Integrand size = 28, antiderivative size = 233 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=-\frac {x (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {d+c^2 d x^2}}-\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{c^3 d \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{3 b c^3 d \sqrt {d+c^2 d x^2}}+\frac {2 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c^3 d \sqrt {d+c^2 d x^2}}+\frac {b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{c^3 d \sqrt {d+c^2 d x^2}} \]
-x*(a+b*arcsinh(c*x))^2/c^2/d/(c^2*d*x^2+d)^(1/2)-(a+b*arcsinh(c*x))^2*(c^ 2*x^2+1)^(1/2)/c^3/d/(c^2*d*x^2+d)^(1/2)+1/3*(a+b*arcsinh(c*x))^3*(c^2*x^2 +1)^(1/2)/b/c^3/d/(c^2*d*x^2+d)^(1/2)+2*b*(a+b*arcsinh(c*x))*ln(1+(c*x+(c^ 2*x^2+1)^(1/2))^2)*(c^2*x^2+1)^(1/2)/c^3/d/(c^2*d*x^2+d)^(1/2)+b^2*polylog (2,-(c*x+(c^2*x^2+1)^(1/2))^2)*(c^2*x^2+1)^(1/2)/c^3/d/(c^2*d*x^2+d)^(1/2)
Time = 1.35 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.92 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {-3 a^2 c d x-3 a b d \left (2 c x \text {arcsinh}(c x)-\sqrt {1+c^2 x^2} \left (\text {arcsinh}(c x)^2+\log \left (1+c^2 x^2\right )\right )\right )+3 a^2 \sqrt {d} \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+b^2 d \left (\text {arcsinh}(c x) \left (-3 c x \text {arcsinh}(c x)+\sqrt {1+c^2 x^2} \left (\text {arcsinh}(c x) (3+\text {arcsinh}(c x))+6 \log \left (1+e^{-2 \text {arcsinh}(c x)}\right )\right )\right )-3 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{-2 \text {arcsinh}(c x)}\right )\right )}{3 c^3 d^2 \sqrt {d+c^2 d x^2}} \]
(-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])
Result contains complex when optimal does not.
Time = 0.97 (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 |
\(\displaystyle -\frac {x (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}-\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c^3 d \sqrt {c^2 d x^2+d}}+\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^3}{3 b c^3 d \sqrt {c^2 d x^2+d}}\) |
-((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])
3.4.3.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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]
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]
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]
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]
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]
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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(477\) vs. \(2(231)=462\).
Time = 0.25 (sec) , antiderivative size = 478, normalized size of antiderivative = 2.05
method | result | size |
default | \(-\frac {a^{2} x}{c^{2} d \sqrt {c^{2} d \,x^{2}+d}}+\frac {a^{2} \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{c^{2} d \sqrt {c^{2} d}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{3}}{3 \sqrt {c^{2} x^{2}+1}\, c^{3} d^{2}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2} x}{c^{2} d^{2} \left (c^{2} x^{2}+1\right )}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2}}{c^{3} d^{2} \sqrt {c^{2} x^{2}+1}}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, c^{3} d^{2}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, c^{3} d^{2}}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2}}{\sqrt {c^{2} x^{2}+1}\, c^{3} d^{2}}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{3} d^{2}}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) x}{c^{2} d^{2} \left (c^{2} x^{2}+1\right )}+\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, c^{3} d^{2}}\) | \(478\) |
parts | \(-\frac {a^{2} x}{c^{2} d \sqrt {c^{2} d \,x^{2}+d}}+\frac {a^{2} \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{c^{2} d \sqrt {c^{2} d}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{3}}{3 \sqrt {c^{2} x^{2}+1}\, c^{3} d^{2}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2} x}{c^{2} d^{2} \left (c^{2} x^{2}+1\right )}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2}}{c^{3} d^{2} \sqrt {c^{2} x^{2}+1}}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, c^{3} d^{2}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, c^{3} d^{2}}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2}}{\sqrt {c^{2} x^{2}+1}\, c^{3} d^{2}}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{3} d^{2}}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) x}{c^{2} d^{2} \left (c^{2} x^{2}+1\right )}+\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, c^{3} d^{2}}\) | \(478\) |
-a^2*x/c^2/d/(c^2*d*x^2+d)^(1/2)+a^2/c^2/d*ln(c^2*d*x/(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)/c^3/d^2*arcsinh(c*x)^3-b^2*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)^2/c^2/d^2 /(c^2*x^2+1)*x-b^2*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)^2/c^3/d^2/(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)/c^3/d^2*arcsinh(c*x) *ln(1+(c*x+(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)/c^3/d^2*polylog(2,-(c*x+(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)/c^3/d^2*arcsinh(c*x)^2-2*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2 *x^2+1)^(1/2)/c^3/d^2*arcsinh(c*x)-2*a*b*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x )/c^2/d^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)/c^3/ d^2*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)
\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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)
\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^{2} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
-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)
\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d\,c^2\,x^2+d\right )}^{3/2}} \,d x \]