\(\int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{(d+c^2 d x^2)^{5/2}} \, dx\) [311]

Optimal result

Integrand size = 28, antiderivative size = 398 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=-\frac {b^2 x}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {b^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {b x^2 (a+b \text {arcsinh}(c x))}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {x (a+b \text {arcsinh}(c x))^2}{c^4 d^2 \sqrt {d+c^2 d x^2}}-\frac {4 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{3 b c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {8 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {4 b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{3 c^5 d^2 \sqrt {d+c^2 d x^2}} \]

[Out]

-1/3*x^3*(a+b*arcsinh(c*x))^2/c^2/d/(c^2*d*x^2+d)^(3/2)-1/3*b^2*x/c^4/d^2/(c^2*d*x^2+d)^(1/2)-x*(a+b*arcsinh(c
*x))^2/c^4/d^2/(c^2*d*x^2+d)^(1/2)-1/3*b*x^2*(a+b*arcsinh(c*x))/c^3/d^2/(c^2*x^2+1)^(1/2)/(c^2*d*x^2+d)^(1/2)+
1/3*b^2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)/c^5/d^2/(c^2*d*x^2+d)^(1/2)-4/3*(a+b*arcsinh(c*x))^2*(c^2*x^2+1)^(1/2)/
c^5/d^2/(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^5/d^2/(c^2*d*x^2+d)^(1/2)+8/3*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^5/d^2/(c^2*d*x^2+d)^(1/2)+4/3*b^2*polylog(
2,-(c*x+(c^2*x^2+1)^(1/2))^2)*(c^2*x^2+1)^(1/2)/c^5/d^2/(c^2*d*x^2+d)^(1/2)

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {5810, 5783, 5797, 3799, 2221, 2317, 2438, 294, 221} \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}+\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^3}{3 b c^5 d^2 \sqrt {c^2 d x^2+d}}-\frac {4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{3 c^5 d^2 \sqrt {c^2 d x^2+d}}+\frac {8 b \sqrt {c^2 x^2+1} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))}{3 c^5 d^2 \sqrt {c^2 d x^2+d}}-\frac {x (a+b \text {arcsinh}(c x))^2}{c^4 d^2 \sqrt {c^2 d x^2+d}}-\frac {b x^2 (a+b \text {arcsinh}(c x))}{3 c^3 d^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}}+\frac {4 b^2 \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{3 c^5 d^2 \sqrt {c^2 d x^2+d}}+\frac {b^2 \sqrt {c^2 x^2+1} \text {arcsinh}(c x)}{3 c^5 d^2 \sqrt {c^2 d x^2+d}}-\frac {b^2 x}{3 c^4 d^2 \sqrt {c^2 d x^2+d}} \]

[In]

Int[(x^4*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(5/2),x]

[Out]

-1/3*(b^2*x)/(c^4*d^2*Sqrt[d + c^2*d*x^2]) + (b^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(3*c^5*d^2*Sqrt[d + c^2*d*x^
2]) - (b*x^2*(a + b*ArcSinh[c*x]))/(3*c^3*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]) - (x^3*(a + b*ArcSinh[c*x
])^2)/(3*c^2*d*(d + c^2*d*x^2)^(3/2)) - (x*(a + b*ArcSinh[c*x])^2)/(c^4*d^2*Sqrt[d + c^2*d*x^2]) - (4*Sqrt[1 +
c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(3*c^5*d^2*Sqrt[d + c^2*d*x^2]) + (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^3)
/(3*b*c^5*d^2*Sqrt[d + c^2*d*x^2]) + (8*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/
(3*c^5*d^2*Sqrt[d + c^2*d*x^2]) + (4*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(2*ArcSinh[c*x])])/(3*c^5*d^2*Sqrt[d
+ c^2*d*x^2])

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 294

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*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 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 3799

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] + Dist[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 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 5797

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[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 5810

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

Rubi steps \begin{align*} \text {integral}& = -\frac {x^3 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {\int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx}{c^2 d}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (1+c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt {d+c^2 d x^2}} \\ & = -\frac {b x^2 (a+b \text {arcsinh}(c x))}{3 c^3 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 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {x (a+b \text {arcsinh}(c x))^2}{c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx}{c^4 d^2}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {x (a+b \text {arcsinh}(c x))}{1+c^2 x^2} \, dx}{3 c^3 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {x (a+b \text {arcsinh}(c x))}{1+c^2 x^2} \, dx}{c^3 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x^2}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{3 c^2 d^2 \sqrt {d+c^2 d x^2}} \\ & = -\frac {b^2 x}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}-\frac {b x^2 (a+b \text {arcsinh}(c x))}{3 c^3 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 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {x (a+b \text {arcsinh}(c x))^2}{c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{3 b c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \text {Subst}(\int (a+b x) \tanh (x) \, dx,x,\text {arcsinh}(c x))}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \text {Subst}(\int (a+b x) \tanh (x) \, dx,x,\text {arcsinh}(c x))}{c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{3 c^4 d^2 \sqrt {d+c^2 d x^2}} \\ & = -\frac {b^2 x}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {b^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {b x^2 (a+b \text {arcsinh}(c x))}{3 c^3 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 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {x (a+b \text {arcsinh}(c x))^2}{c^4 d^2 \sqrt {d+c^2 d x^2}}-\frac {4 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{3 b c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (4 b \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\text {arcsinh}(c x)\right )}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (4 b \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\text {arcsinh}(c x)\right )}{c^5 d^2 \sqrt {d+c^2 d x^2}} \\ & = -\frac {b^2 x}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {b^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {b x^2 (a+b \text {arcsinh}(c x))}{3 c^3 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 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {x (a+b \text {arcsinh}(c x))^2}{c^4 d^2 \sqrt {d+c^2 d x^2}}-\frac {4 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{3 b c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {8 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^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (2 b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (2 b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c^5 d^2 \sqrt {d+c^2 d x^2}} \\ & = -\frac {b^2 x}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {b^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {b x^2 (a+b \text {arcsinh}(c x))}{3 c^3 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 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {x (a+b \text {arcsinh}(c x))^2}{c^4 d^2 \sqrt {d+c^2 d x^2}}-\frac {4 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{3 b c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {8 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^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {arcsinh}(c x)}\right )}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {arcsinh}(c x)}\right )}{c^5 d^2 \sqrt {d+c^2 d x^2}} \\ & = -\frac {b^2 x}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {b^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {b x^2 (a+b \text {arcsinh}(c x))}{3 c^3 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 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {x (a+b \text {arcsinh}(c x))^2}{c^4 d^2 \sqrt {d+c^2 d x^2}}-\frac {4 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{3 b c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {8 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^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {4 b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{3 c^5 d^2 \sqrt {d+c^2 d x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.22 (sec) , antiderivative size = 359, normalized size of antiderivative = 0.90 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {-a^2 c \sqrt {d} x \left (3+4 c^2 x^2\right )+a b \sqrt {d} \left (\sqrt {1+c^2 x^2}+2 c x \text {arcsinh}(c x)-8 c x \left (1+c^2 x^2\right ) \text {arcsinh}(c x)+\left (1+c^2 x^2\right )^{3/2} \left (3 \text {arcsinh}(c x)^2+4 \log \left (1+c^2 x^2\right )\right )\right )+3 a^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )-b^2 \sqrt {d} \left (c x+c^3 x^3-\sqrt {1+c^2 x^2} \text {arcsinh}(c x)+3 c x \text {arcsinh}(c x)^2+4 c^3 x^3 \text {arcsinh}(c x)^2-4 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)^2-\left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)^3-8 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x) \log \left (1+e^{-2 \text {arcsinh}(c x)}\right )+4 \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-e^{-2 \text {arcsinh}(c x)}\right )\right )}{3 c^5 d^{5/2} \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}} \]

[In]

Integrate[(x^4*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(5/2),x]

[Out]

(-(a^2*c*Sqrt[d]*x*(3 + 4*c^2*x^2)) + a*b*Sqrt[d]*(Sqrt[1 + c^2*x^2] + 2*c*x*ArcSinh[c*x] - 8*c*x*(1 + c^2*x^2
)*ArcSinh[c*x] + (1 + c^2*x^2)^(3/2)*(3*ArcSinh[c*x]^2 + 4*Log[1 + c^2*x^2])) + 3*a^2*(1 + c^2*x^2)*Sqrt[d + c
^2*d*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - b^2*Sqrt[d]*(c*x + c^3*x^3 - Sqrt[1 + c^2*x^2]*ArcSinh[c*
x] + 3*c*x*ArcSinh[c*x]^2 + 4*c^3*x^3*ArcSinh[c*x]^2 - 4*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]^2 - (1 + c^2*x^2)^(3
/2)*ArcSinh[c*x]^3 - 8*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]*Log[1 + E^(-2*ArcSinh[c*x])] + 4*(1 + c^2*x^2)^(3/2)*P
olyLog[2, -E^(-2*ArcSinh[c*x])]))/(3*c^5*d^(5/2)*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(765\) vs. \(2(370)=740\).

Time = 0.31 (sec) , antiderivative size = 766, normalized size of antiderivative = 1.92

method	result	size
default	\(-\frac {a^{2} x^{3}}{3 c^{2} d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {a^{2} x}{c^{4} d^{2} \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^{4} d^{2} \sqrt {c^{2} d}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} x^{2}+1}\, \left (\operatorname {arcsinh}\left (c x \right )^{3} x^{4} c^{4}-4 \operatorname {arcsinh}\left (c x \right )^{2} x^{4} c^{4}+8 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{4} c^{4}-4 \operatorname {arcsinh}\left (c x \right )^{2} \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+4 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{4} c^{4}+c^{4} x^{4}+2 \operatorname {arcsinh}\left (c x \right )^{3} x^{2} c^{2}-c^{3} x^{3} \sqrt {c^{2} x^{2}+1}-8 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}+16 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}-3 \operatorname {arcsinh}\left (c x \right )^{2} \sqrt {c^{2} x^{2}+1}\, c x +\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+8 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}+2 c^{2} x^{2}+\operatorname {arcsinh}\left (c x \right )^{3}-c x \sqrt {c^{2} x^{2}+1}-4 \operatorname {arcsinh}\left (c x \right )^{2}+8 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {arcsinh}\left (c x \right )+4 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+1\right )}{3 \left (c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) c^{5} d^{3}}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} x^{2}+1}\, \left (3 \operatorname {arcsinh}\left (c x \right )^{2} x^{4} c^{4}-8 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}+8 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{4} c^{4}-8 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+6 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}-16 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+16 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}-6 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+c^{2} x^{2}+3 \operatorname {arcsinh}\left (c x \right )^{2}-8 \,\operatorname {arcsinh}\left (c x \right )+8 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+1\right )}{3 \left (c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) c^{5} d^{3}}\)	\(766\)
parts	\(-\frac {a^{2} x^{3}}{3 c^{2} d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {a^{2} x}{c^{4} d^{2} \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^{4} d^{2} \sqrt {c^{2} d}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} x^{2}+1}\, \left (\operatorname {arcsinh}\left (c x \right )^{3} x^{4} c^{4}-4 \operatorname {arcsinh}\left (c x \right )^{2} x^{4} c^{4}+8 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{4} c^{4}-4 \operatorname {arcsinh}\left (c x \right )^{2} \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+4 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{4} c^{4}+c^{4} x^{4}+2 \operatorname {arcsinh}\left (c x \right )^{3} x^{2} c^{2}-c^{3} x^{3} \sqrt {c^{2} x^{2}+1}-8 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}+16 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}-3 \operatorname {arcsinh}\left (c x \right )^{2} \sqrt {c^{2} x^{2}+1}\, c x +\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+8 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}+2 c^{2} x^{2}+\operatorname {arcsinh}\left (c x \right )^{3}-c x \sqrt {c^{2} x^{2}+1}-4 \operatorname {arcsinh}\left (c x \right )^{2}+8 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {arcsinh}\left (c x \right )+4 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+1\right )}{3 \left (c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) c^{5} d^{3}}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} x^{2}+1}\, \left (3 \operatorname {arcsinh}\left (c x \right )^{2} x^{4} c^{4}-8 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}+8 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{4} c^{4}-8 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+6 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}-16 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+16 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}-6 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+c^{2} x^{2}+3 \operatorname {arcsinh}\left (c x \right )^{2}-8 \,\operatorname {arcsinh}\left (c x \right )+8 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+1\right )}{3 \left (c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) c^{5} d^{3}}\)	\(766\)

[In]

int(x^4*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/3*a^2*x^3/c^2/d/(c^2*d*x^2+d)^(3/2)-a^2/c^4/d^2*x/(c^2*d*x^2+d)^(1/2)+a^2/c^4/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)+1/3*b^2*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+1)^(1/2)/(c^6*x^6+3*c^4*x^4+3*c^2*x^
2+1)/c^5/d^3*(arcsinh(c*x)^3*x^4*c^4-4*arcsinh(c*x)^2*x^4*c^4+8*arcsinh(c*x)*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)*x
^4*c^4-4*arcsinh(c*x)^2*(c^2*x^2+1)^(1/2)*x^3*c^3+4*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))^2)*x^4*c^4+c^4*x^4+2*ar
csinh(c*x)^3*x^2*c^2-c^3*x^3*(c^2*x^2+1)^(1/2)-8*arcsinh(c*x)^2*x^2*c^2+16*arcsinh(c*x)*ln(1+(c*x+(c^2*x^2+1)^
(1/2))^2)*x^2*c^2-3*arcsinh(c*x)^2*(c^2*x^2+1)^(1/2)*c*x+arcsinh(c*x)*c^2*x^2+8*polylog(2,-(c*x+(c^2*x^2+1)^(1
/2))^2)*x^2*c^2+2*c^2*x^2+arcsinh(c*x)^3-c*x*(c^2*x^2+1)^(1/2)-4*arcsinh(c*x)^2+8*arcsinh(c*x)*ln(1+(c*x+(c^2*
x^2+1)^(1/2))^2)+arcsinh(c*x)+4*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))^2)+1)+1/3*a*b*(d*(c^2*x^2+1))^(1/2)*(c^2*x^
2+1)^(1/2)/(c^6*x^6+3*c^4*x^4+3*c^2*x^2+1)/c^5/d^3*(3*arcsinh(c*x)^2*x^4*c^4-8*arcsinh(c*x)*c^4*x^4+8*ln(1+(c*
x+(c^2*x^2+1)^(1/2))^2)*x^4*c^4-8*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*x^3*c^3+6*arcsinh(c*x)^2*x^2*c^2-16*arcsinh(c
*x)*c^2*x^2+16*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)*x^2*c^2-6*arcsinh(c*x)*c*x*(c^2*x^2+1)^(1/2)+c^2*x^2+3*arcsinh(
c*x)^2-8*arcsinh(c*x)+8*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)+1)

Fricas [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{4}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x^4*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

integral((b^2*x^4*arcsinh(c*x)^2 + 2*a*b*x^4*arcsinh(c*x) + a^2*x^4)*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]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^{4} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(x**4*(a+b*asinh(c*x))**2/(c**2*d*x**2+d)**(5/2),x)

[Out]

Integral(x**4*(a + b*asinh(c*x))**2/(d*(c**2*x**2 + 1))**(5/2), x)

Maxima [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{4}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x^4*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{4}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x^4*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arcsinh(c*x) + a)^2*x^4/(c^2*d*x^2 + d)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^4\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d\,c^2\,x^2+d\right )}^{5/2}} \,d x \]

[In]

int((x^4*(a + b*asinh(c*x))^2)/(d + c^2*d*x^2)^(5/2),x)

[Out]

int((x^4*(a + b*asinh(c*x))^2)/(d + c^2*d*x^2)^(5/2), x)