\(\int \frac {x \sqrt {a+b x^2}}{c+d x} \, dx\) [1027]

Optimal result

Integrand size = 20, antiderivative size = 137 \[ \int \frac {x \sqrt {a+b x^2}}{c+d x} \, dx=-\frac {c \sqrt {a+b x^2}}{d^2}+\frac {x \sqrt {a+b x^2}}{2 d}+\frac {\left (2 b c^2+a d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b} d^3}+\frac {c \sqrt {b c^2+a d^2} \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.97 \[ \int \frac {x \sqrt {a+b x^2}}{c+d x} \, dx=\frac {d (-2 c+d x) \sqrt {a+b x^2}-4 c \sqrt {-b c^2-a d^2} \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )-\frac {\left (2 b c^2+a d^2\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{\sqrt {b}}}{2 d^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {591, 25, 719, 224, 219, 488, 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.

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

\(\Big \downarrow \) 591

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

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

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 488

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

rule 591

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

rule 719

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + 
Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, 
d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.48

method	result	size
risch	\(-\frac {\left (-d x +2 c \right ) \sqrt {b \,x^{2}+a}}{2 d^{2}}+\frac {\frac {\left (a \,d^{2}+2 b \,c^{2}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d \sqrt {b}}+\frac {2 c \left (a \,d^{2}+b \,c^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}}{2 d^{2}}\)	\(203\)
default	\(\frac {\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}}{d}-\frac {c \left (\sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}-\frac {\sqrt {b}\, c \ln \left (\frac {-\frac {b c}{d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\right )}{d}-\frac {\left (a \,d^{2}+b \,c^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{2}}\)	\(303\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 684, normalized size of antiderivative = 4.99 \[ \int \frac {x \sqrt {a+b x^2}}{c+d x} \, dx=\left [\frac {2 \, \sqrt {b c^{2} + a d^{2}} b c \log \left (\frac {2 \, a b c d x - a b c^{2} - 2 \, a^{2} d^{2} - {\left (2 \, b^{2} c^{2} + a b d^{2}\right )} x^{2} + 2 \, \sqrt {b c^{2} + a d^{2}} {\left (b c x - a d\right )} \sqrt {b x^{2} + a}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + {\left (2 \, b c^{2} + a d^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (b d^{2} x - 2 \, b c d\right )} \sqrt {b x^{2} + a}}{4 \, b d^{3}}, \frac {4 \, \sqrt {-b c^{2} - a d^{2}} b c \arctan \left (\frac {\sqrt {-b c^{2} - a d^{2}} {\left (b c x - a d\right )} \sqrt {b x^{2} + a}}{a b c^{2} + a^{2} d^{2} + {\left (b^{2} c^{2} + a b d^{2}\right )} x^{2}}\right ) + {\left (2 \, b c^{2} + a d^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (b d^{2} x - 2 \, b c d\right )} \sqrt {b x^{2} + a}}{4 \, b d^{3}}, \frac {\sqrt {b c^{2} + a d^{2}} b c \log \left (\frac {2 \, a b c d x - a b c^{2} - 2 \, a^{2} d^{2} - {\left (2 \, b^{2} c^{2} + a b d^{2}\right )} x^{2} + 2 \, \sqrt {b c^{2} + a d^{2}} {\left (b c x - a d\right )} \sqrt {b x^{2} + a}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - {\left (2 \, b c^{2} + a d^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (b d^{2} x - 2 \, b c d\right )} \sqrt {b x^{2} + a}}{2 \, b d^{3}}, \frac {2 \, \sqrt {-b c^{2} - a d^{2}} b c \arctan \left (\frac {\sqrt {-b c^{2} - a d^{2}} {\left (b c x - a d\right )} \sqrt {b x^{2} + a}}{a b c^{2} + a^{2} d^{2} + {\left (b^{2} c^{2} + a b d^{2}\right )} x^{2}}\right ) - {\left (2 \, b c^{2} + a d^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (b d^{2} x - 2 \, b c d\right )} \sqrt {b x^{2} + a}}{2 \, b d^{3}}\right ] \] Input:

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

Output:

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

Sympy [F]

\[ \int \frac {x \sqrt {a+b x^2}}{c+d x} \, dx=\int \frac {x \sqrt {a + b x^{2}}}{c + d x}\, dx \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.89 \[ \int \frac {x \sqrt {a+b x^2}}{c+d x} \, dx=\frac {\sqrt {b x^{2} + a} x}{2 \, d} + \frac {\sqrt {b} c^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{d^{3}} + \frac {a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {b} d} - \frac {\sqrt {a + \frac {b c^{2}}{d^{2}}} c \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{d^{2}} - \frac {\sqrt {b x^{2} + a} c}{d^{2}} \] Input:

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

Output:

1/2*sqrt(b*x^2 + a)*x/d + sqrt(b)*c^2*arcsinh(b*x/sqrt(a*b))/d^3 + 1/2*a*a 
rcsinh(b*x/sqrt(a*b))/(sqrt(b)*d) - sqrt(a + b*c^2/d^2)*c*arcsinh(b*c*x/(s 
qrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/d^2 - sqrt(b*x^2 + 
a)*c/d^2
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x \sqrt {a+b x^2}}{c+d x} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x \sqrt {a+b x^2}}{c+d x} \, dx=\int \frac {x\,\sqrt {b\,x^2+a}}{c+d\,x} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 1125, normalized size of antiderivative = 8.21 \[ \int \frac {x \sqrt {a+b x^2}}{c+d x} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*sqrt(b)*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)*sqr 
t(a*d**2 + b*c**2)*atan((sqrt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sqrt(b)* 
sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2))*b*c**2 + 2*sqrt(2*sqrt(b)*sq 
rt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)*atan((sqrt(a + b*x**2)*d + sqrt 
(b)*d*x)/sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2))*a*b* 
c*d**2 + 2*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)*ata 
n((sqrt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)* 
c - a*d**2 - 2*b*c**2))*b**2*c**3 + sqrt(b)*sqrt(2*sqrt(b)*sqrt(a*d**2 + b 
*c**2)*c + a*d**2 + 2*b*c**2)*sqrt(a*d**2 + b*c**2)*log( - sqrt(2*sqrt(b)* 
sqrt(a*d**2 + b*c**2)*c + a*d**2 + 2*b*c**2) + sqrt(a + b*x**2)*d + sqrt(b 
)*d*x)*b*c**2 - sqrt(b)*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c + a*d**2 + 
2*b*c**2)*sqrt(a*d**2 + b*c**2)*log(sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c 
 + a*d**2 + 2*b*c**2) + sqrt(a + b*x**2)*d + sqrt(b)*d*x)*b*c**2 + sqrt(a* 
d**2 + b*c**2)*log( - sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c + a*d**2 + 2* 
b*c**2) + sqrt(a + b*x**2)*d + sqrt(b)*d*x)*a*b*c*d**2 + sqrt(a*d**2 + b*c 
**2)*log(sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c + a*d**2 + 2*b*c**2) + sqr 
t(a + b*x**2)*d + sqrt(b)*d*x)*a*b*c*d**2 - sqrt(a*d**2 + b*c**2)*log(2*sq 
rt(b)*sqrt(a*d**2 + b*c**2)*c + 2*sqrt(b)*sqrt(a + b*x**2)*d**2*x - 2*b*c* 
*2 + 2*b*d**2*x**2)*a*b*c*d**2 - sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c + 
a*d**2 + 2*b*c**2)*log( - sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c + a*d*...