\(\int \frac {(d+c^2 d x^2)^{5/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx\) [150]

Optimal result

Integrand size = 26, antiderivative size = 293 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\frac {b c^3 d^2 x^2 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}+\frac {b c^5 d^2 x^4 \sqrt {d+c^2 d x^2}}{4 \sqrt {1+c^2 x^2}}-\frac {5}{16} b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2}+\frac {15}{8} c^2 d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{4} c^2 d x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}+\frac {15 c d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{16 b \sqrt {1+c^2 x^2}}+\frac {b c d^2 \sqrt {d+c^2 d x^2} \log (x)}{\sqrt {1+c^2 x^2}} \] Output:

1/16*b*c^3*d^2*x^2*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)+1/4*b*c^5*d^2*x^4 
*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-5/16*b*c*d^2*(c^2*x^2+1)^(3/2)*(c^2 
*d*x^2+d)^(1/2)+15/8*c^2*d^2*x*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))+5/4* 
c^2*d*x*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))-(c^2*d*x^2+d)^(5/2)*(a+b*ar 
csinh(c*x))/x+15/16*c*d^2*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2/b/(c^2* 
x^2+1)^(1/2)+b*c*d^2*(c^2*d*x^2+d)^(1/2)*ln(x)/(c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 1.16 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.92 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\frac {1}{128} d^2 \left (\frac {16 a \sqrt {d+c^2 d x^2} \left (-8+9 c^2 x^2+2 c^4 x^4\right )}{x}+\frac {64 b \sqrt {d+c^2 d x^2} \left (-2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+c x \text {arcsinh}(c x)^2+2 c x \log (c x)\right )}{x \sqrt {1+c^2 x^2}}+240 a c \sqrt {d} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+\frac {32 b c \sqrt {d+c^2 d x^2} (-\cosh (2 \text {arcsinh}(c x))+2 \text {arcsinh}(c x) (\text {arcsinh}(c x)+\sinh (2 \text {arcsinh}(c x))))}{\sqrt {1+c^2 x^2}}-\frac {b c \sqrt {d+c^2 d x^2} \left (8 \text {arcsinh}(c x)^2+\cosh (4 \text {arcsinh}(c x))-4 \text {arcsinh}(c x) \sinh (4 \text {arcsinh}(c x))\right )}{\sqrt {1+c^2 x^2}}\right ) \] Input:

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

Output:

(d^2*((16*a*Sqrt[d + c^2*d*x^2]*(-8 + 9*c^2*x^2 + 2*c^4*x^4))/x + (64*b*Sq 
rt[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]))/(x*Sqrt[1 + c^2*x^2]) + 240*a*c*Sqrt[d]*Log[c*d*x + Sqr 
t[d]*Sqrt[d + c^2*d*x^2]] + (32*b*c*Sqrt[d + c^2*d*x^2]*(-Cosh[2*ArcSinh[c 
*x]] + 2*ArcSinh[c*x]*(ArcSinh[c*x] + Sinh[2*ArcSinh[c*x]])))/Sqrt[1 + c^2 
*x^2] - (b*c*Sqrt[d + c^2*d*x^2]*(8*ArcSinh[c*x]^2 + Cosh[4*ArcSinh[c*x]] 
- 4*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]]))/Sqrt[1 + c^2*x^2]))/128
 

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.99, 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, 6201, 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.

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

\(\Big \downarrow \) 6222

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6201

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

\(\Big \downarrow \) 244

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6200

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6198

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

Input:

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

Output:

-(((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x) + 5*c^2*d*(-1/4*(b*c*d*S 
qrt[d + c^2*d*x^2]*(x^2/2 + (c^2*x^4)/4))/Sqrt[1 + c^2*x^2] + (x*(d + c^2* 
d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/4 + (3*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])))/ 
4) + (b*c*d^2*Sqrt[d + c^2*d*x^2]*(2*c^2*x^2 + (c^4*x^4)/2 + Log[x^2]))/(2 
*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 6201

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), 
x_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcSinh[c*x])^n/(2*p + 1)), x] + 
(Simp[2*d*(p/(2*p + 1))   Int[(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x 
], x] - Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Int[x* 
(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, 
b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 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 = 1.05 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.97

method	result	size
default	\(-\frac {a \left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{d x}+a \,c^{2} x \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}+\frac {5 \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} a \,c^{2} d x}{4}+\frac {15 a \,d^{2} \sqrt {c^{2} d \,x^{2}+d}\, c^{2} x}{8}+\frac {15 a \,c^{2} d^{3} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{8 \sqrt {c^{2} d}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (32 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}-8 x^{5} c^{5}+144 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) x^{2} c^{2}-72 x^{3} c^{3}+120 \operatorname {arcsinh}\left (x c \right )^{2} x c +128 \ln \left (\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) x c -128 x c \,\operatorname {arcsinh}\left (x c \right )-128 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}-33 x c \right ) d^{2}}{128 \sqrt {c^{2} x^{2}+1}\, x}\)	\(285\)
parts	\(-\frac {a \left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{d x}+a \,c^{2} x \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}+\frac {5 \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} a \,c^{2} d x}{4}+\frac {15 a \,d^{2} \sqrt {c^{2} d \,x^{2}+d}\, c^{2} x}{8}+\frac {15 a \,c^{2} d^{3} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{8 \sqrt {c^{2} d}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (32 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}-8 x^{5} c^{5}+144 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) x^{2} c^{2}-72 x^{3} c^{3}+120 \operatorname {arcsinh}\left (x c \right )^{2} x c +128 \ln \left (\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) x c -128 x c \,\operatorname {arcsinh}\left (x c \right )-128 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}-33 x c \right ) d^{2}}{128 \sqrt {c^{2} x^{2}+1}\, x}\)	\(285\)

Input:

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

Output:

-a/d/x*(c^2*d*x^2+d)^(7/2)+a*c^2*x*(c^2*d*x^2+d)^(5/2)+5/4*(c^2*d*x^2+d)^( 
3/2)*a*c^2*d*x+15/8*a*d^2*(c^2*d*x^2+d)^(1/2)*c^2*x+15/8*a*c^2*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)+1/128*b*(d*(c^2*x^2+1 
))^(1/2)/(c^2*x^2+1)^(1/2)/x*(32*arcsinh(x*c)*(c^2*x^2+1)^(1/2)*x^4*c^4-8* 
x^5*c^5+144*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*x^2*c^2-72*x^3*c^3+120*arcsinh( 
x*c)^2*x*c+128*ln((x*c+(c^2*x^2+1)^(1/2))^2-1)*x*c-128*x*c*arcsinh(x*c)-12 
8*arcsinh(x*c)*(c^2*x^2+1)^(1/2)-33*x*c)*d^2
 

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\frac {\sqrt {d}\, d^{2} \left (2 \sqrt {c^{2} x^{2}+1}\, a \,c^{4} x^{4}+9 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}-8 \sqrt {c^{2} x^{2}+1}\, a +8 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{x^{2}}d x \right ) b x +8 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{2}d x \right ) b \,c^{4} x +16 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )d x \right ) b \,c^{2} x +15 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a c x -10 a c x \right )}{8 x} \] Input:

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

Output:

(sqrt(d)*d**2*(2*sqrt(c**2*x**2 + 1)*a*c**4*x**4 + 9*sqrt(c**2*x**2 + 1)*a 
*c**2*x**2 - 8*sqrt(c**2*x**2 + 1)*a + 8*int((sqrt(c**2*x**2 + 1)*asinh(c* 
x))/x**2,x)*b*x + 8*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**2,x)*b*c**4*x + 
16*int(sqrt(c**2*x**2 + 1)*asinh(c*x),x)*b*c**2*x + 15*log(sqrt(c**2*x**2 
+ 1) + c*x)*a*c*x - 10*a*c*x))/(8*x)