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

Optimal result

Integrand size = 22, antiderivative size = 120 \[ \int x^3 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {b d x \sqrt {1+c^2 x^2}}{24 c^3}-\frac {b d x^3 \sqrt {1+c^2 x^2}}{36 c}-\frac {1}{36} b c d x^5 \sqrt {1+c^2 x^2}-\frac {b d \text {arcsinh}(c x)}{24 c^4}+\frac {1}{4} d x^4 (a+b \text {arcsinh}(c x))+\frac {1}{6} c^2 d x^6 (a+b \text {arcsinh}(c x)) \] Output:

1/24*b*d*x*(c^2*x^2+1)^(1/2)/c^3-1/36*b*d*x^3*(c^2*x^2+1)^(1/2)/c-1/36*b*c 
*d*x^5*(c^2*x^2+1)^(1/2)-1/24*b*d*arcsinh(c*x)/c^4+1/4*d*x^4*(a+b*arcsinh( 
c*x))+1/6*c^2*d*x^6*(a+b*arcsinh(c*x))
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.73 \[ \int x^3 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {d \left (6 a c^4 x^4 \left (3+2 c^2 x^2\right )+b c x \sqrt {1+c^2 x^2} \left (3-2 c^2 x^2-2 c^4 x^4\right )+3 b \left (-1+6 c^4 x^4+4 c^6 x^6\right ) \text {arcsinh}(c x)\right )}{72 c^4} \] Input:

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

Output:

(d*(6*a*c^4*x^4*(3 + 2*c^2*x^2) + b*c*x*Sqrt[1 + c^2*x^2]*(3 - 2*c^2*x^2 - 
 2*c^4*x^4) + 3*b*(-1 + 6*c^4*x^4 + 4*c^6*x^6)*ArcSinh[c*x]))/(72*c^4)
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6218, 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.

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

\(\Big \downarrow \) 6218

\(\displaystyle -b c \int \frac {d x^4 \left (2 c^2 x^2+3\right )}{12 \sqrt {c^2 x^2+1}}dx+\frac {1}{6} c^2 d x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} d x^4 (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{12} b c d \int \frac {x^4 \left (2 c^2 x^2+3\right )}{\sqrt {c^2 x^2+1}}dx+\frac {1}{6} c^2 d x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} d x^4 (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 363

\(\displaystyle -\frac {1}{12} b c d \left (\frac {4}{3} \int \frac {x^4}{\sqrt {c^2 x^2+1}}dx+\frac {1}{3} x^5 \sqrt {c^2 x^2+1}\right )+\frac {1}{6} c^2 d x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} d x^4 (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 262

\(\displaystyle -\frac {1}{12} b c d \left (\frac {4}{3} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \int \frac {x^2}{\sqrt {c^2 x^2+1}}dx}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {c^2 x^2+1}\right )+\frac {1}{6} c^2 d x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} d x^4 (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 262

\(\displaystyle -\frac {1}{12} b c d \left (\frac {4}{3} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\int \frac {1}{\sqrt {c^2 x^2+1}}dx}{2 c^2}\right )}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {c^2 x^2+1}\right )+\frac {1}{6} c^2 d x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} d x^4 (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {1}{6} c^2 d x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} d x^4 (a+b \text {arcsinh}(c x))-\frac {1}{12} b c d \left (\frac {4}{3} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {c^2 x^2+1}\right )\)

Input:

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

Output:

(d*x^4*(a + b*ArcSinh[c*x]))/4 + (c^2*d*x^6*(a + b*ArcSinh[c*x]))/6 - (b*c 
*d*((x^5*Sqrt[1 + c^2*x^2])/3 + (4*((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)))/3))/12
 

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 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 = 0.68 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.91

method	result	size
parts	\(a d \left (\frac {1}{6} c^{2} x^{6}+\frac {1}{4} x^{4}\right )+\frac {b d \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{6}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{4}-\frac {\sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}}{36}+\frac {\sqrt {c^{2} x^{2}+1}\, x c}{24}-\frac {\operatorname {arcsinh}\left (x c \right )}{24}-\frac {\sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}}{36}\right )}{c^{4}}\)	\(109\)
derivativedivides	\(\frac {a d \left (\frac {\left (c^{2} x^{2}+1\right )^{3}}{6}-\frac {\left (c^{2} x^{2}+1\right )^{2}}{4}\right )+b d \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{6}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{4}-\frac {\operatorname {arcsinh}\left (x c \right )}{24}+\frac {x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{36}+\frac {\sqrt {c^{2} x^{2}+1}\, x c}{24}-\frac {x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{36}\right )}{c^{4}}\)	\(115\)
default	\(\frac {a d \left (\frac {\left (c^{2} x^{2}+1\right )^{3}}{6}-\frac {\left (c^{2} x^{2}+1\right )^{2}}{4}\right )+b d \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{6}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{4}-\frac {\operatorname {arcsinh}\left (x c \right )}{24}+\frac {x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{36}+\frac {\sqrt {c^{2} x^{2}+1}\, x c}{24}-\frac {x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{36}\right )}{c^{4}}\)	\(115\)
orering	\(\frac {\left (22 c^{6} x^{6}+34 c^{4} x^{4}-9 c^{2} x^{2}-12\right ) \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{72 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {\left (2 c^{4} x^{4}+2 c^{2} x^{2}-3\right ) \left (3 x^{2} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )+2 x^{4} c^{2} d \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )+\frac {x^{3} \left (c^{2} d \,x^{2}+d \right ) b c}{\sqrt {c^{2} x^{2}+1}}\right )}{72 x^{2} c^{4}}\)	\(156\)

Input:

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

Output:

a*d*(1/6*c^2*x^6+1/4*x^4)+b*d/c^4*(1/6*arcsinh(x*c)*x^6*c^6+1/4*arcsinh(x* 
c)*c^4*x^4-1/36*(c^2*x^2+1)^(1/2)*c^3*x^3+1/24*(c^2*x^2+1)^(1/2)*x*c-1/24* 
arcsinh(x*c)-1/36*(c^2*x^2+1)^(1/2)*x^5*c^5)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.91 \[ \int x^3 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {12 \, a c^{6} d x^{6} + 18 \, a c^{4} d x^{4} + 3 \, {\left (4 \, b c^{6} d x^{6} + 6 \, b c^{4} d x^{4} - b d\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (2 \, b c^{5} d x^{5} + 2 \, b c^{3} d x^{3} - 3 \, b c d x\right )} \sqrt {c^{2} x^{2} + 1}}{72 \, c^{4}} \] Input:

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

Output:

1/72*(12*a*c^6*d*x^6 + 18*a*c^4*d*x^4 + 3*(4*b*c^6*d*x^6 + 6*b*c^4*d*x^4 - 
 b*d)*log(c*x + sqrt(c^2*x^2 + 1)) - (2*b*c^5*d*x^5 + 2*b*c^3*d*x^3 - 3*b* 
c*d*x)*sqrt(c^2*x^2 + 1))/c^4
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.15 \[ \int x^3 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {a c^{2} d x^{6}}{6} + \frac {a d x^{4}}{4} + \frac {b c^{2} d x^{6} \operatorname {asinh}{\left (c x \right )}}{6} - \frac {b c d x^{5} \sqrt {c^{2} x^{2} + 1}}{36} + \frac {b d x^{4} \operatorname {asinh}{\left (c x \right )}}{4} - \frac {b d x^{3} \sqrt {c^{2} x^{2} + 1}}{36 c} + \frac {b d x \sqrt {c^{2} x^{2} + 1}}{24 c^{3}} - \frac {b d \operatorname {asinh}{\left (c x \right )}}{24 c^{4}} & \text {for}\: c \neq 0 \\\frac {a d x^{4}}{4} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((a*c**2*d*x**6/6 + a*d*x**4/4 + b*c**2*d*x**6*asinh(c*x)/6 - b*c 
*d*x**5*sqrt(c**2*x**2 + 1)/36 + b*d*x**4*asinh(c*x)/4 - b*d*x**3*sqrt(c** 
2*x**2 + 1)/(36*c) + b*d*x*sqrt(c**2*x**2 + 1)/(24*c**3) - b*d*asinh(c*x)/ 
(24*c**4), Ne(c, 0)), (a*d*x**4/4, True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.38 \[ \int x^3 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{6} \, a c^{2} d x^{6} + \frac {1}{4} \, a d x^{4} + \frac {1}{288} \, {\left (48 \, x^{6} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{5}}{c^{2}} - \frac {10 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \operatorname {arsinh}\left (c x\right )}{c^{7}}\right )} c\right )} b c^{2} d + \frac {1}{32} \, {\left (8 \, x^{4} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac {3 \, \sqrt {c^{2} x^{2} + 1} x}{c^{4}} + \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c\right )} b d \] Input:

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

Output:

1/6*a*c^2*d*x^6 + 1/4*a*d*x^4 + 1/288*(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 + 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
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.02 \[ \int x^3 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {d \left (12 \mathit {asinh} \left (c x \right ) b \,c^{6} x^{6}+18 \mathit {asinh} \left (c x \right ) b \,c^{4} x^{4}-2 \sqrt {c^{2} x^{2}+1}\, b \,c^{5} x^{5}-2 \sqrt {c^{2} x^{2}+1}\, b \,c^{3} x^{3}+3 \sqrt {c^{2} x^{2}+1}\, b c x -3 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) b +12 a \,c^{6} x^{6}+18 a \,c^{4} x^{4}\right )}{72 c^{4}} \] Input:

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

Output:

(d*(12*asinh(c*x)*b*c**6*x**6 + 18*asinh(c*x)*b*c**4*x**4 - 2*sqrt(c**2*x* 
*2 + 1)*b*c**5*x**5 - 2*sqrt(c**2*x**2 + 1)*b*c**3*x**3 + 3*sqrt(c**2*x**2 
 + 1)*b*c*x - 3*log(sqrt(c**2*x**2 + 1) + c*x)*b + 12*a*c**6*x**6 + 18*a*c 
**4*x**4))/(72*c**4)