\(\int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x)) \, dx\) [59]

Optimal result

Integrand size = 21, antiderivative size = 76 \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x)) \, dx=\frac {b e^2 \sqrt {1+(c+d x)^2}}{3 d}-\frac {b e^2 \left (1+(c+d x)^2\right )^{3/2}}{9 d}+\frac {e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))}{3 d} \] Output:

1/3*b*e^2*(1+(d*x+c)^2)^(1/2)/d-1/9*b*e^2*(1+(d*x+c)^2)^(3/2)/d+1/3*e^2*(d 
*x+c)^3*(a+b*arcsinh(d*x+c))/d
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.84 \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x)) \, dx=\frac {e^2 \left (-\frac {1}{9} b \left (-2+c^2+2 c d x+d^2 x^2\right ) \sqrt {1+(c+d x)^2}+\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))\right )}{d} \] Input:

Integrate[(c*e + d*e*x)^2*(a + b*ArcSinh[c + d*x]),x]
 

Output:

(e^2*(-1/9*(b*(-2 + c^2 + 2*c*d*x + d^2*x^2)*Sqrt[1 + (c + d*x)^2]) + ((c 
+ d*x)^3*(a + b*ArcSinh[c + d*x]))/3))/d
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.88, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6274, 27, 6191, 243, 53, 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.

\(\displaystyle \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x)) \, dx\)

\(\Big \downarrow \) 6274

\(\displaystyle \frac {\int e^2 (c+d x)^2 (a+b \text {arcsinh}(c+d x))d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^2 \int (c+d x)^2 (a+b \text {arcsinh}(c+d x))d(c+d x)}{d}\)

\(\Big \downarrow \) 6191

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

\(\Big \downarrow \) 243

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))-\frac {1}{6} b \int \frac {(c+d x)^2}{\sqrt {(c+d x)^2+1}}d(c+d x)^2\right )}{d}\)

\(\Big \downarrow \) 53

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))-\frac {1}{6} b \int \left (\sqrt {(c+d x)^2+1}-\frac {1}{\sqrt {(c+d x)^2+1}}\right )d(c+d x)^2\right )}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))-\frac {1}{6} b \left (\frac {2}{3} \left ((c+d x)^2+1\right )^{3/2}-2 \sqrt {(c+d x)^2+1}\right )\right )}{d}\)

Input:

Int[(c*e + d*e*x)^2*(a + b*ArcSinh[c + d*x]),x]
 

Output:

(e^2*(-1/6*(b*(-2*Sqrt[1 + (c + d*x)^2] + (2*(1 + (c + d*x)^2)^(3/2))/3)) 
+ ((c + d*x)^3*(a + b*ArcSinh[c + d*x]))/3))/d
 

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 53

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6191

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
 :> Simp[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* 
(n/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + 
c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, 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 = 0.25 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.96

method	result	size
derivativedivides	\(\frac {\frac {a \,e^{2} \left (d x +c \right )^{3}}{3}+e^{2} b \left (\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )}{3}-\frac {\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{9}+\frac {2 \sqrt {1+\left (d x +c \right )^{2}}}{9}\right )}{d}\)	\(73\)
default	\(\frac {\frac {a \,e^{2} \left (d x +c \right )^{3}}{3}+e^{2} b \left (\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )}{3}-\frac {\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{9}+\frac {2 \sqrt {1+\left (d x +c \right )^{2}}}{9}\right )}{d}\)	\(73\)
parts	\(\frac {a \,e^{2} \left (d x +c \right )^{3}}{3 d}+\frac {e^{2} b \left (\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )}{3}-\frac {\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{9}+\frac {2 \sqrt {1+\left (d x +c \right )^{2}}}{9}\right )}{d}\)	\(75\)
orering	\(\frac {\left (5 d^{4} x^{4}+20 c \,d^{3} x^{3}+30 c^{2} d^{2} x^{2}+20 c^{3} d x +5 c^{4}-2 d^{2} x^{2}-4 c d x -2 c^{2}-4\right ) \left (d e x +c e \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}{9 d \left (d x +c \right )^{3}}-\frac {\left (d^{2} x^{2}+2 c d x +c^{2}-2\right ) \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) \left (2 \left (d e x +c e \right ) \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right ) d e +\frac {\left (d e x +c e \right )^{2} b d}{\sqrt {1+\left (d x +c \right )^{2}}}\right )}{9 d^{2} \left (d x +c \right )^{2}}\)	\(187\)

Input:

int((d*e*x+c*e)^2*(a+b*arcsinh(d*x+c)),x,method=_RETURNVERBOSE)
 

Output:

1/d*(1/3*a*e^2*(d*x+c)^3+e^2*b*(1/3*(d*x+c)^3*arcsinh(d*x+c)-1/9*(d*x+c)^2 
*(1+(d*x+c)^2)^(1/2)+2/9*(1+(d*x+c)^2)^(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (66) = 132\).

Time = 0.09 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.21 \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x)) \, dx=\frac {3 \, a d^{3} e^{2} x^{3} + 9 \, a c d^{2} e^{2} x^{2} + 9 \, a c^{2} d e^{2} x + 3 \, {\left (b d^{3} e^{2} x^{3} + 3 \, b c d^{2} e^{2} x^{2} + 3 \, b c^{2} d e^{2} x + b c^{3} e^{2}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - {\left (b d^{2} e^{2} x^{2} + 2 \, b c d e^{2} x + {\left (b c^{2} - 2 \, b\right )} e^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{9 \, d} \] Input:

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

Output:

1/9*(3*a*d^3*e^2*x^3 + 9*a*c*d^2*e^2*x^2 + 9*a*c^2*d*e^2*x + 3*(b*d^3*e^2* 
x^3 + 3*b*c*d^2*e^2*x^2 + 3*b*c^2*d*e^2*x + b*c^3*e^2)*log(d*x + c + sqrt( 
d^2*x^2 + 2*c*d*x + c^2 + 1)) - (b*d^2*e^2*x^2 + 2*b*c*d*e^2*x + (b*c^2 - 
2*b)*e^2)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))/d
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (63) = 126\).

Time = 0.18 (sec) , antiderivative size = 258, normalized size of antiderivative = 3.39 \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x)) \, dx=\begin {cases} a c^{2} e^{2} x + a c d e^{2} x^{2} + \frac {a d^{2} e^{2} x^{3}}{3} + \frac {b c^{3} e^{2} \operatorname {asinh}{\left (c + d x \right )}}{3 d} + b c^{2} e^{2} x \operatorname {asinh}{\left (c + d x \right )} - \frac {b c^{2} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{9 d} + b c d e^{2} x^{2} \operatorname {asinh}{\left (c + d x \right )} - \frac {2 b c e^{2} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{9} + \frac {b d^{2} e^{2} x^{3} \operatorname {asinh}{\left (c + d x \right )}}{3} - \frac {b d e^{2} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{9} + \frac {2 b e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{9 d} & \text {for}\: d \neq 0 \\c^{2} e^{2} x \left (a + b \operatorname {asinh}{\left (c \right )}\right ) & \text {otherwise} \end {cases} \] Input:

integrate((d*e*x+c*e)**2*(a+b*asinh(d*x+c)),x)
 

Output:

Piecewise((a*c**2*e**2*x + a*c*d*e**2*x**2 + a*d**2*e**2*x**3/3 + b*c**3*e 
**2*asinh(c + d*x)/(3*d) + b*c**2*e**2*x*asinh(c + d*x) - b*c**2*e**2*sqrt 
(c**2 + 2*c*d*x + d**2*x**2 + 1)/(9*d) + b*c*d*e**2*x**2*asinh(c + d*x) - 
2*b*c*e**2*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/9 + b*d**2*e**2*x**3*asi 
nh(c + d*x)/3 - b*d*e**2*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/9 + 2*b 
*e**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(9*d), Ne(d, 0)), (c**2*e**2*x* 
(a + b*asinh(c)), True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 445 vs. \(2 (66) = 132\).

Time = 0.04 (sec) , antiderivative size = 445, normalized size of antiderivative = 5.86 \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x)) \, dx=\frac {1}{3} \, a d^{2} e^{2} x^{3} + a c d e^{2} x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {arsinh}\left (d x + c\right ) - d {\left (\frac {3 \, c^{2} \operatorname {arsinh}\left (\frac {2 \, {\left (d^{2} x + c d\right )}}{\sqrt {-4 \, c^{2} d^{2} + 4 \, {\left (c^{2} + 1\right )} d^{2}}}\right )}{d^{3}} + \frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} x}{d^{2}} - \frac {{\left (c^{2} + 1\right )} \operatorname {arsinh}\left (\frac {2 \, {\left (d^{2} x + c d\right )}}{\sqrt {-4 \, c^{2} d^{2} + 4 \, {\left (c^{2} + 1\right )} d^{2}}}\right )}{d^{3}} - \frac {3 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} c}{d^{3}}\right )}\right )} b c d e^{2} + \frac {1}{18} \, {\left (6 \, x^{3} \operatorname {arsinh}\left (d x + c\right ) - d {\left (\frac {2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} x^{2}}{d^{2}} - \frac {15 \, c^{3} \operatorname {arsinh}\left (\frac {2 \, {\left (d^{2} x + c d\right )}}{\sqrt {-4 \, c^{2} d^{2} + 4 \, {\left (c^{2} + 1\right )} d^{2}}}\right )}{d^{4}} - \frac {5 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} c x}{d^{3}} + \frac {9 \, {\left (c^{2} + 1\right )} c \operatorname {arsinh}\left (\frac {2 \, {\left (d^{2} x + c d\right )}}{\sqrt {-4 \, c^{2} d^{2} + 4 \, {\left (c^{2} + 1\right )} d^{2}}}\right )}{d^{4}} + \frac {15 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} c^{2}}{d^{4}} - \frac {4 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} {\left (c^{2} + 1\right )}}{d^{4}}\right )}\right )} b d^{2} e^{2} + a c^{2} e^{2} x + \frac {{\left ({\left (d x + c\right )} \operatorname {arsinh}\left (d x + c\right ) - \sqrt {{\left (d x + c\right )}^{2} + 1}\right )} b c^{2} e^{2}}{d} \] Input:

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

Output:

1/3*a*d^2*e^2*x^3 + a*c*d*e^2*x^2 + 1/2*(2*x^2*arcsinh(d*x + c) - d*(3*c^2 
*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^3 + sqrt(d^ 
2*x^2 + 2*c*d*x + c^2 + 1)*x/d^2 - (c^2 + 1)*arcsinh(2*(d^2*x + c*d)/sqrt( 
-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^3 - 3*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c 
/d^3))*b*c*d*e^2 + 1/18*(6*x^3*arcsinh(d*x + c) - d*(2*sqrt(d^2*x^2 + 2*c* 
d*x + c^2 + 1)*x^2/d^2 - 15*c^3*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 
4*(c^2 + 1)*d^2))/d^4 - 5*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c*x/d^3 + 9*(c 
^2 + 1)*c*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^4 
+ 15*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c^2/d^4 - 4*sqrt(d^2*x^2 + 2*c*d*x 
+ c^2 + 1)*(c^2 + 1)/d^4))*b*d^2*e^2 + a*c^2*e^2*x + ((d*x + c)*arcsinh(d* 
x + c) - sqrt((d*x + c)^2 + 1))*b*c^2*e^2/d
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 419 vs. \(2 (66) = 132\).

Time = 0.47 (sec) , antiderivative size = 419, normalized size of antiderivative = 5.51 \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x)) \, dx=\frac {1}{3} \, a d^{2} e^{2} x^{3} + a c d e^{2} x^{2} - {\left (d {\left (\frac {c \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} {\left | d \right |} \right |}\right )}{d {\left | d \right |}} + \frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{d^{2}}\right )} - x \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )\right )} b c^{2} e^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - {\left (\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} {\left (\frac {x}{d^{2}} - \frac {3 \, c}{d^{3}}\right )} - \frac {{\left (2 \, c^{2} - 1\right )} \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} {\left | d \right |} \right |}\right )}{d^{2} {\left | d \right |}}\right )} d\right )} b c d e^{2} + \frac {1}{18} \, {\left (6 \, x^{3} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - {\left (\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} {\left (x {\left (\frac {2 \, x}{d^{2}} - \frac {5 \, c}{d^{3}}\right )} + \frac {11 \, c^{2} d - 4 \, d}{d^{5}}\right )} + \frac {3 \, {\left (2 \, c^{3} - 3 \, c\right )} \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} {\left | d \right |} \right |}\right )}{d^{3} {\left | d \right |}}\right )} d\right )} b d^{2} e^{2} + a c^{2} e^{2} x \] Input:

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

Output:

1/3*a*d^2*e^2*x^3 + a*c*d*e^2*x^2 - (d*(c*log(abs(-c*d - (x*abs(d) - sqrt( 
d^2*x^2 + 2*c*d*x + c^2 + 1))*abs(d)))/(d*abs(d)) + sqrt(d^2*x^2 + 2*c*d*x 
 + c^2 + 1)/d^2) - x*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)))*b*c 
^2*e^2 + 1/2*(2*x^2*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)) - (sq 
rt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(x/d^2 - 3*c/d^3) - (2*c^2 - 1)*log(abs(-c 
*d - (x*abs(d) - sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))*abs(d)))/(d^2*abs(d))) 
*d)*b*c*d*e^2 + 1/18*(6*x^3*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1 
)) - (sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(x*(2*x/d^2 - 5*c/d^3) + (11*c^2*d 
 - 4*d)/d^5) + 3*(2*c^3 - 3*c)*log(abs(-c*d - (x*abs(d) - sqrt(d^2*x^2 + 2 
*c*d*x + c^2 + 1))*abs(d)))/(d^3*abs(d)))*d)*b*d^2*e^2 + a*c^2*e^2*x
 

Mupad [F(-1)]

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

int((c*e + d*e*x)^2*(a + b*asinh(c + d*x)),x)
                                                                                    
                                                                                    
 

Output:

int((c*e + d*e*x)^2*(a + b*asinh(c + d*x)), x)
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.88 \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x)) \, dx=\frac {e^{2} \left (9 \mathit {asinh} \left (d x +c \right ) b \,c^{3}+9 \mathit {asinh} \left (d x +c \right ) b \,c^{2} d x +9 \mathit {asinh} \left (d x +c \right ) b c \,d^{2} x^{2}+3 \mathit {asinh} \left (d x +c \right ) b \,d^{3} x^{3}-\sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, b \,c^{2}-2 \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, b c d x -\sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, b \,d^{2} x^{2}+2 \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, b -6 \,\mathrm {log}\left (\sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}+c +d x \right ) b \,c^{3}+9 a \,c^{2} d x +9 a c \,d^{2} x^{2}+3 a \,d^{3} x^{3}\right )}{9 d} \] Input:

int((d*e*x+c*e)^2*(a+b*asinh(d*x+c)),x)
 

Output:

(e**2*(9*asinh(c + d*x)*b*c**3 + 9*asinh(c + d*x)*b*c**2*d*x + 9*asinh(c + 
 d*x)*b*c*d**2*x**2 + 3*asinh(c + d*x)*b*d**3*x**3 - sqrt(c**2 + 2*c*d*x + 
 d**2*x**2 + 1)*b*c**2 - 2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*b*c*d*x - 
sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*b*d**2*x**2 + 2*sqrt(c**2 + 2*c*d*x + 
 d**2*x**2 + 1)*b - 6*log(sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1) + c + d*x)* 
b*c**3 + 9*a*c**2*d*x + 9*a*c*d**2*x**2 + 3*a*d**3*x**3))/(9*d)