\(\int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{(d+c^2 d x^2)^3} \, dx\) [256]

Optimal result

Integrand size = 26, antiderivative size = 167 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=-\frac {b^2}{12 c^4 d^3 \left (1+c^2 x^2\right )}+\frac {b x^3 (a+b \text {arcsinh}(c x))}{6 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b x (a+b \text {arcsinh}(c x))}{2 c^3 d^3 \sqrt {1+c^2 x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{4 c^4 d^3}+\frac {x^4 (a+b \text {arcsinh}(c x))^2}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {b^2 \log \left (1+c^2 x^2\right )}{3 c^4 d^3} \] Output:

-1/12*b^2/c^4/d^3/(c^2*x^2+1)+1/6*b*x^3*(a+b*arcsinh(c*x))/c/d^3/(c^2*x^2+ 
1)^(3/2)+1/2*b*x*(a+b*arcsinh(c*x))/c^3/d^3/(c^2*x^2+1)^(1/2)-1/4*(a+b*arc 
sinh(c*x))^2/c^4/d^3+1/4*x^4*(a+b*arcsinh(c*x))^2/d^3/(c^2*x^2+1)^2-1/3*b^ 
2*ln(c^2*x^2+1)/c^4/d^3
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.11 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=-\frac {3 a^2+b^2+6 a^2 c^2 x^2+b^2 c^2 x^2-6 a b c x \sqrt {1+c^2 x^2}-8 a b c^3 x^3 \sqrt {1+c^2 x^2}+2 b \left (-b c x \sqrt {1+c^2 x^2} \left (3+4 c^2 x^2\right )+a \left (3+6 c^2 x^2\right )\right ) \text {arcsinh}(c x)+3 b^2 \left (1+2 c^2 x^2\right ) \text {arcsinh}(c x)^2+4 \left (b+b c^2 x^2\right )^2 \log \left (1+c^2 x^2\right )}{12 c^4 d^3 \left (1+c^2 x^2\right )^2} \] Input:

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

Output:

-1/12*(3*a^2 + b^2 + 6*a^2*c^2*x^2 + b^2*c^2*x^2 - 6*a*b*c*x*Sqrt[1 + c^2* 
x^2] - 8*a*b*c^3*x^3*Sqrt[1 + c^2*x^2] + 2*b*(-(b*c*x*Sqrt[1 + c^2*x^2]*(3 
 + 4*c^2*x^2)) + a*(3 + 6*c^2*x^2))*ArcSinh[c*x] + 3*b^2*(1 + 2*c^2*x^2)*A 
rcSinh[c*x]^2 + 4*(b + b*c^2*x^2)^2*Log[1 + c^2*x^2])/(c^4*d^3*(1 + c^2*x^ 
2)^2)
 

Rubi [A] (verified)

Time = 1.24 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6215, 6225, 243, 49, 2009, 6225, 240, 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.

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

\(\Big \downarrow \) 6215

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

\(\Big \downarrow \) 6225

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6225

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

\(\Big \downarrow \) 240

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

\(\Big \downarrow \) 6198

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 49

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] 
&& IGtQ[m, 0] && IGtQ[m + n + 2, 0]
 

rule 240

Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x 
^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

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 6215

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

rule 6225

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a 
+ b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) 
   Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] - S 
imp[b*f*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Int[(f*x)^( 
m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; Fre 
eQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && IG 
tQ[m, 1]
 

Maple [A] (verified)

Time = 1.22 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.63

method	result	size
derivativedivides	\(\frac {\frac {a^{2} \left (\frac {1}{4 \left (c^{2} x^{2}+1\right )^{2}}-\frac {1}{2 \left (c^{2} x^{2}+1\right )}\right )}{d^{3}}+\frac {b^{2} \left (\frac {4 \,\operatorname {arcsinh}\left (x c \right )}{3}-\frac {-8 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+8 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}+6 \operatorname {arcsinh}\left (x c \right )^{2} x^{2} c^{2}-6 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c +16 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+3 \operatorname {arcsinh}\left (x c \right )^{2}+c^{2} x^{2}+8 \,\operatorname {arcsinh}\left (x c \right )+1}{12 \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}-\frac {2 \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3}\right )}{d^{3}}+\frac {2 a b \left (\frac {\operatorname {arcsinh}\left (x c \right )}{4 \left (c^{2} x^{2}+1\right )^{2}}-\frac {\operatorname {arcsinh}\left (x c \right )}{2 \left (c^{2} x^{2}+1\right )}-\frac {x c}{12 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {x c}{3 \sqrt {c^{2} x^{2}+1}}\right )}{d^{3}}}{c^{4}}\)	\(272\)
default	\(\frac {\frac {a^{2} \left (\frac {1}{4 \left (c^{2} x^{2}+1\right )^{2}}-\frac {1}{2 \left (c^{2} x^{2}+1\right )}\right )}{d^{3}}+\frac {b^{2} \left (\frac {4 \,\operatorname {arcsinh}\left (x c \right )}{3}-\frac {-8 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+8 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}+6 \operatorname {arcsinh}\left (x c \right )^{2} x^{2} c^{2}-6 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c +16 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+3 \operatorname {arcsinh}\left (x c \right )^{2}+c^{2} x^{2}+8 \,\operatorname {arcsinh}\left (x c \right )+1}{12 \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}-\frac {2 \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3}\right )}{d^{3}}+\frac {2 a b \left (\frac {\operatorname {arcsinh}\left (x c \right )}{4 \left (c^{2} x^{2}+1\right )^{2}}-\frac {\operatorname {arcsinh}\left (x c \right )}{2 \left (c^{2} x^{2}+1\right )}-\frac {x c}{12 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {x c}{3 \sqrt {c^{2} x^{2}+1}}\right )}{d^{3}}}{c^{4}}\)	\(272\)
parts	\(\frac {a^{2} \left (-\frac {1}{2 c^{4} \left (c^{2} x^{2}+1\right )}+\frac {1}{4 c^{4} \left (c^{2} x^{2}+1\right )^{2}}\right )}{d^{3}}+\frac {b^{2} \left (\frac {4 \,\operatorname {arcsinh}\left (x c \right )}{3}-\frac {-8 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+8 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}+6 \operatorname {arcsinh}\left (x c \right )^{2} x^{2} c^{2}-6 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c +16 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+3 \operatorname {arcsinh}\left (x c \right )^{2}+c^{2} x^{2}+8 \,\operatorname {arcsinh}\left (x c \right )+1}{12 \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}-\frac {2 \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3}\right )}{d^{3} c^{4}}+\frac {2 a b \left (\frac {\operatorname {arcsinh}\left (x c \right )}{4 \left (c^{2} x^{2}+1\right )^{2}}-\frac {\operatorname {arcsinh}\left (x c \right )}{2 \left (c^{2} x^{2}+1\right )}-\frac {x c}{12 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {x c}{3 \sqrt {c^{2} x^{2}+1}}\right )}{d^{3} c^{4}}\)	\(280\)

Input:

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

Output:

1/c^4*(a^2/d^3*(1/4/(c^2*x^2+1)^2-1/2/(c^2*x^2+1))+b^2/d^3*(4/3*arcsinh(x* 
c)-1/12*(-8*arcsinh(x*c)*(c^2*x^2+1)^(1/2)*x^3*c^3+8*arcsinh(x*c)*c^4*x^4+ 
6*arcsinh(x*c)^2*x^2*c^2-6*arcsinh(x*c)*(c^2*x^2+1)^(1/2)*x*c+16*arcsinh(x 
*c)*c^2*x^2+3*arcsinh(x*c)^2+c^2*x^2+8*arcsinh(x*c)+1)/(c^4*x^4+2*c^2*x^2+ 
1)-2/3*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2))+2*a*b/d^3*(1/4*arcsinh(x*c)/(c^2*x 
^2+1)^2-1/2/(c^2*x^2+1)*arcsinh(x*c)-1/12/(c^2*x^2+1)^(3/2)*x*c+1/3*x*c/(c 
^2*x^2+1)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.69 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {8 \, a b c^{4} x^{4} - {\left (6 \, a^{2} - 16 \, a b + b^{2}\right )} c^{2} x^{2} - 3 \, {\left (2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} - 3 \, a^{2} + 8 \, a b - b^{2} - 4 \, {\left (b^{2} c^{4} x^{4} + 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \log \left (c^{2} x^{2} + 1\right ) + 2 \, {\left (3 \, a b c^{4} x^{4} + {\left (4 \, b^{2} c^{3} x^{3} + 3 \, b^{2} c x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 6 \, {\left (a b c^{4} x^{4} + 2 \, a b c^{2} x^{2} + a b\right )} \log \left (-c x + \sqrt {c^{2} x^{2} + 1}\right ) + 2 \, {\left (4 \, a b c^{3} x^{3} + 3 \, a b c x\right )} \sqrt {c^{2} x^{2} + 1}}{12 \, {\left (c^{8} d^{3} x^{4} + 2 \, c^{6} d^{3} x^{2} + c^{4} d^{3}\right )}} \] Input:

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

Output:

1/12*(8*a*b*c^4*x^4 - (6*a^2 - 16*a*b + b^2)*c^2*x^2 - 3*(2*b^2*c^2*x^2 + 
b^2)*log(c*x + sqrt(c^2*x^2 + 1))^2 - 3*a^2 + 8*a*b - b^2 - 4*(b^2*c^4*x^4 
 + 2*b^2*c^2*x^2 + b^2)*log(c^2*x^2 + 1) + 2*(3*a*b*c^4*x^4 + (4*b^2*c^3*x 
^3 + 3*b^2*c*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + 6*(a*b*c 
^4*x^4 + 2*a*b*c^2*x^2 + a*b)*log(-c*x + sqrt(c^2*x^2 + 1)) + 2*(4*a*b*c^3 
*x^3 + 3*a*b*c*x)*sqrt(c^2*x^2 + 1))/(c^8*d^3*x^4 + 2*c^6*d^3*x^2 + c^4*d^ 
3)
 

Sympy [F]

\[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {\int \frac {a^{2} x^{3}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx + \int \frac {b^{2} x^{3} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx + \int \frac {2 a b x^{3} \operatorname {asinh}{\left (c x \right )}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx}{d^{3}} \] Input:

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

Output:

(Integral(a**2*x**3/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x**2 + 1), x) + Inte 
gral(b**2*x**3*asinh(c*x)**2/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x**2 + 1), 
x) + Integral(2*a*b*x**3*asinh(c*x)/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x**2 
 + 1), x))/d**3
 

Maxima [F]

\[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{3}}{{\left (c^{2} d x^{2} + d\right )}^{3}} \,d x } \] Input:

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

Output:

-1/4*(2*c^2*x^2 + 1)*a^2/(c^8*d^3*x^4 + 2*c^6*d^3*x^2 + c^4*d^3) - 1/4*(2* 
b^2*c^2*x^2 + b^2)*log(c*x + sqrt(c^2*x^2 + 1))^2/(c^8*d^3*x^4 + 2*c^6*d^3 
*x^2 + c^4*d^3) + integrate(1/2*(3*b^2*c^2*x^2 + 2*(2*a*b*c^4 + b^2*c^4)*x 
^4 + b^2 + (b^2*c*x + 2*(2*a*b*c^3 + b^2*c^3)*x^3)*sqrt(c^2*x^2 + 1))*log( 
c*x + sqrt(c^2*x^2 + 1))/(c^10*d^3*x^7 + 3*c^8*d^3*x^5 + 3*c^6*d^3*x^3 + c 
^4*d^3*x + (c^9*d^3*x^6 + 3*c^7*d^3*x^4 + 3*c^5*d^3*x^2 + c^3*d^3)*sqrt(c^ 
2*x^2 + 1)), x)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {8 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{3}}{c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1}d x \right ) a b \,c^{4} x^{4}+16 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{3}}{c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1}d x \right ) a b \,c^{2} x^{2}+8 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{3}}{c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1}d x \right ) a b +4 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2} x^{3}}{c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1}d x \right ) b^{2} c^{4} x^{4}+8 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2} x^{3}}{c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1}d x \right ) b^{2} c^{2} x^{2}+4 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2} x^{3}}{c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1}d x \right ) b^{2}+a^{2} x^{4}}{4 d^{3} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )} \] Input:

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

Output:

(8*int((asinh(c*x)*x**3)/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x**2 + 1),x)*a* 
b*c**4*x**4 + 16*int((asinh(c*x)*x**3)/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x 
**2 + 1),x)*a*b*c**2*x**2 + 8*int((asinh(c*x)*x**3)/(c**6*x**6 + 3*c**4*x* 
*4 + 3*c**2*x**2 + 1),x)*a*b + 4*int((asinh(c*x)**2*x**3)/(c**6*x**6 + 3*c 
**4*x**4 + 3*c**2*x**2 + 1),x)*b**2*c**4*x**4 + 8*int((asinh(c*x)**2*x**3) 
/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x**2 + 1),x)*b**2*c**2*x**2 + 4*int((as 
inh(c*x)**2*x**3)/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x**2 + 1),x)*b**2 + a* 
*2*x**4)/(4*d**3*(c**4*x**4 + 2*c**2*x**2 + 1))