\(\int x^3 (d+c^2 d x^2)^2 (a+b \text {arcsinh}(c x)) \, dx\) [11]

Optimal result

Integrand size = 24, antiderivative size = 180 \[ \int x^3 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {73 b d^2 x \sqrt {1+c^2 x^2}}{3072 c^3}-\frac {73 b d^2 x^3 \sqrt {1+c^2 x^2}}{4608 c}-\frac {43 b c d^2 x^5 \sqrt {1+c^2 x^2}}{1152}-\frac {1}{64} b c^3 d^2 x^7 \sqrt {1+c^2 x^2}-\frac {73 b d^2 \text {arcsinh}(c x)}{3072 c^4}+\frac {1}{4} d^2 x^4 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 d^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arcsinh}(c x)) \] Output:

73/3072*b*d^2*x*(c^2*x^2+1)^(1/2)/c^3-73/4608*b*d^2*x^3*(c^2*x^2+1)^(1/2)/ 
c-43/1152*b*c*d^2*x^5*(c^2*x^2+1)^(1/2)-1/64*b*c^3*d^2*x^7*(c^2*x^2+1)^(1/ 
2)-73/3072*b*d^2*arcsinh(c*x)/c^4+1/4*d^2*x^4*(a+b*arcsinh(c*x))+1/3*c^2*d 
^2*x^6*(a+b*arcsinh(c*x))+1/8*c^4*d^2*x^8*(a+b*arcsinh(c*x))
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.64 \[ \int x^3 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {d^2 \left (384 a c^4 x^4 \left (6+8 c^2 x^2+3 c^4 x^4\right )-b c x \sqrt {1+c^2 x^2} \left (-219+146 c^2 x^2+344 c^4 x^4+144 c^6 x^6\right )+3 b \left (-73+768 c^4 x^4+1024 c^6 x^6+384 c^8 x^8\right ) \text {arcsinh}(c x)\right )}{9216 c^4} \] Input:

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

Output:

(d^2*(384*a*c^4*x^4*(6 + 8*c^2*x^2 + 3*c^4*x^4) - b*c*x*Sqrt[1 + c^2*x^2]* 
(-219 + 146*c^2*x^2 + 344*c^4*x^4 + 144*c^6*x^6) + 3*b*(-73 + 768*c^4*x^4 
+ 1024*c^6*x^6 + 384*c^8*x^8)*ArcSinh[c*x]))/(9216*c^4)
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6218, 27, 1590, 27, 363, 262, 262, 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.

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

\(\Big \downarrow \) 6218

\(\displaystyle -b c \int \frac {d^2 x^4 \left (3 c^4 x^4+8 c^2 x^2+6\right )}{24 \sqrt {c^2 x^2+1}}dx+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 d^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} d^2 x^4 (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{24} b c d^2 \int \frac {x^4 \left (3 c^4 x^4+8 c^2 x^2+6\right )}{\sqrt {c^2 x^2+1}}dx+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 d^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} d^2 x^4 (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 1590

\(\displaystyle -\frac {1}{24} b c d^2 \left (\frac {\int \frac {c^2 x^4 \left (43 c^2 x^2+48\right )}{\sqrt {c^2 x^2+1}}dx}{8 c^2}+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2+1}\right )+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 d^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} d^2 x^4 (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{24} b c d^2 \left (\frac {1}{8} \int \frac {x^4 \left (43 c^2 x^2+48\right )}{\sqrt {c^2 x^2+1}}dx+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2+1}\right )+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 d^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} d^2 x^4 (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 363

\(\displaystyle -\frac {1}{24} b c d^2 \left (\frac {1}{8} \left (\frac {73}{6} \int \frac {x^4}{\sqrt {c^2 x^2+1}}dx+\frac {43}{6} x^5 \sqrt {c^2 x^2+1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2+1}\right )+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 d^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} d^2 x^4 (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 262

\(\displaystyle -\frac {1}{24} b c d^2 \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \int \frac {x^2}{\sqrt {c^2 x^2+1}}dx}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {c^2 x^2+1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2+1}\right )+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 d^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} d^2 x^4 (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 262

\(\displaystyle -\frac {1}{24} b c d^2 \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\int \frac {1}{\sqrt {c^2 x^2+1}}dx}{2 c^2}\right )}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {c^2 x^2+1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2+1}\right )+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 d^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} d^2 x^4 (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {1}{8} c^4 d^2 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 d^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} d^2 x^4 (a+b \text {arcsinh}(c x))-\frac {1}{24} b c d^2 \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {c^2 x^2+1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2+1}\right )\)

Input:

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

Output:

(d^2*x^4*(a + b*ArcSinh[c*x]))/4 + (c^2*d^2*x^6*(a + b*ArcSinh[c*x]))/3 + 
(c^4*d^2*x^8*(a + b*ArcSinh[c*x]))/8 - (b*c*d^2*((3*c^2*x^7*Sqrt[1 + c^2*x 
^2])/8 + ((43*x^5*Sqrt[1 + c^2*x^2])/6 + (73*((x^3*Sqrt[1 + c^2*x^2])/(4*c 
^2) - (3*((x*Sqrt[1 + c^2*x^2])/(2*c^2) - ArcSinh[c*x]/(2*c^3)))/(4*c^2))) 
/6)/8))/24
 

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 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 363

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), 
 x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3))   Int[(e*x)^ 
m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d 
, 0] && NeQ[m + 2*p + 3, 0]
 

rule 1590

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( 
c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[c^p*(f*x)^(m + 4*p - 1)*((d + e*x^2)^ 
(q + 1)/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1))), x] + Simp[1/(e*(m + 4*p + 2*q 
 + 1))   Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + 
b*x^2 + c*x^4)^p - c^p*x^(4*p)) - d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], 
x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 
0] &&  !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 0]
 

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 = 0.77 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.81

method	result	size
derivativedivides	\(\frac {a \,d^{2} \left (\frac {\left (c^{2} x^{2}+1\right )^{4}}{8}-\frac {\left (c^{2} x^{2}+1\right )^{3}}{6}\right )+d^{2} b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{8} c^{8}}{8}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{3}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{4}-\frac {73 \,\operatorname {arcsinh}\left (x c \right )}{3072}+\frac {11 x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{1152}+\frac {55 x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{4608}+\frac {55 \sqrt {c^{2} x^{2}+1}\, x c}{3072}-\frac {x c \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{64}\right )}{c^{4}}\)	\(146\)
default	\(\frac {a \,d^{2} \left (\frac {\left (c^{2} x^{2}+1\right )^{4}}{8}-\frac {\left (c^{2} x^{2}+1\right )^{3}}{6}\right )+d^{2} b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{8} c^{8}}{8}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{3}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{4}-\frac {73 \,\operatorname {arcsinh}\left (x c \right )}{3072}+\frac {11 x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{1152}+\frac {55 x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{4608}+\frac {55 \sqrt {c^{2} x^{2}+1}\, x c}{3072}-\frac {x c \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{64}\right )}{c^{4}}\)	\(146\)
parts	\(a \,d^{2} \left (\frac {1}{8} c^{4} x^{8}+\frac {1}{3} c^{2} x^{6}+\frac {1}{4} x^{4}\right )+\frac {d^{2} b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{8} c^{8}}{8}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{3}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{4}-\frac {73 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}}{4608}+\frac {73 \sqrt {c^{2} x^{2}+1}\, x c}{3072}-\frac {73 \,\operatorname {arcsinh}\left (x c \right )}{3072}-\frac {43 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}}{1152}-\frac {x^{7} c^{7} \sqrt {c^{2} x^{2}+1}}{64}\right )}{c^{4}}\)	\(152\)
orering	\(\frac {\left (2160 c^{8} x^{8}+5912 c^{6} x^{6}+4358 c^{4} x^{4}-1095 c^{2} x^{2}-876\right ) \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{9216 c^{4} \left (c^{2} x^{2}+1\right )^{2}}-\frac {\left (144 c^{6} x^{6}+344 c^{4} x^{4}+146 c^{2} x^{2}-219\right ) \left (3 x^{2} \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )+4 x^{4} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) c^{2} d +\frac {x^{3} \left (c^{2} d \,x^{2}+d \right )^{2} b c}{\sqrt {c^{2} x^{2}+1}}\right )}{9216 x^{2} c^{4} \left (c^{2} x^{2}+1\right )}\)	\(199\)

Input:

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

Output:

1/c^4*(a*d^2*(1/8*(c^2*x^2+1)^4-1/6*(c^2*x^2+1)^3)+d^2*b*(1/8*arcsinh(x*c) 
*x^8*c^8+1/3*arcsinh(x*c)*x^6*c^6+1/4*arcsinh(x*c)*c^4*x^4-73/3072*arcsinh 
(x*c)+11/1152*x*c*(c^2*x^2+1)^(5/2)+55/4608*x*c*(c^2*x^2+1)^(3/2)+55/3072* 
(c^2*x^2+1)^(1/2)*x*c-1/64*x*c*(c^2*x^2+1)^(7/2)))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.89 \[ \int x^3 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {1152 \, a c^{8} d^{2} x^{8} + 3072 \, a c^{6} d^{2} x^{6} + 2304 \, a c^{4} d^{2} x^{4} + 3 \, {\left (384 \, b c^{8} d^{2} x^{8} + 1024 \, b c^{6} d^{2} x^{6} + 768 \, b c^{4} d^{2} x^{4} - 73 \, b d^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (144 \, b c^{7} d^{2} x^{7} + 344 \, b c^{5} d^{2} x^{5} + 146 \, b c^{3} d^{2} x^{3} - 219 \, b c d^{2} x\right )} \sqrt {c^{2} x^{2} + 1}}{9216 \, c^{4}} \] Input:

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

Output:

1/9216*(1152*a*c^8*d^2*x^8 + 3072*a*c^6*d^2*x^6 + 2304*a*c^4*d^2*x^4 + 3*( 
384*b*c^8*d^2*x^8 + 1024*b*c^6*d^2*x^6 + 768*b*c^4*d^2*x^4 - 73*b*d^2)*log 
(c*x + sqrt(c^2*x^2 + 1)) - (144*b*c^7*d^2*x^7 + 344*b*c^5*d^2*x^5 + 146*b 
*c^3*d^2*x^3 - 219*b*c*d^2*x)*sqrt(c^2*x^2 + 1))/c^4
 

Sympy [A] (verification not implemented)

Time = 0.94 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.21 \[ \int x^3 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {a c^{4} d^{2} x^{8}}{8} + \frac {a c^{2} d^{2} x^{6}}{3} + \frac {a d^{2} x^{4}}{4} + \frac {b c^{4} d^{2} x^{8} \operatorname {asinh}{\left (c x \right )}}{8} - \frac {b c^{3} d^{2} x^{7} \sqrt {c^{2} x^{2} + 1}}{64} + \frac {b c^{2} d^{2} x^{6} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {43 b c d^{2} x^{5} \sqrt {c^{2} x^{2} + 1}}{1152} + \frac {b d^{2} x^{4} \operatorname {asinh}{\left (c x \right )}}{4} - \frac {73 b d^{2} x^{3} \sqrt {c^{2} x^{2} + 1}}{4608 c} + \frac {73 b d^{2} x \sqrt {c^{2} x^{2} + 1}}{3072 c^{3}} - \frac {73 b d^{2} \operatorname {asinh}{\left (c x \right )}}{3072 c^{4}} & \text {for}\: c \neq 0 \\\frac {a d^{2} x^{4}}{4} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((a*c**4*d**2*x**8/8 + a*c**2*d**2*x**6/3 + a*d**2*x**4/4 + b*c** 
4*d**2*x**8*asinh(c*x)/8 - b*c**3*d**2*x**7*sqrt(c**2*x**2 + 1)/64 + b*c** 
2*d**2*x**6*asinh(c*x)/3 - 43*b*c*d**2*x**5*sqrt(c**2*x**2 + 1)/1152 + b*d 
**2*x**4*asinh(c*x)/4 - 73*b*d**2*x**3*sqrt(c**2*x**2 + 1)/(4608*c) + 73*b 
*d**2*x*sqrt(c**2*x**2 + 1)/(3072*c**3) - 73*b*d**2*asinh(c*x)/(3072*c**4) 
, Ne(c, 0)), (a*d**2*x**4/4, True))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.62 \[ \int x^3 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{8} \, a c^{4} d^{2} x^{8} + \frac {1}{3} \, a c^{2} d^{2} x^{6} + \frac {1}{3072} \, {\left (384 \, x^{8} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {48 \, \sqrt {c^{2} x^{2} + 1} x^{7}}{c^{2}} - \frac {56 \, \sqrt {c^{2} x^{2} + 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{6}} - \frac {105 \, \sqrt {c^{2} x^{2} + 1} x}{c^{8}} + \frac {105 \, \operatorname {arsinh}\left (c x\right )}{c^{9}}\right )} c\right )} b c^{4} d^{2} + \frac {1}{4} \, a d^{2} x^{4} + \frac {1}{144} \, {\left (48 \, x^{6} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{5}}{c^{2}} - \frac {10 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \operatorname {arsinh}\left (c x\right )}{c^{7}}\right )} c\right )} b c^{2} d^{2} + \frac {1}{32} \, {\left (8 \, x^{4} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac {3 \, \sqrt {c^{2} x^{2} + 1} x}{c^{4}} + \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c\right )} b d^{2} \] Input:

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

Output:

1/8*a*c^4*d^2*x^8 + 1/3*a*c^2*d^2*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^4*d^2 + 1/4*a*d^2*x^4 + 1/144*(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^2*d^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*d^2
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.92 \[ \int x^3 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {d^{2} \left (1152 \mathit {asinh} \left (c x \right ) b \,c^{8} x^{8}+3072 \mathit {asinh} \left (c x \right ) b \,c^{6} x^{6}+2304 \mathit {asinh} \left (c x \right ) b \,c^{4} x^{4}-144 \sqrt {c^{2} x^{2}+1}\, b \,c^{7} x^{7}-344 \sqrt {c^{2} x^{2}+1}\, b \,c^{5} x^{5}-146 \sqrt {c^{2} x^{2}+1}\, b \,c^{3} x^{3}+219 \sqrt {c^{2} x^{2}+1}\, b c x -219 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) b +1152 a \,c^{8} x^{8}+3072 a \,c^{6} x^{6}+2304 a \,c^{4} x^{4}\right )}{9216 c^{4}} \] Input:

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

Output:

(d**2*(1152*asinh(c*x)*b*c**8*x**8 + 3072*asinh(c*x)*b*c**6*x**6 + 2304*as 
inh(c*x)*b*c**4*x**4 - 144*sqrt(c**2*x**2 + 1)*b*c**7*x**7 - 344*sqrt(c**2 
*x**2 + 1)*b*c**5*x**5 - 146*sqrt(c**2*x**2 + 1)*b*c**3*x**3 + 219*sqrt(c* 
*2*x**2 + 1)*b*c*x - 219*log(sqrt(c**2*x**2 + 1) + c*x)*b + 1152*a*c**8*x* 
*8 + 3072*a*c**6*x**6 + 2304*a*c**4*x**4))/(9216*c**4)