Integrand size = 12, antiderivative size = 331 \[ \int x^3 \text {arcsinh}(a+b x)^2 \, dx=\frac {4 a x}{3 b^3}-\frac {2 a^3 x}{b^3}-\frac {3 (a+b x)^2}{32 b^4}+\frac {3 a^2 (a+b x)^2}{4 b^4}-\frac {2 a (a+b x)^3}{9 b^4}+\frac {(a+b x)^4}{32 b^4}-\frac {4 a \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{3 b^4}+\frac {2 a^3 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{b^4}+\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{16 b^4}-\frac {3 a^2 (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{2 b^4}+\frac {2 a (a+b x)^2 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{3 b^4}-\frac {(a+b x)^3 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{8 b^4}-\frac {3 \text {arcsinh}(a+b x)^2}{32 b^4}+\frac {3 a^2 \text {arcsinh}(a+b x)^2}{4 b^4}-\frac {a^4 \text {arcsinh}(a+b x)^2}{4 b^4}+\frac {1}{4} x^4 \text {arcsinh}(a+b x)^2 \]
4/3*a*x/b^3-2*a^3*x/b^3-3/32*(b*x+a)^2/b^4+3/4*a^2*(b*x+a)^2/b^4-2/9*a*(b* x+a)^3/b^4+1/32*(b*x+a)^4/b^4-3/32*arcsinh(b*x+a)^2/b^4+3/4*a^2*arcsinh(b* x+a)^2/b^4-1/4*a^4*arcsinh(b*x+a)^2/b^4+1/4*x^4*arcsinh(b*x+a)^2-4/3*a*arc sinh(b*x+a)*(1+(b*x+a)^2)^(1/2)/b^4+2*a^3*arcsinh(b*x+a)*(1+(b*x+a)^2)^(1/ 2)/b^4+3/16*(b*x+a)*arcsinh(b*x+a)*(1+(b*x+a)^2)^(1/2)/b^4-3/2*a^2*(b*x+a) *arcsinh(b*x+a)*(1+(b*x+a)^2)^(1/2)/b^4+2/3*a*(b*x+a)^2*arcsinh(b*x+a)*(1+ (b*x+a)^2)^(1/2)/b^4-1/8*(b*x+a)^3*arcsinh(b*x+a)*(1+(b*x+a)^2)^(1/2)/b^4
Time = 0.50 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.44 \[ \int x^3 \text {arcsinh}(a+b x)^2 \, dx=\frac {b x \left (-300 a^3+78 a^2 b x+a \left (330-28 b^2 x^2\right )+9 b x \left (-3+b^2 x^2\right )\right )+6 \sqrt {1+a^2+2 a b x+b^2 x^2} \left (50 a^3+9 b x-26 a^2 b x-6 b^3 x^3+a \left (-55+14 b^2 x^2\right )\right ) \text {arcsinh}(a+b x)-9 \left (3-24 a^2+8 a^4-8 b^4 x^4\right ) \text {arcsinh}(a+b x)^2}{288 b^4} \]
(b*x*(-300*a^3 + 78*a^2*b*x + a*(330 - 28*b^2*x^2) + 9*b*x*(-3 + b^2*x^2)) + 6*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*(50*a^3 + 9*b*x - 26*a^2*b*x - 6*b^ 3*x^3 + a*(-55 + 14*b^2*x^2))*ArcSinh[a + b*x] - 9*(3 - 24*a^2 + 8*a^4 - 8 *b^4*x^4)*ArcSinh[a + b*x]^2)/(288*b^4)
Time = 0.73 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.92, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6274, 25, 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.
| \(\displaystyle \int x^3 \text {arcsinh}(a+b x)^2 \, dx\) |
| \(\Big \downarrow \) 6274 |
| \(\displaystyle \frac {\int x^3 \text {arcsinh}(a+b x)^2d(a+b x)}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -x^3 \text {arcsinh}(a+b x)^2d(a+b x)}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int -b^3 x^3 \text {arcsinh}(a+b x)^2d(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 6243 |
| \(\displaystyle -\frac {\frac {1}{2} \int \frac {b^4 x^4 \text {arcsinh}(a+b x)}{\sqrt {(a+b x)^2+1}}d(a+b x)-\frac {1}{4} b^4 x^4 \text {arcsinh}(a+b x)^2}{b^4}\) |
| \(\Big \downarrow \) 6253 |
| \(\displaystyle -\frac {\frac {1}{2} \int \left (\frac {\text {arcsinh}(a+b x) a^4}{\sqrt {(a+b x)^2+1}}-\frac {4 (a+b x) \text {arcsinh}(a+b x) a^3}{\sqrt {(a+b x)^2+1}}+\frac {6 (a+b x)^2 \text {arcsinh}(a+b x) a^2}{\sqrt {(a+b x)^2+1}}-\frac {4 (a+b x)^3 \text {arcsinh}(a+b x) a}{\sqrt {(a+b x)^2+1}}+\frac {(a+b x)^4 \text {arcsinh}(a+b x)}{\sqrt {(a+b x)^2+1}}\right )d(a+b x)-\frac {1}{4} b^4 x^4 \text {arcsinh}(a+b x)^2}{b^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{2} a^4 \text {arcsinh}(a+b x)^2-4 a^3 \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)+4 a^3 (a+b x)-\frac {3}{2} a^2 \text {arcsinh}(a+b x)^2+3 a^2 (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)-\frac {3}{2} a^2 (a+b x)^2-\frac {4}{3} a (a+b x)^2 \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)+\frac {8}{3} a \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)+\frac {3}{16} \text {arcsinh}(a+b x)^2+\frac {1}{4} (a+b x)^3 \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)-\frac {3}{8} (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)+\frac {4}{9} a (a+b x)^3-\frac {8}{3} a (a+b x)-\frac {1}{16} (a+b x)^4+\frac {3}{16} (a+b x)^2\right )-\frac {1}{4} b^4 x^4 \text {arcsinh}(a+b x)^2}{b^4}\) |
-((-1/4*(b^4*x^4*ArcSinh[a + b*x]^2) + ((-8*a*(a + b*x))/3 + 4*a^3*(a + b* x) + (3*(a + b*x)^2)/16 - (3*a^2*(a + b*x)^2)/2 + (4*a*(a + b*x)^3)/9 - (a + b*x)^4/16 + (8*a*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/3 - 4*a^3*Sqrt [1 + (a + b*x)^2]*ArcSinh[a + b*x] - (3*(a + b*x)*Sqrt[1 + (a + b*x)^2]*Ar cSinh[a + b*x])/8 + 3*a^2*(a + b*x)*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x] - (4*a*(a + b*x)^2*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/3 + ((a + b*x) ^3*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/4 + (3*ArcSinh[a + b*x]^2)/16 - (3*a^2*ArcSinh[a + b*x]^2)/2 + (a^4*ArcSinh[a + b*x]^2)/2)/2)/b^4)
3.1.67.3.1 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 = 0.20 (sec) , antiderivative size = 293, normalized size of antiderivative = 0.89
| method | result | size |
| derivativedivides | \(\frac {\frac {\operatorname {arcsinh}\left (b x +a \right )^{2} \left (b x +a \right )^{4}}{4}-\frac {\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )^{3}}{8}+\frac {3 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )}{16}-\frac {3 \operatorname {arcsinh}\left (b x +a \right )^{2}}{32}+\frac {\left (b x +a \right )^{4}}{32}-\frac {3 \left (b x +a \right )^{2}}{32}-\frac {3}{32}-\frac {a \left (9 \left (b x +a \right )^{3} \operatorname {arcsinh}\left (b x +a \right )^{2}-6 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )^{2}+2 \left (b x +a \right )^{3}+12 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}-12 b x -12 a \right )}{9}+\frac {3 a^{2} \left (2 \operatorname {arcsinh}\left (b x +a \right )^{2} \left (b x +a \right )^{2}-2 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )+\operatorname {arcsinh}\left (b x +a \right )^{2}+\left (b x +a \right )^{2}+1\right )}{4}-a^{3} \left (\operatorname {arcsinh}\left (b x +a \right )^{2} \left (b x +a \right )-2 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}+2 b x +2 a \right )}{b^{4}}\) | \(293\) |
| default | \(\frac {\frac {\operatorname {arcsinh}\left (b x +a \right )^{2} \left (b x +a \right )^{4}}{4}-\frac {\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )^{3}}{8}+\frac {3 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )}{16}-\frac {3 \operatorname {arcsinh}\left (b x +a \right )^{2}}{32}+\frac {\left (b x +a \right )^{4}}{32}-\frac {3 \left (b x +a \right )^{2}}{32}-\frac {3}{32}-\frac {a \left (9 \left (b x +a \right )^{3} \operatorname {arcsinh}\left (b x +a \right )^{2}-6 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )^{2}+2 \left (b x +a \right )^{3}+12 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}-12 b x -12 a \right )}{9}+\frac {3 a^{2} \left (2 \operatorname {arcsinh}\left (b x +a \right )^{2} \left (b x +a \right )^{2}-2 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )+\operatorname {arcsinh}\left (b x +a \right )^{2}+\left (b x +a \right )^{2}+1\right )}{4}-a^{3} \left (\operatorname {arcsinh}\left (b x +a \right )^{2} \left (b x +a \right )-2 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}+2 b x +2 a \right )}{b^{4}}\) | \(293\) |
1/b^4*(1/4*arcsinh(b*x+a)^2*(b*x+a)^4-1/8*arcsinh(b*x+a)*(1+(b*x+a)^2)^(1/ 2)*(b*x+a)^3+3/16*arcsinh(b*x+a)*(1+(b*x+a)^2)^(1/2)*(b*x+a)-3/32*arcsinh( b*x+a)^2+1/32*(b*x+a)^4-3/32*(b*x+a)^2-3/32-1/9*a*(9*(b*x+a)^3*arcsinh(b*x +a)^2-6*arcsinh(b*x+a)*(1+(b*x+a)^2)^(1/2)*(b*x+a)^2+2*(b*x+a)^3+12*arcsin h(b*x+a)*(1+(b*x+a)^2)^(1/2)-12*b*x-12*a)+3/4*a^2*(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^3*(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 = 0.27 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.55 \[ \int x^3 \text {arcsinh}(a+b x)^2 \, dx=\frac {9 \, b^{4} x^{4} - 28 \, a b^{3} x^{3} + 3 \, {\left (26 \, a^{2} - 9\right )} b^{2} x^{2} - 30 \, {\left (10 \, a^{3} - 11 \, a\right )} b x + 9 \, {\left (8 \, b^{4} x^{4} - 8 \, a^{4} + 24 \, a^{2} - 3\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2} - 6 \, {\left (6 \, b^{3} x^{3} - 14 \, a b^{2} x^{2} - 50 \, a^{3} + {\left (26 \, a^{2} - 9\right )} b x + 55 \, a\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}{288 \, b^{4}} \]
1/288*(9*b^4*x^4 - 28*a*b^3*x^3 + 3*(26*a^2 - 9)*b^2*x^2 - 30*(10*a^3 - 11 *a)*b*x + 9*(8*b^4*x^4 - 8*a^4 + 24*a^2 - 3)*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2 - 6*(6*b^3*x^3 - 14*a*b^2*x^2 - 50*a^3 + (26*a^2 - 9 )*b*x + 55*a)*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^4
Time = 0.44 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.11 \[ \int x^3 \text {arcsinh}(a+b x)^2 \, dx=\begin {cases} - \frac {a^{4} \operatorname {asinh}^{2}{\left (a + b x \right )}}{4 b^{4}} - \frac {25 a^{3} x}{24 b^{3}} + \frac {25 a^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{24 b^{4}} + \frac {13 a^{2} x^{2}}{48 b^{2}} - \frac {13 a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{24 b^{3}} + \frac {3 a^{2} \operatorname {asinh}^{2}{\left (a + b x \right )}}{4 b^{4}} - \frac {7 a x^{3}}{72 b} + \frac {7 a x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{24 b^{2}} + \frac {55 a x}{48 b^{3}} - \frac {55 a \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{48 b^{4}} + \frac {x^{4} \operatorname {asinh}^{2}{\left (a + b x \right )}}{4} + \frac {x^{4}}{32} - \frac {x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{8 b} - \frac {3 x^{2}}{32 b^{2}} + \frac {3 x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{16 b^{3}} - \frac {3 \operatorname {asinh}^{2}{\left (a + b x \right )}}{32 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} \operatorname {asinh}^{2}{\left (a \right )}}{4} & \text {otherwise} \end {cases} \]
Piecewise((-a**4*asinh(a + b*x)**2/(4*b**4) - 25*a**3*x/(24*b**3) + 25*a** 3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)/(24*b**4) + 13*a**2* x**2/(48*b**2) - 13*a**2*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)/(24*b**3) + 3*a**2*asinh(a + b*x)**2/(4*b**4) - 7*a*x**3/(72*b) + 7*a *x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)/(24*b**2) + 55*a *x/(48*b**3) - 55*a*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)/(4 8*b**4) + x**4*asinh(a + b*x)**2/4 + x**4/32 - x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)/(8*b) - 3*x**2/(32*b**2) + 3*x*sqrt(a**2 + 2 *a*b*x + b**2*x**2 + 1)*asinh(a + b*x)/(16*b**3) - 3*asinh(a + b*x)**2/(32 *b**4), Ne(b, 0)), (x**4*asinh(a)**2/4, True))
\[ \int x^3 \text {arcsinh}(a+b x)^2 \, dx=\int { x^{3} \operatorname {arsinh}\left (b x + a\right )^{2} \,d x } \]
1/4*x^4*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2 - integrate(1/2 *(b^3*x^6 + 2*a*b^2*x^5 + (a^2*b + b)*x^4 + (b^2*x^5 + a*b*x^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))/ (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)
\[ \int x^3 \text {arcsinh}(a+b x)^2 \, dx=\int { x^{3} \operatorname {arsinh}\left (b x + a\right )^{2} \,d x } \]
Timed out. \[ \int x^3 \text {arcsinh}(a+b x)^2 \, dx=\int x^3\,{\mathrm {asinh}\left (a+b\,x\right )}^2 \,d x \]