\(\int x^3 \text {arcsinh}(a+b x) \, dx\) [1]

Optimal result

Integrand size = 10, antiderivative size = 145 \[ \int x^3 \text {arcsinh}(a+b x) \, dx=-\frac {a \left (1-a^2\right ) \sqrt {1+(a+b x)^2}}{b^4}+\frac {3 \left (1-8 a^2\right ) (a+b x) \sqrt {1+(a+b x)^2}}{32 b^4}-\frac {(a+b x)^3 \sqrt {1+(a+b x)^2}}{16 b^4}+\frac {a \left (1+(a+b x)^2\right )^{3/2}}{3 b^4}-\frac {\left (3-24 a^2+8 a^4\right ) \text {arcsinh}(a+b x)}{32 b^4}+\frac {1}{4} x^4 \text {arcsinh}(a+b x) \] Output:

-a*(-a^2+1)*(1+(b*x+a)^2)^(1/2)/b^4+3/32*(-8*a^2+1)*(b*x+a)*(1+(b*x+a)^2)^ 
(1/2)/b^4-1/16*(b*x+a)^3*(1+(b*x+a)^2)^(1/2)/b^4+1/3*a*(1+(b*x+a)^2)^(3/2) 
/b^4-1/32*(8*a^4-24*a^2+3)*arcsinh(b*x+a)/b^4+1/4*x^4*arcsinh(b*x+a)
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.66 \[ \int x^3 \text {arcsinh}(a+b x) \, dx=\frac {\sqrt {1+a^2+2 a b x+b^2 x^2} \left (50 a^3+9 b x-26 a^2 b x-6 b^3 x^3+a \left (-55+14 b^2 x^2\right )\right )-3 \left (3-24 a^2+8 a^4-8 b^4 x^4\right ) \text {arcsinh}(a+b x)}{96 b^4} \] Input:

Integrate[x^3*ArcSinh[a + b*x],x]
 

Output:

(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)) - 3*(3 - 24*a^2 + 8*a^4 - 8*b^4*x^4)*ArcSinh[a + 
 b*x])/(96*b^4)
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {6274, 25, 27, 6243, 497, 25, 687, 676, 222}

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.

\(\displaystyle \int x^3 \text {arcsinh}(a+b x) \, dx\)

\(\Big \downarrow \) 6274

\(\displaystyle \frac {\int x^3 \text {arcsinh}(a+b x)d(a+b x)}{b}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int -x^3 \text {arcsinh}(a+b x)d(a+b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int -b^3 x^3 \text {arcsinh}(a+b x)d(a+b x)}{b^4}\)

\(\Big \downarrow \) 6243

\(\displaystyle -\frac {\frac {1}{4} \int \frac {b^4 x^4}{\sqrt {(a+b x)^2+1}}d(a+b x)-\frac {1}{4} b^4 x^4 \text {arcsinh}(a+b x)}{b^4}\)

\(\Big \downarrow \) 497

\(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{4} \int -\frac {b^2 x^2 \left (-4 a^2+7 (a+b x) a+3\right )}{\sqrt {(a+b x)^2+1}}d(a+b x)+\frac {1}{4} b^3 x^3 \sqrt {(a+b x)^2+1}\right )-\frac {1}{4} b^4 x^4 \text {arcsinh}(a+b x)}{b^4}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{4} b^3 x^3 \sqrt {(a+b x)^2+1}-\frac {1}{4} \int \frac {b^2 x^2 \left (-4 a^2+7 (a+b x) a+3\right )}{\sqrt {(a+b x)^2+1}}d(a+b x)\right )-\frac {1}{4} b^4 x^4 \text {arcsinh}(a+b x)}{b^4}\)

\(\Big \downarrow \) 687

\(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{4} \left (-\frac {1}{3} \int -\frac {b x \left (a \left (23-12 a^2\right )-\left (9-26 a^2\right ) (a+b x)\right )}{\sqrt {(a+b x)^2+1}}d(a+b x)-\frac {7}{3} a b^2 x^2 \sqrt {(a+b x)^2+1}\right )+\frac {1}{4} b^3 x^3 \sqrt {(a+b x)^2+1}\right )-\frac {1}{4} b^4 x^4 \text {arcsinh}(a+b x)}{b^4}\)

\(\Big \downarrow \) 676

\(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (8 a^4-24 a^2+3\right ) \int \frac {1}{\sqrt {(a+b x)^2+1}}d(a+b x)+2 a \left (16-19 a^2\right ) \sqrt {(a+b x)^2+1}-\frac {1}{2} \left (9-26 a^2\right ) (a+b x) \sqrt {(a+b x)^2+1}\right )-\frac {7}{3} a b^2 x^2 \sqrt {(a+b x)^2+1}\right )+\frac {1}{4} b^3 x^3 \sqrt {(a+b x)^2+1}\right )-\frac {1}{4} b^4 x^4 \text {arcsinh}(a+b x)}{b^4}\)

\(\Big \downarrow \) 222

\(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{4} \left (\frac {1}{3} \left (2 a \left (16-19 a^2\right ) \sqrt {(a+b x)^2+1}-\frac {1}{2} \left (9-26 a^2\right ) (a+b x) \sqrt {(a+b x)^2+1}+\frac {3}{2} \left (8 a^4-24 a^2+3\right ) \text {arcsinh}(a+b x)\right )-\frac {7}{3} a b^2 x^2 \sqrt {(a+b x)^2+1}\right )+\frac {1}{4} b^3 x^3 \sqrt {(a+b x)^2+1}\right )-\frac {1}{4} b^4 x^4 \text {arcsinh}(a+b x)}{b^4}\)

Input:

Int[x^3*ArcSinh[a + b*x],x]
 

Output:

-((-1/4*(b^4*x^4*ArcSinh[a + b*x]) + ((b^3*x^3*Sqrt[1 + (a + b*x)^2])/4 + 
((-7*a*b^2*x^2*Sqrt[1 + (a + b*x)^2])/3 + (2*a*(16 - 19*a^2)*Sqrt[1 + (a + 
 b*x)^2] - ((9 - 26*a^2)*(a + b*x)*Sqrt[1 + (a + b*x)^2])/2 + (3*(3 - 24*a 
^2 + 8*a^4)*ArcSinh[a + b*x])/2)/3)/4)/4)/b^4)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt 
[a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
 

rule 497

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[1/(b 
*(n + 2*p + 1))   Int[(c + d*x)^(n - 2)*(a + b*x^2)^p*Simp[b*c^2*(n + 2*p + 
 1) - a*d^2*(n - 1) + 2*b*c*d*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, n 
, p}, x] && If[RationalQ[n], GtQ[n, 1], SumSimplerQ[n, -2]] && NeQ[n + 2*p 
+ 1, 0] && IntQuadraticQ[a, 0, b, c, d, n, p, x]
 

rule 676

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x 
_Symbol] :> Simp[(e*f + d*g)*((a + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + (Sim 
p[e*g*x*((a + c*x^2)^(p + 1)/(c*(2*p + 3))), x] - Simp[(a*e*g - c*d*f*(2*p 
+ 3))/(c*(2*p + 3))   Int[(a + c*x^2)^p, x], x]) /; FreeQ[{a, c, d, e, f, g 
, p}, x] &&  !LeQ[p, -1]
 

rule 687

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) 
), x] + Simp[1/(c*(m + 2*p + 2))   Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp 
[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x 
] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && 
 (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) &&  !(IGtQ[m, 0] && Eq 
Q[f, 0])
 

rule 6243

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]
 

rule 6274

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]
 

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.34

method	result	size
orering	\(-\frac {\left (-14 b^{5} x^{5}+2 a \,b^{4} x^{4}-4 a^{2} b^{3} x^{3}+12 a^{3} b^{2} x^{2}+82 b \,a^{4} x +3 b^{3} x^{3}+50 a^{5}-23 a \,b^{2} x^{2}-151 b \,a^{2} x -5 a^{3}+12 b x -55 a \right ) \operatorname {arcsinh}\left (b x +a \right )}{32 b^{5} x}+\frac {\left (-6 b^{3} x^{3}+14 a \,b^{2} x^{2}-26 b \,a^{2} x +50 a^{3}+9 b x -55 a \right ) \left (x^{2} b^{2}+2 a b x +a^{2}+1\right ) \left (3 x^{2} \operatorname {arcsinh}\left (b x +a \right )+\frac {x^{3} b}{\sqrt {1+\left (b x +a \right )^{2}}}\right )}{96 b^{5} x^{3}}\)	\(194\)
derivativedivides	\(\frac {-\operatorname {arcsinh}\left (b x +a \right ) a^{3} \left (b x +a \right )+\frac {3 \,\operatorname {arcsinh}\left (b x +a \right ) a^{2} \left (b x +a \right )^{2}}{2}-\operatorname {arcsinh}\left (b x +a \right ) a \left (b x +a \right )^{3}+\frac {\operatorname {arcsinh}\left (b x +a \right ) \left (b x +a \right )^{4}}{4}-\frac {\left (b x +a \right )^{3} \sqrt {1+\left (b x +a \right )^{2}}}{16}+\frac {3 \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{32}-\frac {3 \,\operatorname {arcsinh}\left (b x +a \right )}{32}+a \left (\frac {\left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}}{3}-\frac {2 \sqrt {1+\left (b x +a \right )^{2}}}{3}\right )-\frac {3 a^{2} \left (\frac {\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{2}-\frac {\operatorname {arcsinh}\left (b x +a \right )}{2}\right )}{2}+a^{3} \sqrt {1+\left (b x +a \right )^{2}}}{b^{4}}\)	\(200\)
default	\(\frac {-\operatorname {arcsinh}\left (b x +a \right ) a^{3} \left (b x +a \right )+\frac {3 \,\operatorname {arcsinh}\left (b x +a \right ) a^{2} \left (b x +a \right )^{2}}{2}-\operatorname {arcsinh}\left (b x +a \right ) a \left (b x +a \right )^{3}+\frac {\operatorname {arcsinh}\left (b x +a \right ) \left (b x +a \right )^{4}}{4}-\frac {\left (b x +a \right )^{3} \sqrt {1+\left (b x +a \right )^{2}}}{16}+\frac {3 \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{32}-\frac {3 \,\operatorname {arcsinh}\left (b x +a \right )}{32}+a \left (\frac {\left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}}{3}-\frac {2 \sqrt {1+\left (b x +a \right )^{2}}}{3}\right )-\frac {3 a^{2} \left (\frac {\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{2}-\frac {\operatorname {arcsinh}\left (b x +a \right )}{2}\right )}{2}+a^{3} \sqrt {1+\left (b x +a \right )^{2}}}{b^{4}}\)	\(200\)
parts	\(\frac {x^{4} \operatorname {arcsinh}\left (b x +a \right )}{4}-\frac {b \left (\frac {x^{3} \sqrt {x^{2} b^{2}+2 a b x +a^{2}+1}}{4 b^{2}}-\frac {7 a \left (\frac {x^{2} \sqrt {x^{2} b^{2}+2 a b x +a^{2}+1}}{3 b^{2}}-\frac {5 a \left (\frac {x \sqrt {x^{2} b^{2}+2 a b x +a^{2}+1}}{2 b^{2}}-\frac {3 a \left (\frac {\sqrt {x^{2} b^{2}+2 a b x +a^{2}+1}}{b^{2}}-\frac {a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {x^{2} b^{2}+2 a b x +a^{2}+1}\right )}{b \sqrt {b^{2}}}\right )}{2 b}-\frac {\left (a^{2}+1\right ) \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {x^{2} b^{2}+2 a b x +a^{2}+1}\right )}{2 b^{2} \sqrt {b^{2}}}\right )}{3 b}-\frac {2 \left (a^{2}+1\right ) \left (\frac {\sqrt {x^{2} b^{2}+2 a b x +a^{2}+1}}{b^{2}}-\frac {a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {x^{2} b^{2}+2 a b x +a^{2}+1}\right )}{b \sqrt {b^{2}}}\right )}{3 b^{2}}\right )}{4 b}-\frac {3 \left (a^{2}+1\right ) \left (\frac {x \sqrt {x^{2} b^{2}+2 a b x +a^{2}+1}}{2 b^{2}}-\frac {3 a \left (\frac {\sqrt {x^{2} b^{2}+2 a b x +a^{2}+1}}{b^{2}}-\frac {a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {x^{2} b^{2}+2 a b x +a^{2}+1}\right )}{b \sqrt {b^{2}}}\right )}{2 b}-\frac {\left (a^{2}+1\right ) \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {x^{2} b^{2}+2 a b x +a^{2}+1}\right )}{2 b^{2} \sqrt {b^{2}}}\right )}{4 b^{2}}\right )}{4}\)	\(483\)

Input:

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

Output:

-1/32*(-14*b^5*x^5+2*a*b^4*x^4-4*a^2*b^3*x^3+12*a^3*b^2*x^2+82*a^4*b*x+3*b 
^3*x^3+50*a^5-23*a*b^2*x^2-151*a^2*b*x-5*a^3+12*b*x-55*a)/b^5/x*arcsinh(b* 
x+a)+1/96*(-6*b^3*x^3+14*a*b^2*x^2-26*a^2*b*x+50*a^3+9*b*x-55*a)/b^5/x^3*( 
b^2*x^2+2*a*b*x+a^2+1)*(3*x^2*arcsinh(b*x+a)+x^3*b/(1+(b*x+a)^2)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.76 \[ \int x^3 \text {arcsinh}(a+b x) \, dx=\frac {3 \, {\left (8 \, b^{4} x^{4} - 8 \, a^{4} + 24 \, a^{2} - 3\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - {\left (6 \, b^{3} x^{3} - 14 \, a b^{2} x^{2} - 50 \, a^{3} + {\left (26 \, a^{2} - 9\right )} b x + 55 \, a\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{96 \, b^{4}} \] Input:

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

Output:

1/96*(3*(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)) - (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))/b^4
 

Sympy [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.76 \[ \int x^3 \text {arcsinh}(a+b x) \, dx=\begin {cases} - \frac {a^{4} \operatorname {asinh}{\left (a + b x \right )}}{4 b^{4}} + \frac {25 a^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{48 b^{4}} - \frac {13 a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{48 b^{3}} + \frac {3 a^{2} \operatorname {asinh}{\left (a + b x \right )}}{4 b^{4}} + \frac {7 a x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{48 b^{2}} - \frac {55 a \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{96 b^{4}} + \frac {x^{4} \operatorname {asinh}{\left (a + b x \right )}}{4} - \frac {x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{16 b} + \frac {3 x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{32 b^{3}} - \frac {3 \operatorname {asinh}{\left (a + b x \right )}}{32 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} \operatorname {asinh}{\left (a \right )}}{4} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((-a**4*asinh(a + b*x)/(4*b**4) + 25*a**3*sqrt(a**2 + 2*a*b*x + b 
**2*x**2 + 1)/(48*b**4) - 13*a**2*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)/( 
48*b**3) + 3*a**2*asinh(a + b*x)/(4*b**4) + 7*a*x**2*sqrt(a**2 + 2*a*b*x + 
 b**2*x**2 + 1)/(48*b**2) - 55*a*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)/(96* 
b**4) + x**4*asinh(a + b*x)/4 - x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)/ 
(16*b) + 3*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)/(32*b**3) - 3*asinh(a + 
b*x)/(32*b**4), Ne(b, 0)), (x**4*asinh(a)/4, True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (124) = 248\).

Time = 0.04 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.19 \[ \int x^3 \text {arcsinh}(a+b x) \, dx=\frac {1}{4} \, x^{4} \operatorname {arsinh}\left (b x + a\right ) - \frac {1}{96} \, {\left (\frac {6 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} x^{3}}{b^{2}} - \frac {14 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a x^{2}}{b^{3}} + \frac {105 \, a^{4} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{b^{5}} + \frac {35 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2} x}{b^{4}} - \frac {90 \, {\left (a^{2} + 1\right )} a^{2} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{b^{5}} - \frac {105 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{3}}{b^{5}} - \frac {9 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )} x}{b^{4}} + \frac {9 \, {\left (a^{2} + 1\right )}^{2} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{b^{5}} + \frac {55 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )} a}{b^{5}}\right )} b \] Input:

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

Output:

1/4*x^4*arcsinh(b*x + a) - 1/96*(6*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*x^3/b 
^2 - 14*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a*x^2/b^3 + 105*a^4*arcsinh(2*(b 
^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^5 + 35*sqrt(b^2*x^2 + 2* 
a*b*x + a^2 + 1)*a^2*x/b^4 - 90*(a^2 + 1)*a^2*arcsinh(2*(b^2*x + a*b)/sqrt 
(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^5 - 105*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1 
)*a^3/b^5 - 9*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)*x/b^4 + 9*(a^2 + 
 1)^2*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^5 + 55 
*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)*a/b^5)*b
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.12 \[ \int x^3 \text {arcsinh}(a+b x) \, dx=\frac {1}{4} \, x^{4} \log \left (b x + a + \sqrt {{\left (b x + a\right )}^{2} + 1}\right ) - \frac {1}{96} \, {\left (\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left ({\left (2 \, x {\left (\frac {3 \, x}{b^{2}} - \frac {7 \, a}{b^{3}}\right )} + \frac {26 \, a^{2} b^{3} - 9 \, b^{3}}{b^{7}}\right )} x - \frac {5 \, {\left (10 \, a^{3} b^{2} - 11 \, a b^{2}\right )}}{b^{7}}\right )} - \frac {3 \, {\left (8 \, a^{4} - 24 \, a^{2} + 3\right )} \log \left ({\left | -a b - {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} {\left | b \right |} \right |}\right )}{b^{4} {\left | b \right |}}\right )} b \] Input:

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

Output:

1/4*x^4*log(b*x + a + sqrt((b*x + a)^2 + 1)) - 1/96*(sqrt(b^2*x^2 + 2*a*b* 
x + a^2 + 1)*((2*x*(3*x/b^2 - 7*a/b^3) + (26*a^2*b^3 - 9*b^3)/b^7)*x - 5*( 
10*a^3*b^2 - 11*a*b^2)/b^7) - 3*(8*a^4 - 24*a^2 + 3)*log(abs(-a*b - (x*abs 
(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))*abs(b)))/(b^4*abs(b)))*b
 

Mupad [F(-1)]

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

int(x^3*asinh(a + b*x),x)
 

Output:

int(x^3*asinh(a + b*x), x)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.71 \[ \int x^3 \text {arcsinh}(a+b x) \, dx=\frac {24 \mathit {asinh} \left (b x +a \right ) b^{4} x^{4}+50 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{3}-26 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{2} b x +14 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a \,b^{2} x^{2}-55 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a -6 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, b^{3} x^{3}+9 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, b x -24 \,\mathrm {log}\left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}+a +b x \right ) a^{4}+72 \,\mathrm {log}\left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}+a +b x \right ) a^{2}-9 \,\mathrm {log}\left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}+a +b x \right )}{96 b^{4}} \] Input:

int(x^3*asinh(b*x+a),x)
 

Output:

(24*asinh(a + b*x)*b**4*x**4 + 50*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*a** 
3 - 26*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*a**2*b*x + 14*sqrt(a**2 + 2*a* 
b*x + b**2*x**2 + 1)*a*b**2*x**2 - 55*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) 
*a - 6*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*b**3*x**3 + 9*sqrt(a**2 + 2*a* 
b*x + b**2*x**2 + 1)*b*x - 24*log(sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + a 
 + b*x)*a**4 + 72*log(sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + a + b*x)*a**2 
 - 9*log(sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + a + b*x))/(96*b**4)