\(\int x^5 \text {arcsinh}(a x)^4 \, dx\) [32]

Optimal result

Integrand size = 10, antiderivative size = 276 \[ \int x^5 \text {arcsinh}(a x)^4 \, dx=\frac {245 x^2}{1152 a^4}-\frac {65 x^4}{3456 a^2}+\frac {x^6}{324}-\frac {245 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{576 a^5}+\frac {65 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{864 a^3}-\frac {x^5 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{54 a}+\frac {245 \text {arcsinh}(a x)^2}{1152 a^6}+\frac {5 x^2 \text {arcsinh}(a x)^2}{16 a^4}-\frac {5 x^4 \text {arcsinh}(a x)^2}{48 a^2}+\frac {1}{18} x^6 \text {arcsinh}(a x)^2-\frac {5 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{24 a^5}+\frac {5 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{36 a^3}-\frac {x^5 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{9 a}+\frac {5 \text {arcsinh}(a x)^4}{96 a^6}+\frac {1}{6} x^6 \text {arcsinh}(a x)^4 \]

[Out]

245/1152*x^2/a^4-65/3456*x^4/a^2+1/324*x^6+245/1152*arcsinh(a*x)^2/a^6+5/16*x^2*arcsinh(a*x)^2/a^4-5/48*x^4*ar
csinh(a*x)^2/a^2+1/18*x^6*arcsinh(a*x)^2+5/96*arcsinh(a*x)^4/a^6+1/6*x^6*arcsinh(a*x)^4-245/576*x*arcsinh(a*x)
*(a^2*x^2+1)^(1/2)/a^5+65/864*x^3*arcsinh(a*x)*(a^2*x^2+1)^(1/2)/a^3-1/54*x^5*arcsinh(a*x)*(a^2*x^2+1)^(1/2)/a
-5/24*x*arcsinh(a*x)^3*(a^2*x^2+1)^(1/2)/a^5+5/36*x^3*arcsinh(a*x)^3*(a^2*x^2+1)^(1/2)/a^3-1/9*x^5*arcsinh(a*x
)^3*(a^2*x^2+1)^(1/2)/a

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5776, 5812, 5783, 30} \[ \int x^5 \text {arcsinh}(a x)^4 \, dx=\frac {5 \text {arcsinh}(a x)^4}{96 a^6}+\frac {245 \text {arcsinh}(a x)^2}{1152 a^6}+\frac {5 x^2 \text {arcsinh}(a x)^2}{16 a^4}+\frac {245 x^2}{1152 a^4}-\frac {5 x^4 \text {arcsinh}(a x)^2}{48 a^2}-\frac {x^5 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{9 a}-\frac {x^5 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{54 a}-\frac {65 x^4}{3456 a^2}-\frac {5 x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{24 a^5}-\frac {245 x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{576 a^5}+\frac {5 x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{36 a^3}+\frac {65 x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{864 a^3}+\frac {1}{6} x^6 \text {arcsinh}(a x)^4+\frac {1}{18} x^6 \text {arcsinh}(a x)^2+\frac {x^6}{324} \]

[In]

Int[x^5*ArcSinh[a*x]^4,x]

[Out]

(245*x^2)/(1152*a^4) - (65*x^4)/(3456*a^2) + x^6/324 - (245*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(576*a^5) + (65*
x^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(864*a^3) - (x^5*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(54*a) + (245*ArcSinh[a*x
]^2)/(1152*a^6) + (5*x^2*ArcSinh[a*x]^2)/(16*a^4) - (5*x^4*ArcSinh[a*x]^2)/(48*a^2) + (x^6*ArcSinh[a*x]^2)/18
- (5*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(24*a^5) + (5*x^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(36*a^3) - (x^5*S
qrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(9*a) + (5*ArcSinh[a*x]^4)/(96*a^6) + (x^6*ArcSinh[a*x]^4)/6

Rule 30

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

Rule 5776

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcS
inh[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[
1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5783

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*S
imp[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 5812

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] + (-Dist[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] - Dist[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
]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} x^6 \text {arcsinh}(a x)^4-\frac {1}{3} (2 a) \int \frac {x^6 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {x^5 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{9 a}+\frac {1}{6} x^6 \text {arcsinh}(a x)^4+\frac {1}{3} \int x^5 \text {arcsinh}(a x)^2 \, dx+\frac {5 \int \frac {x^4 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{9 a} \\ & = \frac {1}{18} x^6 \text {arcsinh}(a x)^2+\frac {5 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{36 a^3}-\frac {x^5 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{9 a}+\frac {1}{6} x^6 \text {arcsinh}(a x)^4-\frac {5 \int \frac {x^2 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{12 a^3}-\frac {5 \int x^3 \text {arcsinh}(a x)^2 \, dx}{12 a^2}-\frac {1}{9} a \int \frac {x^6 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {x^5 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{54 a}-\frac {5 x^4 \text {arcsinh}(a x)^2}{48 a^2}+\frac {1}{18} x^6 \text {arcsinh}(a x)^2-\frac {5 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{24 a^5}+\frac {5 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{36 a^3}-\frac {x^5 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{9 a}+\frac {1}{6} x^6 \text {arcsinh}(a x)^4+\frac {\int x^5 \, dx}{54}+\frac {5 \int \frac {\text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{24 a^5}+\frac {5 \int x \text {arcsinh}(a x)^2 \, dx}{8 a^4}+\frac {5 \int \frac {x^4 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{54 a}+\frac {5 \int \frac {x^4 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{24 a} \\ & = \frac {x^6}{324}+\frac {65 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{864 a^3}-\frac {x^5 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{54 a}+\frac {5 x^2 \text {arcsinh}(a x)^2}{16 a^4}-\frac {5 x^4 \text {arcsinh}(a x)^2}{48 a^2}+\frac {1}{18} x^6 \text {arcsinh}(a x)^2-\frac {5 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{24 a^5}+\frac {5 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{36 a^3}-\frac {x^5 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{9 a}+\frac {5 \text {arcsinh}(a x)^4}{96 a^6}+\frac {1}{6} x^6 \text {arcsinh}(a x)^4-\frac {5 \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{72 a^3}-\frac {5 \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{32 a^3}-\frac {5 \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{8 a^3}-\frac {5 \int x^3 \, dx}{216 a^2}-\frac {5 \int x^3 \, dx}{96 a^2} \\ & = -\frac {65 x^4}{3456 a^2}+\frac {x^6}{324}-\frac {245 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{576 a^5}+\frac {65 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{864 a^3}-\frac {x^5 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{54 a}+\frac {5 x^2 \text {arcsinh}(a x)^2}{16 a^4}-\frac {5 x^4 \text {arcsinh}(a x)^2}{48 a^2}+\frac {1}{18} x^6 \text {arcsinh}(a x)^2-\frac {5 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{24 a^5}+\frac {5 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{36 a^3}-\frac {x^5 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{9 a}+\frac {5 \text {arcsinh}(a x)^4}{96 a^6}+\frac {1}{6} x^6 \text {arcsinh}(a x)^4+\frac {5 \int \frac {\text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{144 a^5}+\frac {5 \int \frac {\text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{64 a^5}+\frac {5 \int \frac {\text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{16 a^5}+\frac {5 \int x \, dx}{144 a^4}+\frac {5 \int x \, dx}{64 a^4}+\frac {5 \int x \, dx}{16 a^4} \\ & = \frac {245 x^2}{1152 a^4}-\frac {65 x^4}{3456 a^2}+\frac {x^6}{324}-\frac {245 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{576 a^5}+\frac {65 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{864 a^3}-\frac {x^5 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{54 a}+\frac {245 \text {arcsinh}(a x)^2}{1152 a^6}+\frac {5 x^2 \text {arcsinh}(a x)^2}{16 a^4}-\frac {5 x^4 \text {arcsinh}(a x)^2}{48 a^2}+\frac {1}{18} x^6 \text {arcsinh}(a x)^2-\frac {5 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{24 a^5}+\frac {5 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{36 a^3}-\frac {x^5 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{9 a}+\frac {5 \text {arcsinh}(a x)^4}{96 a^6}+\frac {1}{6} x^6 \text {arcsinh}(a x)^4 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.60 \[ \int x^5 \text {arcsinh}(a x)^4 \, dx=\frac {a^2 x^2 \left (2205-195 a^2 x^2+32 a^4 x^4\right )-6 a x \sqrt {1+a^2 x^2} \left (735-130 a^2 x^2+32 a^4 x^4\right ) \text {arcsinh}(a x)+9 \left (245+360 a^2 x^2-120 a^4 x^4+64 a^6 x^6\right ) \text {arcsinh}(a x)^2-144 a x \sqrt {1+a^2 x^2} \left (15-10 a^2 x^2+8 a^4 x^4\right ) \text {arcsinh}(a x)^3+108 \left (5+16 a^6 x^6\right ) \text {arcsinh}(a x)^4}{10368 a^6} \]

[In]

Integrate[x^5*ArcSinh[a*x]^4,x]

[Out]

(a^2*x^2*(2205 - 195*a^2*x^2 + 32*a^4*x^4) - 6*a*x*Sqrt[1 + a^2*x^2]*(735 - 130*a^2*x^2 + 32*a^4*x^4)*ArcSinh[
a*x] + 9*(245 + 360*a^2*x^2 - 120*a^4*x^4 + 64*a^6*x^6)*ArcSinh[a*x]^2 - 144*a*x*Sqrt[1 + a^2*x^2]*(15 - 10*a^
2*x^2 + 8*a^4*x^4)*ArcSinh[a*x]^3 + 108*(5 + 16*a^6*x^6)*ArcSinh[a*x]^4)/(10368*a^6)

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(\frac {\frac {a^{6} x^{6} \operatorname {arcsinh}\left (a x \right )^{4}}{6}-\frac {a^{5} x^{5} \operatorname {arcsinh}\left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}}{9}+\frac {5 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}}{36}-\frac {5 \operatorname {arcsinh}\left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}\, a x}{24}+\frac {5 \operatorname {arcsinh}\left (a x \right )^{4}}{96}+\frac {\operatorname {arcsinh}\left (a x \right )^{2} a^{6} x^{6}}{18}-\frac {\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{5} x^{5}}{54}+\frac {65 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}}{864}-\frac {245 \,\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x}{576}-\frac {115 \operatorname {arcsinh}\left (a x \right )^{2}}{1152}+\frac {a^{6} x^{6}}{324}-\frac {65 a^{4} x^{4}}{3456}+\frac {245 a^{2} x^{2}}{1152}+\frac {245}{1152}-\frac {5 a^{4} x^{4} \operatorname {arcsinh}\left (a x \right )^{2}}{48}+\frac {5 \operatorname {arcsinh}\left (a x \right )^{2} \left (a^{2} x^{2}+1\right )}{16}}{a^{6}}\)	\(242\)
default	\(\frac {\frac {a^{6} x^{6} \operatorname {arcsinh}\left (a x \right )^{4}}{6}-\frac {a^{5} x^{5} \operatorname {arcsinh}\left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}}{9}+\frac {5 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}}{36}-\frac {5 \operatorname {arcsinh}\left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}\, a x}{24}+\frac {5 \operatorname {arcsinh}\left (a x \right )^{4}}{96}+\frac {\operatorname {arcsinh}\left (a x \right )^{2} a^{6} x^{6}}{18}-\frac {\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{5} x^{5}}{54}+\frac {65 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}}{864}-\frac {245 \,\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x}{576}-\frac {115 \operatorname {arcsinh}\left (a x \right )^{2}}{1152}+\frac {a^{6} x^{6}}{324}-\frac {65 a^{4} x^{4}}{3456}+\frac {245 a^{2} x^{2}}{1152}+\frac {245}{1152}-\frac {5 a^{4} x^{4} \operatorname {arcsinh}\left (a x \right )^{2}}{48}+\frac {5 \operatorname {arcsinh}\left (a x \right )^{2} \left (a^{2} x^{2}+1\right )}{16}}{a^{6}}\)	\(242\)

[In]

int(x^5*arcsinh(a*x)^4,x,method=_RETURNVERBOSE)

[Out]

1/a^6*(1/6*a^6*x^6*arcsinh(a*x)^4-1/9*a^5*x^5*arcsinh(a*x)^3*(a^2*x^2+1)^(1/2)+5/36*a^3*x^3*arcsinh(a*x)^3*(a^
2*x^2+1)^(1/2)-5/24*arcsinh(a*x)^3*(a^2*x^2+1)^(1/2)*a*x+5/96*arcsinh(a*x)^4+1/18*arcsinh(a*x)^2*a^6*x^6-1/54*
arcsinh(a*x)*(a^2*x^2+1)^(1/2)*a^5*x^5+65/864*a^3*x^3*arcsinh(a*x)*(a^2*x^2+1)^(1/2)-245/576*arcsinh(a*x)*(a^2
*x^2+1)^(1/2)*a*x-115/1152*arcsinh(a*x)^2+1/324*a^6*x^6-65/3456*a^4*x^4+245/1152*a^2*x^2+245/1152-5/48*a^4*x^4
*arcsinh(a*x)^2+5/16*arcsinh(a*x)^2*(a^2*x^2+1))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.75 \[ \int x^5 \text {arcsinh}(a x)^4 \, dx=\frac {32 \, a^{6} x^{6} - 195 \, a^{4} x^{4} + 108 \, {\left (16 \, a^{6} x^{6} + 5\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{4} - 144 \, {\left (8 \, a^{5} x^{5} - 10 \, a^{3} x^{3} + 15 \, a x\right )} \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} + 2205 \, a^{2} x^{2} + 9 \, {\left (64 \, a^{6} x^{6} - 120 \, a^{4} x^{4} + 360 \, a^{2} x^{2} + 245\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} - 6 \, {\left (32 \, a^{5} x^{5} - 130 \, a^{3} x^{3} + 735 \, a x\right )} \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{10368 \, a^{6}} \]

[In]

integrate(x^5*arcsinh(a*x)^4,x, algorithm="fricas")

[Out]

1/10368*(32*a^6*x^6 - 195*a^4*x^4 + 108*(16*a^6*x^6 + 5)*log(a*x + sqrt(a^2*x^2 + 1))^4 - 144*(8*a^5*x^5 - 10*
a^3*x^3 + 15*a*x)*sqrt(a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 + 1))^3 + 2205*a^2*x^2 + 9*(64*a^6*x^6 - 120*a^4*x^
4 + 360*a^2*x^2 + 245)*log(a*x + sqrt(a^2*x^2 + 1))^2 - 6*(32*a^5*x^5 - 130*a^3*x^3 + 735*a*x)*sqrt(a^2*x^2 +
1)*log(a*x + sqrt(a^2*x^2 + 1)))/a^6

Sympy [A] (verification not implemented)

Time = 1.32 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.97 \[ \int x^5 \text {arcsinh}(a x)^4 \, dx=\begin {cases} \frac {x^{6} \operatorname {asinh}^{4}{\left (a x \right )}}{6} + \frac {x^{6} \operatorname {asinh}^{2}{\left (a x \right )}}{18} + \frac {x^{6}}{324} - \frac {x^{5} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a x \right )}}{9 a} - \frac {x^{5} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{54 a} - \frac {5 x^{4} \operatorname {asinh}^{2}{\left (a x \right )}}{48 a^{2}} - \frac {65 x^{4}}{3456 a^{2}} + \frac {5 x^{3} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a x \right )}}{36 a^{3}} + \frac {65 x^{3} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{864 a^{3}} + \frac {5 x^{2} \operatorname {asinh}^{2}{\left (a x \right )}}{16 a^{4}} + \frac {245 x^{2}}{1152 a^{4}} - \frac {5 x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a x \right )}}{24 a^{5}} - \frac {245 x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{576 a^{5}} + \frac {5 \operatorname {asinh}^{4}{\left (a x \right )}}{96 a^{6}} + \frac {245 \operatorname {asinh}^{2}{\left (a x \right )}}{1152 a^{6}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

integrate(x**5*asinh(a*x)**4,x)

[Out]

Piecewise((x**6*asinh(a*x)**4/6 + x**6*asinh(a*x)**2/18 + x**6/324 - x**5*sqrt(a**2*x**2 + 1)*asinh(a*x)**3/(9
*a) - x**5*sqrt(a**2*x**2 + 1)*asinh(a*x)/(54*a) - 5*x**4*asinh(a*x)**2/(48*a**2) - 65*x**4/(3456*a**2) + 5*x*
*3*sqrt(a**2*x**2 + 1)*asinh(a*x)**3/(36*a**3) + 65*x**3*sqrt(a**2*x**2 + 1)*asinh(a*x)/(864*a**3) + 5*x**2*as
inh(a*x)**2/(16*a**4) + 245*x**2/(1152*a**4) - 5*x*sqrt(a**2*x**2 + 1)*asinh(a*x)**3/(24*a**5) - 245*x*sqrt(a*
*2*x**2 + 1)*asinh(a*x)/(576*a**5) + 5*asinh(a*x)**4/(96*a**6) + 245*asinh(a*x)**2/(1152*a**6), Ne(a, 0)), (0,
True))

Maxima [F]

\[ \int x^5 \text {arcsinh}(a x)^4 \, dx=\int { x^{5} \operatorname {arsinh}\left (a x\right )^{4} \,d x } \]

[In]

integrate(x^5*arcsinh(a*x)^4,x, algorithm="maxima")

[Out]

1/6*x^6*log(a*x + sqrt(a^2*x^2 + 1))^4 - integrate(2/3*(a^3*x^8 + sqrt(a^2*x^2 + 1)*a^2*x^7 + a*x^6)*log(a*x +
sqrt(a^2*x^2 + 1))^3/(a^3*x^3 + a*x + (a^2*x^2 + 1)^(3/2)), x)

Giac [F(-2)]

Exception generated. \[ \int x^5 \text {arcsinh}(a x)^4 \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^5*arcsinh(a*x)^4,x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int x^5 \text {arcsinh}(a x)^4 \, dx=\int x^5\,{\mathrm {asinh}\left (a\,x\right )}^4 \,d x \]

[In]

int(x^5*asinh(a*x)^4,x)

[Out]

int(x^5*asinh(a*x)^4, x)