\(\int \frac {(d+c^2 d x^2)^{3/2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx\) [271]

Optimal result

Integrand size = 28, antiderivative size = 398 \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\frac {1}{4} b^2 c^2 d x \sqrt {d+c^2 d x^2}-\frac {5 b^2 c d \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{4 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 d x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 \sqrt {1+c^2 x^2}}+b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3}{2} c^2 d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {c d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}+\frac {c d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{2 b \sqrt {1+c^2 x^2}}+\frac {2 b c d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {b^2 c d \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}} \]

[Out]

-(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))^2/x+1/4*b^2*c^2*d*x*(c^2*d*x^2+d)^(1/2)+3/2*c^2*d*x*(a+b*arcsinh(c*x))
^2*(c^2*d*x^2+d)^(1/2)-5/4*b^2*c*d*arcsinh(c*x)*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-3/2*b*c^3*d*x^2*(a+b*arc
sinh(c*x))*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)+c*d*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2
)+1/2*c*d*(a+b*arcsinh(c*x))^3*(c^2*d*x^2+d)^(1/2)/b/(c^2*x^2+1)^(1/2)+2*b*c*d*(a+b*arcsinh(c*x))*ln(1-1/(c*x+
(c^2*x^2+1)^(1/2))^2)*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-b^2*c*d*polylog(2,1/(c*x+(c^2*x^2+1)^(1/2))^2)*(c^
2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)+b*c*d*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)*(c^2*d*x^2+d)^(1/2)

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {5807, 5785, 5783, 5776, 327, 221, 5801, 5775, 3797, 2221, 2317, 2438, 201} \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\frac {3}{2} c^2 d x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2+\frac {c d \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{2 b \sqrt {c^2 x^2+1}}+\frac {c d \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}+b c d \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}+\frac {2 b c d \sqrt {c^2 d x^2+d} \log \left (1-e^{-2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {3 b c^3 d x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{2 \sqrt {c^2 x^2+1}}-\frac {b^2 c d \sqrt {c^2 d x^2+d} \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )}{\sqrt {c^2 x^2+1}}-\frac {5 b^2 c d \text {arcsinh}(c x) \sqrt {c^2 d x^2+d}}{4 \sqrt {c^2 x^2+1}}+\frac {1}{4} b^2 c^2 d x \sqrt {c^2 d x^2+d} \]

[In]

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

[Out]

(b^2*c^2*d*x*Sqrt[d + c^2*d*x^2])/4 - (5*b^2*c*d*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(4*Sqrt[1 + c^2*x^2]) - (3*
b*c^3*d*x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(2*Sqrt[1 + c^2*x^2]) + b*c*d*Sqrt[1 + c^2*x^2]*Sqrt[d +
c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (3*c^2*d*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/2 + (c*d*Sqrt[d + c^
2*d*x^2]*(a + b*ArcSinh[c*x])^2)/Sqrt[1 + c^2*x^2] - ((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/x + (c*d*S
qrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^3)/(2*b*Sqrt[1 + c^2*x^2]) + (2*b*c*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSi
nh[c*x])*Log[1 - E^(-2*ArcSinh[c*x])])/Sqrt[1 + c^2*x^2] - (b^2*c*d*Sqrt[d + c^2*d*x^2]*PolyLog[2, E^(-2*ArcSi
nh[c*x])])/Sqrt[1 + c^2*x^2]

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 221

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 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 2221

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]
- Dist[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 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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 2438

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 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((
c + d*x)^(m + 1)/(d*(m + 1))), x] + Dist[2*I, Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 5775

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Dist[1/b, Subst[Int[x^n*Coth[-a/b + x/b], x],
x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 5776

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcS
inh[c*x])^n/(d*(m + 1))), x] - Dist[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 5783

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*S
imp[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 5785

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

Rule 5801

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

Rule 5807

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*((a + b*ArcSinh[c*x])^n/(f*(m + 1))), x] + (-Dist[2*e*(p/(f^2*(m + 1))), Int[(f*x
)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x], x] - Dist[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}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}+\left (3 c^2 d\right ) \int \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx+\frac {\left (2 b c d \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{x} \, dx}{\sqrt {1+c^2 x^2}} \\ & = b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3}{2} c^2 d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2-\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}+\frac {\left (2 b c d \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{x} \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (3 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}-\frac {\left (b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \sqrt {1+c^2 x^2} \, dx}{\sqrt {1+c^2 x^2}}-\frac {\left (3 b c^3 d \sqrt {d+c^2 d x^2}\right ) \int x (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}} \\ & = -\frac {1}{2} b^2 c^2 d x \sqrt {d+c^2 d x^2}-\frac {3 b c^3 d x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 \sqrt {1+c^2 x^2}}+b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3}{2} c^2 d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2-\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}+\frac {c d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{2 b \sqrt {1+c^2 x^2}}-\frac {\left (2 c d \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int x \coth \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}+\frac {\left (3 b^2 c^4 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}} \\ & = \frac {1}{4} b^2 c^2 d x \sqrt {d+c^2 d x^2}-\frac {b^2 c d \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{2 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 d x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 \sqrt {1+c^2 x^2}}+b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3}{2} c^2 d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {c d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}+\frac {c d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{2 b \sqrt {1+c^2 x^2}}+\frac {\left (4 c d \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )} x}{1-e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (3 b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{4 \sqrt {1+c^2 x^2}} \\ & = \frac {1}{4} b^2 c^2 d x \sqrt {d+c^2 d x^2}-\frac {5 b^2 c d \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{4 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 d x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 \sqrt {1+c^2 x^2}}+b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3}{2} c^2 d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {c d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}+\frac {c d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{2 b \sqrt {1+c^2 x^2}}+\frac {2 b c d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (2 b c d \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1-e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{\sqrt {1+c^2 x^2}} \\ & = \frac {1}{4} b^2 c^2 d x \sqrt {d+c^2 d x^2}-\frac {5 b^2 c d \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{4 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 d x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 \sqrt {1+c^2 x^2}}+b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3}{2} c^2 d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {c d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}+\frac {c d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{2 b \sqrt {1+c^2 x^2}}+\frac {2 b c d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {\left (b^2 c d \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}\right )}{\sqrt {1+c^2 x^2}} \\ & = \frac {1}{4} b^2 c^2 d x \sqrt {d+c^2 d x^2}-\frac {5 b^2 c d \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{4 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 d x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 \sqrt {1+c^2 x^2}}+b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3}{2} c^2 d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {c d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}+\frac {c d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{2 b \sqrt {1+c^2 x^2}}+\frac {2 b c d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {b^2 c d \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}\right )}{\sqrt {1+c^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.50 (sec) , antiderivative size = 369, normalized size of antiderivative = 0.93 \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\frac {12 a^2 d \left (-2+c^2 x^2\right ) \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+24 a b d \sqrt {d+c^2 d x^2} \left (-2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+c x \text {arcsinh}(c x)^2+2 c x \log (c x)\right )+36 a^2 c d^{3/2} x \sqrt {1+c^2 x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )-8 b^2 d \sqrt {d+c^2 d x^2} \left (\text {arcsinh}(c x) \left (3 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)-c x \text {arcsinh}(c x) (3+\text {arcsinh}(c x))-6 c x \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )\right )+3 c x \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )\right )+b^2 c d x \sqrt {d+c^2 d x^2} \left (4 \text {arcsinh}(c x)^3-6 \text {arcsinh}(c x) \cosh (2 \text {arcsinh}(c x))+\left (3+6 \text {arcsinh}(c x)^2\right ) \sinh (2 \text {arcsinh}(c x))\right )-6 a b c d x \sqrt {d+c^2 d x^2} (\cosh (2 \text {arcsinh}(c x))-2 \text {arcsinh}(c x) (\text {arcsinh}(c x)+\sinh (2 \text {arcsinh}(c x))))}{24 x \sqrt {1+c^2 x^2}} \]

[In]

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

[Out]

(12*a^2*d*(-2 + c^2*x^2)*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 24*a*b*d*Sqrt[d + c^2*d*x^2]*(-2*Sqrt[1 + c^2
*x^2]*ArcSinh[c*x] + c*x*ArcSinh[c*x]^2 + 2*c*x*Log[c*x]) + 36*a^2*c*d^(3/2)*x*Sqrt[1 + c^2*x^2]*Log[c*d*x + S
qrt[d]*Sqrt[d + c^2*d*x^2]] - 8*b^2*d*Sqrt[d + c^2*d*x^2]*(ArcSinh[c*x]*(3*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - c*
x*ArcSinh[c*x]*(3 + ArcSinh[c*x]) - 6*c*x*Log[1 - E^(-2*ArcSinh[c*x])]) + 3*c*x*PolyLog[2, E^(-2*ArcSinh[c*x])
]) + b^2*c*d*x*Sqrt[d + c^2*d*x^2]*(4*ArcSinh[c*x]^3 - 6*ArcSinh[c*x]*Cosh[2*ArcSinh[c*x]] + (3 + 6*ArcSinh[c*
x]^2)*Sinh[2*ArcSinh[c*x]]) - 6*a*b*c*d*x*Sqrt[d + c^2*d*x^2]*(Cosh[2*ArcSinh[c*x]] - 2*ArcSinh[c*x]*(ArcSinh[
c*x] + Sinh[2*ArcSinh[c*x]])))/(24*x*Sqrt[1 + c^2*x^2])

Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.19

method	result	size
default	\(-\frac {a^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{d x}+a^{2} c^{2} x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}+\frac {3 \sqrt {c^{2} d \,x^{2}+d}\, a^{2} c^{2} d x}{2}+\frac {3 a^{2} c^{2} d^{2} \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 \sqrt {c^{2} d}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 \operatorname {arcsinh}\left (c x \right )^{2} \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}-2 \,\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}+2 \operatorname {arcsinh}\left (c x \right )^{3} x c +c^{2} x^{2} \sqrt {c^{2} x^{2}+1}-4 \operatorname {arcsinh}\left (c x \right )^{2} x c +8 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) x c +8 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) x c -4 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right )^{2}-\operatorname {arcsinh}\left (c x \right ) c x +8 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) x c +8 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) x c \right ) d}{4 \sqrt {c^{2} x^{2}+1}\, x}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}-2 c^{3} x^{3}+6 \operatorname {arcsinh}\left (c x \right )^{2} x c -8 \,\operatorname {arcsinh}\left (c x \right ) c x +8 \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) x c -8 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-c x \right ) d}{4 \sqrt {c^{2} x^{2}+1}\, x}\)	\(473\)
parts	\(-\frac {a^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{d x}+a^{2} c^{2} x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}+\frac {3 \sqrt {c^{2} d \,x^{2}+d}\, a^{2} c^{2} d x}{2}+\frac {3 a^{2} c^{2} d^{2} \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 \sqrt {c^{2} d}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 \operatorname {arcsinh}\left (c x \right )^{2} \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}-2 \,\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}+2 \operatorname {arcsinh}\left (c x \right )^{3} x c +c^{2} x^{2} \sqrt {c^{2} x^{2}+1}-4 \operatorname {arcsinh}\left (c x \right )^{2} x c +8 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) x c +8 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) x c -4 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right )^{2}-\operatorname {arcsinh}\left (c x \right ) c x +8 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) x c +8 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) x c \right ) d}{4 \sqrt {c^{2} x^{2}+1}\, x}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}-2 c^{3} x^{3}+6 \operatorname {arcsinh}\left (c x \right )^{2} x c -8 \,\operatorname {arcsinh}\left (c x \right ) c x +8 \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) x c -8 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-c x \right ) d}{4 \sqrt {c^{2} x^{2}+1}\, x}\)	\(473\)

[In]

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

[Out]

-a^2/d/x*(c^2*d*x^2+d)^(5/2)+a^2*c^2*x*(c^2*d*x^2+d)^(3/2)+3/2*(c^2*d*x^2+d)^(1/2)*a^2*c^2*d*x+3/2*a^2*c^2*d^2
*ln(c^2*d*x/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)+1/4*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/x
*(2*arcsinh(c*x)^2*(c^2*x^2+1)^(1/2)*x^2*c^2-2*arcsinh(c*x)*c^3*x^3+2*arcsinh(c*x)^3*x*c+c^2*x^2*(c^2*x^2+1)^(
1/2)-4*arcsinh(c*x)^2*x*c+8*arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2))*x*c+8*arcsinh(c*x)*ln(1-c*x-(c^2*x^2+1)^(
1/2))*x*c-4*(c^2*x^2+1)^(1/2)*arcsinh(c*x)^2-arcsinh(c*x)*c*x+8*polylog(2,-c*x-(c^2*x^2+1)^(1/2))*x*c+8*polylo
g(2,c*x+(c^2*x^2+1)^(1/2))*x*c)*d+1/4*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/x*(4*arcsinh(c*x)*(c^2*x^2+1
)^(1/2)*x^2*c^2-2*c^3*x^3+6*arcsinh(c*x)^2*x*c-8*arcsinh(c*x)*c*x+8*ln((c*x+(c^2*x^2+1)^(1/2))^2-1)*x*c-8*arcs
inh(c*x)*(c^2*x^2+1)^(1/2)-c*x)*d

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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