\(\int x (d+c^2 d x^2) (a+b \text {arcsinh}(c x))^2 \, dx\) [213]

Optimal result

Integrand size = 22, antiderivative size = 143 \[ \int x \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {3}{32} b^2 d x^2+\frac {b^2 d \left (1+c^2 x^2\right )^2}{32 c^2}-\frac {3 b d x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{16 c}-\frac {b d x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{8 c}-\frac {3 d (a+b \text {arcsinh}(c x))^2}{32 c^2}+\frac {d \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{4 c^2} \] Output:

3/32*b^2*d*x^2+1/32*b^2*d*(c^2*x^2+1)^2/c^2-3/16*b*d*x*(c^2*x^2+1)^(1/2)*( 
a+b*arcsinh(c*x))/c-1/8*b*d*x*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))/c-3/32* 
d*(a+b*arcsinh(c*x))^2/c^2+1/4*d*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))^2/c^2
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.08 \[ \int x \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d \left (c x \left (8 a^2 c x \left (2+c^2 x^2\right )+b^2 c x \left (5+c^2 x^2\right )-2 a b \sqrt {1+c^2 x^2} \left (5+2 c^2 x^2\right )\right )+2 b \left (-b c x \sqrt {1+c^2 x^2} \left (5+2 c^2 x^2\right )+a \left (5+16 c^2 x^2+8 c^4 x^4\right )\right ) \text {arcsinh}(c x)+b^2 \left (5+16 c^2 x^2+8 c^4 x^4\right ) \text {arcsinh}(c x)^2\right )}{32 c^2} \] Input:

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

Output:

(d*(c*x*(8*a^2*c*x*(2 + c^2*x^2) + b^2*c*x*(5 + c^2*x^2) - 2*a*b*Sqrt[1 + 
c^2*x^2]*(5 + 2*c^2*x^2)) + 2*b*(-(b*c*x*Sqrt[1 + c^2*x^2]*(5 + 2*c^2*x^2) 
) + a*(5 + 16*c^2*x^2 + 8*c^4*x^4))*ArcSinh[c*x] + b^2*(5 + 16*c^2*x^2 + 8 
*c^4*x^4)*ArcSinh[c*x]^2))/(32*c^2)
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6213, 6201, 244, 2009, 6200, 15, 6198}

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.

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

\(\Big \downarrow \) 6213

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

\(\Big \downarrow \) 6201

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

\(\Big \downarrow \) 244

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6200

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6198

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 244

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p 
, 0]
 

rule 2009

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

rule 6198

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( 
a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c 
^2*d] && NeQ[n, -1]
 

rule 6200

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSinh[c*x])^n/2), x] + (Simp[(1 
/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]   Int[(a + b*ArcSinh[c*x])^n/Sq 
rt[1 + c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2* 
x^2]]   Int[x*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[e, c^2*d] && GtQ[n, 0]
 

rule 6201

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), 
x_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcSinh[c*x])^n/(2*p + 1)), x] + 
(Simp[2*d*(p/(2*p + 1))   Int[(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x 
], x] - Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Int[x* 
(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, 
b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0]
 

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 = 1.54 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.27

method	result	size
derivativedivides	\(\frac {\frac {a^{2} d \left (c^{2} x^{2}+1\right )^{2}}{4}+b^{2} d \left (\frac {\operatorname {arcsinh}\left (x c \right )^{2} \left (c^{2} x^{2}+1\right )^{2}}{4}-\frac {\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{8}-\frac {3 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c}{16}-\frac {3 \operatorname {arcsinh}\left (x c \right )^{2}}{32}+\frac {\left (c^{2} x^{2}+1\right )^{2}}{32}+\frac {3 c^{2} x^{2}}{32}+\frac {3}{32}\right )+2 a b d \left (\frac {\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{4}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}}{2}+\frac {5 \,\operatorname {arcsinh}\left (x c \right )}{32}-\frac {x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{16}-\frac {3 \sqrt {c^{2} x^{2}+1}\, x c}{32}\right )}{c^{2}}\)	\(182\)
default	\(\frac {\frac {a^{2} d \left (c^{2} x^{2}+1\right )^{2}}{4}+b^{2} d \left (\frac {\operatorname {arcsinh}\left (x c \right )^{2} \left (c^{2} x^{2}+1\right )^{2}}{4}-\frac {\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{8}-\frac {3 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c}{16}-\frac {3 \operatorname {arcsinh}\left (x c \right )^{2}}{32}+\frac {\left (c^{2} x^{2}+1\right )^{2}}{32}+\frac {3 c^{2} x^{2}}{32}+\frac {3}{32}\right )+2 a b d \left (\frac {\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{4}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}}{2}+\frac {5 \,\operatorname {arcsinh}\left (x c \right )}{32}-\frac {x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{16}-\frac {3 \sqrt {c^{2} x^{2}+1}\, x c}{32}\right )}{c^{2}}\)	\(182\)
parts	\(\frac {a^{2} d \left (c^{2} x^{2}+1\right )^{2}}{4 c^{2}}+\frac {b^{2} d \left (\frac {\operatorname {arcsinh}\left (x c \right )^{2} \left (c^{2} x^{2}+1\right )^{2}}{4}-\frac {\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{8}-\frac {3 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c}{16}-\frac {3 \operatorname {arcsinh}\left (x c \right )^{2}}{32}+\frac {\left (c^{2} x^{2}+1\right )^{2}}{32}+\frac {3 c^{2} x^{2}}{32}+\frac {3}{32}\right )}{c^{2}}+\frac {2 a b d \left (\frac {\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{4}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}}{2}+\frac {5 \,\operatorname {arcsinh}\left (x c \right )}{32}-\frac {x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{16}-\frac {3 \sqrt {c^{2} x^{2}+1}\, x c}{32}\right )}{c^{2}}\)	\(187\)
orering	\(\frac {\left (37 c^{6} x^{6}+144 c^{4} x^{4}+113 c^{2} x^{2}+30\right ) \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}}{64 c^{2} \left (c^{2} x^{2}+1\right )^{2}}-\frac {\left (9 c^{4} x^{4}+41 c^{2} x^{2}+20\right ) \left (\left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+2 x^{2} c^{2} d \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+\frac {2 x \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} x^{2}+1}}\right )}{64 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {x \left (c^{2} x^{2}+5\right ) \left (6 c^{2} d x \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+\frac {4 \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} x^{2}+1}}+\frac {8 x^{2} c^{3} d \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b}{\sqrt {c^{2} x^{2}+1}}+\frac {2 x \left (c^{2} d \,x^{2}+d \right ) b^{2} c^{2}}{c^{2} x^{2}+1}-\frac {2 x^{2} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b \,c^{3}}{\left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{64 c^{2}}\)	\(336\)

Input:

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

Output:

1/c^2*(1/4*a^2*d*(c^2*x^2+1)^2+b^2*d*(1/4*arcsinh(x*c)^2*(c^2*x^2+1)^2-1/8 
*arcsinh(x*c)*x*c*(c^2*x^2+1)^(3/2)-3/16*arcsinh(x*c)*(c^2*x^2+1)^(1/2)*x* 
c-3/32*arcsinh(x*c)^2+1/32*(c^2*x^2+1)^2+3/32*c^2*x^2+3/32)+2*a*b*d*(1/4*a 
rcsinh(x*c)*c^4*x^4+1/2*arcsinh(x*c)*c^2*x^2+5/32*arcsinh(x*c)-1/16*x*c*(c 
^2*x^2+1)^(3/2)-3/32*(c^2*x^2+1)^(1/2)*x*c))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.43 \[ \int x \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {{\left (8 \, a^{2} + b^{2}\right )} c^{4} d x^{4} + {\left (16 \, a^{2} + 5 \, b^{2}\right )} c^{2} d x^{2} + {\left (8 \, b^{2} c^{4} d x^{4} + 16 \, b^{2} c^{2} d x^{2} + 5 \, b^{2} d\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 2 \, {\left (8 \, a b c^{4} d x^{4} + 16 \, a b c^{2} d x^{2} + 5 \, a b d - {\left (2 \, b^{2} c^{3} d x^{3} + 5 \, b^{2} c d x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 2 \, {\left (2 \, a b c^{3} d x^{3} + 5 \, a b c d x\right )} \sqrt {c^{2} x^{2} + 1}}{32 \, c^{2}} \] Input:

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

Output:

1/32*((8*a^2 + b^2)*c^4*d*x^4 + (16*a^2 + 5*b^2)*c^2*d*x^2 + (8*b^2*c^4*d* 
x^4 + 16*b^2*c^2*d*x^2 + 5*b^2*d)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 2*(8*a* 
b*c^4*d*x^4 + 16*a*b*c^2*d*x^2 + 5*a*b*d - (2*b^2*c^3*d*x^3 + 5*b^2*c*d*x) 
*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) - 2*(2*a*b*c^3*d*x^3 + 5* 
a*b*c*d*x)*sqrt(c^2*x^2 + 1))/c^2
 

Sympy [A] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.88 \[ \int x \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\begin {cases} \frac {a^{2} c^{2} d x^{4}}{4} + \frac {a^{2} d x^{2}}{2} + \frac {a b c^{2} d x^{4} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {a b c d x^{3} \sqrt {c^{2} x^{2} + 1}}{8} + a b d x^{2} \operatorname {asinh}{\left (c x \right )} - \frac {5 a b d x \sqrt {c^{2} x^{2} + 1}}{16 c} + \frac {5 a b d \operatorname {asinh}{\left (c x \right )}}{16 c^{2}} + \frac {b^{2} c^{2} d x^{4} \operatorname {asinh}^{2}{\left (c x \right )}}{4} + \frac {b^{2} c^{2} d x^{4}}{32} - \frac {b^{2} c d x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{8} + \frac {b^{2} d x^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{2} + \frac {5 b^{2} d x^{2}}{32} - \frac {5 b^{2} d x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{16 c} + \frac {5 b^{2} d \operatorname {asinh}^{2}{\left (c x \right )}}{32 c^{2}} & \text {for}\: c \neq 0 \\\frac {a^{2} d x^{2}}{2} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((a**2*c**2*d*x**4/4 + a**2*d*x**2/2 + a*b*c**2*d*x**4*asinh(c*x) 
/2 - a*b*c*d*x**3*sqrt(c**2*x**2 + 1)/8 + a*b*d*x**2*asinh(c*x) - 5*a*b*d* 
x*sqrt(c**2*x**2 + 1)/(16*c) + 5*a*b*d*asinh(c*x)/(16*c**2) + b**2*c**2*d* 
x**4*asinh(c*x)**2/4 + b**2*c**2*d*x**4/32 - b**2*c*d*x**3*sqrt(c**2*x**2 
+ 1)*asinh(c*x)/8 + b**2*d*x**2*asinh(c*x)**2/2 + 5*b**2*d*x**2/32 - 5*b** 
2*d*x*sqrt(c**2*x**2 + 1)*asinh(c*x)/(16*c) + 5*b**2*d*asinh(c*x)**2/(32*c 
**2), Ne(c, 0)), (a**2*d*x**2/2, True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (127) = 254\).

Time = 0.05 (sec) , antiderivative size = 347, normalized size of antiderivative = 2.43 \[ \int x \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{4} \, b^{2} c^{2} d x^{4} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{4} \, a^{2} c^{2} d x^{4} + \frac {1}{2} \, b^{2} d x^{2} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{16} \, {\left (8 \, x^{4} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac {3 \, \sqrt {c^{2} x^{2} + 1} x}{c^{4}} + \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c\right )} a b c^{2} d + \frac {1}{32} \, {\left ({\left (\frac {x^{4}}{c^{2}} - \frac {3 \, x^{2}}{c^{4}} + \frac {3 \, \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{c^{6}}\right )} c^{2} - 2 \, {\left (\frac {2 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac {3 \, \sqrt {c^{2} x^{2} + 1} x}{c^{4}} + \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c \operatorname {arsinh}\left (c x\right )\right )} b^{2} c^{2} d + \frac {1}{2} \, a^{2} d x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\right )}\right )} a b d + \frac {1}{4} \, {\left (c^{2} {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{c^{4}}\right )} - 2 \, c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\right )} \operatorname {arsinh}\left (c x\right )\right )} b^{2} d \] Input:

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

Output:

1/4*b^2*c^2*d*x^4*arcsinh(c*x)^2 + 1/4*a^2*c^2*d*x^4 + 1/2*b^2*d*x^2*arcsi 
nh(c*x)^2 + 1/16*(8*x^4*arcsinh(c*x) - (2*sqrt(c^2*x^2 + 1)*x^3/c^2 - 3*sq 
rt(c^2*x^2 + 1)*x/c^4 + 3*arcsinh(c*x)/c^5)*c)*a*b*c^2*d + 1/32*((x^4/c^2 
- 3*x^2/c^4 + 3*log(c*x + sqrt(c^2*x^2 + 1))^2/c^6)*c^2 - 2*(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*arcsinh 
(c*x))*b^2*c^2*d + 1/2*a^2*d*x^2 + 1/2*(2*x^2*arcsinh(c*x) - c*(sqrt(c^2*x 
^2 + 1)*x/c^2 - arcsinh(c*x)/c^3))*a*b*d + 1/4*(c^2*(x^2/c^2 - log(c*x + s 
qrt(c^2*x^2 + 1))^2/c^4) - 2*c*(sqrt(c^2*x^2 + 1)*x/c^2 - arcsinh(c*x)/c^3 
)*arcsinh(c*x))*b^2*d
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int x \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d \left (8 \mathit {asinh} \left (c x \right )^{2} b^{2} c^{2} x^{2}+4 \mathit {asinh} \left (c x \right )^{2} b^{2}-8 \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) b^{2} c x +8 \mathit {asinh} \left (c x \right ) a b \,c^{4} x^{4}+16 \mathit {asinh} \left (c x \right ) a b \,c^{2} x^{2}-2 \sqrt {c^{2} x^{2}+1}\, a b \,c^{3} x^{3}-5 \sqrt {c^{2} x^{2}+1}\, a b c x +16 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{3}d x \right ) b^{2} c^{4}+5 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a b +4 a^{2} c^{4} x^{4}+8 a^{2} c^{2} x^{2}+4 b^{2} c^{2} x^{2}\right )}{16 c^{2}} \] Input:

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

Output:

(d*(8*asinh(c*x)**2*b**2*c**2*x**2 + 4*asinh(c*x)**2*b**2 - 8*sqrt(c**2*x* 
*2 + 1)*asinh(c*x)*b**2*c*x + 8*asinh(c*x)*a*b*c**4*x**4 + 16*asinh(c*x)*a 
*b*c**2*x**2 - 2*sqrt(c**2*x**2 + 1)*a*b*c**3*x**3 - 5*sqrt(c**2*x**2 + 1) 
*a*b*c*x + 16*int(asinh(c*x)**2*x**3,x)*b**2*c**4 + 5*log(sqrt(c**2*x**2 + 
 1) + c*x)*a*b + 4*a**2*c**4*x**4 + 8*a**2*c**2*x**2 + 4*b**2*c**2*x**2))/ 
(16*c**2)