\(\int \frac {(\pi +c^2 \pi x^2)^{5/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx\) [81]

Optimal result

Integrand size = 26, antiderivative size = 205 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=-\frac {b c \pi ^{5/2}}{2 x}-\frac {7}{3} b c^3 \pi ^{5/2} x-\frac {1}{9} b c^5 \pi ^{5/2} x^3+\frac {5}{2} c^2 \pi ^2 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))+\frac {5}{6} c^2 \pi \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}-5 c^2 \pi ^{5/2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-\frac {5}{2} b c^2 \pi ^{5/2} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )+\frac {5}{2} b c^2 \pi ^{5/2} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) \] Output:

-1/2*b*c*Pi^(5/2)/x-7/3*b*c^3*Pi^(5/2)*x-1/9*b*c^5*Pi^(5/2)*x^3+5/2*c^2*Pi 
^2*(Pi*c^2*x^2+Pi)^(1/2)*(a+b*arcsinh(c*x))+5/6*c^2*Pi*(Pi*c^2*x^2+Pi)^(3/ 
2)*(a+b*arcsinh(c*x))-1/2*(Pi*c^2*x^2+Pi)^(5/2)*(a+b*arcsinh(c*x))/x^2-5*c 
^2*Pi^(5/2)*(a+b*arcsinh(c*x))*arctanh(c*x+(c^2*x^2+1)^(1/2))-5/2*b*c^2*Pi 
^(5/2)*polylog(2,-c*x-(c^2*x^2+1)^(1/2))+5/2*b*c^2*Pi^(5/2)*polylog(2,c*x+ 
(c^2*x^2+1)^(1/2))
 

Mathematica [A] (verified)

Time = 1.40 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.70 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=\frac {\pi ^{5/2} \left (-168 b c^3 x^3-8 b c^5 x^5-36 a \sqrt {1+c^2 x^2}+168 a c^2 x^2 \sqrt {1+c^2 x^2}+24 a c^4 x^4 \sqrt {1+c^2 x^2}+168 b c^2 x^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+24 b c^4 x^4 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)-9 b c^3 x^3 \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )-9 b c^2 x^2 \text {arcsinh}(c x) \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )+180 b c^2 x^2 \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-180 b c^2 x^2 \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+180 a c^2 x^2 \log (x)-180 a c^2 x^2 \log \left (\pi \left (1+\sqrt {1+c^2 x^2}\right )\right )+180 b c^2 x^2 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-180 b c^2 x^2 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )+36 b c x \sinh ^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )-36 b \text {arcsinh}(c x) \sinh ^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}{72 x^2} \] Input:

Integrate[((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x^3,x]
 

Output:

(Pi^(5/2)*(-168*b*c^3*x^3 - 8*b*c^5*x^5 - 36*a*Sqrt[1 + c^2*x^2] + 168*a*c 
^2*x^2*Sqrt[1 + c^2*x^2] + 24*a*c^4*x^4*Sqrt[1 + c^2*x^2] + 168*b*c^2*x^2* 
Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + 24*b*c^4*x^4*Sqrt[1 + c^2*x^2]*ArcSinh[c* 
x] - 9*b*c^3*x^3*Csch[ArcSinh[c*x]/2]^2 - 9*b*c^2*x^2*ArcSinh[c*x]*Csch[Ar 
cSinh[c*x]/2]^2 + 180*b*c^2*x^2*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - 
180*b*c^2*x^2*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 180*a*c^2*x^2*Log[ 
x] - 180*a*c^2*x^2*Log[Pi*(1 + Sqrt[1 + c^2*x^2])] + 180*b*c^2*x^2*PolyLog 
[2, -E^(-ArcSinh[c*x])] - 180*b*c^2*x^2*PolyLog[2, E^(-ArcSinh[c*x])] + 36 
*b*c*x*Sinh[ArcSinh[c*x]/2]^2 - 36*b*ArcSinh[c*x]*Sinh[ArcSinh[c*x]/2]^2)) 
/(72*x^2)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.29 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6222, 244, 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.

\(\displaystyle \int \frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx\)

\(\Big \downarrow \) 6222

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

\(\Big \downarrow \) 244

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6223

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6221

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 6231

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

Int[((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x^3,x]
 

Output:

(b*c*Pi^(5/2)*(-x^(-1) + 2*c^2*x + (c^4*x^3)/3))/2 - ((Pi + c^2*Pi*x^2)^(5 
/2)*(a + b*ArcSinh[c*x]))/(2*x^2) + (5*c^2*Pi*(-1/3*(b*c*Pi^(3/2)*(x + (c^ 
2*x^3)/3)) + ((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/3 + Pi*(-(b*c* 
Sqrt[Pi]*x) + Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]) + I*Sqrt[Pi]*((2* 
I)*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]] + I*b*PolyLog[2, -E^ArcSin 
h[c*x]] - I*b*PolyLog[2, E^ArcSinh[c*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 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 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 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]
 

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 = 0.99 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.70

method	result	size
default	\(a \left (-\frac {\left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {7}{2}}}{2 \pi \,x^{2}}+\frac {5 c^{2} \left (\frac {\left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{5}+\pi \left (\frac {\left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{3}+\pi \left (\sqrt {\pi \,c^{2} x^{2}+\pi }-\sqrt {\pi }\, \operatorname {arctanh}\left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )\right )\right )\right )}{2}\right )-\frac {5 b \,c^{2} \pi ^{\frac {5}{2}} \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )}{2}-\frac {b c \,\pi ^{\frac {5}{2}}}{2 x}-\frac {b \,\pi ^{\frac {5}{2}} \operatorname {arcsinh}\left (x c \right )}{2 \sqrt {c^{2} x^{2}+1}\, x^{2}}-\frac {b \,\pi ^{\frac {5}{2}} \operatorname {arcsinh}\left (x c \right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}+\frac {b \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) \pi ^{\frac {5}{2}} x^{2} c^{4}}{3}+\frac {7 b \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) \pi ^{\frac {5}{2}} c^{2}}{3}+\frac {5 b \,\pi ^{\frac {5}{2}} c^{2} \operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )}{2}-\frac {5 b \,\pi ^{\frac {5}{2}} c^{2} \operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )}{2}-\frac {b \,c^{5} \pi ^{\frac {5}{2}} x^{3}}{9}-\frac {7 b \,c^{3} \pi ^{\frac {5}{2}} x}{3}+\frac {5 b \,c^{2} \pi ^{\frac {5}{2}} \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )}{2}\)	\(348\)
parts	\(a \left (-\frac {\left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {7}{2}}}{2 \pi \,x^{2}}+\frac {5 c^{2} \left (\frac {\left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{5}+\pi \left (\frac {\left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{3}+\pi \left (\sqrt {\pi \,c^{2} x^{2}+\pi }-\sqrt {\pi }\, \operatorname {arctanh}\left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )\right )\right )\right )}{2}\right )-\frac {5 b \,c^{2} \pi ^{\frac {5}{2}} \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )}{2}-\frac {b c \,\pi ^{\frac {5}{2}}}{2 x}-\frac {b \,\pi ^{\frac {5}{2}} \operatorname {arcsinh}\left (x c \right )}{2 \sqrt {c^{2} x^{2}+1}\, x^{2}}-\frac {b \,\pi ^{\frac {5}{2}} \operatorname {arcsinh}\left (x c \right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}+\frac {b \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) \pi ^{\frac {5}{2}} x^{2} c^{4}}{3}+\frac {7 b \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) \pi ^{\frac {5}{2}} c^{2}}{3}+\frac {5 b \,\pi ^{\frac {5}{2}} c^{2} \operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )}{2}-\frac {5 b \,\pi ^{\frac {5}{2}} c^{2} \operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )}{2}-\frac {b \,c^{5} \pi ^{\frac {5}{2}} x^{3}}{9}-\frac {7 b \,c^{3} \pi ^{\frac {5}{2}} x}{3}+\frac {5 b \,c^{2} \pi ^{\frac {5}{2}} \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )}{2}\)	\(348\)

Input:

int((Pi*c^2*x^2+Pi)^(5/2)*(a+b*arcsinh(x*c))/x^3,x,method=_RETURNVERBOSE)
 

Output:

a*(-1/2/Pi/x^2*(Pi*c^2*x^2+Pi)^(7/2)+5/2*c^2*(1/5*(Pi*c^2*x^2+Pi)^(5/2)+Pi 
*(1/3*(Pi*c^2*x^2+Pi)^(3/2)+Pi*((Pi*c^2*x^2+Pi)^(1/2)-Pi^(1/2)*arctanh(Pi^ 
(1/2)/(Pi*c^2*x^2+Pi)^(1/2))))))-5/2*b*c^2*Pi^(5/2)*polylog(2,-x*c-(c^2*x^ 
2+1)^(1/2))-1/2*b*c*Pi^(5/2)/x-1/2*b*Pi^(5/2)/(c^2*x^2+1)^(1/2)/x^2*arcsin 
h(x*c)-1/2*b*Pi^(5/2)/(c^2*x^2+1)^(1/2)*arcsinh(x*c)*c^2+1/3*b*(c^2*x^2+1) 
^(1/2)*arcsinh(x*c)*Pi^(5/2)*x^2*c^4+7/3*b*(c^2*x^2+1)^(1/2)*arcsinh(x*c)* 
Pi^(5/2)*c^2+5/2*b*Pi^(5/2)*c^2*arcsinh(x*c)*ln(1-x*c-(c^2*x^2+1)^(1/2))-5 
/2*b*Pi^(5/2)*c^2*arcsinh(x*c)*ln(1+x*c+(c^2*x^2+1)^(1/2))-1/9*b*c^5*Pi^(5 
/2)*x^3-7/3*b*c^3*Pi^(5/2)*x+5/2*b*c^2*Pi^(5/2)*polylog(2,x*c+(c^2*x^2+1)^ 
(1/2))
 

Fricas [F]

\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=\int { \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \] Input:

integrate((pi*c^2*x^2+pi)^(5/2)*(a+b*arcsinh(c*x))/x^3,x, algorithm="frica 
s")
 

Output:

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

Sympy [F]

\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=\pi ^{\frac {5}{2}} \left (\int \frac {a \sqrt {c^{2} x^{2} + 1}}{x^{3}}\, dx + \int \frac {2 a c^{2} \sqrt {c^{2} x^{2} + 1}}{x}\, dx + \int a c^{4} x \sqrt {c^{2} x^{2} + 1}\, dx + \int \frac {b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {2 b c^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{x}\, dx + \int b c^{4} x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \] Input:

integrate((pi*c**2*x**2+pi)**(5/2)*(a+b*asinh(c*x))/x**3,x)
 

Output:

pi**(5/2)*(Integral(a*sqrt(c**2*x**2 + 1)/x**3, x) + Integral(2*a*c**2*sqr 
t(c**2*x**2 + 1)/x, x) + Integral(a*c**4*x*sqrt(c**2*x**2 + 1), x) + Integ 
ral(b*sqrt(c**2*x**2 + 1)*asinh(c*x)/x**3, x) + Integral(2*b*c**2*sqrt(c** 
2*x**2 + 1)*asinh(c*x)/x, x) + Integral(b*c**4*x*sqrt(c**2*x**2 + 1)*asinh 
(c*x), x))
 

Maxima [F]

\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=\int { \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \] Input:

integrate((pi*c^2*x^2+pi)^(5/2)*(a+b*arcsinh(c*x))/x^3,x, algorithm="maxim 
a")
 

Output:

-1/6*(15*pi^(5/2)*c^2*arcsinh(1/(c*abs(x))) - 15*pi^2*sqrt(pi + pi*c^2*x^2 
)*c^2 - 5*pi*(pi + pi*c^2*x^2)^(3/2)*c^2 - 3*(pi + pi*c^2*x^2)^(5/2)*c^2 + 
 3*(pi + pi*c^2*x^2)^(7/2)/(pi*x^2))*a + b*integrate((pi + pi*c^2*x^2)^(5/ 
2)*log(c*x + sqrt(c^2*x^2 + 1))/x^3, x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((pi*c^2*x^2+pi)^(5/2)*(a+b*arcsinh(c*x))/x^3,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. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{5/2}}{x^3} \,d x \] Input:

int(((a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(5/2))/x^3,x)
 

Output:

int(((a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(5/2))/x^3, x)
 

Reduce [F]

\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=\frac {\sqrt {\pi }\, \pi ^{2} \left (2 \sqrt {c^{2} x^{2}+1}\, a \,c^{4} x^{4}+14 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, a +6 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{x^{3}}d x \right ) b \,x^{2}+12 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{x}d x \right ) b \,c^{2} x^{2}+6 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x d x \right ) b \,c^{4} x^{2}+15 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x -1\right ) a \,c^{2} x^{2}-15 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x +1\right ) a \,c^{2} x^{2}\right )}{6 x^{2}} \] Input:

int((Pi*c^2*x^2+Pi)^(5/2)*(a+b*asinh(c*x))/x^3,x)
 

Output:

(sqrt(pi)*pi**2*(2*sqrt(c**2*x**2 + 1)*a*c**4*x**4 + 14*sqrt(c**2*x**2 + 1 
)*a*c**2*x**2 - 3*sqrt(c**2*x**2 + 1)*a + 6*int((sqrt(c**2*x**2 + 1)*asinh 
(c*x))/x**3,x)*b*x**2 + 12*int((sqrt(c**2*x**2 + 1)*asinh(c*x))/x,x)*b*c** 
2*x**2 + 6*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x,x)*b*c**4*x**2 + 15*log(sq 
rt(c**2*x**2 + 1) + c*x - 1)*a*c**2*x**2 - 15*log(sqrt(c**2*x**2 + 1) + c* 
x + 1)*a*c**2*x**2))/(6*x**2)