\(\int \frac {(d+c^2 d x^2)^3 (a+b \text {arcsinh}(c x))}{x^4} \, dx\) [27]

Optimal result

Integrand size = 24, antiderivative size = 174 \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x^4} \, dx=-\frac {8}{3} b c^3 d^3 \sqrt {1+c^2 x^2}-\frac {b c d^3 \sqrt {1+c^2 x^2}}{6 x^2}-\frac {1}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2}-\frac {d^3 (a+b \text {arcsinh}(c x))}{3 x^3}-\frac {3 c^2 d^3 (a+b \text {arcsinh}(c x))}{x}+3 c^4 d^3 x (a+b \text {arcsinh}(c x))+\frac {1}{3} c^6 d^3 x^3 (a+b \text {arcsinh}(c x))-\frac {17}{6} b c^3 d^3 \text {arctanh}\left (\sqrt {1+c^2 x^2}\right ) \] Output:

-8/3*b*c^3*d^3*(c^2*x^2+1)^(1/2)-1/6*b*c*d^3*(c^2*x^2+1)^(1/2)/x^2-1/9*b*c 
^3*d^3*(c^2*x^2+1)^(3/2)-1/3*d^3*(a+b*arcsinh(c*x))/x^3-3*c^2*d^3*(a+b*arc 
sinh(c*x))/x+3*c^4*d^3*x*(a+b*arcsinh(c*x))+1/3*c^6*d^3*x^3*(a+b*arcsinh(c 
*x))-17/6*b*c^3*d^3*arctanh((c^2*x^2+1)^(1/2))
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.98 \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x^4} \, dx=\frac {d^3 \left (-6 a-54 a c^2 x^2+54 a c^4 x^4+6 a c^6 x^6-3 b c x \sqrt {1+c^2 x^2}-50 b c^3 x^3 \sqrt {1+c^2 x^2}-2 b c^5 x^5 \sqrt {1+c^2 x^2}+6 b \left (-1-9 c^2 x^2+9 c^4 x^4+c^6 x^6\right ) \text {arcsinh}(c x)+51 b c^3 x^3 \log (x)-51 b c^3 x^3 \log \left (1+\sqrt {1+c^2 x^2}\right )\right )}{18 x^3} \] Input:

Integrate[((d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x]))/x^4,x]
 

Output:

(d^3*(-6*a - 54*a*c^2*x^2 + 54*a*c^4*x^4 + 6*a*c^6*x^6 - 3*b*c*x*Sqrt[1 + 
c^2*x^2] - 50*b*c^3*x^3*Sqrt[1 + c^2*x^2] - 2*b*c^5*x^5*Sqrt[1 + c^2*x^2] 
+ 6*b*(-1 - 9*c^2*x^2 + 9*c^4*x^4 + c^6*x^6)*ArcSinh[c*x] + 51*b*c^3*x^3*L 
og[x] - 51*b*c^3*x^3*Log[1 + Sqrt[1 + c^2*x^2]]))/(18*x^3)
 

Rubi [A] (warning: unable to verify)

Time = 0.62 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.90, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6218, 27, 2331, 2124, 27, 1192, 25, 1467, 2009}

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.

\(\displaystyle \int \frac {\left (c^2 d x^2+d\right )^3 (a+b \text {arcsinh}(c x))}{x^4} \, dx\)

\(\Big \downarrow \) 6218

\(\displaystyle -b c \int -\frac {d^3 \left (-c^6 x^6-9 c^4 x^4+9 c^2 x^2+1\right )}{3 x^3 \sqrt {c^2 x^2+1}}dx+\frac {1}{3} c^6 d^3 x^3 (a+b \text {arcsinh}(c x))+3 c^4 d^3 x (a+b \text {arcsinh}(c x))-\frac {3 c^2 d^3 (a+b \text {arcsinh}(c x))}{x}-\frac {d^3 (a+b \text {arcsinh}(c x))}{3 x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} b c d^3 \int \frac {-c^6 x^6-9 c^4 x^4+9 c^2 x^2+1}{x^3 \sqrt {c^2 x^2+1}}dx+\frac {1}{3} c^6 d^3 x^3 (a+b \text {arcsinh}(c x))+3 c^4 d^3 x (a+b \text {arcsinh}(c x))-\frac {3 c^2 d^3 (a+b \text {arcsinh}(c x))}{x}-\frac {d^3 (a+b \text {arcsinh}(c x))}{3 x^3}\)

\(\Big \downarrow \) 2331

\(\displaystyle \frac {1}{6} b c d^3 \int \frac {-c^6 x^6-9 c^4 x^4+9 c^2 x^2+1}{x^4 \sqrt {c^2 x^2+1}}dx^2+\frac {1}{3} c^6 d^3 x^3 (a+b \text {arcsinh}(c x))+3 c^4 d^3 x (a+b \text {arcsinh}(c x))-\frac {3 c^2 d^3 (a+b \text {arcsinh}(c x))}{x}-\frac {d^3 (a+b \text {arcsinh}(c x))}{3 x^3}\)

\(\Big \downarrow \) 2124

\(\displaystyle \frac {1}{6} b c d^3 \left (-\int -\frac {-2 x^4 c^6-18 x^2 c^4+17 c^2}{2 x^2 \sqrt {c^2 x^2+1}}dx^2-\frac {\sqrt {c^2 x^2+1}}{x^2}\right )+\frac {1}{3} c^6 d^3 x^3 (a+b \text {arcsinh}(c x))+3 c^4 d^3 x (a+b \text {arcsinh}(c x))-\frac {3 c^2 d^3 (a+b \text {arcsinh}(c x))}{x}-\frac {d^3 (a+b \text {arcsinh}(c x))}{3 x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} b c d^3 \left (\frac {1}{2} \int \frac {-2 x^4 c^6-18 x^2 c^4+17 c^2}{x^2 \sqrt {c^2 x^2+1}}dx^2-\frac {\sqrt {c^2 x^2+1}}{x^2}\right )+\frac {1}{3} c^6 d^3 x^3 (a+b \text {arcsinh}(c x))+3 c^4 d^3 x (a+b \text {arcsinh}(c x))-\frac {3 c^2 d^3 (a+b \text {arcsinh}(c x))}{x}-\frac {d^3 (a+b \text {arcsinh}(c x))}{3 x^3}\)

\(\Big \downarrow \) 1192

\(\displaystyle \frac {1}{6} b c d^3 \left (\frac {\int -\frac {-2 c^6 x^8-14 c^6 x^4+33 c^6}{1-x^4}d\sqrt {c^2 x^2+1}}{c^4}-\frac {\sqrt {c^2 x^2+1}}{x^2}\right )+\frac {1}{3} c^6 d^3 x^3 (a+b \text {arcsinh}(c x))+3 c^4 d^3 x (a+b \text {arcsinh}(c x))-\frac {3 c^2 d^3 (a+b \text {arcsinh}(c x))}{x}-\frac {d^3 (a+b \text {arcsinh}(c x))}{3 x^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{6} b c d^3 \left (-\frac {\int \frac {-2 c^6 x^8-14 c^6 x^4+33 c^6}{1-x^4}d\sqrt {c^2 x^2+1}}{c^4}-\frac {\sqrt {c^2 x^2+1}}{x^2}\right )+\frac {1}{3} c^6 d^3 x^3 (a+b \text {arcsinh}(c x))+3 c^4 d^3 x (a+b \text {arcsinh}(c x))-\frac {3 c^2 d^3 (a+b \text {arcsinh}(c x))}{x}-\frac {d^3 (a+b \text {arcsinh}(c x))}{3 x^3}\)

\(\Big \downarrow \) 1467

\(\displaystyle \frac {1}{6} b c d^3 \left (-\frac {\int \left (2 x^4 c^6+\frac {17 c^6}{1-x^4}+16 c^6\right )d\sqrt {c^2 x^2+1}}{c^4}-\frac {\sqrt {c^2 x^2+1}}{x^2}\right )+\frac {1}{3} c^6 d^3 x^3 (a+b \text {arcsinh}(c x))+3 c^4 d^3 x (a+b \text {arcsinh}(c x))-\frac {3 c^2 d^3 (a+b \text {arcsinh}(c x))}{x}-\frac {d^3 (a+b \text {arcsinh}(c x))}{3 x^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} c^6 d^3 x^3 (a+b \text {arcsinh}(c x))+3 c^4 d^3 x (a+b \text {arcsinh}(c x))-\frac {3 c^2 d^3 (a+b \text {arcsinh}(c x))}{x}-\frac {d^3 (a+b \text {arcsinh}(c x))}{3 x^3}+\frac {1}{6} b c d^3 \left (\frac {-17 c^6 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )-\frac {2}{3} c^6 x^6-16 c^6 \sqrt {c^2 x^2+1}}{c^4}-\frac {\sqrt {c^2 x^2+1}}{x^2}\right )\)

Input:

Int[((d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x]))/x^4,x]
 

Output:

-1/3*(d^3*(a + b*ArcSinh[c*x]))/x^3 - (3*c^2*d^3*(a + b*ArcSinh[c*x]))/x + 
 3*c^4*d^3*x*(a + b*ArcSinh[c*x]) + (c^6*d^3*x^3*(a + b*ArcSinh[c*x]))/3 + 
 (b*c*d^3*(-(Sqrt[1 + c^2*x^2]/x^2) + ((-2*c^6*x^6)/3 - 16*c^6*Sqrt[1 + c^ 
2*x^2] - 17*c^6*ArcTanh[Sqrt[1 + c^2*x^2]])/c^4))/6
 

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 1192

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2/e^(n + 2*p + 1)   Subst[Int[x^( 
2*m + 1)*(e*f - d*g + g*x^2)^n*(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + 
 c*x^4)^p, x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && 
IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m + 1/2]
 

rule 1467

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), 
 x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], 
x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e 
 + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
 

rule 2009

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

rule 2124

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] : 
> With[{Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px 
, a + b*x, x]}, Simp[R*(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((m + 1)*(b*c - 
 a*d))), x] + Simp[1/((m + 1)*(b*c - a*d))   Int[(a + b*x)^(m + 1)*(c + d*x 
)^n*ExpandToSum[(m + 1)*(b*c - a*d)*Qx - d*R*(m + n + 2), x], x], x]] /; Fr 
eeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && LtQ[m, -1] && (IntegerQ[m] ||  ! 
ILtQ[n, -1])
 

rule 2331

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2   S 
ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; 
 FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
 

rule 6218

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_ 
)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Simp 
[(a + b*ArcSinh[c*x])   u, x] - Simp[b*c   Int[SimplifyIntegrand[u/Sqrt[1 + 
 c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] 
&& IGtQ[p, 0]
 

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.88

method	result	size
parts	\(d^{3} a \left (\frac {c^{6} x^{3}}{3}+3 c^{4} x -\frac {1}{3 x^{3}}-\frac {3 c^{2}}{x}\right )+d^{3} b \,c^{3} \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}+3 x c \,\operatorname {arcsinh}\left (x c \right )-\frac {3 \,\operatorname {arcsinh}\left (x c \right )}{x c}-\frac {\operatorname {arcsinh}\left (x c \right )}{3 x^{3} c^{3}}-\frac {x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {25 \sqrt {c^{2} x^{2}+1}}{9}-\frac {\sqrt {c^{2} x^{2}+1}}{6 x^{2} c^{2}}-\frac {17 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}\right )\)	\(153\)
derivativedivides	\(c^{3} \left (d^{3} a \left (\frac {x^{3} c^{3}}{3}+3 x c -\frac {3}{x c}-\frac {1}{3 x^{3} c^{3}}\right )+d^{3} b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}+3 x c \,\operatorname {arcsinh}\left (x c \right )-\frac {3 \,\operatorname {arcsinh}\left (x c \right )}{x c}-\frac {\operatorname {arcsinh}\left (x c \right )}{3 x^{3} c^{3}}-\frac {x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {25 \sqrt {c^{2} x^{2}+1}}{9}-\frac {\sqrt {c^{2} x^{2}+1}}{6 x^{2} c^{2}}-\frac {17 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}\right )\right )\)	\(155\)
default	\(c^{3} \left (d^{3} a \left (\frac {x^{3} c^{3}}{3}+3 x c -\frac {3}{x c}-\frac {1}{3 x^{3} c^{3}}\right )+d^{3} b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}+3 x c \,\operatorname {arcsinh}\left (x c \right )-\frac {3 \,\operatorname {arcsinh}\left (x c \right )}{x c}-\frac {\operatorname {arcsinh}\left (x c \right )}{3 x^{3} c^{3}}-\frac {x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {25 \sqrt {c^{2} x^{2}+1}}{9}-\frac {\sqrt {c^{2} x^{2}+1}}{6 x^{2} c^{2}}-\frac {17 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}\right )\right )\)	\(155\)

Input:

int((c^2*d*x^2+d)^3*(a+b*arcsinh(x*c))/x^4,x,method=_RETURNVERBOSE)
 

Output:

d^3*a*(1/3*c^6*x^3+3*c^4*x-1/3/x^3-3*c^2/x)+d^3*b*c^3*(1/3*arcsinh(x*c)*x^ 
3*c^3+3*x*c*arcsinh(x*c)-3*arcsinh(x*c)/x/c-1/3*arcsinh(x*c)/x^3/c^3-1/9*x 
^2*c^2*(c^2*x^2+1)^(1/2)-25/9*(c^2*x^2+1)^(1/2)-1/6/x^2/c^2*(c^2*x^2+1)^(1 
/2)-17/6*arctanh(1/(c^2*x^2+1)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.66 \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x^4} \, dx=\frac {6 \, a c^{6} d^{3} x^{6} + 54 \, a c^{4} d^{3} x^{4} - 51 \, b c^{3} d^{3} x^{3} \log \left (-c x + \sqrt {c^{2} x^{2} + 1} + 1\right ) + 51 \, b c^{3} d^{3} x^{3} \log \left (-c x + \sqrt {c^{2} x^{2} + 1} - 1\right ) - 54 \, a c^{2} d^{3} x^{2} - 6 \, {\left (b c^{6} + 9 \, b c^{4} - 9 \, b c^{2} - b\right )} d^{3} x^{3} \log \left (-c x + \sqrt {c^{2} x^{2} + 1}\right ) - 6 \, a d^{3} + 6 \, {\left (b c^{6} d^{3} x^{6} + 9 \, b c^{4} d^{3} x^{4} - 9 \, b c^{2} d^{3} x^{2} - {\left (b c^{6} + 9 \, b c^{4} - 9 \, b c^{2} - b\right )} d^{3} x^{3} - b d^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (2 \, b c^{5} d^{3} x^{5} + 50 \, b c^{3} d^{3} x^{3} + 3 \, b c d^{3} x\right )} \sqrt {c^{2} x^{2} + 1}}{18 \, x^{3}} \] Input:

integrate((c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))/x^4,x, algorithm="fricas")
 

Output:

1/18*(6*a*c^6*d^3*x^6 + 54*a*c^4*d^3*x^4 - 51*b*c^3*d^3*x^3*log(-c*x + sqr 
t(c^2*x^2 + 1) + 1) + 51*b*c^3*d^3*x^3*log(-c*x + sqrt(c^2*x^2 + 1) - 1) - 
 54*a*c^2*d^3*x^2 - 6*(b*c^6 + 9*b*c^4 - 9*b*c^2 - b)*d^3*x^3*log(-c*x + s 
qrt(c^2*x^2 + 1)) - 6*a*d^3 + 6*(b*c^6*d^3*x^6 + 9*b*c^4*d^3*x^4 - 9*b*c^2 
*d^3*x^2 - (b*c^6 + 9*b*c^4 - 9*b*c^2 - b)*d^3*x^3 - b*d^3)*log(c*x + sqrt 
(c^2*x^2 + 1)) - (2*b*c^5*d^3*x^5 + 50*b*c^3*d^3*x^3 + 3*b*c*d^3*x)*sqrt(c 
^2*x^2 + 1))/x^3
 

Sympy [F]

\[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x^4} \, dx=d^{3} \left (\int 3 a c^{4}\, dx + \int \frac {a}{x^{4}}\, dx + \int \frac {3 a c^{2}}{x^{2}}\, dx + \int a c^{6} x^{2}\, dx + \int 3 b c^{4} \operatorname {asinh}{\left (c x \right )}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {3 b c^{2} \operatorname {asinh}{\left (c x \right )}}{x^{2}}\, dx + \int b c^{6} x^{2} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \] Input:

integrate((c**2*d*x**2+d)**3*(a+b*asinh(c*x))/x**4,x)
 

Output:

d**3*(Integral(3*a*c**4, x) + Integral(a/x**4, x) + Integral(3*a*c**2/x**2 
, x) + Integral(a*c**6*x**2, x) + Integral(3*b*c**4*asinh(c*x), x) + Integ 
ral(b*asinh(c*x)/x**4, x) + Integral(3*b*c**2*asinh(c*x)/x**2, x) + Integr 
al(b*c**6*x**2*asinh(c*x), x))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.20 \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x^4} \, dx=\frac {1}{3} \, a c^{6} d^{3} x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b c^{6} d^{3} + 3 \, a c^{4} d^{3} x + 3 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b c^{3} d^{3} - 3 \, {\left (c \operatorname {arsinh}\left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arsinh}\left (c x\right )}{x}\right )} b c^{2} d^{3} + \frac {1}{6} \, {\left ({\left (c^{2} \operatorname {arsinh}\left (\frac {1}{c {\left | x \right |}}\right ) - \frac {\sqrt {c^{2} x^{2} + 1}}{x^{2}}\right )} c - \frac {2 \, \operatorname {arsinh}\left (c x\right )}{x^{3}}\right )} b d^{3} - \frac {3 \, a c^{2} d^{3}}{x} - \frac {a d^{3}}{3 \, x^{3}} \] Input:

integrate((c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))/x^4,x, algorithm="maxima")
 

Output:

1/3*a*c^6*d^3*x^3 + 1/9*(3*x^3*arcsinh(c*x) - c*(sqrt(c^2*x^2 + 1)*x^2/c^2 
 - 2*sqrt(c^2*x^2 + 1)/c^4))*b*c^6*d^3 + 3*a*c^4*d^3*x + 3*(c*x*arcsinh(c* 
x) - sqrt(c^2*x^2 + 1))*b*c^3*d^3 - 3*(c*arcsinh(1/(c*abs(x))) + arcsinh(c 
*x)/x)*b*c^2*d^3 + 1/6*((c^2*arcsinh(1/(c*abs(x))) - sqrt(c^2*x^2 + 1)/x^2 
)*c - 2*arcsinh(c*x)/x^3)*b*d^3 - 3*a*c^2*d^3/x - 1/3*a*d^3/x^3
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x^4} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))/x^4,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 [F(-1)]

Timed out. \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x^4} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^3}{x^4} \,d x \] Input:

int(((a + b*asinh(c*x))*(d + c^2*d*x^2)^3)/x^4,x)
 

Output:

int(((a + b*asinh(c*x))*(d + c^2*d*x^2)^3)/x^4, x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.08 \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x^4} \, dx=\frac {d^{3} \left (6 \mathit {asinh} \left (c x \right ) b \,c^{6} x^{6}+54 \mathit {asinh} \left (c x \right ) b \,c^{4} x^{4}-54 \mathit {asinh} \left (c x \right ) b \,c^{2} x^{2}-6 \mathit {asinh} \left (c x \right ) b -2 \sqrt {c^{2} x^{2}+1}\, b \,c^{5} x^{5}-50 \sqrt {c^{2} x^{2}+1}\, b \,c^{3} x^{3}-3 \sqrt {c^{2} x^{2}+1}\, b c x +51 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x -1\right ) b \,c^{3} x^{3}-51 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x +1\right ) b \,c^{3} x^{3}+6 a \,c^{6} x^{6}+54 a \,c^{4} x^{4}-54 a \,c^{2} x^{2}-6 a \right )}{18 x^{3}} \] Input:

int((c^2*d*x^2+d)^3*(a+b*asinh(c*x))/x^4,x)
 

Output:

(d**3*(6*asinh(c*x)*b*c**6*x**6 + 54*asinh(c*x)*b*c**4*x**4 - 54*asinh(c*x 
)*b*c**2*x**2 - 6*asinh(c*x)*b - 2*sqrt(c**2*x**2 + 1)*b*c**5*x**5 - 50*sq 
rt(c**2*x**2 + 1)*b*c**3*x**3 - 3*sqrt(c**2*x**2 + 1)*b*c*x + 51*log(sqrt( 
c**2*x**2 + 1) + c*x - 1)*b*c**3*x**3 - 51*log(sqrt(c**2*x**2 + 1) + c*x + 
 1)*b*c**3*x**3 + 6*a*c**6*x**6 + 54*a*c**4*x**4 - 54*a*c**2*x**2 - 6*a))/ 
(18*x**3)