\(\int x^4 (d+c^2 d x^2)^2 (a+b \text {arcsinh}(c x)) \, dx\) [10]

Optimal result

Integrand size = 24, antiderivative size = 181 \[ \int x^4 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=-\frac {8 b d^2 \sqrt {1+c^2 x^2}}{315 c^5}-\frac {4 b d^2 \left (1+c^2 x^2\right )^{3/2}}{945 c^5}-\frac {b d^2 \left (1+c^2 x^2\right )^{5/2}}{525 c^5}+\frac {10 b d^2 \left (1+c^2 x^2\right )^{7/2}}{441 c^5}-\frac {b d^2 \left (1+c^2 x^2\right )^{9/2}}{81 c^5}+\frac {1}{5} d^2 x^5 (a+b \text {arcsinh}(c x))+\frac {2}{7} c^2 d^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{9} c^4 d^2 x^9 (a+b \text {arcsinh}(c x)) \] Output:

-8/315*b*d^2*(c^2*x^2+1)^(1/2)/c^5-4/945*b*d^2*(c^2*x^2+1)^(3/2)/c^5-1/525 
*b*d^2*(c^2*x^2+1)^(5/2)/c^5+10/441*b*d^2*(c^2*x^2+1)^(7/2)/c^5-1/81*b*d^2 
*(c^2*x^2+1)^(9/2)/c^5+1/5*d^2*x^5*(a+b*arcsinh(c*x))+2/7*c^2*d^2*x^7*(a+b 
*arcsinh(c*x))+1/9*c^4*d^2*x^9*(a+b*arcsinh(c*x))
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.66 \[ \int x^4 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {d^2 \left (315 a c^5 x^5 \left (63+90 c^2 x^2+35 c^4 x^4\right )-b \sqrt {1+c^2 x^2} \left (2104-1052 c^2 x^2+789 c^4 x^4+2650 c^6 x^6+1225 c^8 x^8\right )+315 b c^5 x^5 \left (63+90 c^2 x^2+35 c^4 x^4\right ) \text {arcsinh}(c x)\right )}{99225 c^5} \] Input:

Integrate[x^4*(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x]),x]
 

Output:

(d^2*(315*a*c^5*x^5*(63 + 90*c^2*x^2 + 35*c^4*x^4) - b*Sqrt[1 + c^2*x^2]*( 
2104 - 1052*c^2*x^2 + 789*c^4*x^4 + 2650*c^6*x^6 + 1225*c^8*x^8) + 315*b*c 
^5*x^5*(63 + 90*c^2*x^2 + 35*c^4*x^4)*ArcSinh[c*x]))/(99225*c^5)
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6218, 27, 1578, 1195, 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.

\(\displaystyle \int x^4 \left (c^2 d x^2+d\right )^2 (a+b \text {arcsinh}(c x)) \, dx\)

\(\Big \downarrow \) 6218

\(\displaystyle -b c \int \frac {d^2 x^5 \left (35 c^4 x^4+90 c^2 x^2+63\right )}{315 \sqrt {c^2 x^2+1}}dx+\frac {1}{9} c^4 d^2 x^9 (a+b \text {arcsinh}(c x))+\frac {2}{7} c^2 d^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} d^2 x^5 (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{315} b c d^2 \int \frac {x^5 \left (35 c^4 x^4+90 c^2 x^2+63\right )}{\sqrt {c^2 x^2+1}}dx+\frac {1}{9} c^4 d^2 x^9 (a+b \text {arcsinh}(c x))+\frac {2}{7} c^2 d^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} d^2 x^5 (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 1578

\(\displaystyle -\frac {1}{630} b c d^2 \int \frac {x^4 \left (35 c^4 x^4+90 c^2 x^2+63\right )}{\sqrt {c^2 x^2+1}}dx^2+\frac {1}{9} c^4 d^2 x^9 (a+b \text {arcsinh}(c x))+\frac {2}{7} c^2 d^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} d^2 x^5 (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 1195

\(\displaystyle -\frac {1}{630} b c d^2 \int \left (\frac {35 \left (c^2 x^2+1\right )^{7/2}}{c^4}-\frac {50 \left (c^2 x^2+1\right )^{5/2}}{c^4}+\frac {3 \left (c^2 x^2+1\right )^{3/2}}{c^4}+\frac {4 \sqrt {c^2 x^2+1}}{c^4}+\frac {8}{c^4 \sqrt {c^2 x^2+1}}\right )dx^2+\frac {1}{9} c^4 d^2 x^9 (a+b \text {arcsinh}(c x))+\frac {2}{7} c^2 d^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} d^2 x^5 (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{9} c^4 d^2 x^9 (a+b \text {arcsinh}(c x))+\frac {2}{7} c^2 d^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} d^2 x^5 (a+b \text {arcsinh}(c x))-\frac {1}{630} b c d^2 \left (\frac {70 \left (c^2 x^2+1\right )^{9/2}}{9 c^6}-\frac {100 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {8 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2+1}}{c^6}\right )\)

Input:

Int[x^4*(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x]),x]
 

Output:

-1/630*(b*c*d^2*((16*Sqrt[1 + c^2*x^2])/c^6 + (8*(1 + c^2*x^2)^(3/2))/(3*c 
^6) + (6*(1 + c^2*x^2)^(5/2))/(5*c^6) - (100*(1 + c^2*x^2)^(7/2))/(7*c^6) 
+ (70*(1 + c^2*x^2)^(9/2))/(9*c^6))) + (d^2*x^5*(a + b*ArcSinh[c*x]))/5 + 
(2*c^2*d^2*x^7*(a + b*ArcSinh[c*x]))/7 + (c^4*d^2*x^9*(a + b*ArcSinh[c*x]) 
)/9
 

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 1195

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + 
 g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x 
] && IGtQ[p, 0]
 

rule 1578

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ 
)^4)^(p_.), x_Symbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a 
+ b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int 
egerQ[(m - 1)/2]
 

rule 2009

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

rule 6218

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_ 
)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Simp 
[(a + b*ArcSinh[c*x])   u, x] - Simp[b*c   Int[SimplifyIntegrand[u/Sqrt[1 + 
 c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] 
&& IGtQ[p, 0]
 

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.90

method	result	size
parts	\(a \,d^{2} \left (\frac {1}{9} c^{4} x^{9}+\frac {2}{7} c^{2} x^{7}+\frac {1}{5} x^{5}\right )+\frac {d^{2} b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{9} c^{9}}{9}+\frac {2 \,\operatorname {arcsinh}\left (x c \right ) x^{7} c^{7}}{7}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}-\frac {263 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{33075}+\frac {1052 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{99225}-\frac {2104 \sqrt {c^{2} x^{2}+1}}{99225}-\frac {106 x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{3969}-\frac {x^{8} c^{8} \sqrt {c^{2} x^{2}+1}}{81}\right )}{c^{5}}\)	\(163\)
derivativedivides	\(\frac {a \,d^{2} \left (\frac {1}{9} c^{9} x^{9}+\frac {2}{7} x^{7} c^{7}+\frac {1}{5} x^{5} c^{5}\right )+d^{2} b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{9} c^{9}}{9}+\frac {2 \,\operatorname {arcsinh}\left (x c \right ) x^{7} c^{7}}{7}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}-\frac {263 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{33075}+\frac {1052 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{99225}-\frac {2104 \sqrt {c^{2} x^{2}+1}}{99225}-\frac {106 x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{3969}-\frac {x^{8} c^{8} \sqrt {c^{2} x^{2}+1}}{81}\right )}{c^{5}}\)	\(167\)
default	\(\frac {a \,d^{2} \left (\frac {1}{9} c^{9} x^{9}+\frac {2}{7} x^{7} c^{7}+\frac {1}{5} x^{5} c^{5}\right )+d^{2} b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{9} c^{9}}{9}+\frac {2 \,\operatorname {arcsinh}\left (x c \right ) x^{7} c^{7}}{7}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}-\frac {263 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{33075}+\frac {1052 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{99225}-\frac {2104 \sqrt {c^{2} x^{2}+1}}{99225}-\frac {106 x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{3969}-\frac {x^{8} c^{8} \sqrt {c^{2} x^{2}+1}}{81}\right )}{c^{5}}\)	\(167\)
orering	\(\frac {\left (20825 c^{10} x^{10}+54450 c^{8} x^{8}+36757 c^{6} x^{6}-5260 c^{4} x^{4}+12624 c^{2} x^{2}+8416\right ) \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{99225 c^{6} \left (c^{2} x^{2}+1\right )^{2} x}-\frac {\left (1225 c^{8} x^{8}+2650 c^{6} x^{6}+789 c^{4} x^{4}-1052 c^{2} x^{2}+2104\right ) \left (4 x^{3} \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )+4 x^{5} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) c^{2} d +\frac {x^{4} \left (c^{2} d \,x^{2}+d \right )^{2} b c}{\sqrt {c^{2} x^{2}+1}}\right )}{99225 c^{6} \left (c^{2} x^{2}+1\right ) x^{4}}\)	\(218\)

Input:

int(x^4*(c^2*d*x^2+d)^2*(a+b*arcsinh(x*c)),x,method=_RETURNVERBOSE)
 

Output:

a*d^2*(1/9*c^4*x^9+2/7*c^2*x^7+1/5*x^5)+d^2*b/c^5*(1/9*arcsinh(x*c)*x^9*c^ 
9+2/7*arcsinh(x*c)*x^7*c^7+1/5*arcsinh(x*c)*x^5*c^5-263/33075*x^4*c^4*(c^2 
*x^2+1)^(1/2)+1052/99225*x^2*c^2*(c^2*x^2+1)^(1/2)-2104/99225*(c^2*x^2+1)^ 
(1/2)-106/3969*x^6*c^6*(c^2*x^2+1)^(1/2)-1/81*x^8*c^8*(c^2*x^2+1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.91 \[ \int x^4 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {11025 \, a c^{9} d^{2} x^{9} + 28350 \, a c^{7} d^{2} x^{7} + 19845 \, a c^{5} d^{2} x^{5} + 315 \, {\left (35 \, b c^{9} d^{2} x^{9} + 90 \, b c^{7} d^{2} x^{7} + 63 \, b c^{5} d^{2} x^{5}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (1225 \, b c^{8} d^{2} x^{8} + 2650 \, b c^{6} d^{2} x^{6} + 789 \, b c^{4} d^{2} x^{4} - 1052 \, b c^{2} d^{2} x^{2} + 2104 \, b d^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{99225 \, c^{5}} \] Input:

integrate(x^4*(c^2*d*x^2+d)^2*(a+b*arcsinh(c*x)),x, algorithm="fricas")
 

Output:

1/99225*(11025*a*c^9*d^2*x^9 + 28350*a*c^7*d^2*x^7 + 19845*a*c^5*d^2*x^5 + 
 315*(35*b*c^9*d^2*x^9 + 90*b*c^7*d^2*x^7 + 63*b*c^5*d^2*x^5)*log(c*x + sq 
rt(c^2*x^2 + 1)) - (1225*b*c^8*d^2*x^8 + 2650*b*c^6*d^2*x^6 + 789*b*c^4*d^ 
2*x^4 - 1052*b*c^2*d^2*x^2 + 2104*b*d^2)*sqrt(c^2*x^2 + 1))/c^5
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 1.29 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.27 \[ \int x^4 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {a c^{4} d^{2} x^{9}}{9} + \frac {2 a c^{2} d^{2} x^{7}}{7} + \frac {a d^{2} x^{5}}{5} + \frac {b c^{4} d^{2} x^{9} \operatorname {asinh}{\left (c x \right )}}{9} - \frac {b c^{3} d^{2} x^{8} \sqrt {c^{2} x^{2} + 1}}{81} + \frac {2 b c^{2} d^{2} x^{7} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {106 b c d^{2} x^{6} \sqrt {c^{2} x^{2} + 1}}{3969} + \frac {b d^{2} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {263 b d^{2} x^{4} \sqrt {c^{2} x^{2} + 1}}{33075 c} + \frac {1052 b d^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{99225 c^{3}} - \frac {2104 b d^{2} \sqrt {c^{2} x^{2} + 1}}{99225 c^{5}} & \text {for}\: c \neq 0 \\\frac {a d^{2} x^{5}}{5} & \text {otherwise} \end {cases} \] Input:

integrate(x**4*(c**2*d*x**2+d)**2*(a+b*asinh(c*x)),x)
 

Output:

Piecewise((a*c**4*d**2*x**9/9 + 2*a*c**2*d**2*x**7/7 + a*d**2*x**5/5 + b*c 
**4*d**2*x**9*asinh(c*x)/9 - b*c**3*d**2*x**8*sqrt(c**2*x**2 + 1)/81 + 2*b 
*c**2*d**2*x**7*asinh(c*x)/7 - 106*b*c*d**2*x**6*sqrt(c**2*x**2 + 1)/3969 
+ b*d**2*x**5*asinh(c*x)/5 - 263*b*d**2*x**4*sqrt(c**2*x**2 + 1)/(33075*c) 
 + 1052*b*d**2*x**2*sqrt(c**2*x**2 + 1)/(99225*c**3) - 2104*b*d**2*sqrt(c* 
*2*x**2 + 1)/(99225*c**5), Ne(c, 0)), (a*d**2*x**5/5, True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 319 vs. \(2 (155) = 310\).

Time = 0.04 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.76 \[ \int x^4 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{9} \, a c^{4} d^{2} x^{9} + \frac {2}{7} \, a c^{2} d^{2} x^{7} + \frac {1}{2835} \, {\left (315 \, x^{9} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {35 \, \sqrt {c^{2} x^{2} + 1} x^{8}}{c^{2}} - \frac {40 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{6}} - \frac {64 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} + 1}}{c^{10}}\right )} c\right )} b c^{4} d^{2} + \frac {1}{5} \, a d^{2} x^{5} + \frac {2}{245} \, {\left (35 \, x^{7} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{2}} - \frac {6 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{6}} - \frac {16 \, \sqrt {c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b c^{2} d^{2} + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d^{2} \] Input:

integrate(x^4*(c^2*d*x^2+d)^2*(a+b*arcsinh(c*x)),x, algorithm="maxima")
 

Output:

1/9*a*c^4*d^2*x^9 + 2/7*a*c^2*d^2*x^7 + 1/2835*(315*x^9*arcsinh(c*x) - (35 
*sqrt(c^2*x^2 + 1)*x^8/c^2 - 40*sqrt(c^2*x^2 + 1)*x^6/c^4 + 48*sqrt(c^2*x^ 
2 + 1)*x^4/c^6 - 64*sqrt(c^2*x^2 + 1)*x^2/c^8 + 128*sqrt(c^2*x^2 + 1)/c^10 
)*c)*b*c^4*d^2 + 1/5*a*d^2*x^5 + 2/245*(35*x^7*arcsinh(c*x) - (5*sqrt(c^2* 
x^2 + 1)*x^6/c^2 - 6*sqrt(c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(c^2*x^2 + 1)*x^2/c 
^6 - 16*sqrt(c^2*x^2 + 1)/c^8)*c)*b*c^2*d^2 + 1/75*(15*x^5*arcsinh(c*x) - 
(3*sqrt(c^2*x^2 + 1)*x^4/c^2 - 4*sqrt(c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(c^2*x^ 
2 + 1)/c^6)*c)*b*d^2
 

Giac [F(-2)]

Exception generated. \[ \int x^4 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x^4*(c^2*d*x^2+d)^2*(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. \[ \int x^4 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\int x^4\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^2 \,d x \] Input:

int(x^4*(a + b*asinh(c*x))*(d + c^2*d*x^2)^2,x)
 

Output:

int(x^4*(a + b*asinh(c*x))*(d + c^2*d*x^2)^2, x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.91 \[ \int x^4 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {d^{2} \left (11025 \mathit {asinh} \left (c x \right ) b \,c^{9} x^{9}+28350 \mathit {asinh} \left (c x \right ) b \,c^{7} x^{7}+19845 \mathit {asinh} \left (c x \right ) b \,c^{5} x^{5}-1225 \sqrt {c^{2} x^{2}+1}\, b \,c^{8} x^{8}-2650 \sqrt {c^{2} x^{2}+1}\, b \,c^{6} x^{6}-789 \sqrt {c^{2} x^{2}+1}\, b \,c^{4} x^{4}+1052 \sqrt {c^{2} x^{2}+1}\, b \,c^{2} x^{2}-2104 \sqrt {c^{2} x^{2}+1}\, b +11025 a \,c^{9} x^{9}+28350 a \,c^{7} x^{7}+19845 a \,c^{5} x^{5}\right )}{99225 c^{5}} \] Input:

int(x^4*(c^2*d*x^2+d)^2*(a+b*asinh(c*x)),x)
 

Output:

(d**2*(11025*asinh(c*x)*b*c**9*x**9 + 28350*asinh(c*x)*b*c**7*x**7 + 19845 
*asinh(c*x)*b*c**5*x**5 - 1225*sqrt(c**2*x**2 + 1)*b*c**8*x**8 - 2650*sqrt 
(c**2*x**2 + 1)*b*c**6*x**6 - 789*sqrt(c**2*x**2 + 1)*b*c**4*x**4 + 1052*s 
qrt(c**2*x**2 + 1)*b*c**2*x**2 - 2104*sqrt(c**2*x**2 + 1)*b + 11025*a*c**9 
*x**9 + 28350*a*c**7*x**7 + 19845*a*c**5*x**5))/(99225*c**5)