\(\int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{(d+c^2 d x^2)^{5/2}} \, dx\) [320]

Optimal result

Integrand size = 28, antiderivative size = 312 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {b^2 x}{3 c^2 d^2 \sqrt {d+c^2 d x^2}}-\frac {b^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{3 c^3 d^2 \sqrt {d+c^2 d x^2}}+\frac {b x^2 (a+b \text {arcsinh}(c x))}{3 c d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {x^3 (a+b \text {arcsinh}(c x))^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 c^3 d^2 \sqrt {d+c^2 d x^2}}-\frac {2 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{3 c^3 d^2 \sqrt {d+c^2 d x^2}}-\frac {b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{3 c^3 d^2 \sqrt {d+c^2 d x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.30 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.90 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {b^2 c x+a^2 c^3 x^3+b^2 c^3 x^3-a b \sqrt {1+c^2 x^2}-b^2 \left (-c^3 x^3+\sqrt {1+c^2 x^2}+c^2 x^2 \sqrt {1+c^2 x^2}\right ) \text {arcsinh}(c x)^2-b \text {arcsinh}(c x) \left (-2 a c^3 x^3+b \sqrt {1+c^2 x^2}+2 b \left (1+c^2 x^2\right )^{3/2} \log \left (1+e^{-2 \text {arcsinh}(c x)}\right )\right )-a b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )-a b c^2 x^2 \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )+b^2 \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-e^{-2 \text {arcsinh}(c x)}\right )}{3 c^3 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}} \] Input:

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

Output:

(b^2*c*x + a^2*c^3*x^3 + b^2*c^3*x^3 - a*b*Sqrt[1 + c^2*x^2] - b^2*(-(c^3* 
x^3) + Sqrt[1 + c^2*x^2] + c^2*x^2*Sqrt[1 + c^2*x^2])*ArcSinh[c*x]^2 - b*A 
rcSinh[c*x]*(-2*a*c^3*x^3 + b*Sqrt[1 + c^2*x^2] + 2*b*(1 + c^2*x^2)^(3/2)* 
Log[1 + E^(-2*ArcSinh[c*x])]) - a*b*Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^2] - a 
*b*c^2*x^2*Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^2] + b^2*(1 + c^2*x^2)^(3/2)*Po 
lyLog[2, -E^(-2*ArcSinh[c*x])])/(3*c^3*d^2*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^ 
2])
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.18 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.67, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {6215, 6225, 252, 222, 6212, 3042, 26, 4201, 2620, 2715, 2838}

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.

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

\(\Big \downarrow \) 6215

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

\(\Big \downarrow \) 6225

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

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 222

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

\(\Big \downarrow \) 6212

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

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 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 252

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x 
)^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* 
(p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c 
}, x] && LtQ[p, -1] && GtQ[m, 1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi 
alQ[a, b, c, 2, m, p, 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 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 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2206\) vs. \(2(292)=584\).

Time = 1.32 (sec) , antiderivative size = 2207, normalized size of antiderivative = 7.07

method	result	size
default	\(\text {Expression too large to display}\)	\(2207\)
parts	\(\text {Expression too large to display}\)	\(2207\)

Input:

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

Output:

2/3*b^2/(c^2*x^2+1)^(1/2)*(d*(c^2*x^2+1))^(1/2)/d^3/c^3*arcsinh(x*c)^2-1/3 
*b^2/(c^2*x^2+1)^(1/2)*(d*(c^2*x^2+1))^(1/2)/d^3/c^3*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^8*x^8+9*c^6*x^6+10*c^4 
*x^4+5*c^2*x^2+1)/d^3*c^4*x^7+b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^8*x^8+9*c^6*x 
^6+10*c^4*x^4+5*c^2*x^2+1)/d^3*c^2*x^5-1/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^ 
8*x^8+9*c^6*x^6+10*c^4*x^4+5*c^2*x^2+1)/d^3/c^3*(c^2*x^2+1)^(1/2)+2/3*b^2* 
(d*(c^2*x^2+1))^(1/2)/(3*c^8*x^8+9*c^6*x^6+10*c^4*x^4+5*c^2*x^2+1)/d^3*(c^ 
2*x^2+1)*x^3+a^2*(-1/2*x/c^2/d/(c^2*d*x^2+d)^(3/2)+1/2/c^2*(1/3*x/d/(c^2*d 
*x^2+d)^(3/2)+2/3/d^2*x/(c^2*d*x^2+d)^(1/2)))-2*b^2*(d*(c^2*x^2+1))^(1/2)/ 
(3*c^8*x^8+9*c^6*x^6+10*c^4*x^4+5*c^2*x^2+1)/d^3*c*(c^2*x^2+1)^(1/2)*arcsi 
nh(x*c)^2*x^4-b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^8*x^8+9*c^6*x^6+10*c^4*x^4+5* 
c^2*x^2+1)/d^3*c*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*x^4-4/3*b^2*(d*(c^2*x^2+1) 
)^(1/2)/(3*c^8*x^8+9*c^6*x^6+10*c^4*x^4+5*c^2*x^2+1)/d^3/c*(c^2*x^2+1)^(1/ 
2)*arcsinh(x*c)^2*x^2-b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^8*x^8+9*c^6*x^6+10*c^ 
4*x^4+5*c^2*x^2+1)/d^3/c*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*x^2-b^2*(d*(c^2*x^ 
2+1))^(1/2)/(3*c^8*x^8+9*c^6*x^6+10*c^4*x^4+5*c^2*x^2+1)/d^3*c^3*(c^2*x^2+ 
1)^(1/2)*arcsinh(x*c)^2*x^6-1/3*a*b*(c^2*x^2+1)^(1/2)*(d*(c^2*x^2+1))^(1/2 
)*(2*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)*x^4*c^4-2*arcsinh(x*c)*c^4*x^4-2*arcs 
inh(x*c)*(c^2*x^2+1)^(1/2)*x^3*c^3+4*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)*x^2*c 
^2-4*arcsinh(x*c)*c^2*x^2+c^2*x^2+2*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)-2*a...
 

Fricas [F]

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

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

Output:

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

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {\sqrt {c^{2} x^{2}+1}\, a^{2} c^{3} x^{3}+6 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{2}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) a b \,c^{7} x^{4}+12 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{2}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) a b \,c^{5} x^{2}+6 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{2}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) a b \,c^{3}+3 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2} x^{2}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b^{2} c^{7} x^{4}+6 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2} x^{2}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b^{2} c^{5} x^{2}+3 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2} x^{2}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b^{2} c^{3}+a^{2} c^{4} x^{4}+2 a^{2} c^{2} x^{2}+a^{2}}{3 \sqrt {d}\, c^{3} d^{2} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )} \] Input:

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

Output:

(sqrt(c**2*x**2 + 1)*a**2*c**3*x**3 + 6*int((asinh(c*x)*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)*a*b*c**7*x**4 + 12*int((asinh(c*x)*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)*a*b*c**5*x* 
*2 + 6*int((asinh(c*x)*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)*a*b*c**3 + 3*int((asinh(c*x) 
**2*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**7*x**4 + 6*int((asinh(c*x)**2*x**2)/(sq 
rt(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**2 + 3*int((asinh(c*x)**2*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 + a**2*c**4*x**4 + 2*a**2*c**2*x**2 + a**2)/(3*sqrt(d)*c**3*d** 
2*(c**4*x**4 + 2*c**2*x**2 + 1))