\(\int \frac {x^4 (a+b \text {arcsinh}(c x))}{(d+c^2 d x^2)^{5/2}} \, dx\) [175]

Optimal result

Integrand size = 26, antiderivative size = 203 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {b}{6 c^5 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {x (a+b \text {arcsinh}(c x))}{c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{2 b c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {2 b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{3 c^5 d^2 \sqrt {d+c^2 d x^2}} \] Output:

1/6*b/c^5/d^2/(c^2*x^2+1)^(1/2)/(c^2*d*x^2+d)^(1/2)-1/3*x^3*(a+b*arcsinh(c 
*x))/c^2/d/(c^2*d*x^2+d)^(3/2)-x*(a+b*arcsinh(c*x))/c^4/d^2/(c^2*d*x^2+d)^ 
(1/2)+1/2*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))^2/b/c^5/d^2/(c^2*d*x^2+d)^( 
1/2)+2/3*b*(c^2*x^2+1)^(1/2)*ln(c^2*x^2+1)/c^5/d^2/(c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.94 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {-2 a c \sqrt {d} x \left (3+4 c^2 x^2\right )+b \sqrt {d} \left (\sqrt {1+c^2 x^2}+2 c x \text {arcsinh}(c x)-8 c x \left (1+c^2 x^2\right ) \text {arcsinh}(c x)+\left (1+c^2 x^2\right )^{3/2} \left (3 \text {arcsinh}(c x)^2+4 \log \left (1+c^2 x^2\right )\right )\right )+6 a \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )}{6 c^5 d^{5/2} \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}} \] Input:

Integrate[(x^4*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^(5/2),x]
 

Output:

(-2*a*c*Sqrt[d]*x*(3 + 4*c^2*x^2) + b*Sqrt[d]*(Sqrt[1 + c^2*x^2] + 2*c*x*A 
rcSinh[c*x] - 8*c*x*(1 + c^2*x^2)*ArcSinh[c*x] + (1 + c^2*x^2)^(3/2)*(3*Ar 
cSinh[c*x]^2 + 4*Log[1 + c^2*x^2])) + 6*a*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2 
]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]])/(6*c^5*d^(5/2)*(1 + c^2*x^2)*S 
qrt[d + c^2*d*x^2])
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.19, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {6225, 243, 49, 2009, 6225, 240, 6198}

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.

\(\displaystyle \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (c^2 d x^2+d\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 6225

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

\(\Big \downarrow \) 243

\(\displaystyle \frac {\int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (c^2 d x^2+d\right )^{3/2}}dx}{c^2 d}+\frac {b \sqrt {c^2 x^2+1} \int \frac {x^2}{\left (c^2 x^2+1\right )^2}dx^2}{6 c d^2 \sqrt {c^2 d x^2+d}}-\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}\)

\(\Big \downarrow \) 49

\(\displaystyle \frac {\int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (c^2 d x^2+d\right )^{3/2}}dx}{c^2 d}+\frac {b \sqrt {c^2 x^2+1} \int \left (\frac {1}{c^2 \left (c^2 x^2+1\right )}-\frac {1}{c^2 \left (c^2 x^2+1\right )^2}\right )dx^2}{6 c d^2 \sqrt {c^2 d x^2+d}}-\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (c^2 d x^2+d\right )^{3/2}}dx}{c^2 d}-\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}+\frac {b \sqrt {c^2 x^2+1} \left (\frac {1}{c^4 \left (c^2 x^2+1\right )}+\frac {\log \left (c^2 x^2+1\right )}{c^4}\right )}{6 c d^2 \sqrt {c^2 d x^2+d}}\)

\(\Big \downarrow \) 6225

\(\displaystyle \frac {\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 d x^2+d}}dx}{c^2 d}+\frac {b \sqrt {c^2 x^2+1} \int \frac {x}{c^2 x^2+1}dx}{c d \sqrt {c^2 d x^2+d}}-\frac {x (a+b \text {arcsinh}(c x))}{c^2 d \sqrt {c^2 d x^2+d}}}{c^2 d}-\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}+\frac {b \sqrt {c^2 x^2+1} \left (\frac {1}{c^4 \left (c^2 x^2+1\right )}+\frac {\log \left (c^2 x^2+1\right )}{c^4}\right )}{6 c d^2 \sqrt {c^2 d x^2+d}}\)

\(\Big \downarrow \) 240

\(\displaystyle \frac {\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 d x^2+d}}dx}{c^2 d}-\frac {x (a+b \text {arcsinh}(c x))}{c^2 d \sqrt {c^2 d x^2+d}}+\frac {b \sqrt {c^2 x^2+1} \log \left (c^2 x^2+1\right )}{2 c^3 d \sqrt {c^2 d x^2+d}}}{c^2 d}-\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}+\frac {b \sqrt {c^2 x^2+1} \left (\frac {1}{c^4 \left (c^2 x^2+1\right )}+\frac {\log \left (c^2 x^2+1\right )}{c^4}\right )}{6 c d^2 \sqrt {c^2 d x^2+d}}\)

\(\Big \downarrow \) 6198

\(\displaystyle -\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}+\frac {-\frac {x (a+b \text {arcsinh}(c x))}{c^2 d \sqrt {c^2 d x^2+d}}+\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{2 b c^3 d \sqrt {c^2 d x^2+d}}+\frac {b \sqrt {c^2 x^2+1} \log \left (c^2 x^2+1\right )}{2 c^3 d \sqrt {c^2 d x^2+d}}}{c^2 d}+\frac {b \sqrt {c^2 x^2+1} \left (\frac {1}{c^4 \left (c^2 x^2+1\right )}+\frac {\log \left (c^2 x^2+1\right )}{c^4}\right )}{6 c d^2 \sqrt {c^2 d x^2+d}}\)

Input:

Int[(x^4*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^(5/2),x]
 

Output:

-1/3*(x^3*(a + b*ArcSinh[c*x]))/(c^2*d*(d + c^2*d*x^2)^(3/2)) + (b*Sqrt[1 
+ c^2*x^2]*(1/(c^4*(1 + c^2*x^2)) + Log[1 + c^2*x^2]/c^4))/(6*c*d^2*Sqrt[d 
 + c^2*d*x^2]) + (-((x*(a + b*ArcSinh[c*x]))/(c^2*d*Sqrt[d + c^2*d*x^2])) 
+ (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c^3*d*Sqrt[d + c^2*d*x^2 
]) + (b*Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^2])/(2*c^3*d*Sqrt[d + c^2*d*x^2])) 
/(c^2*d)
 

Defintions of rubi rules used

rule 49

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

rule 240

Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x 
^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6198

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( 
a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c 
^2*d] && NeQ[n, -1]
 

rule 6225

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a 
+ b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) 
   Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] - S 
imp[b*f*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Int[(f*x)^( 
m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; Fre 
eQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && IG 
tQ[m, 1]
 

Maple [A] (verified)

Time = 1.03 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.70

method	result	size
default	\(-\frac {a \,x^{3}}{3 c^{2} d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {a x}{c^{4} d^{2} \sqrt {c^{2} d \,x^{2}+d}}+\frac {a \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{c^{4} d^{2} \sqrt {c^{2} d}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} x^{2}+1}\, \left (3 \operatorname {arcsinh}\left (x c \right )^{2} x^{4} c^{4}+8 \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{4} c^{4}-8 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}-8 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+6 \operatorname {arcsinh}\left (x c \right )^{2} x^{2} c^{2}+16 \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}-16 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}-6 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c +c^{2} x^{2}+3 \operatorname {arcsinh}\left (x c \right )^{2}+8 \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-8 \,\operatorname {arcsinh}\left (x c \right )+1\right )}{6 \left (c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) d^{3} c^{5}}\)	\(345\)
parts	\(-\frac {a \,x^{3}}{3 c^{2} d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {a x}{c^{4} d^{2} \sqrt {c^{2} d \,x^{2}+d}}+\frac {a \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{c^{4} d^{2} \sqrt {c^{2} d}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} x^{2}+1}\, \left (3 \operatorname {arcsinh}\left (x c \right )^{2} x^{4} c^{4}+8 \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{4} c^{4}-8 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}-8 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+6 \operatorname {arcsinh}\left (x c \right )^{2} x^{2} c^{2}+16 \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}-16 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}-6 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c +c^{2} x^{2}+3 \operatorname {arcsinh}\left (x c \right )^{2}+8 \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-8 \,\operatorname {arcsinh}\left (x c \right )+1\right )}{6 \left (c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) d^{3} c^{5}}\)	\(345\)

Input:

int(x^4*(a+b*arcsinh(x*c))/(c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/3*a*x^3/c^2/d/(c^2*d*x^2+d)^(3/2)-a/c^4/d^2*x/(c^2*d*x^2+d)^(1/2)+a/c^4 
/d^2*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)+1/6*b*(d* 
(c^2*x^2+1))^(1/2)*(c^2*x^2+1)^(1/2)/(c^6*x^6+3*c^4*x^4+3*c^2*x^2+1)/d^3/c 
^5*(3*arcsinh(x*c)^2*x^4*c^4+8*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)*x^4*c^4-8*a 
rcsinh(x*c)*c^4*x^4-8*arcsinh(x*c)*(c^2*x^2+1)^(1/2)*x^3*c^3+6*arcsinh(x*c 
)^2*x^2*c^2+16*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)*x^2*c^2-16*arcsinh(x*c)*c^2 
*x^2-6*arcsinh(x*c)*(c^2*x^2+1)^(1/2)*x*c+c^2*x^2+3*arcsinh(x*c)^2+8*ln(1+ 
(x*c+(c^2*x^2+1)^(1/2))^2)-8*arcsinh(x*c)+1)
 

Fricas [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^4*(a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(5/2),x, algorithm="fricas" 
)
                                                                                    
                                                                                    
 

Output:

integral((b*x^4*arcsinh(c*x) + a*x^4)*sqrt(c^2*d*x^2 + d)/(c^6*d^3*x^6 + 3 
*c^4*d^3*x^4 + 3*c^2*d^3*x^2 + d^3), x)
 

Sympy [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^{4} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(x**4*(a+b*asinh(c*x))/(c**2*d*x**2+d)**(5/2),x)
 

Output:

Integral(x**4*(a + b*asinh(c*x))/(d*(c**2*x**2 + 1))**(5/2), x)
 

Maxima [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^4*(a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(5/2),x, algorithm="maxima" 
)
 

Output:

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

Giac [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^4*(a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(5/2),x, algorithm="giac")
 

Output:

integrate((b*arcsinh(c*x) + a)*x^4/(c^2*d*x^2 + d)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (d\,c^2\,x^2+d\right )}^{5/2}} \,d x \] Input:

int((x^4*(a + b*asinh(c*x)))/(d + c^2*d*x^2)^(5/2),x)
 

Output:

int((x^4*(a + b*asinh(c*x)))/(d + c^2*d*x^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {-4 \sqrt {c^{2} x^{2}+1}\, a \,c^{3} x^{3}-3 \sqrt {c^{2} x^{2}+1}\, a c x +3 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{4}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b \,c^{9} x^{4}+6 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{4}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b \,c^{7} x^{2}+3 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{4}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b \,c^{5}+3 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a \,c^{4} x^{4}+6 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a \,c^{2} x^{2}+3 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a}{3 \sqrt {d}\, c^{5} d^{2} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )} \] Input:

int(x^4*(a+b*asinh(c*x))/(c^2*d*x^2+d)^(5/2),x)
 

Output:

( - 4*sqrt(c**2*x**2 + 1)*a*c**3*x**3 - 3*sqrt(c**2*x**2 + 1)*a*c*x + 3*in 
t((asinh(c*x)*x**4)/(sqrt(c**2*x**2 + 1)*c**4*x**4 + 2*sqrt(c**2*x**2 + 1) 
*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*b*c**9*x**4 + 6*int((asinh(c*x)*x**4) 
/(sqrt(c**2*x**2 + 1)*c**4*x**4 + 2*sqrt(c**2*x**2 + 1)*c**2*x**2 + sqrt(c 
**2*x**2 + 1)),x)*b*c**7*x**2 + 3*int((asinh(c*x)*x**4)/(sqrt(c**2*x**2 + 
1)*c**4*x**4 + 2*sqrt(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*b 
*c**5 + 3*log(sqrt(c**2*x**2 + 1) + c*x)*a*c**4*x**4 + 6*log(sqrt(c**2*x** 
2 + 1) + c*x)*a*c**2*x**2 + 3*log(sqrt(c**2*x**2 + 1) + c*x)*a)/(3*sqrt(d) 
*c**5*d**2*(c**4*x**4 + 2*c**2*x**2 + 1))