\(\int x^2 \text {arcsinh}(a+b x)^3 \, dx\) [75]

Optimal result

Integrand size = 12, antiderivative size = 355 \[ \int x^2 \text {arcsinh}(a+b x)^3 \, dx=\frac {14 \sqrt {1+(a+b x)^2}}{9 b^3}-\frac {6 a^2 \sqrt {1+(a+b x)^2}}{b^3}+\frac {3 a (a+b x) \sqrt {1+(a+b x)^2}}{4 b^3}-\frac {2 \left (1+(a+b x)^2\right )^{3/2}}{27 b^3}-\frac {3 a \text {arcsinh}(a+b x)}{4 b^3}-\frac {4 (a+b x) \text {arcsinh}(a+b x)}{3 b^3}+\frac {6 a^2 (a+b x) \text {arcsinh}(a+b x)}{b^3}-\frac {3 a (a+b x)^2 \text {arcsinh}(a+b x)}{2 b^3}+\frac {2 (a+b x)^3 \text {arcsinh}(a+b x)}{9 b^3}+\frac {2 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{3 b^3}-\frac {3 a^2 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{b^3}+\frac {3 a (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{2 b^3}-\frac {(a+b x)^2 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{3 b^3}-\frac {a \text {arcsinh}(a+b x)^3}{2 b^3}+\frac {a^3 \text {arcsinh}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \text {arcsinh}(a+b x)^3 \]

[Out]

-2/27*(1+(b*x+a)^2)^(3/2)/b^3-3/4*a*arcsinh(b*x+a)/b^3-4/3*(b*x+a)*arcsinh(b*x+a)/b^3+6*a^2*(b*x+a)*arcsinh(b*
x+a)/b^3-3/2*a*(b*x+a)^2*arcsinh(b*x+a)/b^3+2/9*(b*x+a)^3*arcsinh(b*x+a)/b^3-1/2*a*arcsinh(b*x+a)^3/b^3+1/3*a^
3*arcsinh(b*x+a)^3/b^3+1/3*x^3*arcsinh(b*x+a)^3+14/9*(1+(b*x+a)^2)^(1/2)/b^3-6*a^2*(1+(b*x+a)^2)^(1/2)/b^3+3/4
*a*(b*x+a)*(1+(b*x+a)^2)^(1/2)/b^3+2/3*arcsinh(b*x+a)^2*(1+(b*x+a)^2)^(1/2)/b^3-3*a^2*arcsinh(b*x+a)^2*(1+(b*x
+a)^2)^(1/2)/b^3+3/2*a*(b*x+a)*arcsinh(b*x+a)^2*(1+(b*x+a)^2)^(1/2)/b^3-1/3*(b*x+a)^2*arcsinh(b*x+a)^2*(1+(b*x
+a)^2)^(1/2)/b^3

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {5859, 5828, 5843, 3398, 3377, 2718, 3392, 30, 2715, 8, 2713} \[ \int x^2 \text {arcsinh}(a+b x)^3 \, dx=\frac {a^3 \text {arcsinh}(a+b x)^3}{3 b^3}+\frac {6 a^2 (a+b x) \text {arcsinh}(a+b x)}{b^3}-\frac {3 a^2 \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^2}{b^3}-\frac {6 a^2 \sqrt {(a+b x)^2+1}}{b^3}+\frac {2 (a+b x)^3 \text {arcsinh}(a+b x)}{9 b^3}-\frac {\sqrt {(a+b x)^2+1} (a+b x)^2 \text {arcsinh}(a+b x)^2}{3 b^3}-\frac {3 a (a+b x)^2 \text {arcsinh}(a+b x)}{2 b^3}+\frac {3 a \sqrt {(a+b x)^2+1} (a+b x) \text {arcsinh}(a+b x)^2}{2 b^3}-\frac {4 (a+b x) \text {arcsinh}(a+b x)}{3 b^3}-\frac {a \text {arcsinh}(a+b x)^3}{2 b^3}+\frac {2 \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^2}{3 b^3}-\frac {3 a \text {arcsinh}(a+b x)}{4 b^3}+\frac {1}{3} x^3 \text {arcsinh}(a+b x)^3+\frac {3 a \sqrt {(a+b x)^2+1} (a+b x)}{4 b^3}-\frac {2 \left ((a+b x)^2+1\right )^{3/2}}{27 b^3}+\frac {14 \sqrt {(a+b x)^2+1}}{9 b^3} \]

[In]

Int[x^2*ArcSinh[a + b*x]^3,x]

[Out]

(14*Sqrt[1 + (a + b*x)^2])/(9*b^3) - (6*a^2*Sqrt[1 + (a + b*x)^2])/b^3 + (3*a*(a + b*x)*Sqrt[1 + (a + b*x)^2])
/(4*b^3) - (2*(1 + (a + b*x)^2)^(3/2))/(27*b^3) - (3*a*ArcSinh[a + b*x])/(4*b^3) - (4*(a + b*x)*ArcSinh[a + b*
x])/(3*b^3) + (6*a^2*(a + b*x)*ArcSinh[a + b*x])/b^3 - (3*a*(a + b*x)^2*ArcSinh[a + b*x])/(2*b^3) + (2*(a + b*
x)^3*ArcSinh[a + b*x])/(9*b^3) + (2*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x]^2)/(3*b^3) - (3*a^2*Sqrt[1 + (a + b
*x)^2]*ArcSinh[a + b*x]^2)/b^3 + (3*a*(a + b*x)*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x]^2)/(2*b^3) - ((a + b*x)
^2*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x]^2)/(3*b^3) - (a*ArcSinh[a + b*x]^3)/(2*b^3) + (a^3*ArcSinh[a + b*x]^
3)/(3*b^3) + (x^3*ArcSinh[a + b*x]^3)/3

Rule 8

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

Rule 30

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

Rule 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3392

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*m*(c + d*x)^(m - 1)*((
b*Sin[e + f*x])^n/(f^2*n^2)), x] + (Dist[b^2*((n - 1)/n), Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[d^2*m*((m - 1)/(f^2*n^2)), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[b*(c + d*x)^m*Cos[e + f
*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

Rule 3398

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

Rule 5828

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

Rule 5843

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

Rule 5859

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (-\frac {a}{b}+\frac {x}{b}\right )^2 \text {arcsinh}(x)^3 \, dx,x,a+b x\right )}{b} \\ & = \frac {1}{3} x^3 \text {arcsinh}(a+b x)^3-\text {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^3 \text {arcsinh}(x)^2}{\sqrt {1+x^2}} \, dx,x,a+b x\right ) \\ & = \frac {1}{3} x^3 \text {arcsinh}(a+b x)^3-\text {Subst}\left (\int x^2 \left (-\frac {a}{b}+\frac {\sinh (x)}{b}\right )^3 \, dx,x,\text {arcsinh}(a+b x)\right ) \\ & = \frac {1}{3} x^3 \text {arcsinh}(a+b x)^3-\text {Subst}\left (\int \left (-\frac {a^3 x^2}{b^3}+\frac {3 a^2 x^2 \sinh (x)}{b^3}-\frac {3 a x^2 \sinh ^2(x)}{b^3}+\frac {x^2 \sinh ^3(x)}{b^3}\right ) \, dx,x,\text {arcsinh}(a+b x)\right ) \\ & = \frac {a^3 \text {arcsinh}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \text {arcsinh}(a+b x)^3-\frac {\text {Subst}\left (\int x^2 \sinh ^3(x) \, dx,x,\text {arcsinh}(a+b x)\right )}{b^3}+\frac {(3 a) \text {Subst}\left (\int x^2 \sinh ^2(x) \, dx,x,\text {arcsinh}(a+b x)\right )}{b^3}-\frac {\left (3 a^2\right ) \text {Subst}\left (\int x^2 \sinh (x) \, dx,x,\text {arcsinh}(a+b x)\right )}{b^3} \\ & = -\frac {3 a (a+b x)^2 \text {arcsinh}(a+b x)}{2 b^3}+\frac {2 (a+b x)^3 \text {arcsinh}(a+b x)}{9 b^3}-\frac {3 a^2 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{b^3}+\frac {3 a (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{2 b^3}-\frac {(a+b x)^2 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{3 b^3}+\frac {a^3 \text {arcsinh}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \text {arcsinh}(a+b x)^3-\frac {2 \text {Subst}\left (\int \sinh ^3(x) \, dx,x,\text {arcsinh}(a+b x)\right )}{9 b^3}+\frac {2 \text {Subst}\left (\int x^2 \sinh (x) \, dx,x,\text {arcsinh}(a+b x)\right )}{3 b^3}-\frac {(3 a) \text {Subst}\left (\int x^2 \, dx,x,\text {arcsinh}(a+b x)\right )}{2 b^3}+\frac {(3 a) \text {Subst}\left (\int \sinh ^2(x) \, dx,x,\text {arcsinh}(a+b x)\right )}{2 b^3}+\frac {\left (6 a^2\right ) \text {Subst}(\int x \cosh (x) \, dx,x,\text {arcsinh}(a+b x))}{b^3} \\ & = \frac {3 a (a+b x) \sqrt {1+(a+b x)^2}}{4 b^3}+\frac {6 a^2 (a+b x) \text {arcsinh}(a+b x)}{b^3}-\frac {3 a (a+b x)^2 \text {arcsinh}(a+b x)}{2 b^3}+\frac {2 (a+b x)^3 \text {arcsinh}(a+b x)}{9 b^3}+\frac {2 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{3 b^3}-\frac {3 a^2 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{b^3}+\frac {3 a (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{2 b^3}-\frac {(a+b x)^2 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{3 b^3}-\frac {a \text {arcsinh}(a+b x)^3}{2 b^3}+\frac {a^3 \text {arcsinh}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \text {arcsinh}(a+b x)^3+\frac {2 \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\sqrt {1+(a+b x)^2}\right )}{9 b^3}-\frac {4 \text {Subst}(\int x \cosh (x) \, dx,x,\text {arcsinh}(a+b x))}{3 b^3}-\frac {(3 a) \text {Subst}(\int 1 \, dx,x,\text {arcsinh}(a+b x))}{4 b^3}-\frac {\left (6 a^2\right ) \text {Subst}(\int \sinh (x) \, dx,x,\text {arcsinh}(a+b x))}{b^3} \\ & = \frac {2 \sqrt {1+(a+b x)^2}}{9 b^3}-\frac {6 a^2 \sqrt {1+(a+b x)^2}}{b^3}+\frac {3 a (a+b x) \sqrt {1+(a+b x)^2}}{4 b^3}-\frac {2 \left (1+(a+b x)^2\right )^{3/2}}{27 b^3}-\frac {3 a \text {arcsinh}(a+b x)}{4 b^3}-\frac {4 (a+b x) \text {arcsinh}(a+b x)}{3 b^3}+\frac {6 a^2 (a+b x) \text {arcsinh}(a+b x)}{b^3}-\frac {3 a (a+b x)^2 \text {arcsinh}(a+b x)}{2 b^3}+\frac {2 (a+b x)^3 \text {arcsinh}(a+b x)}{9 b^3}+\frac {2 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{3 b^3}-\frac {3 a^2 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{b^3}+\frac {3 a (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{2 b^3}-\frac {(a+b x)^2 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{3 b^3}-\frac {a \text {arcsinh}(a+b x)^3}{2 b^3}+\frac {a^3 \text {arcsinh}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \text {arcsinh}(a+b x)^3+\frac {4 \text {Subst}(\int \sinh (x) \, dx,x,\text {arcsinh}(a+b x))}{3 b^3} \\ & = \frac {14 \sqrt {1+(a+b x)^2}}{9 b^3}-\frac {6 a^2 \sqrt {1+(a+b x)^2}}{b^3}+\frac {3 a (a+b x) \sqrt {1+(a+b x)^2}}{4 b^3}-\frac {2 \left (1+(a+b x)^2\right )^{3/2}}{27 b^3}-\frac {3 a \text {arcsinh}(a+b x)}{4 b^3}-\frac {4 (a+b x) \text {arcsinh}(a+b x)}{3 b^3}+\frac {6 a^2 (a+b x) \text {arcsinh}(a+b x)}{b^3}-\frac {3 a (a+b x)^2 \text {arcsinh}(a+b x)}{2 b^3}+\frac {2 (a+b x)^3 \text {arcsinh}(a+b x)}{9 b^3}+\frac {2 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{3 b^3}-\frac {3 a^2 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{b^3}+\frac {3 a (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{2 b^3}-\frac {(a+b x)^2 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{3 b^3}-\frac {a \text {arcsinh}(a+b x)^3}{2 b^3}+\frac {a^3 \text {arcsinh}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \text {arcsinh}(a+b x)^3 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.49 \[ \int x^2 \text {arcsinh}(a+b x)^3 \, dx=\frac {\left (160-575 a^2+65 a b x-8 b^2 x^2\right ) \sqrt {1+a^2+2 a b x+b^2 x^2}+3 \left (170 a^3+132 a^2 b x+8 b x \left (-6+b^2 x^2\right )-15 a \left (5+2 b^2 x^2\right )\right ) \text {arcsinh}(a+b x)-18 \sqrt {1+a^2+2 a b x+b^2 x^2} \left (-4+11 a^2-5 a b x+2 b^2 x^2\right ) \text {arcsinh}(a+b x)^2+18 \left (-3 a+2 a^3+2 b^3 x^3\right ) \text {arcsinh}(a+b x)^3}{108 b^3} \]

[In]

Integrate[x^2*ArcSinh[a + b*x]^3,x]

[Out]

((160 - 575*a^2 + 65*a*b*x - 8*b^2*x^2)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2] + 3*(170*a^3 + 132*a^2*b*x + 8*b*x*(
-6 + b^2*x^2) - 15*a*(5 + 2*b^2*x^2))*ArcSinh[a + b*x] - 18*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*(-4 + 11*a^2 - 5
*a*b*x + 2*b^2*x^2)*ArcSinh[a + b*x]^2 + 18*(-3*a + 2*a^3 + 2*b^3*x^3)*ArcSinh[a + b*x]^3)/(108*b^3)

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.84

method	result	size
derivativedivides	\(\frac {\frac {\operatorname {arcsinh}\left (b x +a \right )^{3} \left (b x +a \right )^{3}}{3}+\frac {2 \operatorname {arcsinh}\left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}}{3}-\frac {\operatorname {arcsinh}\left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )^{2}}{3}-\frac {4 \left (b x +a \right ) \operatorname {arcsinh}\left (b x +a \right )}{3}+\frac {40 \sqrt {1+\left (b x +a \right )^{2}}}{27}+\frac {2 \,\operatorname {arcsinh}\left (b x +a \right ) \left (b x +a \right )^{3}}{9}-\frac {2 \left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}}{27}-\frac {a \left (4 \operatorname {arcsinh}\left (b x +a \right )^{3} \left (b x +a \right )^{2}-6 \operatorname {arcsinh}\left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )+2 \operatorname {arcsinh}\left (b x +a \right )^{3}+6 \,\operatorname {arcsinh}\left (b x +a \right ) \left (b x +a \right )^{2}-3 \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}+3 \,\operatorname {arcsinh}\left (b x +a \right )\right )}{4}+a^{2} \left (\operatorname {arcsinh}\left (b x +a \right )^{3} \left (b x +a \right )-3 \operatorname {arcsinh}\left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}+6 \left (b x +a \right ) \operatorname {arcsinh}\left (b x +a \right )-6 \sqrt {1+\left (b x +a \right )^{2}}\right )}{b^{3}}\)	\(297\)
default	\(\frac {\frac {\operatorname {arcsinh}\left (b x +a \right )^{3} \left (b x +a \right )^{3}}{3}+\frac {2 \operatorname {arcsinh}\left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}}{3}-\frac {\operatorname {arcsinh}\left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )^{2}}{3}-\frac {4 \left (b x +a \right ) \operatorname {arcsinh}\left (b x +a \right )}{3}+\frac {40 \sqrt {1+\left (b x +a \right )^{2}}}{27}+\frac {2 \,\operatorname {arcsinh}\left (b x +a \right ) \left (b x +a \right )^{3}}{9}-\frac {2 \left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}}{27}-\frac {a \left (4 \operatorname {arcsinh}\left (b x +a \right )^{3} \left (b x +a \right )^{2}-6 \operatorname {arcsinh}\left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )+2 \operatorname {arcsinh}\left (b x +a \right )^{3}+6 \,\operatorname {arcsinh}\left (b x +a \right ) \left (b x +a \right )^{2}-3 \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}+3 \,\operatorname {arcsinh}\left (b x +a \right )\right )}{4}+a^{2} \left (\operatorname {arcsinh}\left (b x +a \right )^{3} \left (b x +a \right )-3 \operatorname {arcsinh}\left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}+6 \left (b x +a \right ) \operatorname {arcsinh}\left (b x +a \right )-6 \sqrt {1+\left (b x +a \right )^{2}}\right )}{b^{3}}\)	\(297\)

[In]

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

[Out]

1/b^3*(1/3*arcsinh(b*x+a)^3*(b*x+a)^3+2/3*arcsinh(b*x+a)^2*(1+(b*x+a)^2)^(1/2)-1/3*arcsinh(b*x+a)^2*(1+(b*x+a)
^2)^(1/2)*(b*x+a)^2-4/3*(b*x+a)*arcsinh(b*x+a)+40/27*(1+(b*x+a)^2)^(1/2)+2/9*arcsinh(b*x+a)*(b*x+a)^3-2/27*(b*
x+a)^2*(1+(b*x+a)^2)^(1/2)-1/4*a*(4*arcsinh(b*x+a)^3*(b*x+a)^2-6*arcsinh(b*x+a)^2*(1+(b*x+a)^2)^(1/2)*(b*x+a)+
2*arcsinh(b*x+a)^3+6*arcsinh(b*x+a)*(b*x+a)^2-3*(b*x+a)*(1+(b*x+a)^2)^(1/2)+3*arcsinh(b*x+a))+a^2*(arcsinh(b*x
+a)^3*(b*x+a)-3*arcsinh(b*x+a)^2*(1+(b*x+a)^2)^(1/2)+6*(b*x+a)*arcsinh(b*x+a)-6*(1+(b*x+a)^2)^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.63 \[ \int x^2 \text {arcsinh}(a+b x)^3 \, dx=\frac {18 \, {\left (2 \, b^{3} x^{3} + 2 \, a^{3} - 3 \, a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{3} - 18 \, {\left (2 \, b^{2} x^{2} - 5 \, a b x + 11 \, a^{2} - 4\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2} + 3 \, {\left (8 \, b^{3} x^{3} - 30 \, a b^{2} x^{2} + 170 \, a^{3} + 12 \, {\left (11 \, a^{2} - 4\right )} b x - 75 \, a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - {\left (8 \, b^{2} x^{2} - 65 \, a b x + 575 \, a^{2} - 160\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{108 \, b^{3}} \]

[In]

integrate(x^2*arcsinh(b*x+a)^3,x, algorithm="fricas")

[Out]

1/108*(18*(2*b^3*x^3 + 2*a^3 - 3*a)*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^3 - 18*(2*b^2*x^2 - 5*a*b
*x + 11*a^2 - 4)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2 + 3*(8*b
^3*x^3 - 30*a*b^2*x^2 + 170*a^3 + 12*(11*a^2 - 4)*b*x - 75*a)*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))
- (8*b^2*x^2 - 65*a*b*x + 575*a^2 - 160)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))/b^3

Sympy [A] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.22 \[ \int x^2 \text {arcsinh}(a+b x)^3 \, dx=\begin {cases} \frac {a^{3} \operatorname {asinh}^{3}{\left (a + b x \right )}}{3 b^{3}} + \frac {85 a^{3} \operatorname {asinh}{\left (a + b x \right )}}{18 b^{3}} + \frac {11 a^{2} x \operatorname {asinh}{\left (a + b x \right )}}{3 b^{2}} - \frac {11 a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a + b x \right )}}{6 b^{3}} - \frac {575 a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{108 b^{3}} - \frac {5 a x^{2} \operatorname {asinh}{\left (a + b x \right )}}{6 b} + \frac {5 a x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a + b x \right )}}{6 b^{2}} + \frac {65 a x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{108 b^{2}} - \frac {a \operatorname {asinh}^{3}{\left (a + b x \right )}}{2 b^{3}} - \frac {25 a \operatorname {asinh}{\left (a + b x \right )}}{12 b^{3}} + \frac {x^{3} \operatorname {asinh}^{3}{\left (a + b x \right )}}{3} + \frac {2 x^{3} \operatorname {asinh}{\left (a + b x \right )}}{9} - \frac {x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a + b x \right )}}{3 b} - \frac {2 x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{27 b} - \frac {4 x \operatorname {asinh}{\left (a + b x \right )}}{3 b^{2}} + \frac {2 \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a + b x \right )}}{3 b^{3}} + \frac {40 \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{27 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \operatorname {asinh}^{3}{\left (a \right )}}{3} & \text {otherwise} \end {cases} \]

[In]

integrate(x**2*asinh(b*x+a)**3,x)

[Out]

Piecewise((a**3*asinh(a + b*x)**3/(3*b**3) + 85*a**3*asinh(a + b*x)/(18*b**3) + 11*a**2*x*asinh(a + b*x)/(3*b*
*2) - 11*a**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)**2/(6*b**3) - 575*a**2*sqrt(a**2 + 2*a*b*x +
b**2*x**2 + 1)/(108*b**3) - 5*a*x**2*asinh(a + b*x)/(6*b) + 5*a*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(
a + b*x)**2/(6*b**2) + 65*a*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)/(108*b**2) - a*asinh(a + b*x)**3/(2*b**3) -
25*a*asinh(a + b*x)/(12*b**3) + x**3*asinh(a + b*x)**3/3 + 2*x**3*asinh(a + b*x)/9 - x**2*sqrt(a**2 + 2*a*b*x
+ b**2*x**2 + 1)*asinh(a + b*x)**2/(3*b) - 2*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)/(27*b) - 4*x*asinh(a +
b*x)/(3*b**2) + 2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)**2/(3*b**3) + 40*sqrt(a**2 + 2*a*b*x +
b**2*x**2 + 1)/(27*b**3), Ne(b, 0)), (x**3*asinh(a)**3/3, True))

Maxima [F]

\[ \int x^2 \text {arcsinh}(a+b x)^3 \, dx=\int { x^{2} \operatorname {arsinh}\left (b x + a\right )^{3} \,d x } \]

[In]

integrate(x^2*arcsinh(b*x+a)^3,x, algorithm="maxima")

[Out]

1/3*x^3*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^3 - integrate((b^3*x^5 + 2*a*b^2*x^4 + (a^2*b + b)*x^
3 + (b^2*x^4 + a*b*x^3)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2/
(b^3*x^3 + 3*a*b^2*x^2 + a^3 + (3*a^2*b + b)*x + (b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2) + a), x)

Giac [F]

\[ \int x^2 \text {arcsinh}(a+b x)^3 \, dx=\int { x^{2} \operatorname {arsinh}\left (b x + a\right )^{3} \,d x } \]

[In]

integrate(x^2*arcsinh(b*x+a)^3,x, algorithm="giac")

[Out]

integrate(x^2*arcsinh(b*x + a)^3, x)

Mupad [F(-1)]

Timed out. \[ \int x^2 \text {arcsinh}(a+b x)^3 \, dx=\int x^2\,{\mathrm {asinh}\left (a+b\,x\right )}^3 \,d x \]

[In]

int(x^2*asinh(a + b*x)^3,x)

[Out]

int(x^2*asinh(a + b*x)^3, x)