\(\int x (d+c^2 d x^2)^3 (a+b \text {arcsinh}(c x)) \, dx\) [22]

Optimal result

Integrand size = 22, antiderivative size = 145 \[ \int x \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=-\frac {35 b d^3 x \sqrt {1+c^2 x^2}}{1024 c}-\frac {35 b d^3 x \left (1+c^2 x^2\right )^{3/2}}{1536 c}-\frac {7 b d^3 x \left (1+c^2 x^2\right )^{5/2}}{384 c}-\frac {b d^3 x \left (1+c^2 x^2\right )^{7/2}}{64 c}-\frac {35 b d^3 \text {arcsinh}(c x)}{1024 c^2}+\frac {d^3 \left (1+c^2 x^2\right )^4 (a+b \text {arcsinh}(c x))}{8 c^2} \] Output:

-35/1024*b*d^3*x*(c^2*x^2+1)^(1/2)/c-35/1536*b*d^3*x*(c^2*x^2+1)^(3/2)/c-7 
/384*b*d^3*x*(c^2*x^2+1)^(5/2)/c-1/64*b*d^3*x*(c^2*x^2+1)^(7/2)/c-35/1024* 
b*d^3*arcsinh(c*x)/c^2+1/8*d^3*(c^2*x^2+1)^4*(a+b*arcsinh(c*x))/c^2
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.88 \[ \int x \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {d^3 \left (c x \left (384 a c x \left (4+6 c^2 x^2+4 c^4 x^4+c^6 x^6\right )-b \sqrt {1+c^2 x^2} \left (279+326 c^2 x^2+200 c^4 x^4+48 c^6 x^6\right )\right )+3 b \left (93+512 c^2 x^2+768 c^4 x^4+512 c^6 x^6+128 c^8 x^8\right ) \text {arcsinh}(c x)\right )}{3072 c^2} \] Input:

Integrate[x*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x]),x]
 

Output:

(d^3*(c*x*(384*a*c*x*(4 + 6*c^2*x^2 + 4*c^4*x^4 + c^6*x^6) - b*Sqrt[1 + c^ 
2*x^2]*(279 + 326*c^2*x^2 + 200*c^4*x^4 + 48*c^6*x^6)) + 3*b*(93 + 512*c^2 
*x^2 + 768*c^4*x^4 + 512*c^6*x^6 + 128*c^8*x^8)*ArcSinh[c*x]))/(3072*c^2)
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6213, 211, 211, 211, 211, 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.

\(\displaystyle \int x \left (c^2 d x^2+d\right )^3 (a+b \text {arcsinh}(c x)) \, dx\)

\(\Big \downarrow \) 6213

\(\displaystyle \frac {d^3 \left (c^2 x^2+1\right )^4 (a+b \text {arcsinh}(c x))}{8 c^2}-\frac {b d^3 \int \left (c^2 x^2+1\right )^{7/2}dx}{8 c}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {d^3 \left (c^2 x^2+1\right )^4 (a+b \text {arcsinh}(c x))}{8 c^2}-\frac {b d^3 \left (\frac {7}{8} \int \left (c^2 x^2+1\right )^{5/2}dx+\frac {1}{8} x \left (c^2 x^2+1\right )^{7/2}\right )}{8 c}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {d^3 \left (c^2 x^2+1\right )^4 (a+b \text {arcsinh}(c x))}{8 c^2}-\frac {b d^3 \left (\frac {7}{8} \left (\frac {5}{6} \int \left (c^2 x^2+1\right )^{3/2}dx+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )+\frac {1}{8} x \left (c^2 x^2+1\right )^{7/2}\right )}{8 c}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {d^3 \left (c^2 x^2+1\right )^4 (a+b \text {arcsinh}(c x))}{8 c^2}-\frac {b d^3 \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {c^2 x^2+1}dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )+\frac {1}{8} x \left (c^2 x^2+1\right )^{7/2}\right )}{8 c}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {d^3 \left (c^2 x^2+1\right )^4 (a+b \text {arcsinh}(c x))}{8 c^2}-\frac {b d^3 \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )+\frac {1}{8} x \left (c^2 x^2+1\right )^{7/2}\right )}{8 c}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {d^3 \left (c^2 x^2+1\right )^4 (a+b \text {arcsinh}(c x))}{8 c^2}-\frac {b d^3 \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )+\frac {1}{8} x \left (c^2 x^2+1\right )^{7/2}\right )}{8 c}\)

Input:

Int[x*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x]),x]
 

Output:

(d^3*(1 + c^2*x^2)^4*(a + b*ArcSinh[c*x]))/(8*c^2) - (b*d^3*((x*(1 + c^2*x 
^2)^(7/2))/8 + (7*((x*(1 + c^2*x^2)^(5/2))/6 + (5*((x*(1 + c^2*x^2)^(3/2)) 
/4 + (3*((x*Sqrt[1 + c^2*x^2])/2 + ArcSinh[c*x]/(2*c)))/4))/6))/8))/(8*c)
 

Defintions of rubi rules used

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

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 6213

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p 
+ 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] 
 Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ 
{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
 

Maple [A] (verified)

Time = 0.74 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.99

method	result	size
derivativedivides	\(\frac {\frac {d^{3} a \left (c^{2} x^{2}+1\right )^{4}}{8}+d^{3} b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{8} c^{8}}{8}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{2}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{4}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}}{2}+\frac {93 \,\operatorname {arcsinh}\left (x c \right )}{1024}-\frac {x c \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{64}-\frac {7 x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{384}-\frac {35 x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{1536}-\frac {35 \sqrt {c^{2} x^{2}+1}\, x c}{1024}\right )}{c^{2}}\)	\(143\)
default	\(\frac {\frac {d^{3} a \left (c^{2} x^{2}+1\right )^{4}}{8}+d^{3} b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{8} c^{8}}{8}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{2}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{4}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}}{2}+\frac {93 \,\operatorname {arcsinh}\left (x c \right )}{1024}-\frac {x c \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{64}-\frac {7 x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{384}-\frac {35 x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{1536}-\frac {35 \sqrt {c^{2} x^{2}+1}\, x c}{1024}\right )}{c^{2}}\)	\(143\)
parts	\(\frac {d^{3} a \left (c^{2} x^{2}+1\right )^{4}}{8 c^{2}}+\frac {d^{3} b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{8} c^{8}}{8}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{2}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{4}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}}{2}+\frac {93 \,\operatorname {arcsinh}\left (x c \right )}{1024}-\frac {x c \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{64}-\frac {7 x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{384}-\frac {35 x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{1536}-\frac {35 \sqrt {c^{2} x^{2}+1}\, x c}{1024}\right )}{c^{2}}\)	\(145\)
orering	\(\frac {\left (720 c^{8} x^{8}+2984 c^{6} x^{6}+4786 c^{4} x^{4}+3815 c^{2} x^{2}+558\right ) \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{3072 c^{2} \left (c^{2} x^{2}+1\right )^{3}}-\frac {\left (48 c^{6} x^{6}+200 c^{4} x^{4}+326 c^{2} x^{2}+279\right ) \left (\left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )+6 x^{2} \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) c^{2} d +\frac {x \left (c^{2} d \,x^{2}+d \right )^{3} b c}{\sqrt {c^{2} x^{2}+1}}\right )}{3072 c^{2} \left (c^{2} x^{2}+1\right )^{2}}\)	\(192\)

Input:

int(x*(c^2*d*x^2+d)^3*(a+b*arcsinh(x*c)),x,method=_RETURNVERBOSE)
 

Output:

1/c^2*(1/8*d^3*a*(c^2*x^2+1)^4+d^3*b*(1/8*arcsinh(x*c)*x^8*c^8+1/2*arcsinh 
(x*c)*x^6*c^6+3/4*arcsinh(x*c)*c^4*x^4+1/2*arcsinh(x*c)*c^2*x^2+93/1024*ar 
csinh(x*c)-1/64*x*c*(c^2*x^2+1)^(7/2)-7/384*x*c*(c^2*x^2+1)^(5/2)-35/1536* 
x*c*(c^2*x^2+1)^(3/2)-35/1024*(c^2*x^2+1)^(1/2)*x*c))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.28 \[ \int x \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {384 \, a c^{8} d^{3} x^{8} + 1536 \, a c^{6} d^{3} x^{6} + 2304 \, a c^{4} d^{3} x^{4} + 1536 \, a c^{2} d^{3} x^{2} + 3 \, {\left (128 \, b c^{8} d^{3} x^{8} + 512 \, b c^{6} d^{3} x^{6} + 768 \, b c^{4} d^{3} x^{4} + 512 \, b c^{2} d^{3} x^{2} + 93 \, b d^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (48 \, b c^{7} d^{3} x^{7} + 200 \, b c^{5} d^{3} x^{5} + 326 \, b c^{3} d^{3} x^{3} + 279 \, b c d^{3} x\right )} \sqrt {c^{2} x^{2} + 1}}{3072 \, c^{2}} \] Input:

integrate(x*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x)),x, algorithm="fricas")
 

Output:

1/3072*(384*a*c^8*d^3*x^8 + 1536*a*c^6*d^3*x^6 + 2304*a*c^4*d^3*x^4 + 1536 
*a*c^2*d^3*x^2 + 3*(128*b*c^8*d^3*x^8 + 512*b*c^6*d^3*x^6 + 768*b*c^4*d^3* 
x^4 + 512*b*c^2*d^3*x^2 + 93*b*d^3)*log(c*x + sqrt(c^2*x^2 + 1)) - (48*b*c 
^7*d^3*x^7 + 200*b*c^5*d^3*x^5 + 326*b*c^3*d^3*x^3 + 279*b*c*d^3*x)*sqrt(c 
^2*x^2 + 1))/c^2
 

Sympy [A] (verification not implemented)

Time = 0.93 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.74 \[ \int x \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {a c^{6} d^{3} x^{8}}{8} + \frac {a c^{4} d^{3} x^{6}}{2} + \frac {3 a c^{2} d^{3} x^{4}}{4} + \frac {a d^{3} x^{2}}{2} + \frac {b c^{6} d^{3} x^{8} \operatorname {asinh}{\left (c x \right )}}{8} - \frac {b c^{5} d^{3} x^{7} \sqrt {c^{2} x^{2} + 1}}{64} + \frac {b c^{4} d^{3} x^{6} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {25 b c^{3} d^{3} x^{5} \sqrt {c^{2} x^{2} + 1}}{384} + \frac {3 b c^{2} d^{3} x^{4} \operatorname {asinh}{\left (c x \right )}}{4} - \frac {163 b c d^{3} x^{3} \sqrt {c^{2} x^{2} + 1}}{1536} + \frac {b d^{3} x^{2} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {93 b d^{3} x \sqrt {c^{2} x^{2} + 1}}{1024 c} + \frac {93 b d^{3} \operatorname {asinh}{\left (c x \right )}}{1024 c^{2}} & \text {for}\: c \neq 0 \\\frac {a d^{3} x^{2}}{2} & \text {otherwise} \end {cases} \] Input:

integrate(x*(c**2*d*x**2+d)**3*(a+b*asinh(c*x)),x)
 

Output:

Piecewise((a*c**6*d**3*x**8/8 + a*c**4*d**3*x**6/2 + 3*a*c**2*d**3*x**4/4 
+ a*d**3*x**2/2 + b*c**6*d**3*x**8*asinh(c*x)/8 - b*c**5*d**3*x**7*sqrt(c* 
*2*x**2 + 1)/64 + b*c**4*d**3*x**6*asinh(c*x)/2 - 25*b*c**3*d**3*x**5*sqrt 
(c**2*x**2 + 1)/384 + 3*b*c**2*d**3*x**4*asinh(c*x)/4 - 163*b*c*d**3*x**3* 
sqrt(c**2*x**2 + 1)/1536 + b*d**3*x**2*asinh(c*x)/2 - 93*b*d**3*x*sqrt(c** 
2*x**2 + 1)/(1024*c) + 93*b*d**3*asinh(c*x)/(1024*c**2), Ne(c, 0)), (a*d** 
3*x**2/2, True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 352 vs. \(2 (125) = 250\).

Time = 0.04 (sec) , antiderivative size = 352, normalized size of antiderivative = 2.43 \[ \int x \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{8} \, a c^{6} d^{3} x^{8} + \frac {1}{2} \, a c^{4} d^{3} x^{6} + \frac {1}{3072} \, {\left (384 \, x^{8} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {48 \, \sqrt {c^{2} x^{2} + 1} x^{7}}{c^{2}} - \frac {56 \, \sqrt {c^{2} x^{2} + 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{6}} - \frac {105 \, \sqrt {c^{2} x^{2} + 1} x}{c^{8}} + \frac {105 \, \operatorname {arsinh}\left (c x\right )}{c^{9}}\right )} c\right )} b c^{6} d^{3} + \frac {3}{4} \, a c^{2} d^{3} x^{4} + \frac {1}{96} \, {\left (48 \, x^{6} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{5}}{c^{2}} - \frac {10 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \operatorname {arsinh}\left (c x\right )}{c^{7}}\right )} c\right )} b c^{4} d^{3} + \frac {3}{32} \, {\left (8 \, x^{4} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac {3 \, \sqrt {c^{2} x^{2} + 1} x}{c^{4}} + \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c\right )} b c^{2} d^{3} + \frac {1}{2} \, a d^{3} x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\right )}\right )} b d^{3} \] Input:

integrate(x*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x)),x, algorithm="maxima")
 

Output:

1/8*a*c^6*d^3*x^8 + 1/2*a*c^4*d^3*x^6 + 1/3072*(384*x^8*arcsinh(c*x) - (48 
*sqrt(c^2*x^2 + 1)*x^7/c^2 - 56*sqrt(c^2*x^2 + 1)*x^5/c^4 + 70*sqrt(c^2*x^ 
2 + 1)*x^3/c^6 - 105*sqrt(c^2*x^2 + 1)*x/c^8 + 105*arcsinh(c*x)/c^9)*c)*b* 
c^6*d^3 + 3/4*a*c^2*d^3*x^4 + 1/96*(48*x^6*arcsinh(c*x) - (8*sqrt(c^2*x^2 
+ 1)*x^5/c^2 - 10*sqrt(c^2*x^2 + 1)*x^3/c^4 + 15*sqrt(c^2*x^2 + 1)*x/c^6 - 
 15*arcsinh(c*x)/c^7)*c)*b*c^4*d^3 + 3/32*(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* 
c^2*d^3 + 1/2*a*d^3*x^2 + 1/4*(2*x^2*arcsinh(c*x) - c*(sqrt(c^2*x^2 + 1)*x 
/c^2 - arcsinh(c*x)/c^3))*b*d^3
 

Giac [F(-2)]

Exception generated. \[ \int x \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x)),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int x \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\int x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^3 \,d x \] Input:

int(x*(a + b*asinh(c*x))*(d + c^2*d*x^2)^3,x)
 

Output:

int(x*(a + b*asinh(c*x))*(d + c^2*d*x^2)^3, x)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.29 \[ \int x \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {d^{3} \left (384 \mathit {asinh} \left (c x \right ) b \,c^{8} x^{8}+1536 \mathit {asinh} \left (c x \right ) b \,c^{6} x^{6}+2304 \mathit {asinh} \left (c x \right ) b \,c^{4} x^{4}+1536 \mathit {asinh} \left (c x \right ) b \,c^{2} x^{2}-48 \sqrt {c^{2} x^{2}+1}\, b \,c^{7} x^{7}-200 \sqrt {c^{2} x^{2}+1}\, b \,c^{5} x^{5}-326 \sqrt {c^{2} x^{2}+1}\, b \,c^{3} x^{3}-279 \sqrt {c^{2} x^{2}+1}\, b c x +279 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) b +384 a \,c^{8} x^{8}+1536 a \,c^{6} x^{6}+2304 a \,c^{4} x^{4}+1536 a \,c^{2} x^{2}\right )}{3072 c^{2}} \] Input:

int(x*(c^2*d*x^2+d)^3*(a+b*asinh(c*x)),x)
 

Output:

(d**3*(384*asinh(c*x)*b*c**8*x**8 + 1536*asinh(c*x)*b*c**6*x**6 + 2304*asi 
nh(c*x)*b*c**4*x**4 + 1536*asinh(c*x)*b*c**2*x**2 - 48*sqrt(c**2*x**2 + 1) 
*b*c**7*x**7 - 200*sqrt(c**2*x**2 + 1)*b*c**5*x**5 - 326*sqrt(c**2*x**2 + 
1)*b*c**3*x**3 - 279*sqrt(c**2*x**2 + 1)*b*c*x + 279*log(sqrt(c**2*x**2 + 
1) + c*x)*b + 384*a*c**8*x**8 + 1536*a*c**6*x**6 + 2304*a*c**4*x**4 + 1536 
*a*c**2*x**2))/(3072*c**2)