\(\int x^2 (d+c^2 d x^2)^{5/2} (a+b \text {arcsinh}(c x)) \, dx\) [146]

Optimal result

Integrand size = 26, antiderivative size = 337 \[ \int x^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {5 b d^2 x^2 \sqrt {d+c^2 d x^2}}{256 c \sqrt {1+c^2 x^2}}-\frac {59 b c d^2 x^4 \sqrt {d+c^2 d x^2}}{768 \sqrt {1+c^2 x^2}}-\frac {17 b c^3 d^2 x^6 \sqrt {d+c^2 d x^2}}{288 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^8 \sqrt {d+c^2 d x^2}}{64 \sqrt {1+c^2 x^2}}+\frac {5 d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{48} d x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{8} x^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {5 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{256 b c^3 \sqrt {1+c^2 x^2}} \] Output:

-5/256*b*d^2*x^2*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-59/768*b*c*d^2*x^ 
4*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-17/288*b*c^3*d^2*x^6*(c^2*d*x^2+d) 
^(1/2)/(c^2*x^2+1)^(1/2)-1/64*b*c^5*d^2*x^8*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1 
)^(1/2)+5/128*d^2*x*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))/c^2+5/64*d^2*x^ 
3*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))+5/48*d*x^3*(c^2*d*x^2+d)^(3/2)*(a 
+b*arcsinh(c*x))+1/8*x^3*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))-5/256*d^2* 
(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2/b/c^3/(c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.79 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.15 \[ \int x^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {d^2 \left (2880 a c x \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+22656 a c^3 x^3 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+26112 a c^5 x^5 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+9216 a c^7 x^7 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}-1440 b \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)^2+576 b \sqrt {d+c^2 d x^2} \cosh (2 \text {arcsinh}(c x))-144 b \sqrt {d+c^2 d x^2} \cosh (4 \text {arcsinh}(c x))-64 b \sqrt {d+c^2 d x^2} \cosh (6 \text {arcsinh}(c x))-9 b \sqrt {d+c^2 d x^2} \cosh (8 \text {arcsinh}(c x))-2880 a \sqrt {d} \sqrt {1+c^2 x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+24 b \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) (-48 \sinh (2 \text {arcsinh}(c x))+24 \sinh (4 \text {arcsinh}(c x))+16 \sinh (6 \text {arcsinh}(c x))+3 \sinh (8 \text {arcsinh}(c x)))\right )}{73728 c^3 \sqrt {1+c^2 x^2}} \] Input:

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

Output:

(d^2*(2880*a*c*x*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 22656*a*c^3*x^3*S 
qrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 26112*a*c^5*x^5*Sqrt[1 + c^2*x^2]*S 
qrt[d + c^2*d*x^2] + 9216*a*c^7*x^7*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] 
- 1440*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^2 + 576*b*Sqrt[d + c^2*d*x^2]*Co 
sh[2*ArcSinh[c*x]] - 144*b*Sqrt[d + c^2*d*x^2]*Cosh[4*ArcSinh[c*x]] - 64*b 
*Sqrt[d + c^2*d*x^2]*Cosh[6*ArcSinh[c*x]] - 9*b*Sqrt[d + c^2*d*x^2]*Cosh[8 
*ArcSinh[c*x]] - 2880*a*Sqrt[d]*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt 
[d + c^2*d*x^2]] + 24*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*(-48*Sinh[2*ArcSi 
nh[c*x]] + 24*Sinh[4*ArcSinh[c*x]] + 16*Sinh[6*ArcSinh[c*x]] + 3*Sinh[8*Ar 
cSinh[c*x]])))/(73728*c^3*Sqrt[1 + c^2*x^2])
 

Rubi [A] (verified)

Time = 1.66 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.02, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {6223, 243, 49, 2009, 6223, 244, 2009, 6221, 15, 6227, 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.

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

\(\Big \downarrow \) 6223

\(\displaystyle \frac {5}{8} d \int x^2 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))dx-\frac {b c d^2 \sqrt {c^2 d x^2+d} \int x^3 \left (c^2 x^2+1\right )^2dx}{8 \sqrt {c^2 x^2+1}}+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {5}{8} d \int x^2 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))dx-\frac {b c d^2 \sqrt {c^2 d x^2+d} \int x^2 \left (c^2 x^2+1\right )^2dx^2}{16 \sqrt {c^2 x^2+1}}+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 49

\(\displaystyle \frac {5}{8} d \int x^2 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))dx-\frac {b c d^2 \sqrt {c^2 d x^2+d} \int \left (c^4 x^6+2 c^2 x^4+x^2\right )dx^2}{16 \sqrt {c^2 x^2+1}}+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {5}{8} d \int x^2 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))dx+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}+\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {c^2 d x^2+d}}{16 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 6223

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))dx-\frac {b c d \sqrt {c^2 d x^2+d} \int x^3 \left (c^2 x^2+1\right )dx}{6 \sqrt {c^2 x^2+1}}+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))\right )+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}+\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {c^2 d x^2+d}}{16 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 244

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))dx-\frac {b c d \sqrt {c^2 d x^2+d} \int \left (c^2 x^5+x^3\right )dx}{6 \sqrt {c^2 x^2+1}}+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))\right )+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}+\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {c^2 d x^2+d}}{16 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))dx+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {b c d \left (\frac {c^2 x^6}{6}+\frac {x^4}{4}\right ) \sqrt {c^2 d x^2+d}}{6 \sqrt {c^2 x^2+1}}\right )+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}+\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {c^2 d x^2+d}}{16 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 6221

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{4 \sqrt {c^2 x^2+1}}-\frac {b c \sqrt {c^2 d x^2+d} \int x^3dx}{4 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))\right )+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {b c d \left (\frac {c^2 x^6}{6}+\frac {x^4}{4}\right ) \sqrt {c^2 d x^2+d}}{6 \sqrt {c^2 x^2+1}}\right )+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}+\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {c^2 d x^2+d}}{16 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{4 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))-\frac {b c x^4 \sqrt {c^2 d x^2+d}}{16 \sqrt {c^2 x^2+1}}\right )+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {b c d \left (\frac {c^2 x^6}{6}+\frac {x^4}{4}\right ) \sqrt {c^2 d x^2+d}}{6 \sqrt {c^2 x^2+1}}\right )+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}+\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {c^2 d x^2+d}}{16 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 6227

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {c^2 d x^2+d} \left (-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx}{2 c^2}-\frac {b \int xdx}{2 c}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}\right )}{4 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))-\frac {b c x^4 \sqrt {c^2 d x^2+d}}{16 \sqrt {c^2 x^2+1}}\right )+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {b c d \left (\frac {c^2 x^6}{6}+\frac {x^4}{4}\right ) \sqrt {c^2 d x^2+d}}{6 \sqrt {c^2 x^2+1}}\right )+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}+\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {c^2 d x^2+d}}{16 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {c^2 d x^2+d} \left (-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))-\frac {b c x^4 \sqrt {c^2 d x^2+d}}{16 \sqrt {c^2 x^2+1}}\right )+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {b c d \left (\frac {c^2 x^6}{6}+\frac {x^4}{4}\right ) \sqrt {c^2 d x^2+d}}{6 \sqrt {c^2 x^2+1}}\right )+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}+\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {c^2 d x^2+d}}{16 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 6198

\(\displaystyle \frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{8} d \left (\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{2} d \left (\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))+\frac {\sqrt {c^2 d x^2+d} \left (-\frac {(a+b \text {arcsinh}(c x))^2}{4 b c^3}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 \sqrt {c^2 x^2+1}}-\frac {b c x^4 \sqrt {c^2 d x^2+d}}{16 \sqrt {c^2 x^2+1}}\right )-\frac {b c d \left (\frac {c^2 x^6}{6}+\frac {x^4}{4}\right ) \sqrt {c^2 d x^2+d}}{6 \sqrt {c^2 x^2+1}}\right )-\frac {b c d^2 \left (\frac {c^4 x^8}{4}+\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {c^2 d x^2+d}}{16 \sqrt {c^2 x^2+1}}\)

Input:

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

Output:

-1/16*(b*c*d^2*Sqrt[d + c^2*d*x^2]*(x^4/2 + (2*c^2*x^6)/3 + (c^4*x^8)/4))/ 
Sqrt[1 + c^2*x^2] + (x^3*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/8 + ( 
5*d*(-1/6*(b*c*d*Sqrt[d + c^2*d*x^2]*(x^4/4 + (c^2*x^6)/6))/Sqrt[1 + c^2*x 
^2] + (x^3*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/6 + (d*(-1/16*(b*c* 
x^4*Sqrt[d + c^2*d*x^2])/Sqrt[1 + c^2*x^2] + (x^3*Sqrt[d + c^2*d*x^2]*(a + 
 b*ArcSinh[c*x]))/4 + (Sqrt[d + c^2*d*x^2]*(-1/4*(b*x^2)/c + (x*Sqrt[1 + c 
^2*x^2]*(a + b*ArcSinh[c*x]))/(2*c^2) - (a + b*ArcSinh[c*x])^2/(4*b*c^3))) 
/(4*Sqrt[1 + c^2*x^2])))/2))/8
 

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 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 6227

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1164\) vs. \(2(291)=582\).

Time = 0.64 (sec) , antiderivative size = 1165, normalized size of antiderivative = 3.46

method	result	size
default	\(\text {Expression too large to display}\)	\(1165\)
parts	\(\text {Expression too large to display}\)	\(1165\)

Input:

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

Output:

1/8*a*x*(c^2*d*x^2+d)^(7/2)/c^2/d-1/48*a/c^2*x*(c^2*d*x^2+d)^(5/2)-5/192*a 
/c^2*d*x*(c^2*d*x^2+d)^(3/2)-5/128*a/c^2*d^2*x*(c^2*d*x^2+d)^(1/2)-5/128*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)+b*(-5 
/256*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^3*arcsinh(x*c)^2*d^2+1/1638 
4*(d*(c^2*x^2+1))^(1/2)*(128*c^9*x^9+128*(c^2*x^2+1)^(1/2)*x^8*c^8+320*x^7 
*c^7+256*x^6*c^6*(c^2*x^2+1)^(1/2)+272*x^5*c^5+160*x^4*c^4*(c^2*x^2+1)^(1/ 
2)+88*x^3*c^3+32*x^2*c^2*(c^2*x^2+1)^(1/2)+8*x*c+(c^2*x^2+1)^(1/2))*(-1+8* 
arcsinh(x*c))*d^2/c^3/(c^2*x^2+1)+1/2304*(d*(c^2*x^2+1))^(1/2)*(32*x^7*c^7 
+32*x^6*c^6*(c^2*x^2+1)^(1/2)+64*x^5*c^5+48*x^4*c^4*(c^2*x^2+1)^(1/2)+38*x 
^3*c^3+18*x^2*c^2*(c^2*x^2+1)^(1/2)+6*x*c+(c^2*x^2+1)^(1/2))*(-1+6*arcsinh 
(x*c))*d^2/c^3/(c^2*x^2+1)+1/1024*(d*(c^2*x^2+1))^(1/2)*(8*x^5*c^5+8*x^4*c 
^4*(c^2*x^2+1)^(1/2)+12*x^3*c^3+8*x^2*c^2*(c^2*x^2+1)^(1/2)+4*x*c+(c^2*x^2 
+1)^(1/2))*(-1+4*arcsinh(x*c))*d^2/c^3/(c^2*x^2+1)-1/256*(d*(c^2*x^2+1))^( 
1/2)*(2*x^3*c^3+2*x^2*c^2*(c^2*x^2+1)^(1/2)+2*x*c+(c^2*x^2+1)^(1/2))*(-1+2 
*arcsinh(x*c))*d^2/c^3/(c^2*x^2+1)-1/256*(d*(c^2*x^2+1))^(1/2)*(2*x^3*c^3- 
2*x^2*c^2*(c^2*x^2+1)^(1/2)+2*x*c-(c^2*x^2+1)^(1/2))*(1+2*arcsinh(x*c))*d^ 
2/c^3/(c^2*x^2+1)+1/1024*(d*(c^2*x^2+1))^(1/2)*(8*x^5*c^5-8*x^4*c^4*(c^2*x 
^2+1)^(1/2)+12*x^3*c^3-8*x^2*c^2*(c^2*x^2+1)^(1/2)+4*x*c-(c^2*x^2+1)^(1/2) 
)*(1+4*arcsinh(x*c))*d^2/c^3/(c^2*x^2+1)+1/2304*(d*(c^2*x^2+1))^(1/2)*(32* 
x^7*c^7-32*x^6*c^6*(c^2*x^2+1)^(1/2)+64*x^5*c^5-48*x^4*c^4*(c^2*x^2+1)^...
 

Fricas [F]

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

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

Output:

integral((a*c^4*d^2*x^6 + 2*a*c^2*d^2*x^4 + a*d^2*x^2 + (b*c^4*d^2*x^6 + 2 
*b*c^2*d^2*x^4 + b*d^2*x^2)*arcsinh(c*x))*sqrt(c^2*d*x^2 + d), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\sqrt {d}\, d^{2} \left (48 \sqrt {c^{2} x^{2}+1}\, a \,c^{7} x^{7}+136 \sqrt {c^{2} x^{2}+1}\, a \,c^{5} x^{5}+118 \sqrt {c^{2} x^{2}+1}\, a \,c^{3} x^{3}+15 \sqrt {c^{2} x^{2}+1}\, a c x +384 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{6}d x \right ) b \,c^{7}+768 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{4}d x \right ) b \,c^{5}+384 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{2}d x \right ) b \,c^{3}-15 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a \right )}{384 c^{3}} \] Input:

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

Output:

(sqrt(d)*d**2*(48*sqrt(c**2*x**2 + 1)*a*c**7*x**7 + 136*sqrt(c**2*x**2 + 1 
)*a*c**5*x**5 + 118*sqrt(c**2*x**2 + 1)*a*c**3*x**3 + 15*sqrt(c**2*x**2 + 
1)*a*c*x + 384*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**6,x)*b*c**7 + 768*int 
(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**4,x)*b*c**5 + 384*int(sqrt(c**2*x**2 + 
1)*asinh(c*x)*x**2,x)*b*c**3 - 15*log(sqrt(c**2*x**2 + 1) + c*x)*a))/(384* 
c**3)