\(\int \frac {A+B x+C x^2+D x^3}{(c+d x) \sqrt {a+b x^2}} \, dx\) [105]

Optimal result

Integrand size = 34, antiderivative size = 190 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x) \sqrt {a+b x^2}} \, dx=\frac {(C d-c D) \sqrt {a+b x^2}}{b d^2}+\frac {D x \sqrt {a+b x^2}}{2 b d}-\frac {\left (a d^2 D+2 b \left (c C d-B d^2-c^2 D\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2} d^3}-\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d^3 \sqrt {b c^2+a d^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.96 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x) \sqrt {a+b x^2}} \, dx=\frac {\frac {d (2 C d-2 c D+d D x) \sqrt {a+b x^2}}{b}+\frac {4 \left (-c^2 C d+B c d^2-A d^3+c^3 D\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\sqrt {-b c^2-a d^2}}+\frac {\left (a d^2 D-2 b \left (-c C d+B d^2+c^2 D\right )\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{b^{3/2}}}{2 d^3} \] Input:

Integrate[(A + B*x + C*x^2 + D*x^3)/((c + d*x)*Sqrt[a + b*x^2]),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2185, 2185, 27, 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.

\(\displaystyle \int \frac {A+B x+C x^2+D x^3}{\sqrt {a+b x^2} (c+d x)} \, dx\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\int \frac {b (2 C d-3 c D) x^2 d^2+(2 A b d-a c D) d^2+\left (-b D c^2+2 b B d^2-a d^2 D\right ) x d}{(c+d x) \sqrt {b x^2+a}}dx}{2 b d^3}+\frac {D \sqrt {a+b x^2} (c+d x)}{2 b d^2}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\frac {\int \frac {b d^3 \left (d (2 A b d-a c D)-\left (a D d^2+2 b \left (-D c^2+C d c-B d^2\right )\right ) x\right )}{(c+d x) \sqrt {b x^2+a}}dx}{b d^2}+d \sqrt {a+b x^2} (2 C d-3 c D)}{2 b d^3}+\frac {D \sqrt {a+b x^2} (c+d x)}{2 b d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d \int \frac {d (2 A b d-a c D)-\left (a D d^2+2 b \left (-D c^2+C d c-B d^2\right )\right ) x}{(c+d x) \sqrt {b x^2+a}}dx+d \sqrt {a+b x^2} (2 C d-3 c D)}{2 b d^3}+\frac {D \sqrt {a+b x^2} (c+d x)}{2 b d^2}\)

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

Int[(A + B*x + C*x^2 + D*x^3)/((c + d*x)*Sqrt[a + b*x^2]),x]
 

Output:

(D*(c + d*x)*Sqrt[a + b*x^2])/(2*b*d^2) + (d*(2*C*d - 3*c*D)*Sqrt[a + b*x^ 
2] + d*(-(((a*d^2*D + 2*b*(c*C*d - B*d^2 - c^2*D))*ArcTanh[(Sqrt[b]*x)/Sqr 
t[a + b*x^2]])/(Sqrt[b]*d)) - (2*b*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*Arc 
Tanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/(d*Sqrt[b*c^2 + 
 a*d^2])))/(2*b*d^3)
 

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 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 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]
 

rule 2185

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) 
^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si 
mp[1/(b*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ 
b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x 
)^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p 
)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d 
, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && 
True) &&  !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 
 1/2, 0]))
 

Maple [A] (verified)

Time = 1.43 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.55

method	result	size
default	\(\frac {\frac {B \,d^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}+\frac {D c^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}+\frac {d \left (C d -D c \right ) \sqrt {b \,x^{2}+a}}{b}+D d^{2} \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )-\frac {C c d \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}}{d^{3}}-\frac {\left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{4} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\)	\(295\)

Input:

int((D*x^3+C*x^2+B*x+A)/(d*x+c)/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/d^3*(B*d^2*ln(b^(1/2)*x+(b*x^2+a)^(1/2))/b^(1/2)+D*c^2*ln(b^(1/2)*x+(b*x 
^2+a)^(1/2))/b^(1/2)+d*(C*d-D*c)/b*(b*x^2+a)^(1/2)+D*d^2*(1/2*x/b*(b*x^2+a 
)^(1/2)-1/2*a/b^(3/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2)))-C*c*d*ln(b^(1/2)*x+(b 
*x^2+a)^(1/2))/b^(1/2))-(A*d^3-B*c*d^2+C*c^2*d-D*c^3)/d^4/((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 [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x) \sqrt {a+b x^2}} \, dx=\text {Timed out} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)/(b*x^2+a)^(1/2),x, algorithm="fricas 
")
 

Output:

Timed out
 

Sympy [F]

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

integrate((D*x**3+C*x**2+B*x+A)/(d*x+c)/(b*x**2+a)**(1/2),x)
 

Output:

Integral((A + B*x + C*x**2 + D*x**3)/(sqrt(a + b*x**2)*(c + d*x)), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 352 vs. \(2 (170) = 340\).

Time = 0.08 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.85 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x) \sqrt {a+b x^2}} \, dx=\frac {\sqrt {b x^{2} + a} D x}{2 \, b d} + \frac {D c^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b} d^{3}} - \frac {C c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b} d^{2}} - \frac {D a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {3}{2}} d} + \frac {B \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b} d} - \frac {D c^{3} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{\sqrt {a + \frac {b c^{2}}{d^{2}}} d^{4}} + \frac {C c^{2} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{\sqrt {a + \frac {b c^{2}}{d^{2}}} d^{3}} - \frac {B c \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{\sqrt {a + \frac {b c^{2}}{d^{2}}} d^{2}} + \frac {A \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{\sqrt {a + \frac {b c^{2}}{d^{2}}} d} - \frac {\sqrt {b x^{2} + a} D c}{b d^{2}} + \frac {\sqrt {b x^{2} + a} C}{b d} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)/(b*x^2+a)^(1/2),x, algorithm="maxima 
")
 

Output:

1/2*sqrt(b*x^2 + a)*D*x/(b*d) + D*c^2*arcsinh(b*x/sqrt(a*b))/(sqrt(b)*d^3) 
 - C*c*arcsinh(b*x/sqrt(a*b))/(sqrt(b)*d^2) - 1/2*D*a*arcsinh(b*x/sqrt(a*b 
))/(b^(3/2)*d) + B*arcsinh(b*x/sqrt(a*b))/(sqrt(b)*d) - D*c^3*arcsinh(b*c* 
x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/(sqrt(a + b*c^2 
/d^2)*d^4) + C*c^2*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b) 
*abs(d*x + c)))/(sqrt(a + b*c^2/d^2)*d^3) - B*c*arcsinh(b*c*x/(sqrt(a*b)*a 
bs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/(sqrt(a + b*c^2/d^2)*d^2) + A 
*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/(s 
qrt(a + b*c^2/d^2)*d) - sqrt(b*x^2 + a)*D*c/(b*d^2) + sqrt(b*x^2 + a)*C/(b 
*d)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x) \sqrt {a+b x^2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)/(b*x^2+a)^(1/2),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. \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x) \sqrt {a+b x^2}} \, dx=\int \frac {A+B\,x+C\,x^2+x^3\,D}{\sqrt {b\,x^2+a}\,\left (c+d\,x\right )} \,d x \] Input:

int((A + B*x + C*x^2 + x^3*D)/((a + b*x^2)^(1/2)*(c + d*x)),x)
 

Output:

int((A + B*x + C*x^2 + x^3*D)/((a + b*x^2)^(1/2)*(c + d*x)), x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 2284, normalized size of antiderivative = 12.02 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x) \sqrt {a+b x^2}} \, dx =\text {Too large to display} \] Input:

int((D*x^3+C*x^2+B*x+A)/(d*x+c)/(b*x^2+a)^(1/2),x)
 

Output:

( - 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)*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**2*c*d + 2*sqrt(b)*sq 
rt(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)*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**3*c**2 - 2*sqrt(2*sqrt(b)*sqrt(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)/s 
qrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2))*a**2*b**2*d**3 
 - 2*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)*atan((sqr 
t(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**3*c**2*d + 2*sqrt(2*sqrt(b)*sqrt(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**3*c*d**2 + 2*sqrt(2* 
sqrt(b)*sqrt(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))*b**4*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)*a*b**2*c*d 
+ 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**...