\(\int \frac {(d+c^2 d x^2)^{5/2} (a+b \text {arcsinh}(c x))}{x} \, dx\) [149]

Optimal result

Integrand size = 26, antiderivative size = 333 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x} \, dx=-\frac {23 b c d^2 x \sqrt {d+c^2 d x^2}}{15 \sqrt {1+c^2 x^2}}-\frac {11 b c^3 d^2 x^3 \sqrt {d+c^2 d x^2}}{45 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^5 \sqrt {d+c^2 d x^2}}{25 \sqrt {1+c^2 x^2}}+\frac {1}{3} d \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{5} \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))+\sqrt {d+c^2 d x^2} \left (a d^2+b d^2 \text {arcsinh}(c x)\right )-\frac {2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {b d^2 \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {b d^2 \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}} \] Output:

-23/15*b*c*d^2*x*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-11/45*b*c^3*d^2*x^3 
*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/25*b*c^5*d^2*x^5*(c^2*d*x^2+d)^(1 
/2)/(c^2*x^2+1)^(1/2)+1/3*d*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))+1/5*(c^ 
2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))+(c^2*d*x^2+d)^(1/2)*(a*d^2+b*d^2*arcsi 
nh(c*x))-2*d^2*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))*arctanh(c*x+(c^2*x^2 
+1)^(1/2))/(c^2*x^2+1)^(1/2)-b*d^2*(c^2*d*x^2+d)^(1/2)*polylog(2,-c*x-(c^2 
*x^2+1)^(1/2))/(c^2*x^2+1)^(1/2)+b*d^2*(c^2*d*x^2+d)^(1/2)*polylog(2,c*x+( 
c^2*x^2+1)^(1/2))/(c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.02 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x} \, dx=\frac {-40 b c d^3 x \sqrt {1+c^2 x^2} \left (3+c^2 x^2\right )-3 b c^3 d^3 x^3 \sqrt {1+c^2 x^2} \left (5+3 c^2 x^2\right )+15 a d^3 \left (1+c^2 x^2\right ) \left (23+11 c^2 x^2+3 c^4 x^4\right )+150 b d^3 \left (1+c^2 x^2\right )^2 \text {arcsinh}(c x)+15 b d^3 \left (1+c^2 x^2\right )^2 \left (-2+3 c^2 x^2\right ) \text {arcsinh}(c x)+225 a d^{5/2} \sqrt {d+c^2 d x^2} \log (x)-225 a d^{5/2} \sqrt {d+c^2 d x^2} \log \left (d+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+225 b d^3 \sqrt {1+c^2 x^2} \left (-c x+\sqrt {1+c^2 x^2} \text {arcsinh}(c x)+\text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-\text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+\operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )\right )}{225 \sqrt {d+c^2 d x^2}} \] Input:

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

Output:

(-40*b*c*d^3*x*Sqrt[1 + c^2*x^2]*(3 + c^2*x^2) - 3*b*c^3*d^3*x^3*Sqrt[1 + 
c^2*x^2]*(5 + 3*c^2*x^2) + 15*a*d^3*(1 + c^2*x^2)*(23 + 11*c^2*x^2 + 3*c^4 
*x^4) + 150*b*d^3*(1 + c^2*x^2)^2*ArcSinh[c*x] + 15*b*d^3*(1 + c^2*x^2)^2* 
(-2 + 3*c^2*x^2)*ArcSinh[c*x] + 225*a*d^(5/2)*Sqrt[d + c^2*d*x^2]*Log[x] - 
 225*a*d^(5/2)*Sqrt[d + c^2*d*x^2]*Log[d + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + 
225*b*d^3*Sqrt[1 + c^2*x^2]*(-(c*x) + Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + Arc 
Sinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c* 
x])] + PolyLog[2, -E^(-ArcSinh[c*x])] - PolyLog[2, E^(-ArcSinh[c*x])]))/(2 
25*Sqrt[d + c^2*d*x^2])
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.41 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.89, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6223, 210, 2009, 6223, 2009, 6221, 24, 6231, 3042, 26, 4670, 2715, 2838}

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.

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

\(\Big \downarrow \) 6223

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

\(\Big \downarrow \) 210

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6223

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6221

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 6231

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

\(\displaystyle d \left (d \left (\frac {i \sqrt {c^2 d x^2+d} \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )\right )}{\sqrt {c^2 x^2+1}}+\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))-\frac {b c x \sqrt {c^2 d x^2+d}}{\sqrt {c^2 x^2+1}}\right )+\frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {b c d \left (\frac {c^2 x^3}{3}+x\right ) \sqrt {c^2 d x^2+d}}{3 \sqrt {c^2 x^2+1}}\right )+\frac {1}{5} \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^5}{5}+\frac {2 c^2 x^3}{3}+x\right ) \sqrt {c^2 d x^2+d}}{5 \sqrt {c^2 x^2+1}}\)

Input:

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

Output:

-1/5*(b*c*d^2*Sqrt[d + c^2*d*x^2]*(x + (2*c^2*x^3)/3 + (c^4*x^5)/5))/Sqrt[ 
1 + c^2*x^2] + ((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/5 + d*(-1/3*(b 
*c*d*Sqrt[d + c^2*d*x^2]*(x + (c^2*x^3)/3))/Sqrt[1 + c^2*x^2] + ((d + c^2* 
d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/3 + d*(-((b*c*x*Sqrt[d + c^2*d*x^2])/Sq 
rt[1 + c^2*x^2]) + Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (I*Sqrt[d + 
c^2*d*x^2]*((2*I)*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]] + I*b*PolyL 
og[2, -E^ArcSinh[c*x]] - I*b*PolyLog[2, E^ArcSinh[c*x]]))/Sqrt[1 + c^2*x^2 
]))
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 210

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

rule 2009

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

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4670

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x 
_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] 
 + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x 
)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e 
+ f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
 

rule 6221

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

rule 6223

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 + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1))   Int[(f* 
x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*c*(n/(f*(m 
 + 2*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]) /; FreeQ[{a, b, c, 
 d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m, -1]
 

rule 6231

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.) 
*(x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e 
*x^2]]   Subst[Int[(a + b*x)^n*Sinh[x]^m, x], x, ArcSinh[c*x]], x] /; FreeQ 
[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && IntegerQ[m]
 

Maple [A] (verified)

Time = 0.96 (sec) , antiderivative size = 540, normalized size of antiderivative = 1.62

method	result	size
default	\(\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} a}{5}+\frac {a d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}-a \,d^{\frac {5}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )+a \,d^{2} \sqrt {c^{2} d \,x^{2}+d}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d^{2} \operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{5 c^{2} x^{2}+5}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d^{2} x^{5} c^{5}}{25 \sqrt {c^{2} x^{2}+1}}+\frac {14 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d^{2} \operatorname {arcsinh}\left (x c \right ) x^{4} c^{4}}{15 \left (c^{2} x^{2}+1\right )}-\frac {11 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d^{2} x^{3} c^{3}}{45 \sqrt {c^{2} x^{2}+1}}+\frac {34 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d^{2} \operatorname {arcsinh}\left (x c \right ) x^{2} c^{2}}{15 \left (c^{2} x^{2}+1\right )}-\frac {23 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d^{2} x c}{15 \sqrt {c^{2} x^{2}+1}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right ) d^{2}}{\sqrt {c^{2} x^{2}+1}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right ) d^{2}}{\sqrt {c^{2} x^{2}+1}}+\frac {23 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d^{2} \operatorname {arcsinh}\left (x c \right )}{15 \left (c^{2} x^{2}+1\right )}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right ) d^{2}}{\sqrt {c^{2} x^{2}+1}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right ) d^{2}}{\sqrt {c^{2} x^{2}+1}}\)	\(540\)
parts	\(\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} a}{5}+\frac {a d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}-a \,d^{\frac {5}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )+a \,d^{2} \sqrt {c^{2} d \,x^{2}+d}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d^{2} \operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{5 c^{2} x^{2}+5}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d^{2} x^{5} c^{5}}{25 \sqrt {c^{2} x^{2}+1}}+\frac {14 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d^{2} \operatorname {arcsinh}\left (x c \right ) x^{4} c^{4}}{15 \left (c^{2} x^{2}+1\right )}-\frac {11 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d^{2} x^{3} c^{3}}{45 \sqrt {c^{2} x^{2}+1}}+\frac {34 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d^{2} \operatorname {arcsinh}\left (x c \right ) x^{2} c^{2}}{15 \left (c^{2} x^{2}+1\right )}-\frac {23 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d^{2} x c}{15 \sqrt {c^{2} x^{2}+1}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right ) d^{2}}{\sqrt {c^{2} x^{2}+1}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right ) d^{2}}{\sqrt {c^{2} x^{2}+1}}+\frac {23 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d^{2} \operatorname {arcsinh}\left (x c \right )}{15 \left (c^{2} x^{2}+1\right )}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right ) d^{2}}{\sqrt {c^{2} x^{2}+1}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right ) d^{2}}{\sqrt {c^{2} x^{2}+1}}\)	\(540\)

Input:

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

Output:

1/5*(c^2*d*x^2+d)^(5/2)*a+1/3*a*d*(c^2*d*x^2+d)^(3/2)-a*d^(5/2)*ln((2*d+2* 
d^(1/2)*(c^2*d*x^2+d)^(1/2))/x)+a*d^2*(c^2*d*x^2+d)^(1/2)+1/5*b*(d*(c^2*x^ 
2+1))^(1/2)*d^2/(c^2*x^2+1)*arcsinh(x*c)*x^6*c^6-1/25*b*(d*(c^2*x^2+1))^(1 
/2)*d^2/(c^2*x^2+1)^(1/2)*x^5*c^5+14/15*b*(d*(c^2*x^2+1))^(1/2)*d^2/(c^2*x 
^2+1)*arcsinh(x*c)*x^4*c^4-11/45*b*(d*(c^2*x^2+1))^(1/2)*d^2/(c^2*x^2+1)^( 
1/2)*x^3*c^3+34/15*b*(d*(c^2*x^2+1))^(1/2)*d^2/(c^2*x^2+1)*arcsinh(x*c)*x^ 
2*c^2-23/15*b*(d*(c^2*x^2+1))^(1/2)*d^2/(c^2*x^2+1)^(1/2)*x*c+b*(d*(c^2*x^ 
2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(x*c)*ln(1-x*c-(c^2*x^2+1)^(1/2))*d^2 
+b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*polylog(2,x*c+(c^2*x^2+1)^(1/2) 
)*d^2+23/15*b*(d*(c^2*x^2+1))^(1/2)*d^2/(c^2*x^2+1)*arcsinh(x*c)-b*(d*(c^2 
*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(x*c)*ln(1+x*c+(c^2*x^2+1)^(1/2))* 
d^2-b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*polylog(2,-x*c-(c^2*x^2+1)^( 
1/2))*d^2
 

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-1/15*(15*d^(5/2)*arcsinh(1/(c*abs(x))) - 3*(c^2*d*x^2 + d)^(5/2) - 5*(c^2 
*d*x^2 + d)^(3/2)*d - 15*sqrt(c^2*d*x^2 + d)*d^2)*a + b*integrate((c^2*d*x 
^2 + d)^(5/2)*log(c*x + sqrt(c^2*x^2 + 1))/x, x)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x} \, dx=\frac {\sqrt {d}\, d^{2} \left (3 \sqrt {c^{2} x^{2}+1}\, a \,c^{4} x^{4}+11 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}+23 \sqrt {c^{2} x^{2}+1}\, a +15 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{x}d x \right ) b +15 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{3}d x \right ) b \,c^{4}+30 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x d x \right ) b \,c^{2}+15 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x -1\right ) a -15 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x +1\right ) a \right )}{15} \] Input:

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

Output:

(sqrt(d)*d**2*(3*sqrt(c**2*x**2 + 1)*a*c**4*x**4 + 11*sqrt(c**2*x**2 + 1)* 
a*c**2*x**2 + 23*sqrt(c**2*x**2 + 1)*a + 15*int((sqrt(c**2*x**2 + 1)*asinh 
(c*x))/x,x)*b + 15*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**3,x)*b*c**4 + 30* 
int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x,x)*b*c**2 + 15*log(sqrt(c**2*x**2 + 1 
) + c*x - 1)*a - 15*log(sqrt(c**2*x**2 + 1) + c*x + 1)*a))/15