\(\int \frac {(d+c^2 d x^2)^{5/2} (a+b \text {arcsinh}(c x))}{x^4} \, dx\) [152]

Optimal result

Integrand size = 26, antiderivative size = 266 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=-\frac {b c d^2 \sqrt {d+c^2 d x^2}}{6 x^2 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^2 \sqrt {d+c^2 d x^2}}{4 \sqrt {1+c^2 x^2}}+\frac {5}{2} c^4 d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))-\frac {5 c^2 d \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 x}-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^3}+\frac {5 c^3 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 b \sqrt {1+c^2 x^2}}+\frac {7 b c^3 d^2 \sqrt {d+c^2 d x^2} \log (x)}{3 \sqrt {1+c^2 x^2}} \] Output:

-1/6*b*c*d^2*(c^2*d*x^2+d)^(1/2)/x^2/(c^2*x^2+1)^(1/2)-1/4*b*c^5*d^2*x^2*( 
c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)+5/2*c^4*d^2*x*(c^2*d*x^2+d)^(1/2)*(a+ 
b*arcsinh(c*x))-5/3*c^2*d*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))/x-1/3*(c^ 
2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))/x^3+5/4*c^3*d^2*(c^2*d*x^2+d)^(1/2)*(a 
+b*arcsinh(c*x))^2/b/(c^2*x^2+1)^(1/2)+7/3*b*c^3*d^2*(c^2*d*x^2+d)^(1/2)*l 
n(x)/(c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.88 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.08 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=\frac {d^2 \left (4 a \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (-2-14 c^2 x^2+3 c^4 x^4\right )+24 b c^2 x^2 \sqrt {d+c^2 d x^2} \left (-2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+c x \text {arcsinh}(c x)^2+2 c x \log (c x)\right )-4 b \sqrt {d+c^2 d x^2} \left (c x+2 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)-2 c^3 x^3 \log (c x)\right )+60 a c^3 \sqrt {d} x^3 \sqrt {1+c^2 x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )-3 b c^3 x^3 \sqrt {d+c^2 d x^2} (\cosh (2 \text {arcsinh}(c x))-2 \text {arcsinh}(c x) (\text {arcsinh}(c x)+\sinh (2 \text {arcsinh}(c x))))\right )}{24 x^3 \sqrt {1+c^2 x^2}} \] Input:

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

Output:

(d^2*(4*a*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(-2 - 14*c^2*x^2 + 3*c^4*x 
^4) + 24*b*c^2*x^2*Sqrt[d + c^2*d*x^2]*(-2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] 
+ c*x*ArcSinh[c*x]^2 + 2*c*x*Log[c*x]) - 4*b*Sqrt[d + c^2*d*x^2]*(c*x + 2* 
(1 + c^2*x^2)^(3/2)*ArcSinh[c*x] - 2*c^3*x^3*Log[c*x]) + 60*a*c^3*Sqrt[d]* 
x^3*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 3*b*c^3*x 
^3*Sqrt[d + c^2*d*x^2]*(Cosh[2*ArcSinh[c*x]] - 2*ArcSinh[c*x]*(ArcSinh[c*x 
] + Sinh[2*ArcSinh[c*x]]))))/(24*x^3*Sqrt[1 + c^2*x^2])
 

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {6222, 243, 49, 2009, 6222, 244, 2009, 6200, 15, 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.

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

\(\Big \downarrow \) 6222

\(\displaystyle \frac {5}{3} c^2 d \int \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2}dx+\frac {b c d^2 \sqrt {c^2 d x^2+d} \int \frac {\left (c^2 x^2+1\right )^2}{x^3}dx}{3 \sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^3}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {5}{3} c^2 d \int \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2}dx+\frac {b c d^2 \sqrt {c^2 d x^2+d} \int \frac {\left (c^2 x^2+1\right )^2}{x^4}dx^2}{6 \sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^3}\)

\(\Big \downarrow \) 49

\(\displaystyle \frac {5}{3} c^2 d \int \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2}dx+\frac {b c d^2 \sqrt {c^2 d x^2+d} \int \left (c^4+\frac {2 c^2}{x^2}+\frac {1}{x^4}\right )dx^2}{6 \sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {5}{3} c^2 d \int \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2}dx-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^3}+\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (c^4 x^2+2 c^2 \log \left (x^2\right )-\frac {1}{x^2}\right )}{6 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 6222

\(\displaystyle \frac {5}{3} c^2 d \left (3 c^2 d \int \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))dx+\frac {b c d \sqrt {c^2 d x^2+d} \int \frac {c^2 x^2+1}{x}dx}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}\right )-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^3}+\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (c^4 x^2+2 c^2 \log \left (x^2\right )-\frac {1}{x^2}\right )}{6 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 244

\(\displaystyle \frac {5}{3} c^2 d \left (3 c^2 d \int \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))dx+\frac {b c d \sqrt {c^2 d x^2+d} \int \left (x c^2+\frac {1}{x}\right )dx}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}\right )-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^3}+\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (c^4 x^2+2 c^2 \log \left (x^2\right )-\frac {1}{x^2}\right )}{6 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {5}{3} c^2 d \left (3 c^2 d \int \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))dx-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {c^2 x^2}{2}+\log (x)\right )}{\sqrt {c^2 x^2+1}}\right )-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^3}+\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (c^4 x^2+2 c^2 \log \left (x^2\right )-\frac {1}{x^2}\right )}{6 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 6200

\(\displaystyle \frac {5}{3} c^2 d \left (3 c^2 d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx}{2 \sqrt {c^2 x^2+1}}-\frac {b c \sqrt {c^2 d x^2+d} \int xdx}{2 \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))\right )-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {c^2 x^2}{2}+\log (x)\right )}{\sqrt {c^2 x^2+1}}\right )-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^3}+\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (c^4 x^2+2 c^2 \log \left (x^2\right )-\frac {1}{x^2}\right )}{6 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {5}{3} c^2 d \left (3 c^2 d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx}{2 \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))-\frac {b c x^2 \sqrt {c^2 d x^2+d}}{4 \sqrt {c^2 x^2+1}}\right )-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {c^2 x^2}{2}+\log (x)\right )}{\sqrt {c^2 x^2+1}}\right )-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^3}+\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (c^4 x^2+2 c^2 \log \left (x^2\right )-\frac {1}{x^2}\right )}{6 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 6198

\(\displaystyle \frac {5}{3} c^2 d \left (3 c^2 d \left (\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))+\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{4 b c \sqrt {c^2 x^2+1}}-\frac {b c x^2 \sqrt {c^2 d x^2+d}}{4 \sqrt {c^2 x^2+1}}\right )-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {c^2 x^2}{2}+\log (x)\right )}{\sqrt {c^2 x^2+1}}\right )-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^3}+\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (c^4 x^2+2 c^2 \log \left (x^2\right )-\frac {1}{x^2}\right )}{6 \sqrt {c^2 x^2+1}}\)

Input:

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

Output:

-1/3*((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x^3 + (5*c^2*d*(-(((d + 
c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x) + 3*c^2*d*(-1/4*(b*c*x^2*Sqrt[d 
+ c^2*d*x^2])/Sqrt[1 + c^2*x^2] + (x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c* 
x]))/2 + (Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(4*b*c*Sqrt[1 + c^2* 
x^2])) + (b*c*d*Sqrt[d + c^2*d*x^2]*((c^2*x^2)/2 + Log[x]))/Sqrt[1 + c^2*x 
^2]))/3 + (b*c*d^2*Sqrt[d + c^2*d*x^2]*(-x^(-2) + c^4*x^2 + 2*c^2*Log[x^2] 
))/(6*Sqrt[1 + c^2*x^2])
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

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 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 244

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p 
, 0]
 

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 6200

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSinh[c*x])^n/2), x] + (Simp[(1 
/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]   Int[(a + b*ArcSinh[c*x])^n/Sq 
rt[1 + c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2* 
x^2]]   Int[x*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[e, c^2*d] && GtQ[n, 0]
 

rule 6222

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

Maple [A] (verified)

Time = 0.96 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.21

method	result	size
default	\(-\frac {a \left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{3 d \,x^{3}}-\frac {4 a \,c^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{3 d x}+\frac {4 a \,c^{4} x \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3}+\frac {5 a \,c^{4} d x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+\frac {5 a \,c^{4} d^{2} x \sqrt {c^{2} d \,x^{2}+d}}{2}+\frac {5 a \,c^{4} d^{3} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 \sqrt {c^{2} d}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (12 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}-6 x^{5} c^{5}+30 \operatorname {arcsinh}\left (x c \right )^{2} x^{3} c^{3}+56 \ln \left (\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) x^{3} c^{3}-56 \,\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}-56 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) x^{2} c^{2}-3 x^{3} c^{3}-8 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}-4 x c \right ) d^{2}}{24 \sqrt {c^{2} x^{2}+1}\, x^{3}}\)	\(322\)
parts	\(-\frac {a \left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{3 d \,x^{3}}-\frac {4 a \,c^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{3 d x}+\frac {4 a \,c^{4} x \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3}+\frac {5 a \,c^{4} d x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+\frac {5 a \,c^{4} d^{2} x \sqrt {c^{2} d \,x^{2}+d}}{2}+\frac {5 a \,c^{4} d^{3} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 \sqrt {c^{2} d}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (12 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}-6 x^{5} c^{5}+30 \operatorname {arcsinh}\left (x c \right )^{2} x^{3} c^{3}+56 \ln \left (\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) x^{3} c^{3}-56 \,\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}-56 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) x^{2} c^{2}-3 x^{3} c^{3}-8 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}-4 x c \right ) d^{2}}{24 \sqrt {c^{2} x^{2}+1}\, x^{3}}\)	\(322\)

Input:

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

Output:

-1/3*a/d/x^3*(c^2*d*x^2+d)^(7/2)-4/3*a*c^2/d/x*(c^2*d*x^2+d)^(7/2)+4/3*a*c 
^4*x*(c^2*d*x^2+d)^(5/2)+5/3*a*c^4*d*x*(c^2*d*x^2+d)^(3/2)+5/2*a*c^4*d^2*x 
*(c^2*d*x^2+d)^(1/2)+5/2*a*c^4*d^3*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d)^ 
(1/2))/(c^2*d)^(1/2)+1/24*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/x^3*(1 
2*arcsinh(x*c)*(c^2*x^2+1)^(1/2)*x^4*c^4-6*x^5*c^5+30*arcsinh(x*c)^2*x^3*c 
^3+56*ln((x*c+(c^2*x^2+1)^(1/2))^2-1)*x^3*c^3-56*arcsinh(x*c)*x^3*c^3-56*( 
c^2*x^2+1)^(1/2)*arcsinh(x*c)*x^2*c^2-3*x^3*c^3-8*arcsinh(x*c)*(c^2*x^2+1) 
^(1/2)-4*x*c)*d^2
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=\frac {\sqrt {d}\, d^{2} \left (6 \sqrt {c^{2} x^{2}+1}\, a \,c^{4} x^{4}-28 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}-4 \sqrt {c^{2} x^{2}+1}\, a +12 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{x^{4}}d x \right ) b \,x^{3}+24 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{x^{2}}d x \right ) b \,c^{2} x^{3}+12 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )d x \right ) b \,c^{4} x^{3}+30 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a \,c^{3} x^{3}+5 a \,c^{3} x^{3}\right )}{12 x^{3}} \] Input:

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

Output:

(sqrt(d)*d**2*(6*sqrt(c**2*x**2 + 1)*a*c**4*x**4 - 28*sqrt(c**2*x**2 + 1)* 
a*c**2*x**2 - 4*sqrt(c**2*x**2 + 1)*a + 12*int((sqrt(c**2*x**2 + 1)*asinh( 
c*x))/x**4,x)*b*x**3 + 24*int((sqrt(c**2*x**2 + 1)*asinh(c*x))/x**2,x)*b*c 
**2*x**3 + 12*int(sqrt(c**2*x**2 + 1)*asinh(c*x),x)*b*c**4*x**3 + 30*log(s 
qrt(c**2*x**2 + 1) + c*x)*a*c**3*x**3 + 5*a*c**3*x**3))/(12*x**3)