Integrand size = 12, antiderivative size = 211 \[ \int x^2 \text {arcsinh}(a+b x)^2 \, dx=-\frac {4 x}{9 b^2}+\frac {2 a^2 x}{b^2}-\frac {a (a+b x)^2}{2 b^3}+\frac {2 (a+b x)^3}{27 b^3}+\frac {4 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{9 b^3}-\frac {2 a^2 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{b^3}+\frac {a (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{b^3}-\frac {2 (a+b x)^2 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{9 b^3}-\frac {a \text {arcsinh}(a+b x)^2}{2 b^3}+\frac {a^3 \text {arcsinh}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \text {arcsinh}(a+b x)^2 \]
-4/9*x/b^2+2*a^2*x/b^2-1/2*a*(b*x+a)^2/b^3+2/27*(b*x+a)^3/b^3-1/2*a*arcsin h(b*x+a)^2/b^3+1/3*a^3*arcsinh(b*x+a)^2/b^3+1/3*x^3*arcsinh(b*x+a)^2+4/9*a rcsinh(b*x+a)*(1+(b*x+a)^2)^(1/2)/b^3-2*a^2*arcsinh(b*x+a)*(1+(b*x+a)^2)^( 1/2)/b^3+a*(b*x+a)*arcsinh(b*x+a)*(1+(b*x+a)^2)^(1/2)/b^3-2/9*(b*x+a)^2*ar csinh(b*x+a)*(1+(b*x+a)^2)^(1/2)/b^3
Time = 0.41 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.51 \[ \int x^2 \text {arcsinh}(a+b x)^2 \, dx=\frac {b x \left (-24+66 a^2-15 a b x+4 b^2 x^2\right )-6 \sqrt {1+a^2+2 a b x+b^2 x^2} \left (-4+11 a^2-5 a b x+2 b^2 x^2\right ) \text {arcsinh}(a+b x)+9 \left (-3 a+2 a^3+2 b^3 x^3\right ) \text {arcsinh}(a+b x)^2}{54 b^3} \]
(b*x*(-24 + 66*a^2 - 15*a*b*x + 4*b^2*x^2) - 6*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] + 9*(-3*a + 2* a^3 + 2*b^3*x^3)*ArcSinh[a + b*x]^2)/(54*b^3)
Time = 0.60 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6274, 27, 6243, 6253, 2009}
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.
| \(\displaystyle \int x^2 \text {arcsinh}(a+b x)^2 \, dx\) |
| \(\Big \downarrow \) 6274 |
| \(\displaystyle \frac {\int x^2 \text {arcsinh}(a+b x)^2d(a+b x)}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int b^2 x^2 \text {arcsinh}(a+b x)^2d(a+b x)}{b^3}\) |
| \(\Big \downarrow \) 6243 |
| \(\displaystyle \frac {\frac {2}{3} \int -\frac {b^3 x^3 \text {arcsinh}(a+b x)}{\sqrt {(a+b x)^2+1}}d(a+b x)+\frac {1}{3} b^3 x^3 \text {arcsinh}(a+b x)^2}{b^3}\) |
| \(\Big \downarrow \) 6253 |
| \(\displaystyle \frac {\frac {2}{3} \int \left (\frac {\text {arcsinh}(a+b x) a^3}{\sqrt {(a+b x)^2+1}}-\frac {3 (a+b x) \text {arcsinh}(a+b x) a^2}{\sqrt {(a+b x)^2+1}}+\frac {3 (a+b x)^2 \text {arcsinh}(a+b x) a}{\sqrt {(a+b x)^2+1}}-\frac {(a+b x)^3 \text {arcsinh}(a+b x)}{\sqrt {(a+b x)^2+1}}\right )d(a+b x)+\frac {1}{3} b^3 x^3 \text {arcsinh}(a+b x)^2}{b^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {1}{2} a^3 \text {arcsinh}(a+b x)^2-3 a^2 \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)+3 a^2 (a+b x)-\frac {3}{4} a \text {arcsinh}(a+b x)^2+\frac {3}{2} a (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)-\frac {1}{3} (a+b x)^2 \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)+\frac {2}{3} \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)-\frac {3}{4} a (a+b x)^2+\frac {1}{9} (a+b x)^3-\frac {2}{3} (a+b x)\right )+\frac {1}{3} b^3 x^3 \text {arcsinh}(a+b x)^2}{b^3}\) |
((b^3*x^3*ArcSinh[a + b*x]^2)/3 + (2*((-2*(a + b*x))/3 + 3*a^2*(a + b*x) - (3*a*(a + b*x)^2)/4 + (a + b*x)^3/9 + (2*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/3 - 3*a^2*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x] + (3*a*(a + b*x)* Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/2 - ((a + b*x)^2*Sqrt[1 + (a + b*x )^2]*ArcSinh[a + b*x])/3 - (3*a*ArcSinh[a + b*x]^2)/4 + (a^3*ArcSinh[a + b *x]^2)/2))/3)/b^3
3.1.68.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] - Simp[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]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d _) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ[n , 0] && ((EqQ[n, 1] && GtQ[p, -1]) || GtQ[p, 0] || EqQ[m, 1] || (EqQ[m, 2] && LtQ[p, -2]))
Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[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]
Time = 0.17 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.90
| method | result | size |
| derivativedivides | \(\frac {\frac {\left (b x +a \right )^{3} \operatorname {arcsinh}\left (b x +a \right )^{2}}{3}+\frac {4 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{9}-\frac {2 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )^{2}}{9}-\frac {4 b x}{9}-\frac {4 a}{9}+\frac {2 \left (b x +a \right )^{3}}{27}-\frac {a \left (2 \operatorname {arcsinh}\left (b x +a \right )^{2} \left (b x +a \right )^{2}-2 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )+\operatorname {arcsinh}\left (b x +a \right )^{2}+\left (b x +a \right )^{2}+1\right )}{2}+a^{2} \left (\operatorname {arcsinh}\left (b x +a \right )^{2} \left (b x +a \right )-2 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}+2 b x +2 a \right )}{b^{3}}\) | \(190\) |
| default | \(\frac {\frac {\left (b x +a \right )^{3} \operatorname {arcsinh}\left (b x +a \right )^{2}}{3}+\frac {4 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{9}-\frac {2 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )^{2}}{9}-\frac {4 b x}{9}-\frac {4 a}{9}+\frac {2 \left (b x +a \right )^{3}}{27}-\frac {a \left (2 \operatorname {arcsinh}\left (b x +a \right )^{2} \left (b x +a \right )^{2}-2 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )+\operatorname {arcsinh}\left (b x +a \right )^{2}+\left (b x +a \right )^{2}+1\right )}{2}+a^{2} \left (\operatorname {arcsinh}\left (b x +a \right )^{2} \left (b x +a \right )-2 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}+2 b x +2 a \right )}{b^{3}}\) | \(190\) |
1/b^3*(1/3*(b*x+a)^3*arcsinh(b*x+a)^2+4/9*arcsinh(b*x+a)*(1+(b*x+a)^2)^(1/ 2)-2/9*arcsinh(b*x+a)*(1+(b*x+a)^2)^(1/2)*(b*x+a)^2-4/9*b*x-4/9*a+2/27*(b* x+a)^3-1/2*a*(2*arcsinh(b*x+a)^2*(b*x+a)^2-2*arcsinh(b*x+a)*(1+(b*x+a)^2)^ (1/2)*(b*x+a)+arcsinh(b*x+a)^2+(b*x+a)^2+1)+a^2*(arcsinh(b*x+a)^2*(b*x+a)- 2*arcsinh(b*x+a)*(1+(b*x+a)^2)^(1/2)+2*b*x+2*a))
Time = 0.25 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.69 \[ \int x^2 \text {arcsinh}(a+b x)^2 \, dx=\frac {4 \, b^{3} x^{3} - 15 \, a b^{2} x^{2} + 6 \, {\left (11 \, a^{2} - 4\right )} b x + 9 \, {\left (2 \, b^{3} x^{3} + 2 \, a^{3} - 3 \, a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2} - 6 \, {\left (2 \, b^{2} x^{2} - 5 \, a b x + 11 \, a^{2} - 4\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}{54 \, b^{3}} \]
1/54*(4*b^3*x^3 - 15*a*b^2*x^2 + 6*(11*a^2 - 4)*b*x + 9*(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))^2 - 6*(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 + sq rt(b^2*x^2 + 2*a*b*x + a^2 + 1)))/b^3
Time = 0.30 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.15 \[ \int x^2 \text {arcsinh}(a+b x)^2 \, dx=\begin {cases} \frac {a^{3} \operatorname {asinh}^{2}{\left (a + b x \right )}}{3 b^{3}} + \frac {11 a^{2} x}{9 b^{2}} - \frac {11 a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{9 b^{3}} - \frac {5 a x^{2}}{18 b} + \frac {5 a x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{9 b^{2}} - \frac {a \operatorname {asinh}^{2}{\left (a + b x \right )}}{2 b^{3}} + \frac {x^{3} \operatorname {asinh}^{2}{\left (a + b x \right )}}{3} + \frac {2 x^{3}}{27} - \frac {2 x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{9 b} - \frac {4 x}{9 b^{2}} + \frac {4 \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{9 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \operatorname {asinh}^{2}{\left (a \right )}}{3} & \text {otherwise} \end {cases} \]
Piecewise((a**3*asinh(a + b*x)**2/(3*b**3) + 11*a**2*x/(9*b**2) - 11*a**2* sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)/(9*b**3) - 5*a*x**2/(1 8*b) + 5*a*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)/(9*b**2) - a*asinh(a + b*x)**2/(2*b**3) + x**3*asinh(a + b*x)**2/3 + 2*x**3/27 - 2* x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)/(9*b) - 4*x/(9*b* *2) + 4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)/(9*b**3), Ne(b , 0)), (x**3*asinh(a)**2/3, True))
\[ \int x^2 \text {arcsinh}(a+b x)^2 \, dx=\int { x^{2} \operatorname {arsinh}\left (b x + a\right )^{2} \,d x } \]
1/3*x^3*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2 - integrate(2/3 *(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))/ (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)
\[ \int x^2 \text {arcsinh}(a+b x)^2 \, dx=\int { x^{2} \operatorname {arsinh}\left (b x + a\right )^{2} \,d x } \]
Timed out. \[ \int x^2 \text {arcsinh}(a+b x)^2 \, dx=\int x^2\,{\mathrm {asinh}\left (a+b\,x\right )}^2 \,d x \]