\(\int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{(d+c^2 d x^2)^{3/2}} \, dx\) [310]

Optimal result

Integrand size = 28, antiderivative size = 233 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=-\frac {x (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {d+c^2 d x^2}}-\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{c^3 d \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{3 b c^3 d \sqrt {d+c^2 d x^2}}+\frac {2 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c^3 d \sqrt {d+c^2 d x^2}}+\frac {b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{c^3 d \sqrt {d+c^2 d x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.23 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.92 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {-3 a^2 c d x-3 a b d \left (2 c x \text {arcsinh}(c x)-\sqrt {1+c^2 x^2} \left (\text {arcsinh}(c x)^2+\log \left (1+c^2 x^2\right )\right )\right )+3 a^2 \sqrt {d} \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+b^2 d \left (\text {arcsinh}(c x) \left (-3 c x \text {arcsinh}(c x)+\sqrt {1+c^2 x^2} \left (\text {arcsinh}(c x) (3+\text {arcsinh}(c x))+6 \log \left (1+e^{-2 \text {arcsinh}(c x)}\right )\right )\right )-3 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{-2 \text {arcsinh}(c x)}\right )\right )}{3 c^3 d^2 \sqrt {d+c^2 d x^2}} \] Input:

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

Output:

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.00 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.80, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {6225, 6198, 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.

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

\(\Big \downarrow \) 6225

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

\(\Big \downarrow \) 6198

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

\(\Big \downarrow \) 6212

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

-((x*(a + b*ArcSinh[c*x])^2)/(c^2*d*Sqrt[d + c^2*d*x^2])) + (Sqrt[1 + c^2* 
x^2]*(a + b*ArcSinh[c*x])^3)/(3*b*c^3*d*Sqrt[d + c^2*d*x^2]) - ((2*I)*b*Sq 
rt[1 + c^2*x^2]*(((-1/2*I)*(a + b*ArcSinh[c*x])^2)/b + (2*I)*(((a + b*ArcS 
inh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/2 + (b*PolyLog[2, -E^(2*ArcSinh[c*x 
])])/4)))/(c^3*d*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 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 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 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 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. \(477\) vs. \(2(231)=462\).

Time = 1.15 (sec) , antiderivative size = 478, normalized size of antiderivative = 2.05

method	result	size
default	\(-\frac {a^{2} x}{c^{2} d \sqrt {c^{2} d \,x^{2}+d}}+\frac {a^{2} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{c^{2} d \sqrt {c^{2} d}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{3}}{3 \sqrt {c^{2} x^{2}+1}\, d^{2} c^{3}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2} x}{d^{2} c^{2} \left (c^{2} x^{2}+1\right )}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2}}{d^{2} c^{3} \sqrt {c^{2} x^{2}+1}}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, d^{2} c^{3}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, d^{2} c^{3}}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2}}{\sqrt {c^{2} x^{2}+1}\, d^{2} c^{3}}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )}{\sqrt {c^{2} x^{2}+1}\, d^{2} c^{3}}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) x}{d^{2} c^{2} \left (c^{2} x^{2}+1\right )}+\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, d^{2} c^{3}}\)	\(478\)
parts	\(-\frac {a^{2} x}{c^{2} d \sqrt {c^{2} d \,x^{2}+d}}+\frac {a^{2} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{c^{2} d \sqrt {c^{2} d}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{3}}{3 \sqrt {c^{2} x^{2}+1}\, d^{2} c^{3}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2} x}{d^{2} c^{2} \left (c^{2} x^{2}+1\right )}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2}}{d^{2} c^{3} \sqrt {c^{2} x^{2}+1}}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, d^{2} c^{3}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, d^{2} c^{3}}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2}}{\sqrt {c^{2} x^{2}+1}\, d^{2} c^{3}}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )}{\sqrt {c^{2} x^{2}+1}\, d^{2} c^{3}}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) x}{d^{2} c^{2} \left (c^{2} x^{2}+1\right )}+\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, d^{2} c^{3}}\)	\(478\)

Input:

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

Output:

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

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-a^2*(x/(sqrt(c^2*d*x^2 + d)*c^2*d) - arcsinh(c*x)/(c^3*d^(3/2))) + integr 
ate(b^2*x^2*log(c*x + sqrt(c^2*x^2 + 1))^2/(c^2*d*x^2 + d)^(3/2) + 2*a*b*x 
^2*log(c*x + sqrt(c^2*x^2 + 1))/(c^2*d*x^2 + d)^(3/2), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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