\(\int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx\) [298]

Optimal result

Integrand size = 28, antiderivative size = 311 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=-\frac {15 b^2 x \sqrt {d+c^2 d x^2}}{64 c^4 d}+\frac {b^2 x^3 \sqrt {d+c^2 d x^2}}{32 c^2 d}+\frac {15 b^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{64 c^5 \sqrt {d+c^2 d x^2}}+\frac {3 b x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{8 c^3 \sqrt {d+c^2 d x^2}}-\frac {b x^4 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{8 c \sqrt {d+c^2 d x^2}}-\frac {3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{8 c^4 d}+\frac {x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 c^2 d}+\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{8 b c^5 \sqrt {d+c^2 d x^2}} \] Output:

-15/64*b^2*x*(c^2*d*x^2+d)^(1/2)/c^4/d+1/32*b^2*x^3*(c^2*d*x^2+d)^(1/2)/c^ 
2/d+15/64*b^2*(c^2*x^2+1)^(1/2)*arcsinh(c*x)/c^5/(c^2*d*x^2+d)^(1/2)+3/8*b 
*x^2*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))/c^3/(c^2*d*x^2+d)^(1/2)-1/8*b*x^ 
4*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))/c/(c^2*d*x^2+d)^(1/2)-3/8*x*(c^2*d* 
x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2/c^4/d+1/4*x^3*(c^2*d*x^2+d)^(1/2)*(a+b*a 
rcsinh(c*x))^2/c^2/d+1/8*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))^3/b/c^5/(c^2 
*d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.99 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.86 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\frac {32 a^2 c \sqrt {d} x \left (1+c^2 x^2\right ) \left (-3+2 c^2 x^2\right )+96 a^2 \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+b^2 \sqrt {d} \sqrt {1+c^2 x^2} \left (32 \text {arcsinh}(c x)^3-4 \text {arcsinh}(c x) (-16 \cosh (2 \text {arcsinh}(c x))+\cosh (4 \text {arcsinh}(c x)))-32 \sinh (2 \text {arcsinh}(c x))+\sinh (4 \text {arcsinh}(c x))+8 \text {arcsinh}(c x)^2 (-8 \sinh (2 \text {arcsinh}(c x))+\sinh (4 \text {arcsinh}(c x)))\right )+4 a b \sqrt {d} \sqrt {1+c^2 x^2} (16 \cosh (2 \text {arcsinh}(c x))-\cosh (4 \text {arcsinh}(c x))+4 \text {arcsinh}(c x) (6 \text {arcsinh}(c x)-8 \sinh (2 \text {arcsinh}(c x))+\sinh (4 \text {arcsinh}(c x))))}{256 c^5 \sqrt {d} \sqrt {d+c^2 d x^2}} \] Input:

Integrate[(x^4*(a + b*ArcSinh[c*x])^2)/Sqrt[d + c^2*d*x^2],x]
 

Output:

(32*a^2*c*Sqrt[d]*x*(1 + c^2*x^2)*(-3 + 2*c^2*x^2) + 96*a^2*Sqrt[d + c^2*d 
*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + b^2*Sqrt[d]*Sqrt[1 + c^2* 
x^2]*(32*ArcSinh[c*x]^3 - 4*ArcSinh[c*x]*(-16*Cosh[2*ArcSinh[c*x]] + Cosh[ 
4*ArcSinh[c*x]]) - 32*Sinh[2*ArcSinh[c*x]] + Sinh[4*ArcSinh[c*x]] + 8*ArcS 
inh[c*x]^2*(-8*Sinh[2*ArcSinh[c*x]] + Sinh[4*ArcSinh[c*x]])) + 4*a*b*Sqrt[ 
d]*Sqrt[1 + c^2*x^2]*(16*Cosh[2*ArcSinh[c*x]] - Cosh[4*ArcSinh[c*x]] + 4*A 
rcSinh[c*x]*(6*ArcSinh[c*x] - 8*Sinh[2*ArcSinh[c*x]] + Sinh[4*ArcSinh[c*x] 
])))/(256*c^5*Sqrt[d]*Sqrt[d + c^2*d*x^2])
 

Rubi [A] (verified)

Time = 1.78 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6227, 6191, 262, 262, 222, 6227, 6191, 262, 222, 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.

\(\displaystyle \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}} \, dx\)

\(\Big \downarrow \) 6227

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

\(\Big \downarrow \) 6191

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 222

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

\(\Big \downarrow \) 6227

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

\(\Big \downarrow \) 6191

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 222

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

\(\Big \downarrow \) 6198

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

Input:

Int[(x^4*(a + b*ArcSinh[c*x])^2)/Sqrt[d + c^2*d*x^2],x]
 

Output:

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

Defintions of rubi rules used

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 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 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 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. \(991\) vs. \(2(271)=542\).

Time = 0.88 (sec) , antiderivative size = 992, normalized size of antiderivative = 3.19

method	result	size
default	\(\frac {a^{2} x^{3} \sqrt {c^{2} d \,x^{2}+d}}{4 c^{2} d}-\frac {3 a^{2} x \sqrt {c^{2} d \,x^{2}+d}}{8 c^{4} d}+\frac {3 a^{2} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{8 c^{4} \sqrt {c^{2} d}}+b^{2} \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{3}}{8 \sqrt {c^{2} x^{2}+1}\, c^{5} d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}+8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}+8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (8 \operatorname {arcsinh}\left (x c \right )^{2}-4 \,\operatorname {arcsinh}\left (x c \right )+1\right )}{512 c^{5} d \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (2 \operatorname {arcsinh}\left (x c \right )^{2}-2 \,\operatorname {arcsinh}\left (x c \right )+1\right )}{16 c^{5} d \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (2 \operatorname {arcsinh}\left (x c \right )^{2}+2 \,\operatorname {arcsinh}\left (x c \right )+1\right )}{16 c^{5} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}-8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}-8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (8 \operatorname {arcsinh}\left (x c \right )^{2}+4 \,\operatorname {arcsinh}\left (x c \right )+1\right )}{512 c^{5} d \left (c^{2} x^{2}+1\right )}\right )+2 a b \left (\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2}}{16 \sqrt {c^{2} x^{2}+1}\, c^{5} d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}+8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}+8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+4 \,\operatorname {arcsinh}\left (x c \right )\right )}{256 c^{5} d \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+2 \,\operatorname {arcsinh}\left (x c \right )\right )}{16 c^{5} d \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (1+2 \,\operatorname {arcsinh}\left (x c \right )\right )}{16 c^{5} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}-8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}-8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (1+4 \,\operatorname {arcsinh}\left (x c \right )\right )}{256 c^{5} d \left (c^{2} x^{2}+1\right )}\right )\)	\(992\)
parts	\(\frac {a^{2} x^{3} \sqrt {c^{2} d \,x^{2}+d}}{4 c^{2} d}-\frac {3 a^{2} x \sqrt {c^{2} d \,x^{2}+d}}{8 c^{4} d}+\frac {3 a^{2} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{8 c^{4} \sqrt {c^{2} d}}+b^{2} \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{3}}{8 \sqrt {c^{2} x^{2}+1}\, c^{5} d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}+8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}+8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (8 \operatorname {arcsinh}\left (x c \right )^{2}-4 \,\operatorname {arcsinh}\left (x c \right )+1\right )}{512 c^{5} d \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (2 \operatorname {arcsinh}\left (x c \right )^{2}-2 \,\operatorname {arcsinh}\left (x c \right )+1\right )}{16 c^{5} d \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (2 \operatorname {arcsinh}\left (x c \right )^{2}+2 \,\operatorname {arcsinh}\left (x c \right )+1\right )}{16 c^{5} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}-8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}-8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (8 \operatorname {arcsinh}\left (x c \right )^{2}+4 \,\operatorname {arcsinh}\left (x c \right )+1\right )}{512 c^{5} d \left (c^{2} x^{2}+1\right )}\right )+2 a b \left (\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2}}{16 \sqrt {c^{2} x^{2}+1}\, c^{5} d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}+8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}+8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+4 \,\operatorname {arcsinh}\left (x c \right )\right )}{256 c^{5} d \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+2 \,\operatorname {arcsinh}\left (x c \right )\right )}{16 c^{5} d \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (1+2 \,\operatorname {arcsinh}\left (x c \right )\right )}{16 c^{5} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}-8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}-8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (1+4 \,\operatorname {arcsinh}\left (x c \right )\right )}{256 c^{5} d \left (c^{2} x^{2}+1\right )}\right )\)	\(992\)

Input:

int(x^4*(a+b*arcsinh(x*c))^2/(c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/4*a^2*x^3/c^2/d*(c^2*d*x^2+d)^(1/2)-3/8*a^2/c^4*x/d*(c^2*d*x^2+d)^(1/2)+ 
3/8*a^2/c^4*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)+b^ 
2*(1/8*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^5/d*arcsinh(x*c)^3+1/512* 
(d*(c^2*x^2+1))^(1/2)*(8*x^5*c^5+8*x^4*c^4*(c^2*x^2+1)^(1/2)+12*x^3*c^3+8* 
x^2*c^2*(c^2*x^2+1)^(1/2)+4*x*c+(c^2*x^2+1)^(1/2))*(8*arcsinh(x*c)^2-4*arc 
sinh(x*c)+1)/c^5/d/(c^2*x^2+1)-1/16*(d*(c^2*x^2+1))^(1/2)*(2*x^3*c^3+2*x^2 
*c^2*(c^2*x^2+1)^(1/2)+2*x*c+(c^2*x^2+1)^(1/2))*(2*arcsinh(x*c)^2-2*arcsin 
h(x*c)+1)/c^5/d/(c^2*x^2+1)-1/16*(d*(c^2*x^2+1))^(1/2)*(2*x^3*c^3-2*x^2*c^ 
2*(c^2*x^2+1)^(1/2)+2*x*c-(c^2*x^2+1)^(1/2))*(2*arcsinh(x*c)^2+2*arcsinh(x 
*c)+1)/c^5/d/(c^2*x^2+1)+1/512*(d*(c^2*x^2+1))^(1/2)*(8*x^5*c^5-8*x^4*c^4* 
(c^2*x^2+1)^(1/2)+12*x^3*c^3-8*x^2*c^2*(c^2*x^2+1)^(1/2)+4*x*c-(c^2*x^2+1) 
^(1/2))*(8*arcsinh(x*c)^2+4*arcsinh(x*c)+1)/c^5/d/(c^2*x^2+1))+2*a*b*(3/16 
*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^5/d*arcsinh(x*c)^2+1/256*(d*(c^ 
2*x^2+1))^(1/2)*(8*x^5*c^5+8*x^4*c^4*(c^2*x^2+1)^(1/2)+12*x^3*c^3+8*x^2*c^ 
2*(c^2*x^2+1)^(1/2)+4*x*c+(c^2*x^2+1)^(1/2))*(-1+4*arcsinh(x*c))/c^5/d/(c^ 
2*x^2+1)-1/16*(d*(c^2*x^2+1))^(1/2)*(2*x^3*c^3+2*x^2*c^2*(c^2*x^2+1)^(1/2) 
+2*x*c+(c^2*x^2+1)^(1/2))*(-1+2*arcsinh(x*c))/c^5/d/(c^2*x^2+1)-1/16*(d*(c 
^2*x^2+1))^(1/2)*(2*x^3*c^3-2*x^2*c^2*(c^2*x^2+1)^(1/2)+2*x*c-(c^2*x^2+1)^ 
(1/2))*(1+2*arcsinh(x*c))/c^5/d/(c^2*x^2+1)+1/256*(d*(c^2*x^2+1))^(1/2)*(8 
*x^5*c^5-8*x^4*c^4*(c^2*x^2+1)^(1/2)+12*x^3*c^3-8*x^2*c^2*(c^2*x^2+1)^(...
 

Fricas [F]

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

integrate(x^4*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(1/2),x, algorithm="frica 
s")
 

Output:

integral((b^2*x^4*arcsinh(c*x)^2 + 2*a*b*x^4*arcsinh(c*x) + a^2*x^4)/sqrt( 
c^2*d*x^2 + d), x)
                                                                                    
                                                                                    
 

Sympy [F]

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

integrate(x**4*(a+b*asinh(c*x))**2/(c**2*d*x**2+d)**(1/2),x)
 

Output:

Integral(x**4*(a + b*asinh(c*x))**2/sqrt(d*(c**2*x**2 + 1)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x^4*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(1/2),x, algorithm="maxim 
a")
 

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [F]

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

integrate(x^4*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(1/2),x, algorithm="giac" 
)
 

Output:

integrate((b*arcsinh(c*x) + a)^2*x^4/sqrt(c^2*d*x^2 + d), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\int \frac {x^4\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{\sqrt {d\,c^2\,x^2+d}} \,d x \] Input:

int((x^4*(a + b*asinh(c*x))^2)/(d + c^2*d*x^2)^(1/2),x)
 

Output:

int((x^4*(a + b*asinh(c*x))^2)/(d + c^2*d*x^2)^(1/2), x)
 

Reduce [F]

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

int(x^4*(a+b*asinh(c*x))^2/(c^2*d*x^2+d)^(1/2),x)
 

Output:

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