\(\int x (d+c^2 d x^2)^2 (a+b \text {arcsinh}(c x)) \, dx\) [13]

Optimal result

Integrand size = 22, antiderivative size = 120 \[ \int x \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=-\frac {5 b d^2 x \sqrt {1+c^2 x^2}}{96 c}-\frac {5 b d^2 x \left (1+c^2 x^2\right )^{3/2}}{144 c}-\frac {b d^2 x \left (1+c^2 x^2\right )^{5/2}}{36 c}-\frac {5 b d^2 \text {arcsinh}(c x)}{96 c^2}+\frac {d^2 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))}{6 c^2} \] Output:

-5/96*b*d^2*x*(c^2*x^2+1)^(1/2)/c-5/144*b*d^2*x*(c^2*x^2+1)^(3/2)/c-1/36*b 
*d^2*x*(c^2*x^2+1)^(5/2)/c-5/96*b*d^2*arcsinh(c*x)/c^2+1/6*d^2*(c^2*x^2+1) 
^3*(a+b*arcsinh(c*x))/c^2
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.87 \[ \int x \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {d^2 \left (c x \left (48 a c x \left (3+3 c^2 x^2+c^4 x^4\right )-b \sqrt {1+c^2 x^2} \left (33+26 c^2 x^2+8 c^4 x^4\right )\right )+3 b \left (11+48 c^2 x^2+48 c^4 x^4+16 c^6 x^6\right ) \text {arcsinh}(c x)\right )}{288 c^2} \] Input:

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

Output:

(d^2*(c*x*(48*a*c*x*(3 + 3*c^2*x^2 + c^4*x^4) - b*Sqrt[1 + c^2*x^2]*(33 + 
26*c^2*x^2 + 8*c^4*x^4)) + 3*b*(11 + 48*c^2*x^2 + 48*c^4*x^4 + 16*c^6*x^6) 
*ArcSinh[c*x]))/(288*c^2)
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6213, 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.

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

\(\Big \downarrow \) 6213

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 222

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

Input:

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

Output:

(d^2*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x]))/(6*c^2) - (b*d^2*((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))/(6*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 = 116, normalized size of antiderivative = 0.97

method	result	size
derivativedivides	\(\frac {\frac {a \,d^{2} \left (c^{2} x^{2}+1\right )^{3}}{6}+d^{2} b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{6}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{2}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}}{2}+\frac {11 \,\operatorname {arcsinh}\left (x c \right )}{96}-\frac {x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{36}-\frac {5 x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{144}-\frac {5 \sqrt {c^{2} x^{2}+1}\, x c}{96}\right )}{c^{2}}\)	\(116\)
default	\(\frac {\frac {a \,d^{2} \left (c^{2} x^{2}+1\right )^{3}}{6}+d^{2} b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{6}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{2}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}}{2}+\frac {11 \,\operatorname {arcsinh}\left (x c \right )}{96}-\frac {x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{36}-\frac {5 x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{144}-\frac {5 \sqrt {c^{2} x^{2}+1}\, x c}{96}\right )}{c^{2}}\)	\(116\)
parts	\(\frac {a \,d^{2} \left (c^{2} x^{2}+1\right )^{3}}{6 c^{2}}+\frac {d^{2} b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{6}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{2}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}}{2}+\frac {11 \,\operatorname {arcsinh}\left (x c \right )}{96}-\frac {x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{36}-\frac {5 x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{144}-\frac {5 \sqrt {c^{2} x^{2}+1}\, x c}{96}\right )}{c^{2}}\)	\(118\)
orering	\(\frac {\left (88 c^{6} x^{6}+282 c^{4} x^{4}+335 c^{2} x^{2}+66\right ) \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{288 c^{2} \left (c^{2} x^{2}+1\right )^{2}}-\frac {\left (8 c^{4} x^{4}+26 c^{2} x^{2}+33\right ) \left (\left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )+4 x^{2} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) c^{2} d +\frac {x \left (c^{2} d \,x^{2}+d \right )^{2} b c}{\sqrt {c^{2} x^{2}+1}}\right )}{288 c^{2} \left (c^{2} x^{2}+1\right )}\)	\(174\)

Input:

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

Output:

1/c^2*(1/6*a*d^2*(c^2*x^2+1)^3+d^2*b*(1/6*arcsinh(x*c)*x^6*c^6+1/2*arcsinh 
(x*c)*c^4*x^4+1/2*arcsinh(x*c)*c^2*x^2+11/96*arcsinh(x*c)-1/36*x*c*(c^2*x^ 
2+1)^(5/2)-5/144*x*c*(c^2*x^2+1)^(3/2)-5/96*(c^2*x^2+1)^(1/2)*x*c))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.24 \[ \int x \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {48 \, a c^{6} d^{2} x^{6} + 144 \, a c^{4} d^{2} x^{4} + 144 \, a c^{2} d^{2} x^{2} + 3 \, {\left (16 \, b c^{6} d^{2} x^{6} + 48 \, b c^{4} d^{2} x^{4} + 48 \, b c^{2} d^{2} x^{2} + 11 \, b d^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (8 \, b c^{5} d^{2} x^{5} + 26 \, b c^{3} d^{2} x^{3} + 33 \, b c d^{2} x\right )} \sqrt {c^{2} x^{2} + 1}}{288 \, c^{2}} \] Input:

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

Output:

1/288*(48*a*c^6*d^2*x^6 + 144*a*c^4*d^2*x^4 + 144*a*c^2*d^2*x^2 + 3*(16*b* 
c^6*d^2*x^6 + 48*b*c^4*d^2*x^4 + 48*b*c^2*d^2*x^2 + 11*b*d^2)*log(c*x + sq 
rt(c^2*x^2 + 1)) - (8*b*c^5*d^2*x^5 + 26*b*c^3*d^2*x^3 + 33*b*c*d^2*x)*sqr 
t(c^2*x^2 + 1))/c^2
 

Sympy [A] (verification not implemented)

Time = 0.58 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.58 \[ \int x \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {a c^{4} d^{2} x^{6}}{6} + \frac {a c^{2} d^{2} x^{4}}{2} + \frac {a d^{2} x^{2}}{2} + \frac {b c^{4} d^{2} x^{6} \operatorname {asinh}{\left (c x \right )}}{6} - \frac {b c^{3} d^{2} x^{5} \sqrt {c^{2} x^{2} + 1}}{36} + \frac {b c^{2} d^{2} x^{4} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {13 b c d^{2} x^{3} \sqrt {c^{2} x^{2} + 1}}{144} + \frac {b d^{2} x^{2} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {11 b d^{2} x \sqrt {c^{2} x^{2} + 1}}{96 c} + \frac {11 b d^{2} \operatorname {asinh}{\left (c x \right )}}{96 c^{2}} & \text {for}\: c \neq 0 \\\frac {a d^{2} x^{2}}{2} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((a*c**4*d**2*x**6/6 + a*c**2*d**2*x**4/2 + a*d**2*x**2/2 + b*c** 
4*d**2*x**6*asinh(c*x)/6 - b*c**3*d**2*x**5*sqrt(c**2*x**2 + 1)/36 + b*c** 
2*d**2*x**4*asinh(c*x)/2 - 13*b*c*d**2*x**3*sqrt(c**2*x**2 + 1)/144 + b*d* 
*2*x**2*asinh(c*x)/2 - 11*b*d**2*x*sqrt(c**2*x**2 + 1)/(96*c) + 11*b*d**2* 
asinh(c*x)/(96*c**2), Ne(c, 0)), (a*d**2*x**2/2, True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (104) = 208\).

Time = 0.04 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.95 \[ \int x \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{6} \, a c^{4} d^{2} x^{6} + \frac {1}{2} \, a c^{2} d^{2} x^{4} + \frac {1}{288} \, {\left (48 \, x^{6} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{5}}{c^{2}} - \frac {10 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \operatorname {arsinh}\left (c x\right )}{c^{7}}\right )} c\right )} b c^{4} d^{2} + \frac {1}{16} \, {\left (8 \, x^{4} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac {3 \, \sqrt {c^{2} x^{2} + 1} x}{c^{4}} + \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c\right )} b c^{2} d^{2} + \frac {1}{2} \, a d^{2} x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\right )}\right )} b d^{2} \] Input:

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

Output:

1/6*a*c^4*d^2*x^6 + 1/2*a*c^2*d^2*x^4 + 1/288*(48*x^6*arcsinh(c*x) - (8*sq 
rt(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^2 + 1/16*(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^2 + 1/2*a*d^2*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^2
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.22 \[ \int x \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {d^{2} \left (48 \mathit {asinh} \left (c x \right ) b \,c^{6} x^{6}+144 \mathit {asinh} \left (c x \right ) b \,c^{4} x^{4}+144 \mathit {asinh} \left (c x \right ) b \,c^{2} x^{2}-8 \sqrt {c^{2} x^{2}+1}\, b \,c^{5} x^{5}-26 \sqrt {c^{2} x^{2}+1}\, b \,c^{3} x^{3}-33 \sqrt {c^{2} x^{2}+1}\, b c x +33 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) b +48 a \,c^{6} x^{6}+144 a \,c^{4} x^{4}+144 a \,c^{2} x^{2}\right )}{288 c^{2}} \] Input:

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

Output:

(d**2*(48*asinh(c*x)*b*c**6*x**6 + 144*asinh(c*x)*b*c**4*x**4 + 144*asinh( 
c*x)*b*c**2*x**2 - 8*sqrt(c**2*x**2 + 1)*b*c**5*x**5 - 26*sqrt(c**2*x**2 + 
 1)*b*c**3*x**3 - 33*sqrt(c**2*x**2 + 1)*b*c*x + 33*log(sqrt(c**2*x**2 + 1 
) + c*x)*b + 48*a*c**6*x**6 + 144*a*c**4*x**4 + 144*a*c**2*x**2))/(288*c** 
2)