\(\int \frac {(a+b \text {arcsinh}(c x))^2}{(d+c^2 d x^2)^{5/2}} \, dx\) [322]

Optimal result

Integrand size = 25, antiderivative size = 292 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=-\frac {b^2 x}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {b (a+b \text {arcsinh}(c x))}{3 c d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {2 x (a+b \text {arcsinh}(c x))^2}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 c d^2 \sqrt {d+c^2 d x^2}}-\frac {4 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{3 c d^2 \sqrt {d+c^2 d x^2}}-\frac {2 b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{3 c d^2 \sqrt {d+c^2 d x^2}} \] Output:

-1/3*b^2*x/d^2/(c^2*d*x^2+d)^(1/2)+1/3*b*(a+b*arcsinh(c*x))/c/d^2/(c^2*x^2 
+1)^(1/2)/(c^2*d*x^2+d)^(1/2)+1/3*x*(a+b*arcsinh(c*x))^2/d/(c^2*d*x^2+d)^( 
3/2)+2/3*x*(a+b*arcsinh(c*x))^2/d^2/(c^2*d*x^2+d)^(1/2)+2/3*(c^2*x^2+1)^(1 
/2)*(a+b*arcsinh(c*x))^2/c/d^2/(c^2*d*x^2+d)^(1/2)-4/3*b*(c^2*x^2+1)^(1/2) 
*(a+b*arcsinh(c*x))*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)/c/d^2/(c^2*d*x^2+d)^(1 
/2)-2/3*b^2*(c^2*x^2+1)^(1/2)*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))^2)/c/d^2/ 
(c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.84 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {a^2 c x \left (3+2 c^2 x^2\right )+a b \left (\left (6 c x+4 c^3 x^3\right ) \text {arcsinh}(c x)+\sqrt {1+c^2 x^2} \left (1-2 \left (1+c^2 x^2\right ) \log \left (1+c^2 x^2\right )\right )\right )-b^2 \left (c x+c^3 x^3-\sqrt {1+c^2 x^2} \text {arcsinh}(c x)-c x \text {arcsinh}(c x)^2-2 c x \left (1+c^2 x^2\right ) \text {arcsinh}(c x)^2+2 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x) \left (\text {arcsinh}(c x)+2 \log \left (1+e^{-2 \text {arcsinh}(c x)}\right )\right )-2 \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-e^{-2 \text {arcsinh}(c x)}\right )\right )}{3 d^2 \left (c+c^3 x^2\right ) \sqrt {d+c^2 d x^2}} \] Input:

Integrate[(a + b*ArcSinh[c*x])^2/(d + c^2*d*x^2)^(5/2),x]
 

Output:

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.84 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.88, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {6203, 6202, 6212, 3042, 26, 4201, 2620, 2715, 2838, 6213, 208}

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.

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

\(\Big \downarrow \) 6203

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

\(\Big \downarrow \) 6202

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

\(\Big \downarrow \) 6212

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 6213

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

\(\Big \downarrow \) 208

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

Input:

Int[(a + b*ArcSinh[c*x])^2/(d + c^2*d*x^2)^(5/2),x]
 

Output:

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

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 208

Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), 
x] /; FreeQ[{a, b}, x]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4201

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x 
_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I   Int[ 
(c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] 
 /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
 

rule 6202

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

rule 6203

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x 
_Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*d*(p + 
 1))), x] + (Simp[(2*p + 3)/(2*d*(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] && LtQ[p, 
 -1] && NeQ[p, -3/2]
 

rule 6212

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), 
 x_Symbol] :> Simp[1/e   Subst[Int[(a + b*x)^n*Tanh[x], x], x, ArcSinh[c*x] 
], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2130\) vs. \(2(274)=548\).

Time = 1.19 (sec) , antiderivative size = 2131, normalized size of antiderivative = 7.30

method	result	size
default	\(\text {Expression too large to display}\)	\(2131\)
parts	\(\text {Expression too large to display}\)	\(2131\)

Input:

int((a+b*arcsinh(x*c))^2/(c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-13/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^6*x^6+10*c^4*x^4+11*c^2*x^2+4)/d^3*c^ 
2*x^3+4*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^6*x^6+10*c^4*x^4+11*c^2*x^2+4)/d^3* 
arcsinh(x*c)^2*x-2*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^6*x^6+10*c^4*x^4+11*c^2* 
x^2+4)/d^3*arcsinh(x*c)*x-2/3*b^2/(c^2*x^2+1)^(1/2)*(d*(c^2*x^2+1))^(1/2)/ 
d^3/c*polylog(2,-(x*c+(c^2*x^2+1)^(1/2))^2)-2/3*b^2*(d*(c^2*x^2+1))^(1/2)/ 
(3*c^6*x^6+10*c^4*x^4+11*c^2*x^2+4)/d^3*c^6*x^7-3*b^2*(d*(c^2*x^2+1))^(1/2 
)/(3*c^6*x^6+10*c^4*x^4+11*c^2*x^2+4)/d^3*c^4*x^5+1/3*a*b*(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)+3*x*c+2*(c^2*x^2+1)^(1/2))*(- 
8*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)*x^6*c^6+8*(c^2*x^2+1)^(1/2)*ln(1+(x*c+(c 
^2*x^2+1)^(1/2))^2)*x^5*c^5-24*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)*x^4*c^4+20* 
(c^2*x^2+1)^(1/2)*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)*x^3*c^3+2*c^4*x^4-2*(c^2 
*x^2+1)^(1/2)*c^3*x^3+6*arcsinh(x*c)*c^2*x^2-24*ln(1+(x*c+(c^2*x^2+1)^(1/2 
))^2)*x^2*c^2+12*(c^2*x^2+1)^(1/2)*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)*x*c+4*c 
^2*x^2-3*(c^2*x^2+1)^(1/2)*x*c+8*arcsinh(x*c)-8*ln(1+(x*c+(c^2*x^2+1)^(1/2 
))^2)+2)/(3*c^6*x^6+10*c^4*x^4+11*c^2*x^2+4)/d^3/c+4/3*b^2/(c^2*x^2+1)^(1/ 
2)*(d*(c^2*x^2+1))^(1/2)/d^3/c*arcsinh(x*c)^2+4/3*b^2*(d*(c^2*x^2+1))^(1/2 
)/(3*c^6*x^6+10*c^4*x^4+11*c^2*x^2+4)/d^3/c*(c^2*x^2+1)^(1/2)+2/3*b^2*(d*( 
c^2*x^2+1))^(1/2)/(3*c^6*x^6+10*c^4*x^4+11*c^2*x^2+4)/d^3*(c^2*x^2+1)*x+10 
/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^6*x^6+10*c^4*x^4+11*c^2*x^2+4)/d^3*c^2*( 
c^2*x^2+1)*arcsinh(x*c)*x^3-2*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^6*x^6+10*c...
 

Fricas [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(5/2),x, algorithm="fricas")
 

Output:

integral(sqrt(c^2*d*x^2 + d)*(b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^ 
2)/(c^6*d^3*x^6 + 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 + d^3), x)
 

Sympy [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((a+b*asinh(c*x))**2/(c**2*d*x**2+d)**(5/2),x)
 

Output:

Integral((a + b*asinh(c*x))**2/(d*(c**2*x**2 + 1))**(5/2), x)
 

Maxima [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(5/2),x, algorithm="maxima")
 

Output:

1/3*a*b*c*(1/(c^4*d^(5/2)*x^2 + c^2*d^(5/2)) - 2*log(c^2*x^2 + 1)/(c^2*d^( 
5/2))) + 2/3*a*b*(2*x/(sqrt(c^2*d*x^2 + d)*d^2) + x/((c^2*d*x^2 + d)^(3/2) 
*d))*arcsinh(c*x) + 1/3*a^2*(2*x/(sqrt(c^2*d*x^2 + d)*d^2) + x/((c^2*d*x^2 
 + d)^(3/2)*d)) + b^2*integrate(log(c*x + sqrt(c^2*x^2 + 1))^2/(c^2*d*x^2 
+ d)^(5/2), x)
 

Giac [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(5/2),x, algorithm="giac")
 

Output:

integrate((b*arcsinh(c*x) + a)^2/(c^2*d*x^2 + d)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d\,c^2\,x^2+d\right )}^{5/2}} \,d x \] Input:

int((a + b*asinh(c*x))^2/(d + c^2*d*x^2)^(5/2),x)
 

Output:

int((a + b*asinh(c*x))^2/(d + c^2*d*x^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {2 \sqrt {c^{2} x^{2}+1}\, a^{2} c^{3} x^{3}+3 \sqrt {c^{2} x^{2}+1}\, a^{2} c x +6 \left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) a b \,c^{5} x^{4}+12 \left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) a b \,c^{3} x^{2}+6 \left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) a b c +3 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b^{2} c^{5} x^{4}+6 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b^{2} c^{3} x^{2}+3 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b^{2} c -2 a^{2} c^{4} x^{4}-4 a^{2} c^{2} x^{2}-2 a^{2}}{3 \sqrt {d}\, c \,d^{2} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )} \] Input:

int((a+b*asinh(c*x))^2/(c^2*d*x^2+d)^(5/2),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 + 6 
*int(asinh(c*x)/(sqrt(c**2*x**2 + 1)*c**4*x**4 + 2*sqrt(c**2*x**2 + 1)*c** 
2*x**2 + sqrt(c**2*x**2 + 1)),x)*a*b*c**5*x**4 + 12*int(asinh(c*x)/(sqrt(c 
**2*x**2 + 1)*c**4*x**4 + 2*sqrt(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 
 + 1)),x)*a*b*c**3*x**2 + 6*int(asinh(c*x)/(sqrt(c**2*x**2 + 1)*c**4*x**4 
+ 2*sqrt(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*a*b*c + 3*int( 
asinh(c*x)**2/(sqrt(c**2*x**2 + 1)*c**4*x**4 + 2*sqrt(c**2*x**2 + 1)*c**2* 
x**2 + sqrt(c**2*x**2 + 1)),x)*b**2*c**5*x**4 + 6*int(asinh(c*x)**2/(sqrt( 
c**2*x**2 + 1)*c**4*x**4 + 2*sqrt(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x** 
2 + 1)),x)*b**2*c**3*x**2 + 3*int(asinh(c*x)**2/(sqrt(c**2*x**2 + 1)*c**4* 
x**4 + 2*sqrt(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*b**2*c - 
2*a**2*c**4*x**4 - 4*a**2*c**2*x**2 - 2*a**2)/(3*sqrt(d)*c*d**2*(c**4*x**4 
 + 2*c**2*x**2 + 1))