\, a \right ) a b d \,x^{2}+3 \sqrt {a}\, \mathrm {log}\left (-\sqrt {a}\, \sqrt {a d \,x^{2}+a c +b}+\sqrt {d \,x^{2}+c}\, a \right ) b^{2}}{2 a^{3} d \left (a d \,x^{2}+a c +b \right )} \] Input:

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

Output:

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