\(\int x^3 (d+c^2 d x^2)^3 (a+b \text {arcsinh}(c x)) \, dx\) [20]

Optimal result

Integrand size = 24, antiderivative size = 199 \[ \int x^3 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {49 b d^3 x \sqrt {1+c^2 x^2}}{5120 c^3}+\frac {49 b d^3 x \left (1+c^2 x^2\right )^{3/2}}{7680 c^3}+\frac {49 b d^3 x \left (1+c^2 x^2\right )^{5/2}}{9600 c^3}+\frac {7 b d^3 x \left (1+c^2 x^2\right )^{7/2}}{1600 c^3}-\frac {b d^3 x \left (1+c^2 x^2\right )^{9/2}}{100 c^3}+\frac {49 b d^3 \text {arcsinh}(c x)}{5120 c^4}-\frac {d^3 \left (1+c^2 x^2\right )^4 (a+b \text {arcsinh}(c x))}{8 c^4}+\frac {d^3 \left (1+c^2 x^2\right )^5 (a+b \text {arcsinh}(c x))}{10 c^4} \] Output:

49/5120*b*d^3*x*(c^2*x^2+1)^(1/2)/c^3+49/7680*b*d^3*x*(c^2*x^2+1)^(3/2)/c^ 
3+49/9600*b*d^3*x*(c^2*x^2+1)^(5/2)/c^3+7/1600*b*d^3*x*(c^2*x^2+1)^(7/2)/c 
^3-1/100*b*d^3*x*(c^2*x^2+1)^(9/2)/c^3+49/5120*b*d^3*arcsinh(c*x)/c^4-1/8* 
d^3*(c^2*x^2+1)^4*(a+b*arcsinh(c*x))/c^4+1/10*d^3*(c^2*x^2+1)^5*(a+b*arcsi 
nh(c*x))/c^4
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.70 \[ \int x^3 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {d^3 \left (1920 a c^4 x^4 \left (10+20 c^2 x^2+15 c^4 x^4+4 c^6 x^6\right )-b c x \sqrt {1+c^2 x^2} \left (-1185+790 c^2 x^2+3208 c^4 x^4+2736 c^6 x^6+768 c^8 x^8\right )+15 b \left (-79+1280 c^4 x^4+2560 c^6 x^6+1920 c^8 x^8+512 c^{10} x^{10}\right ) \text {arcsinh}(c x)\right )}{76800 c^4} \] Input:

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

Output:

(d^3*(1920*a*c^4*x^4*(10 + 20*c^2*x^2 + 15*c^4*x^4 + 4*c^6*x^6) - b*c*x*Sq 
rt[1 + c^2*x^2]*(-1185 + 790*c^2*x^2 + 3208*c^4*x^4 + 2736*c^6*x^6 + 768*c 
^8*x^8) + 15*b*(-79 + 1280*c^4*x^4 + 2560*c^6*x^6 + 1920*c^8*x^8 + 512*c^1 
0*x^10)*ArcSinh[c*x]))/(76800*c^4)
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6218, 27, 299, 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.

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

\(\Big \downarrow \) 6218

\(\displaystyle -b c \int -\frac {d^3 \left (1-4 c^2 x^2\right ) \left (c^2 x^2+1\right )^{7/2}}{40 c^4}dx+\frac {d^3 \left (c^2 x^2+1\right )^5 (a+b \text {arcsinh}(c x))}{10 c^4}-\frac {d^3 \left (c^2 x^2+1\right )^4 (a+b \text {arcsinh}(c x))}{8 c^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b d^3 \int \left (1-4 c^2 x^2\right ) \left (c^2 x^2+1\right )^{7/2}dx}{40 c^3}+\frac {d^3 \left (c^2 x^2+1\right )^5 (a+b \text {arcsinh}(c x))}{10 c^4}-\frac {d^3 \left (c^2 x^2+1\right )^4 (a+b \text {arcsinh}(c x))}{8 c^4}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {b d^3 \left (\frac {7}{5} \int \left (c^2 x^2+1\right )^{7/2}dx-\frac {2}{5} x \left (c^2 x^2+1\right )^{9/2}\right )}{40 c^3}+\frac {d^3 \left (c^2 x^2+1\right )^5 (a+b \text {arcsinh}(c x))}{10 c^4}-\frac {d^3 \left (c^2 x^2+1\right )^4 (a+b \text {arcsinh}(c x))}{8 c^4}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {b d^3 \left (\frac {7}{5} \left (\frac {7}{8} \int \left (c^2 x^2+1\right )^{5/2}dx+\frac {1}{8} x \left (c^2 x^2+1\right )^{7/2}\right )-\frac {2}{5} x \left (c^2 x^2+1\right )^{9/2}\right )}{40 c^3}+\frac {d^3 \left (c^2 x^2+1\right )^5 (a+b \text {arcsinh}(c x))}{10 c^4}-\frac {d^3 \left (c^2 x^2+1\right )^4 (a+b \text {arcsinh}(c x))}{8 c^4}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {b d^3 \left (\frac {7}{5} \left (\frac {7}{8} \left (\frac {5}{6} \int \left (c^2 x^2+1\right )^{3/2}dx+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )+\frac {1}{8} x \left (c^2 x^2+1\right )^{7/2}\right )-\frac {2}{5} x \left (c^2 x^2+1\right )^{9/2}\right )}{40 c^3}+\frac {d^3 \left (c^2 x^2+1\right )^5 (a+b \text {arcsinh}(c x))}{10 c^4}-\frac {d^3 \left (c^2 x^2+1\right )^4 (a+b \text {arcsinh}(c x))}{8 c^4}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {b d^3 \left (\frac {7}{5} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {c^2 x^2+1}dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )+\frac {1}{8} x \left (c^2 x^2+1\right )^{7/2}\right )-\frac {2}{5} x \left (c^2 x^2+1\right )^{9/2}\right )}{40 c^3}+\frac {d^3 \left (c^2 x^2+1\right )^5 (a+b \text {arcsinh}(c x))}{10 c^4}-\frac {d^3 \left (c^2 x^2+1\right )^4 (a+b \text {arcsinh}(c x))}{8 c^4}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {b d^3 \left (\frac {7}{5} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )+\frac {1}{8} x \left (c^2 x^2+1\right )^{7/2}\right )-\frac {2}{5} x \left (c^2 x^2+1\right )^{9/2}\right )}{40 c^3}+\frac {d^3 \left (c^2 x^2+1\right )^5 (a+b \text {arcsinh}(c x))}{10 c^4}-\frac {d^3 \left (c^2 x^2+1\right )^4 (a+b \text {arcsinh}(c x))}{8 c^4}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {d^3 \left (c^2 x^2+1\right )^5 (a+b \text {arcsinh}(c x))}{10 c^4}-\frac {d^3 \left (c^2 x^2+1\right )^4 (a+b \text {arcsinh}(c x))}{8 c^4}+\frac {b d^3 \left (\frac {7}{5} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )+\frac {1}{8} x \left (c^2 x^2+1\right )^{7/2}\right )-\frac {2}{5} x \left (c^2 x^2+1\right )^{9/2}\right )}{40 c^3}\)

Input:

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

Output:

-1/8*(d^3*(1 + c^2*x^2)^4*(a + b*ArcSinh[c*x]))/c^4 + (d^3*(1 + c^2*x^2)^5 
*(a + b*ArcSinh[c*x]))/(10*c^4) + (b*d^3*((-2*x*(1 + c^2*x^2)^(9/2))/5 + ( 
7*((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))/5))/(40*c^3)
 

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 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 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 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 = 0.76 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.87

method	result	size
derivativedivides	\(\frac {d^{3} a \left (\frac {\left (c^{2} x^{2}+1\right )^{5}}{10}-\frac {\left (c^{2} x^{2}+1\right )^{4}}{8}\right )+d^{3} b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{10} c^{10}}{10}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) x^{8} c^{8}}{8}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{2}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{4}-\frac {79 \,\operatorname {arcsinh}\left (x c \right )}{5120}+\frac {7 x c \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{1600}+\frac {49 x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{9600}+\frac {49 x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{7680}+\frac {49 \sqrt {c^{2} x^{2}+1}\, x c}{5120}-\frac {x c \left (c^{2} x^{2}+1\right )^{\frac {9}{2}}}{100}\right )}{c^{4}}\)	\(173\)
default	\(\frac {d^{3} a \left (\frac {\left (c^{2} x^{2}+1\right )^{5}}{10}-\frac {\left (c^{2} x^{2}+1\right )^{4}}{8}\right )+d^{3} b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{10} c^{10}}{10}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) x^{8} c^{8}}{8}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{2}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{4}-\frac {79 \,\operatorname {arcsinh}\left (x c \right )}{5120}+\frac {7 x c \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{1600}+\frac {49 x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{9600}+\frac {49 x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{7680}+\frac {49 \sqrt {c^{2} x^{2}+1}\, x c}{5120}-\frac {x c \left (c^{2} x^{2}+1\right )^{\frac {9}{2}}}{100}\right )}{c^{4}}\)	\(173\)
parts	\(d^{3} a \left (\frac {1}{10} c^{6} x^{10}+\frac {3}{8} c^{4} x^{8}+\frac {1}{2} c^{2} x^{6}+\frac {1}{4} x^{4}\right )+\frac {d^{3} b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{10} c^{10}}{10}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) x^{8} c^{8}}{8}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{2}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{4}-\frac {79 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}}{7680}+\frac {79 \sqrt {c^{2} x^{2}+1}\, x c}{5120}-\frac {79 \,\operatorname {arcsinh}\left (x c \right )}{5120}-\frac {401 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}}{9600}-\frac {57 x^{7} c^{7} \sqrt {c^{2} x^{2}+1}}{1600}-\frac {x^{9} c^{9} \sqrt {c^{2} x^{2}+1}}{100}\right )}{c^{4}}\)	\(191\)
orering	\(\frac {\left (4864 c^{10} x^{10}+18576 c^{8} x^{8}+25160 c^{6} x^{6}+11978 c^{4} x^{4}-2765 c^{2} x^{2}-1580\right ) \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{25600 c^{4} \left (c^{2} x^{2}+1\right )^{3}}-\frac {\left (768 c^{8} x^{8}+2736 c^{6} x^{6}+3208 c^{4} x^{4}+790 c^{2} x^{2}-1185\right ) \left (3 x^{2} \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )+6 x^{4} \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) c^{2} d +\frac {x^{3} \left (c^{2} d \,x^{2}+d \right )^{3} b c}{\sqrt {c^{2} x^{2}+1}}\right )}{76800 x^{2} c^{4} \left (c^{2} x^{2}+1\right )^{2}}\)	\(217\)

Input:

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

Output:

1/c^4*(d^3*a*(1/10*(c^2*x^2+1)^5-1/8*(c^2*x^2+1)^4)+d^3*b*(1/10*arcsinh(x* 
c)*x^10*c^10+3/8*arcsinh(x*c)*x^8*c^8+1/2*arcsinh(x*c)*x^6*c^6+1/4*arcsinh 
(x*c)*c^4*x^4-79/5120*arcsinh(x*c)+7/1600*x*c*(c^2*x^2+1)^(7/2)+49/9600*x* 
c*(c^2*x^2+1)^(5/2)+49/7680*x*c*(c^2*x^2+1)^(3/2)+49/5120*(c^2*x^2+1)^(1/2 
)*x*c-1/100*x*c*(c^2*x^2+1)^(9/2)))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.99 \[ \int x^3 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {7680 \, a c^{10} d^{3} x^{10} + 28800 \, a c^{8} d^{3} x^{8} + 38400 \, a c^{6} d^{3} x^{6} + 19200 \, a c^{4} d^{3} x^{4} + 15 \, {\left (512 \, b c^{10} d^{3} x^{10} + 1920 \, b c^{8} d^{3} x^{8} + 2560 \, b c^{6} d^{3} x^{6} + 1280 \, b c^{4} d^{3} x^{4} - 79 \, b d^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (768 \, b c^{9} d^{3} x^{9} + 2736 \, b c^{7} d^{3} x^{7} + 3208 \, b c^{5} d^{3} x^{5} + 790 \, b c^{3} d^{3} x^{3} - 1185 \, b c d^{3} x\right )} \sqrt {c^{2} x^{2} + 1}}{76800 \, c^{4}} \] Input:

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

Output:

1/76800*(7680*a*c^10*d^3*x^10 + 28800*a*c^8*d^3*x^8 + 38400*a*c^6*d^3*x^6 
+ 19200*a*c^4*d^3*x^4 + 15*(512*b*c^10*d^3*x^10 + 1920*b*c^8*d^3*x^8 + 256 
0*b*c^6*d^3*x^6 + 1280*b*c^4*d^3*x^4 - 79*b*d^3)*log(c*x + sqrt(c^2*x^2 + 
1)) - (768*b*c^9*d^3*x^9 + 2736*b*c^7*d^3*x^7 + 3208*b*c^5*d^3*x^5 + 790*b 
*c^3*d^3*x^3 - 1185*b*c*d^3*x)*sqrt(c^2*x^2 + 1))/c^4
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 1.65 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.41 \[ \int x^3 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {a c^{6} d^{3} x^{10}}{10} + \frac {3 a c^{4} d^{3} x^{8}}{8} + \frac {a c^{2} d^{3} x^{6}}{2} + \frac {a d^{3} x^{4}}{4} + \frac {b c^{6} d^{3} x^{10} \operatorname {asinh}{\left (c x \right )}}{10} - \frac {b c^{5} d^{3} x^{9} \sqrt {c^{2} x^{2} + 1}}{100} + \frac {3 b c^{4} d^{3} x^{8} \operatorname {asinh}{\left (c x \right )}}{8} - \frac {57 b c^{3} d^{3} x^{7} \sqrt {c^{2} x^{2} + 1}}{1600} + \frac {b c^{2} d^{3} x^{6} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {401 b c d^{3} x^{5} \sqrt {c^{2} x^{2} + 1}}{9600} + \frac {b d^{3} x^{4} \operatorname {asinh}{\left (c x \right )}}{4} - \frac {79 b d^{3} x^{3} \sqrt {c^{2} x^{2} + 1}}{7680 c} + \frac {79 b d^{3} x \sqrt {c^{2} x^{2} + 1}}{5120 c^{3}} - \frac {79 b d^{3} \operatorname {asinh}{\left (c x \right )}}{5120 c^{4}} & \text {for}\: c \neq 0 \\\frac {a d^{3} x^{4}}{4} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((a*c**6*d**3*x**10/10 + 3*a*c**4*d**3*x**8/8 + a*c**2*d**3*x**6/ 
2 + a*d**3*x**4/4 + b*c**6*d**3*x**10*asinh(c*x)/10 - b*c**5*d**3*x**9*sqr 
t(c**2*x**2 + 1)/100 + 3*b*c**4*d**3*x**8*asinh(c*x)/8 - 57*b*c**3*d**3*x* 
*7*sqrt(c**2*x**2 + 1)/1600 + b*c**2*d**3*x**6*asinh(c*x)/2 - 401*b*c*d**3 
*x**5*sqrt(c**2*x**2 + 1)/9600 + b*d**3*x**4*asinh(c*x)/4 - 79*b*d**3*x**3 
*sqrt(c**2*x**2 + 1)/(7680*c) + 79*b*d**3*x*sqrt(c**2*x**2 + 1)/(5120*c**3 
) - 79*b*d**3*asinh(c*x)/(5120*c**4), Ne(c, 0)), (a*d**3*x**4/4, True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 429 vs. \(2 (173) = 346\).

Time = 0.04 (sec) , antiderivative size = 429, normalized size of antiderivative = 2.16 \[ \int x^3 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{10} \, a c^{6} d^{3} x^{10} + \frac {3}{8} \, a c^{4} d^{3} x^{8} + \frac {1}{2} \, a c^{2} d^{3} x^{6} + \frac {1}{12800} \, {\left (1280 \, x^{10} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {128 \, \sqrt {c^{2} x^{2} + 1} x^{9}}{c^{2}} - \frac {144 \, \sqrt {c^{2} x^{2} + 1} x^{7}}{c^{4}} + \frac {168 \, \sqrt {c^{2} x^{2} + 1} x^{5}}{c^{6}} - \frac {210 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{8}} + \frac {315 \, \sqrt {c^{2} x^{2} + 1} x}{c^{10}} - \frac {315 \, \operatorname {arsinh}\left (c x\right )}{c^{11}}\right )} c\right )} b c^{6} d^{3} + \frac {1}{1024} \, {\left (384 \, x^{8} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {48 \, \sqrt {c^{2} x^{2} + 1} x^{7}}{c^{2}} - \frac {56 \, \sqrt {c^{2} x^{2} + 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{6}} - \frac {105 \, \sqrt {c^{2} x^{2} + 1} x}{c^{8}} + \frac {105 \, \operatorname {arsinh}\left (c x\right )}{c^{9}}\right )} c\right )} b c^{4} d^{3} + \frac {1}{4} \, a d^{3} x^{4} + \frac {1}{96} \, {\left (48 \, x^{6} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{5}}{c^{2}} - \frac {10 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \operatorname {arsinh}\left (c x\right )}{c^{7}}\right )} c\right )} b c^{2} d^{3} + \frac {1}{32} \, {\left (8 \, x^{4} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac {3 \, \sqrt {c^{2} x^{2} + 1} x}{c^{4}} + \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c\right )} b d^{3} \] Input:

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

Output:

1/10*a*c^6*d^3*x^10 + 3/8*a*c^4*d^3*x^8 + 1/2*a*c^2*d^3*x^6 + 1/12800*(128 
0*x^10*arcsinh(c*x) - (128*sqrt(c^2*x^2 + 1)*x^9/c^2 - 144*sqrt(c^2*x^2 + 
1)*x^7/c^4 + 168*sqrt(c^2*x^2 + 1)*x^5/c^6 - 210*sqrt(c^2*x^2 + 1)*x^3/c^8 
 + 315*sqrt(c^2*x^2 + 1)*x/c^10 - 315*arcsinh(c*x)/c^11)*c)*b*c^6*d^3 + 1/ 
1024*(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^3 + 1/4*a*d^3*x^4 + 1/96*(48*x^6*arc 
sinh(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^3 + 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^3
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.04 \[ \int x^3 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {d^{3} \left (7680 \mathit {asinh} \left (c x \right ) b \,c^{10} x^{10}+28800 \mathit {asinh} \left (c x \right ) b \,c^{8} x^{8}+38400 \mathit {asinh} \left (c x \right ) b \,c^{6} x^{6}+19200 \mathit {asinh} \left (c x \right ) b \,c^{4} x^{4}-768 \sqrt {c^{2} x^{2}+1}\, b \,c^{9} x^{9}-2736 \sqrt {c^{2} x^{2}+1}\, b \,c^{7} x^{7}-3208 \sqrt {c^{2} x^{2}+1}\, b \,c^{5} x^{5}-790 \sqrt {c^{2} x^{2}+1}\, b \,c^{3} x^{3}+1185 \sqrt {c^{2} x^{2}+1}\, b c x -1185 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) b +7680 a \,c^{10} x^{10}+28800 a \,c^{8} x^{8}+38400 a \,c^{6} x^{6}+19200 a \,c^{4} x^{4}\right )}{76800 c^{4}} \] Input:

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

Output:

(d**3*(7680*asinh(c*x)*b*c**10*x**10 + 28800*asinh(c*x)*b*c**8*x**8 + 3840 
0*asinh(c*x)*b*c**6*x**6 + 19200*asinh(c*x)*b*c**4*x**4 - 768*sqrt(c**2*x* 
*2 + 1)*b*c**9*x**9 - 2736*sqrt(c**2*x**2 + 1)*b*c**7*x**7 - 3208*sqrt(c** 
2*x**2 + 1)*b*c**5*x**5 - 790*sqrt(c**2*x**2 + 1)*b*c**3*x**3 + 1185*sqrt( 
c**2*x**2 + 1)*b*c*x - 1185*log(sqrt(c**2*x**2 + 1) + c*x)*b + 7680*a*c**1 
0*x**10 + 28800*a*c**8*x**8 + 38400*a*c**6*x**6 + 19200*a*c**4*x**4))/(768 
00*c**4)