\(\int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx\) [299]

Optimal result

Integrand size = 28, antiderivative size = 216 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=-\frac {14 b^2 \sqrt {d+c^2 d x^2}}{9 c^4 d}+\frac {2 b^2 \left (d+c^2 d x^2\right )^{3/2}}{27 c^4 d^2}+\frac {4 b x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 c^3 \sqrt {d+c^2 d x^2}}-\frac {2 b x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c \sqrt {d+c^2 d x^2}}-\frac {2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 c^4 d}+\frac {x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 c^2 d} \] Output:

-14/9*b^2*(c^2*d*x^2+d)^(1/2)/c^4/d+2/27*b^2*(c^2*d*x^2+d)^(3/2)/c^4/d^2+4 
/3*b*x*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))/c^3/(c^2*d*x^2+d)^(1/2)-2/9*b* 
x^3*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))/c/(c^2*d*x^2+d)^(1/2)-2/3*(c^2*d* 
x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2/c^4/d+1/3*x^2*(c^2*d*x^2+d)^(1/2)*(a+b*a 
rcsinh(c*x))^2/c^2/d
 

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.81 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\frac {-6 a b c x \left (-6+c^2 x^2\right ) \sqrt {1+c^2 x^2}+2 b^2 \left (-20-19 c^2 x^2+c^4 x^4\right )+9 a^2 \left (-2-c^2 x^2+c^4 x^4\right )-6 b \left (b c x \left (-6+c^2 x^2\right ) \sqrt {1+c^2 x^2}+a \left (6+3 c^2 x^2-3 c^4 x^4\right )\right ) \text {arcsinh}(c x)+9 b^2 \left (-2-c^2 x^2+c^4 x^4\right ) \text {arcsinh}(c x)^2}{27 c^4 \sqrt {d+c^2 d x^2}} \] Input:

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

Output:

(-6*a*b*c*x*(-6 + c^2*x^2)*Sqrt[1 + c^2*x^2] + 2*b^2*(-20 - 19*c^2*x^2 + c 
^4*x^4) + 9*a^2*(-2 - c^2*x^2 + c^4*x^4) - 6*b*(b*c*x*(-6 + c^2*x^2)*Sqrt[ 
1 + c^2*x^2] + a*(6 + 3*c^2*x^2 - 3*c^4*x^4))*ArcSinh[c*x] + 9*b^2*(-2 - c 
^2*x^2 + c^4*x^4)*ArcSinh[c*x]^2)/(27*c^4*Sqrt[d + c^2*d*x^2])
 

Rubi [A] (verified)

Time = 1.31 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6227, 6191, 243, 53, 2009, 6213, 2009}

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.

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

\(\Big \downarrow \) 6227

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

\(\Big \downarrow \) 6191

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 53

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6213

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 53

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, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
 

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 6191

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
 :> Simp[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* 
(n/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + 
c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
 

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]
 

rule 6227

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/(e*(m + 2*p + 1))), x] + (-Simp[f^2*((m - 1)/(c^2*(m + 
2*p + 1)))   Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] 
 - Simp[b*f*(n/(c*(m + 2*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] 
) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[ 
m, 1] && NeQ[m + 2*p + 1, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(459\) vs. \(2(188)=376\).

Time = 1.34 (sec) , antiderivative size = 460, normalized size of antiderivative = 2.13

method	result	size
orering	\(\frac {\left (19 c^{6} x^{6}-100 c^{4} x^{4}-380 c^{2} x^{2}-240\right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}}{27 c^{6} x^{2} \sqrt {c^{2} d \,x^{2}+d}}-\frac {2 \left (c^{2} x^{2}+1\right ) \left (c^{4} x^{4}-12 c^{2} x^{2}-20\right ) \left (\frac {3 x^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}}{\sqrt {c^{2} d \,x^{2}+d}}+\frac {2 x^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} d \,x^{2}+d}\, \sqrt {c^{2} x^{2}+1}}-\frac {x^{4} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{2} d}{\left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}\right )}{9 x^{4} c^{6}}+\frac {\left (c^{2} x^{2}-20\right ) \left (c^{2} x^{2}+1\right )^{2} \left (\frac {6 x \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}}{\sqrt {c^{2} d \,x^{2}+d}}+\frac {12 x^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} d \,x^{2}+d}\, \sqrt {c^{2} x^{2}+1}}-\frac {7 x^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{2} d}{\left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 x^{3} b^{2} c^{2}}{\left (c^{2} x^{2}+1\right ) \sqrt {c^{2} d \,x^{2}+d}}-\frac {4 x^{4} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b \,c^{3} d}{\left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {c^{2} x^{2}+1}}-\frac {2 x^{4} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b \,c^{3}}{\sqrt {c^{2} d \,x^{2}+d}\, \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {3 x^{5} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{4} d^{2}}{\left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}\right )}{27 c^{6} x^{3}}\)	\(460\)
default	\(a^{2} \left (\frac {x^{2} \sqrt {c^{2} d \,x^{2}+d}}{3 c^{2} d}-\frac {2 \sqrt {c^{2} d \,x^{2}+d}}{3 d \,c^{4}}\right )+b^{2} \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (9 \operatorname {arcsinh}\left (x c \right )^{2}-6 \,\operatorname {arcsinh}\left (x c \right )+2\right )}{216 c^{4} d \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )^{2}-2 \,\operatorname {arcsinh}\left (x c \right )+2\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )^{2}+2 \,\operatorname {arcsinh}\left (x c \right )+2\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (9 \operatorname {arcsinh}\left (x c \right )^{2}+6 \,\operatorname {arcsinh}\left (x c \right )+2\right )}{216 c^{4} d \left (c^{2} x^{2}+1\right )}\right )+2 a b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (-1+3 \,\operatorname {arcsinh}\left (x c \right )\right )}{72 c^{4} d \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )-1\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )+1\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (1+3 \,\operatorname {arcsinh}\left (x c \right )\right )}{72 c^{4} d \left (c^{2} x^{2}+1\right )}\right )\)	\(706\)
parts	\(a^{2} \left (\frac {x^{2} \sqrt {c^{2} d \,x^{2}+d}}{3 c^{2} d}-\frac {2 \sqrt {c^{2} d \,x^{2}+d}}{3 d \,c^{4}}\right )+b^{2} \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (9 \operatorname {arcsinh}\left (x c \right )^{2}-6 \,\operatorname {arcsinh}\left (x c \right )+2\right )}{216 c^{4} d \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )^{2}-2 \,\operatorname {arcsinh}\left (x c \right )+2\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )^{2}+2 \,\operatorname {arcsinh}\left (x c \right )+2\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (9 \operatorname {arcsinh}\left (x c \right )^{2}+6 \,\operatorname {arcsinh}\left (x c \right )+2\right )}{216 c^{4} d \left (c^{2} x^{2}+1\right )}\right )+2 a b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (-1+3 \,\operatorname {arcsinh}\left (x c \right )\right )}{72 c^{4} d \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )-1\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )+1\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (1+3 \,\operatorname {arcsinh}\left (x c \right )\right )}{72 c^{4} d \left (c^{2} x^{2}+1\right )}\right )\)	\(706\)

Input:

int(x^3*(a+b*arcsinh(x*c))^2/(c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/27*(19*c^6*x^6-100*c^4*x^4-380*c^2*x^2-240)/c^6/x^2*(a+b*arcsinh(x*c))^2 
/(c^2*d*x^2+d)^(1/2)-2/9*(c^2*x^2+1)*(c^4*x^4-12*c^2*x^2-20)/x^4/c^6*(3*x^ 
2*(a+b*arcsinh(x*c))^2/(c^2*d*x^2+d)^(1/2)+2*x^3*(a+b*arcsinh(x*c))/(c^2*d 
*x^2+d)^(1/2)*b*c/(c^2*x^2+1)^(1/2)-x^4*(a+b*arcsinh(x*c))^2/(c^2*d*x^2+d) 
^(3/2)*c^2*d)+1/27*(c^2*x^2-20)/c^6*(c^2*x^2+1)^2/x^3*(6*x*(a+b*arcsinh(x* 
c))^2/(c^2*d*x^2+d)^(1/2)+12*x^2*(a+b*arcsinh(x*c))/(c^2*d*x^2+d)^(1/2)*b* 
c/(c^2*x^2+1)^(1/2)-7*x^3*(a+b*arcsinh(x*c))^2/(c^2*d*x^2+d)^(3/2)*c^2*d+2 
*x^3*b^2*c^2/(c^2*x^2+1)/(c^2*d*x^2+d)^(1/2)-4*x^4*(a+b*arcsinh(x*c))/(c^2 
*d*x^2+d)^(3/2)*b*c^3/(c^2*x^2+1)^(1/2)*d-2*x^4*(a+b*arcsinh(x*c))/(c^2*d* 
x^2+d)^(1/2)*b*c^3/(c^2*x^2+1)^(3/2)+3*x^5*(a+b*arcsinh(x*c))^2/(c^2*d*x^2 
+d)^(5/2)*c^4*d^2)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.18 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\frac {9 \, {\left (b^{2} c^{4} x^{4} - b^{2} c^{2} x^{2} - 2 \, b^{2}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 6 \, {\left (3 \, a b c^{4} x^{4} - 3 \, a b c^{2} x^{2} - 6 \, a b - {\left (b^{2} c^{3} x^{3} - 6 \, b^{2} c x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{4} x^{4} - {\left (9 \, a^{2} + 38 \, b^{2}\right )} c^{2} x^{2} - 18 \, a^{2} - 40 \, b^{2} - 6 \, {\left (a b c^{3} x^{3} - 6 \, a b c x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d}}{27 \, {\left (c^{6} d x^{2} + c^{4} d\right )}} \] Input:

integrate(x^3*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(1/2),x, algorithm="frica 
s")
 

Output:

1/27*(9*(b^2*c^4*x^4 - b^2*c^2*x^2 - 2*b^2)*sqrt(c^2*d*x^2 + d)*log(c*x + 
sqrt(c^2*x^2 + 1))^2 + 6*(3*a*b*c^4*x^4 - 3*a*b*c^2*x^2 - 6*a*b - (b^2*c^3 
*x^3 - 6*b^2*c*x)*sqrt(c^2*x^2 + 1))*sqrt(c^2*d*x^2 + d)*log(c*x + sqrt(c^ 
2*x^2 + 1)) + ((9*a^2 + 2*b^2)*c^4*x^4 - (9*a^2 + 38*b^2)*c^2*x^2 - 18*a^2 
 - 40*b^2 - 6*(a*b*c^3*x^3 - 6*a*b*c*x)*sqrt(c^2*x^2 + 1))*sqrt(c^2*d*x^2 
+ d))/(c^6*d*x^2 + c^4*d)
 

Sympy [F]

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

integrate(x**3*(a+b*asinh(c*x))**2/(c**2*d*x**2+d)**(1/2),x)
 

Output:

Integral(x**3*(a + b*asinh(c*x))**2/sqrt(d*(c**2*x**2 + 1)), x)
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.12 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\frac {1}{3} \, b^{2} {\left (\frac {\sqrt {c^{2} d x^{2} + d} x^{2}}{c^{2} d} - \frac {2 \, \sqrt {c^{2} d x^{2} + d}}{c^{4} d}\right )} \operatorname {arsinh}\left (c x\right )^{2} + \frac {2}{3} \, a b {\left (\frac {\sqrt {c^{2} d x^{2} + d} x^{2}}{c^{2} d} - \frac {2 \, \sqrt {c^{2} d x^{2} + d}}{c^{4} d}\right )} \operatorname {arsinh}\left (c x\right ) + \frac {1}{3} \, a^{2} {\left (\frac {\sqrt {c^{2} d x^{2} + d} x^{2}}{c^{2} d} - \frac {2 \, \sqrt {c^{2} d x^{2} + d}}{c^{4} d}\right )} + \frac {2}{27} \, b^{2} {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2} - \frac {20 \, \sqrt {c^{2} x^{2} + 1}}{c^{2}}}{c^{2} \sqrt {d}} - \frac {3 \, {\left (c^{2} x^{3} - 6 \, x\right )} \operatorname {arsinh}\left (c x\right )}{c^{3} \sqrt {d}}\right )} - \frac {2 \, {\left (c^{2} x^{3} - 6 \, x\right )} a b}{9 \, c^{3} \sqrt {d}} \] Input:

integrate(x^3*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(1/2),x, algorithm="maxim 
a")
 

Output:

1/3*b^2*(sqrt(c^2*d*x^2 + d)*x^2/(c^2*d) - 2*sqrt(c^2*d*x^2 + d)/(c^4*d))* 
arcsinh(c*x)^2 + 2/3*a*b*(sqrt(c^2*d*x^2 + d)*x^2/(c^2*d) - 2*sqrt(c^2*d*x 
^2 + d)/(c^4*d))*arcsinh(c*x) + 1/3*a^2*(sqrt(c^2*d*x^2 + d)*x^2/(c^2*d) - 
 2*sqrt(c^2*d*x^2 + d)/(c^4*d)) + 2/27*b^2*((sqrt(c^2*x^2 + 1)*x^2 - 20*sq 
rt(c^2*x^2 + 1)/c^2)/(c^2*sqrt(d)) - 3*(c^2*x^3 - 6*x)*arcsinh(c*x)/(c^3*s 
qrt(d))) - 2/9*(c^2*x^3 - 6*x)*a*b/(c^3*sqrt(d))
 

Giac [F(-2)]

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

integrate(x^3*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(1/2),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. \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\int \frac {x^3\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{\sqrt {d\,c^2\,x^2+d}} \,d x \] Input:

int((x^3*(a + b*asinh(c*x))^2)/(d + c^2*d*x^2)^(1/2),x)
 

Output:

int((x^3*(a + b*asinh(c*x))^2)/(d + c^2*d*x^2)^(1/2), x)
                                                                                    
                                                                                    
 

Reduce [F]

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

int(x^3*(a+b*asinh(c*x))^2/(c^2*d*x^2+d)^(1/2),x)
 

Output:

(sqrt(c**2*x**2 + 1)*a**2*c**2*x**2 - 2*sqrt(c**2*x**2 + 1)*a**2 + 6*int(( 
asinh(c*x)*x**3)/sqrt(c**2*x**2 + 1),x)*a*b*c**4 + 3*int((asinh(c*x)**2*x* 
*3)/sqrt(c**2*x**2 + 1),x)*b**2*c**4)/(3*sqrt(d)*c**4)