\(\int (d+e x)^3 (a+b \text {arcsinh}(c x)) \, dx\) [4]

Optimal result

Integrand size = 16, antiderivative size = 184 \[ \int (d+e x)^3 (a+b \text {arcsinh}(c x)) \, dx=-\frac {b d (c d-e) (c d+e) \sqrt {1+c^2 x^2}}{c^3}-\frac {3 b e \left (8 c^2 d^2-e^2\right ) x \sqrt {1+c^2 x^2}}{32 c^3}-\frac {b e^3 x^3 \sqrt {1+c^2 x^2}}{16 c}-\frac {b d e^2 \left (1+c^2 x^2\right )^{3/2}}{3 c^3}-\frac {b \left (8 c^4 d^4-24 c^2 d^2 e^2+3 e^4\right ) \text {arcsinh}(c x)}{32 c^4 e}+\frac {(d+e x)^4 (a+b \text {arcsinh}(c x))}{4 e} \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.90 \[ \int (d+e x)^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {24 a c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )-b c \sqrt {1+c^2 x^2} \left (-e^2 (64 d+9 e x)+c^2 \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )\right )+3 b \left (24 c^2 d^2 e-3 e^3+8 c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right ) \text {arcsinh}(c x)}{96 c^4} \] Input:

Integrate[(d + e*x)^3*(a + b*ArcSinh[c*x]),x]
 

Output:

(24*a*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) - b*c*Sqrt[1 + c^2 
*x^2]*(-(e^2*(64*d + 9*e*x)) + c^2*(96*d^3 + 72*d^2*e*x + 32*d*e^2*x^2 + 6 
*e^3*x^3)) + 3*b*(24*c^2*d^2*e - 3*e^3 + 8*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d* 
e^2*x^2 + e^3*x^3))*ArcSinh[c*x])/(96*c^4)
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6243, 497, 687, 27, 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.

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

\(\Big \downarrow \) 6243

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

\(\Big \downarrow \) 497

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

\(\Big \downarrow \) 687

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 676

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

\(\Big \downarrow \) 222

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

Input:

Int[(d + e*x)^3*(a + b*ArcSinh[c*x]),x]
 

Output:

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

Defintions of rubi rules used

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]
 

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.32

method	result	size
parts	\(\frac {a \left (e x +d \right )^{4}}{4 e}+\frac {b \left (\frac {c \,e^{3} \operatorname {arcsinh}\left (x c \right ) x^{4}}{4}+c \,e^{2} \operatorname {arcsinh}\left (x c \right ) x^{3} d +\frac {3 c e \,\operatorname {arcsinh}\left (x c \right ) x^{2} d^{2}}{2}+\operatorname {arcsinh}\left (x c \right ) x c \,d^{3}+\frac {c \,\operatorname {arcsinh}\left (x c \right ) d^{4}}{4 e}-\frac {c^{4} d^{4} \operatorname {arcsinh}\left (x c \right )+e^{4} \left (\frac {\sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}}{4}-\frac {3 \sqrt {c^{2} x^{2}+1}\, x c}{8}+\frac {3 \,\operatorname {arcsinh}\left (x c \right )}{8}\right )+4 d c \,e^{3} \left (\frac {x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )+6 d^{2} c^{2} e^{2} \left (\frac {\sqrt {c^{2} x^{2}+1}\, x c}{2}-\frac {\operatorname {arcsinh}\left (x c \right )}{2}\right )+4 d^{3} c^{3} e \sqrt {c^{2} x^{2}+1}}{4 c^{3} e}\right )}{c}\)	\(242\)
orering	\(\frac {\left (14 c^{4} e^{4} x^{5}+72 c^{4} d \,e^{3} x^{4}+152 c^{4} d^{2} e^{2} x^{3}+176 c^{4} d^{3} e \,x^{2}+32 c^{4} d^{4} x -3 c^{2} e^{4} x^{3}-32 c^{2} d \,e^{3} x^{2}+96 c^{2} d^{2} e^{2} x +120 c^{2} d^{3} e -12 e^{4} x -67 d \,e^{3}\right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{32 c^{4} \left (e x +d \right )}-\frac {\left (6 e^{3} c^{2} x^{3}+32 e^{2} c^{2} x^{2} d +72 e \,c^{2} d^{2} x +96 c^{2} d^{3}-9 e^{3} x -64 e^{2} d \right ) \left (c^{2} x^{2}+1\right ) \left (3 \left (e x +d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) e +\frac {\left (e x +d \right )^{3} b c}{\sqrt {c^{2} x^{2}+1}}\right )}{96 c^{4} \left (e x +d \right )^{3}}\)	\(252\)
derivativedivides	\(\frac {\frac {a \left (c e x +c d \right )^{4}}{4 c^{3} e}+\frac {b \left (\frac {\operatorname {arcsinh}\left (x c \right ) c^{4} d^{4}}{4 e}+\operatorname {arcsinh}\left (x c \right ) c^{4} d^{3} x +\frac {3 e \,\operatorname {arcsinh}\left (x c \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \operatorname {arcsinh}\left (x c \right ) c^{4} d \,x^{3}+\frac {e^{3} \operatorname {arcsinh}\left (x c \right ) x^{4} c^{4}}{4}-\frac {c^{4} d^{4} \operatorname {arcsinh}\left (x c \right )+e^{4} \left (\frac {\sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}}{4}-\frac {3 \sqrt {c^{2} x^{2}+1}\, x c}{8}+\frac {3 \,\operatorname {arcsinh}\left (x c \right )}{8}\right )+4 d c \,e^{3} \left (\frac {x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )+6 d^{2} c^{2} e^{2} \left (\frac {\sqrt {c^{2} x^{2}+1}\, x c}{2}-\frac {\operatorname {arcsinh}\left (x c \right )}{2}\right )+4 d^{3} c^{3} e \sqrt {c^{2} x^{2}+1}}{4 e}\right )}{c^{3}}}{c}\)	\(259\)
default	\(\frac {\frac {a \left (c e x +c d \right )^{4}}{4 c^{3} e}+\frac {b \left (\frac {\operatorname {arcsinh}\left (x c \right ) c^{4} d^{4}}{4 e}+\operatorname {arcsinh}\left (x c \right ) c^{4} d^{3} x +\frac {3 e \,\operatorname {arcsinh}\left (x c \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \operatorname {arcsinh}\left (x c \right ) c^{4} d \,x^{3}+\frac {e^{3} \operatorname {arcsinh}\left (x c \right ) x^{4} c^{4}}{4}-\frac {c^{4} d^{4} \operatorname {arcsinh}\left (x c \right )+e^{4} \left (\frac {\sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}}{4}-\frac {3 \sqrt {c^{2} x^{2}+1}\, x c}{8}+\frac {3 \,\operatorname {arcsinh}\left (x c \right )}{8}\right )+4 d c \,e^{3} \left (\frac {x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )+6 d^{2} c^{2} e^{2} \left (\frac {\sqrt {c^{2} x^{2}+1}\, x c}{2}-\frac {\operatorname {arcsinh}\left (x c \right )}{2}\right )+4 d^{3} c^{3} e \sqrt {c^{2} x^{2}+1}}{4 e}\right )}{c^{3}}}{c}\)	\(259\)

Input:

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

Output:

1/4*a*(e*x+d)^4/e+b/c*(1/4*c*e^3*arcsinh(x*c)*x^4+c*e^2*arcsinh(x*c)*x^3*d 
+3/2*c*e*arcsinh(x*c)*x^2*d^2+arcsinh(x*c)*x*c*d^3+1/4*c/e*arcsinh(x*c)*d^ 
4-1/4/c^3/e*(c^4*d^4*arcsinh(x*c)+e^4*(1/4*(c^2*x^2+1)^(1/2)*c^3*x^3-3/8*( 
c^2*x^2+1)^(1/2)*x*c+3/8*arcsinh(x*c))+4*d*c*e^3*(1/3*x^2*c^2*(c^2*x^2+1)^ 
(1/2)-2/3*(c^2*x^2+1)^(1/2))+6*d^2*c^2*e^2*(1/2*(c^2*x^2+1)^(1/2)*x*c-1/2* 
arcsinh(x*c))+4*d^3*c^3*e*(c^2*x^2+1)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.16 \[ \int (d+e x)^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {24 \, a c^{4} e^{3} x^{4} + 96 \, a c^{4} d e^{2} x^{3} + 144 \, a c^{4} d^{2} e x^{2} + 96 \, a c^{4} d^{3} x + 3 \, {\left (8 \, b c^{4} e^{3} x^{4} + 32 \, b c^{4} d e^{2} x^{3} + 48 \, b c^{4} d^{2} e x^{2} + 32 \, b c^{4} d^{3} x + 24 \, b c^{2} d^{2} e - 3 \, b e^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (6 \, b c^{3} e^{3} x^{3} + 32 \, b c^{3} d e^{2} x^{2} + 96 \, b c^{3} d^{3} - 64 \, b c d e^{2} + 9 \, {\left (8 \, b c^{3} d^{2} e - b c e^{3}\right )} x\right )} \sqrt {c^{2} x^{2} + 1}}{96 \, c^{4}} \] Input:

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

Output:

1/96*(24*a*c^4*e^3*x^4 + 96*a*c^4*d*e^2*x^3 + 144*a*c^4*d^2*e*x^2 + 96*a*c 
^4*d^3*x + 3*(8*b*c^4*e^3*x^4 + 32*b*c^4*d*e^2*x^3 + 48*b*c^4*d^2*e*x^2 + 
32*b*c^4*d^3*x + 24*b*c^2*d^2*e - 3*b*e^3)*log(c*x + sqrt(c^2*x^2 + 1)) - 
(6*b*c^3*e^3*x^3 + 32*b*c^3*d*e^2*x^2 + 96*b*c^3*d^3 - 64*b*c*d*e^2 + 9*(8 
*b*c^3*d^2*e - b*c*e^3)*x)*sqrt(c^2*x^2 + 1))/c^4
 

Sympy [A] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.72 \[ \int (d+e x)^3 (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} a d^{3} x + \frac {3 a d^{2} e x^{2}}{2} + a d e^{2} x^{3} + \frac {a e^{3} x^{4}}{4} + b d^{3} x \operatorname {asinh}{\left (c x \right )} + \frac {3 b d^{2} e x^{2} \operatorname {asinh}{\left (c x \right )}}{2} + b d e^{2} x^{3} \operatorname {asinh}{\left (c x \right )} + \frac {b e^{3} x^{4} \operatorname {asinh}{\left (c x \right )}}{4} - \frac {b d^{3} \sqrt {c^{2} x^{2} + 1}}{c} - \frac {3 b d^{2} e x \sqrt {c^{2} x^{2} + 1}}{4 c} - \frac {b d e^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{3 c} - \frac {b e^{3} x^{3} \sqrt {c^{2} x^{2} + 1}}{16 c} + \frac {3 b d^{2} e \operatorname {asinh}{\left (c x \right )}}{4 c^{2}} + \frac {2 b d e^{2} \sqrt {c^{2} x^{2} + 1}}{3 c^{3}} + \frac {3 b e^{3} x \sqrt {c^{2} x^{2} + 1}}{32 c^{3}} - \frac {3 b e^{3} \operatorname {asinh}{\left (c x \right )}}{32 c^{4}} & \text {for}\: c \neq 0 \\a \left (d^{3} x + \frac {3 d^{2} e x^{2}}{2} + d e^{2} x^{3} + \frac {e^{3} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \] Input:

integrate((e*x+d)**3*(a+b*asinh(c*x)),x)
 

Output:

Piecewise((a*d**3*x + 3*a*d**2*e*x**2/2 + a*d*e**2*x**3 + a*e**3*x**4/4 + 
b*d**3*x*asinh(c*x) + 3*b*d**2*e*x**2*asinh(c*x)/2 + b*d*e**2*x**3*asinh(c 
*x) + b*e**3*x**4*asinh(c*x)/4 - b*d**3*sqrt(c**2*x**2 + 1)/c - 3*b*d**2*e 
*x*sqrt(c**2*x**2 + 1)/(4*c) - b*d*e**2*x**2*sqrt(c**2*x**2 + 1)/(3*c) - b 
*e**3*x**3*sqrt(c**2*x**2 + 1)/(16*c) + 3*b*d**2*e*asinh(c*x)/(4*c**2) + 2 
*b*d*e**2*sqrt(c**2*x**2 + 1)/(3*c**3) + 3*b*e**3*x*sqrt(c**2*x**2 + 1)/(3 
2*c**3) - 3*b*e**3*asinh(c*x)/(32*c**4), Ne(c, 0)), (a*(d**3*x + 3*d**2*e* 
x**2/2 + d*e**2*x**3 + e**3*x**4/4), True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.25 \[ \int (d+e x)^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{4} \, a e^{3} x^{4} + a d e^{2} x^{3} + \frac {3}{2} \, a d^{2} e x^{2} + \frac {3}{4} \, {\left (2 \, x^{2} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\right )}\right )} b d^{2} e + \frac {1}{3} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d e^{2} + \frac {1}{32} \, {\left (8 \, x^{4} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac {3 \, \sqrt {c^{2} x^{2} + 1} x}{c^{4}} + \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c\right )} b e^{3} + a d^{3} x + \frac {{\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b d^{3}}{c} \] Input:

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

Output:

1/4*a*e^3*x^4 + a*d*e^2*x^3 + 3/2*a*d^2*e*x^2 + 3/4*(2*x^2*arcsinh(c*x) - 
c*(sqrt(c^2*x^2 + 1)*x/c^2 - arcsinh(c*x)/c^3))*b*d^2*e + 1/3*(3*x^3*arcsi 
nh(c*x) - c*(sqrt(c^2*x^2 + 1)*x^2/c^2 - 2*sqrt(c^2*x^2 + 1)/c^4))*b*d*e^2 
 + 1/32*(8*x^4*arcsinh(c*x) - (2*sqrt(c^2*x^2 + 1)*x^3/c^2 - 3*sqrt(c^2*x^ 
2 + 1)*x/c^4 + 3*arcsinh(c*x)/c^5)*c)*b*e^3 + a*d^3*x + (c*x*arcsinh(c*x) 
- sqrt(c^2*x^2 + 1))*b*d^3/c
 

Giac [F(-2)]

Exception generated. \[ \int (d+e x)^3 (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:

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

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.55 \[ \int (d+e x)^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {96 \mathit {asinh} \left (c x \right ) b \,c^{4} d^{3} x +144 \mathit {asinh} \left (c x \right ) b \,c^{4} d^{2} e \,x^{2}+96 \mathit {asinh} \left (c x \right ) b \,c^{4} d \,e^{2} x^{3}+24 \mathit {asinh} \left (c x \right ) b \,c^{4} e^{3} x^{4}-96 \sqrt {c^{2} x^{2}+1}\, b \,c^{3} d^{3}-72 \sqrt {c^{2} x^{2}+1}\, b \,c^{3} d^{2} e x -32 \sqrt {c^{2} x^{2}+1}\, b \,c^{3} d \,e^{2} x^{2}-6 \sqrt {c^{2} x^{2}+1}\, b \,c^{3} e^{3} x^{3}+64 \sqrt {c^{2} x^{2}+1}\, b c d \,e^{2}+9 \sqrt {c^{2} x^{2}+1}\, b c \,e^{3} x +72 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) b \,c^{2} d^{2} e -9 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) b \,e^{3}+96 a \,c^{4} d^{3} x +144 a \,c^{4} d^{2} e \,x^{2}+96 a \,c^{4} d \,e^{2} x^{3}+24 a \,c^{4} e^{3} x^{4}}{96 c^{4}} \] Input:

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

Output:

(96*asinh(c*x)*b*c**4*d**3*x + 144*asinh(c*x)*b*c**4*d**2*e*x**2 + 96*asin 
h(c*x)*b*c**4*d*e**2*x**3 + 24*asinh(c*x)*b*c**4*e**3*x**4 - 96*sqrt(c**2* 
x**2 + 1)*b*c**3*d**3 - 72*sqrt(c**2*x**2 + 1)*b*c**3*d**2*e*x - 32*sqrt(c 
**2*x**2 + 1)*b*c**3*d*e**2*x**2 - 6*sqrt(c**2*x**2 + 1)*b*c**3*e**3*x**3 
+ 64*sqrt(c**2*x**2 + 1)*b*c*d*e**2 + 9*sqrt(c**2*x**2 + 1)*b*c*e**3*x + 7 
2*log(sqrt(c**2*x**2 + 1) + c*x)*b*c**2*d**2*e - 9*log(sqrt(c**2*x**2 + 1) 
 + c*x)*b*e**3 + 96*a*c**4*d**3*x + 144*a*c**4*d**2*e*x**2 + 96*a*c**4*d*e 
**2*x**3 + 24*a*c**4*e**3*x**4)/(96*c**4)