\(\int \frac {(A+B x) \sqrt {a+b x^2}}{c+d x} \, dx\) [152]

Optimal result

Integrand size = 24, antiderivative size = 149 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{c+d x} \, dx=-\frac {(2 (B c-A d)-B d x) \sqrt {a+b x^2}}{2 d^2}+\frac {\left (a B d^2+2 b c (B c-A d)\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b} d^3}+\frac {(B c-A d) \sqrt {b c^2+a d^2} \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d^3} \] Output:

-1/2*(-B*d*x-2*A*d+2*B*c)*(b*x^2+a)^(1/2)/d^2+1/2*(a*B*d^2+2*b*c*(-A*d+B*c 
))*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(1/2)/d^3+(-A*d+B*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 = 1.08 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.03 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{c+d x} \, dx=\frac {d (-2 B c+2 A d+B d x) \sqrt {a+b x^2}-4 (B c-A d) \sqrt {-b c^2-a d^2} \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )-\frac {\left (a B d^2+2 b c (B c-A d)\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{\sqrt {b}}}{2 d^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {682, 25, 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.

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

\(\Big \downarrow \) 682

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

-1/2*((2*(B*c - A*d) - B*d*x)*Sqrt[a + b*x^2])/d^2 - (-(((a*B*d^2 + 2*b*c* 
(B*c - A*d))*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*d)) - (2*(B*c 
- A*d)*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 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 682

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p 
+ 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p 
+ 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)))   Int[(d + e*x) 
^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* 
d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x 
], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  ! 
RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Intege 
rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

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 = 1.31 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.57

method	result	size
risch	\(\frac {\left (B d x +2 A d -2 B c \right ) \sqrt {b \,x^{2}+a}}{2 d^{2}}-\frac {\frac {\left (2 A b c d -a B \,d^{2}-2 B b \,c^{2}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d \sqrt {b}}+\frac {2 \left (A \,d^{3} a +A b \,c^{2} d -a B c \,d^{2}-b B \,c^{3}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}}{2 d^{2}}\)	\(234\)
default	\(\frac {B \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{d}+\frac {\left (A d -B c \right ) \left (\sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}-\frac {\sqrt {b}\, c \ln \left (\frac {-\frac {b c}{d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\right )}{d}-\frac {\left (a \,d^{2}+b \,c^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{2}}\)	\(310\)

Input:

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

Output:

1/2*(B*d*x+2*A*d-2*B*c)*(b*x^2+a)^(1/2)/d^2-1/2/d^2*((2*A*b*c*d-B*a*d^2-2* 
B*b*c^2)/d*ln(b^(1/2)*x+(b*x^2+a)^(1/2))/b^(1/2)+2*(A*a*d^3+A*b*c^2*d-B*a* 
c*d^2-B*b*c^3)/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 = 39.35 (sec) , antiderivative size = 784, normalized size of antiderivative = 5.26 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{c+d x} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.43 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{c+d x} \, dx=\frac {\sqrt {b x^{2} + a} B x}{2 \, d} + \frac {B \sqrt {b} c^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{d^{3}} - \frac {A \sqrt {b} c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{d^{2}} + \frac {B a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {b} d} - \frac {B \sqrt {a + \frac {b c^{2}}{d^{2}}} c \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{d^{2}} + \frac {A \sqrt {a + \frac {b c^{2}}{d^{2}}} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{d} - \frac {\sqrt {b x^{2} + a} B c}{d^{2}} + \frac {\sqrt {b x^{2} + a} A}{d} \] Input:

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 2189, normalized size of antiderivative = 14.69 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{c+d x} \, dx =\text {Too large to display} \] Input:

int((B*x+A)*(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)* 
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*c*d + 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))*b*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)/sqrt(2*sq 
rt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2))*a**2*d**3 - 2*sqrt(2*s 
qrt(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*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*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**2*c**3 - s 
qrt(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*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**2 + 2*b*c**2) + sqrt(a +...