3.281 \(\int \frac{(d+c^2 d x^2)^{5/2} (a+b \sinh ^{-1}(c x))^2}{x^4} \, dx\)

Optimal. Leaf size=561 \[ -\frac{7 b^2 c^3 d^2 \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )}{3 \sqrt{c^2 x^2+1}}-\frac{5 b c^5 d^2 x^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{c^2 x^2+1}}+\frac{5}{2} c^4 d^2 x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5 c^3 d^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b \sqrt{c^2 x^2+1}}+\frac{7 c^3 d^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt{c^2 x^2+1}}+\frac{7}{3} b c^3 d^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c d^2 \left (c^2 x^2+1\right )^{3/2} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac{14 b c^3 d^2 \sqrt{c^2 d x^2+d} \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{c^2 x^2+1}}-\frac{5 c^2 d \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac{\left (c^2 d x^2+d\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac{7}{12} b^2 c^4 d^2 x \sqrt{c^2 d x^2+d}-\frac{b^2 c^2 d^2 \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d}}{3 x}-\frac{23 b^2 c^3 d^2 \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{12 \sqrt{c^2 x^2+1}} \]

[Out]

(7*b^2*c^4*d^2*x*Sqrt[d + c^2*d*x^2])/12 - (b^2*c^2*d^2*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2])/(3*x) - (23*b^2*c^3
*d^2*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(12*Sqrt[1 + c^2*x^2]) - (5*b*c^5*d^2*x^2*Sqrt[d + c^2*d*x^2]*(a + b*Ar
cSinh[c*x]))/(2*Sqrt[1 + c^2*x^2]) + (7*b*c^3*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/
3 - (b*c*d^2*(1 + c^2*x^2)^(3/2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(3*x^2) + (5*c^4*d^2*x*Sqrt[d + c^2
*d*x^2]*(a + b*ArcSinh[c*x])^2)/2 + (7*c^3*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(3*Sqrt[1 + c^2*x^2
]) - (5*c^2*d*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/(3*x) - ((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]
)^2)/(3*x^3) + (5*c^3*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^3)/(6*b*Sqrt[1 + c^2*x^2]) + (14*b*c^3*d^2*
Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*Log[1 - E^(-2*ArcSinh[c*x])])/(3*Sqrt[1 + c^2*x^2]) - (7*b^2*c^3*d^2*
Sqrt[d + c^2*d*x^2]*PolyLog[2, E^(-2*ArcSinh[c*x])])/(3*Sqrt[1 + c^2*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.850528, antiderivative size = 561, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 15, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.536, Rules used = {5739, 5682, 5675, 5661, 321, 215, 5726, 5659, 3716, 2190, 2279, 2391, 195, 5728, 277} \[ \frac{7 b^2 c^3 d^2 \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt{c^2 x^2+1}}-\frac{5 b c^5 d^2 x^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{c^2 x^2+1}}+\frac{5}{2} c^4 d^2 x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5 c^3 d^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b \sqrt{c^2 x^2+1}}-\frac{7 c^3 d^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt{c^2 x^2+1}}+\frac{7}{3} b c^3 d^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c d^2 \left (c^2 x^2+1\right )^{3/2} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac{14 b c^3 d^2 \sqrt{c^2 d x^2+d} \log \left (1-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{c^2 x^2+1}}-\frac{5 c^2 d \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac{\left (c^2 d x^2+d\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac{7}{12} b^2 c^4 d^2 x \sqrt{c^2 d x^2+d}-\frac{b^2 c^2 d^2 \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d}}{3 x}-\frac{23 b^2 c^3 d^2 \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{12 \sqrt{c^2 x^2+1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(7*b^2*c^4*d^2*x*Sqrt[d + c^2*d*x^2])/12 - (b^2*c^2*d^2*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2])/(3*x) - (23*b^2*c^3
*d^2*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(12*Sqrt[1 + c^2*x^2]) - (5*b*c^5*d^2*x^2*Sqrt[d + c^2*d*x^2]*(a + b*Ar
cSinh[c*x]))/(2*Sqrt[1 + c^2*x^2]) + (7*b*c^3*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/
3 - (b*c*d^2*(1 + c^2*x^2)^(3/2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(3*x^2) + (5*c^4*d^2*x*Sqrt[d + c^2
*d*x^2]*(a + b*ArcSinh[c*x])^2)/2 - (7*c^3*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(3*Sqrt[1 + c^2*x^2
]) - (5*c^2*d*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/(3*x) - ((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]
)^2)/(3*x^3) + (5*c^3*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^3)/(6*b*Sqrt[1 + c^2*x^2]) + (14*b*c^3*d^2*
Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*Log[1 - E^(2*ArcSinh[c*x])])/(3*Sqrt[1 + c^2*x^2]) + (7*b^2*c^3*d^2*S
qrt[d + c^2*d*x^2]*PolyLog[2, E^(2*ArcSinh[c*x])])/(3*Sqrt[1 + c^2*x^2])

Rule 5739

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*d^IntPart[p]*(d + e*x^2)^FracPart[p
])/(f*(m + 1)*(1 + c^2*x^2)^FracPart[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]

Rule 5682

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[Sqrt[d + e*x^2]/(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*Sqrt[d + e*x^2])/(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 5675

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

Rule 5661

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 321

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 215

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 5726

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 5659

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

Rule 3716

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)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2279

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 2391

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 195

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 5728

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((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]))/(f*(m + 1)), x] + (-Dist[(b*c*d^p)/(f*(m + 1)), Int[(f*x)^(m + 1
)*(1 + c^2*x^2)^(p - 1/2), x], x] - Dist[(2*e*p)/(f^2*(m + 1)), Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*A
rcSinh[c*x]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && ILtQ[(m + 1)/2, 0]

Rule 277

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

Rubi steps

\begin{align*} \int \frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x^4} \, dx &=-\frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac{1}{3} \left (5 c^2 d\right ) \int \frac{\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x^2} \, dx+\frac{\left (2 b c d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{\left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{x^3} \, dx}{3 \sqrt{1+c^2 x^2}}\\ &=-\frac{b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac{5 c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\left (5 c^4 d^2\right ) \int \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+\frac{\left (b^2 c^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{\left (1+c^2 x^2\right )^{3/2}}{x^2} \, dx}{3 \sqrt{1+c^2 x^2}}+\frac{\left (4 b c^3 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{\left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx}{3 \sqrt{1+c^2 x^2}}+\frac{\left (10 b c^3 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{\left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx}{3 \sqrt{1+c^2 x^2}}\\ &=-\frac{b^2 c^2 d^2 \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2}}{3 x}+\frac{7}{3} b c^3 d^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac{5}{2} c^4 d^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{5 c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac{\left (4 b c^3 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x} \, dx}{3 \sqrt{1+c^2 x^2}}+\frac{\left (10 b c^3 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x} \, dx}{3 \sqrt{1+c^2 x^2}}+\frac{\left (5 c^4 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}} \, dx}{2 \sqrt{1+c^2 x^2}}-\frac{\left (2 b^2 c^4 d^2 \sqrt{d+c^2 d x^2}\right ) \int \sqrt{1+c^2 x^2} \, dx}{3 \sqrt{1+c^2 x^2}}+\frac{\left (b^2 c^4 d^2 \sqrt{d+c^2 d x^2}\right ) \int \sqrt{1+c^2 x^2} \, dx}{\sqrt{1+c^2 x^2}}-\frac{\left (5 b^2 c^4 d^2 \sqrt{d+c^2 d x^2}\right ) \int \sqrt{1+c^2 x^2} \, dx}{3 \sqrt{1+c^2 x^2}}-\frac{\left (5 b c^5 d^2 \sqrt{d+c^2 d x^2}\right ) \int x \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt{1+c^2 x^2}}\\ &=-\frac{2}{3} b^2 c^4 d^2 x \sqrt{d+c^2 d x^2}-\frac{b^2 c^2 d^2 \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2}}{3 x}-\frac{5 b c^5 d^2 x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{1+c^2 x^2}}+\frac{7}{3} b c^3 d^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac{5}{2} c^4 d^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{5 c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac{5 c^3 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b \sqrt{1+c^2 x^2}}+\frac{\left (4 b c^3 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \coth (x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 \sqrt{1+c^2 x^2}}+\frac{\left (10 b c^3 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \coth (x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 \sqrt{1+c^2 x^2}}-\frac{\left (b^2 c^4 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx}{3 \sqrt{1+c^2 x^2}}+\frac{\left (b^2 c^4 d^2 \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 (5 b^2 c^4 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx}{6 \sqrt{1+c^2 x^2}}+\frac{\left (5 b^2 c^6 d^2 \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{7}{12} b^2 c^4 d^2 x \sqrt{d+c^2 d x^2}-\frac{b^2 c^2 d^2 \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2}}{3 x}-\frac{2 b^2 c^3 d^2 \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{3 \sqrt{1+c^2 x^2}}-\frac{5 b c^5 d^2 x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{1+c^2 x^2}}+\frac{7}{3} b c^3 d^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac{5}{2} c^4 d^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{7 c^3 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt{1+c^2 x^2}}-\frac{5 c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac{5 c^3 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b \sqrt{1+c^2 x^2}}-\frac{\left (8 b c^3 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )}{3 \sqrt{1+c^2 x^2}}-\frac{\left (20 b c^3 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )}{3 \sqrt{1+c^2 x^2}}-\frac{\left (5 b^2 c^4 d^2 \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{7}{12} b^2 c^4 d^2 x \sqrt{d+c^2 d x^2}-\frac{b^2 c^2 d^2 \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2}}{3 x}-\frac{23 b^2 c^3 d^2 \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{12 \sqrt{1+c^2 x^2}}-\frac{5 b c^5 d^2 x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{1+c^2 x^2}}+\frac{7}{3} b c^3 d^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac{5}{2} c^4 d^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{7 c^3 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt{1+c^2 x^2}}-\frac{5 c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac{5 c^3 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b \sqrt{1+c^2 x^2}}+\frac{14 b c^3 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt{1+c^2 x^2}}-\frac{\left (4 b^2 c^3 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 \sqrt{1+c^2 x^2}}-\frac{\left (10 b^2 c^3 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 \sqrt{1+c^2 x^2}}\\ &=\frac{7}{12} b^2 c^4 d^2 x \sqrt{d+c^2 d x^2}-\frac{b^2 c^2 d^2 \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2}}{3 x}-\frac{23 b^2 c^3 d^2 \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{12 \sqrt{1+c^2 x^2}}-\frac{5 b c^5 d^2 x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{1+c^2 x^2}}+\frac{7}{3} b c^3 d^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac{5}{2} c^4 d^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{7 c^3 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt{1+c^2 x^2}}-\frac{5 c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac{5 c^3 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b \sqrt{1+c^2 x^2}}+\frac{14 b c^3 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt{1+c^2 x^2}}-\frac{\left (2 b^2 c^3 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt{1+c^2 x^2}}-\frac{\left (5 b^2 c^3 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt{1+c^2 x^2}}\\ &=\frac{7}{12} b^2 c^4 d^2 x \sqrt{d+c^2 d x^2}-\frac{b^2 c^2 d^2 \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2}}{3 x}-\frac{23 b^2 c^3 d^2 \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{12 \sqrt{1+c^2 x^2}}-\frac{5 b c^5 d^2 x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{1+c^2 x^2}}+\frac{7}{3} b c^3 d^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac{5}{2} c^4 d^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{7 c^3 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt{1+c^2 x^2}}-\frac{5 c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac{5 c^3 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b \sqrt{1+c^2 x^2}}+\frac{14 b c^3 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt{1+c^2 x^2}}+\frac{7 b^2 c^3 d^2 \sqrt{d+c^2 d x^2} \text{Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt{1+c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 2.37959, size = 616, normalized size = 1.1 \[ \frac{d^2 \left (-56 b^2 c^3 x^3 \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )+12 a^2 c^4 x^4 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}-56 a^2 c^2 x^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}-8 a^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+60 a^2 c^3 \sqrt{d} x^3 \sqrt{c^2 x^2+1} \log \left (\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right )-8 a b c x \sqrt{c^2 d x^2+d}+112 a b c^3 x^3 \sqrt{c^2 d x^2+d} \log (c x)+2 b \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)^2 \left (30 a c^3 x^3-4 b \left (-7 c^3 x^3+7 c^2 x^2 \sqrt{c^2 x^2+1}+\sqrt{c^2 x^2+1}\right )+3 b c^3 x^3 \sinh \left (2 \sinh ^{-1}(c x)\right )\right )-6 a b c^3 x^3 \sqrt{c^2 d x^2+d} \cosh \left (2 \sinh ^{-1}(c x)\right )-2 b \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x) \left (56 a c^2 x^2 \sqrt{c^2 x^2+1}+8 a \sqrt{c^2 x^2+1}-6 a c^3 x^3 \sinh \left (2 \sinh ^{-1}(c x)\right )-56 b c^3 x^3 \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )+3 b c^3 x^3 \cosh \left (2 \sinh ^{-1}(c x)\right )+4 b c x\right )-8 b^2 c^2 x^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+20 b^2 c^3 x^3 \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)^3+3 b^2 c^3 x^3 \sqrt{c^2 d x^2+d} \sinh \left (2 \sinh ^{-1}(c x)\right )\right )}{24 x^3 \sqrt{c^2 x^2+1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(d^2*(-8*a*b*c*x*Sqrt[d + c^2*d*x^2] - 8*a^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] - 56*a^2*c^2*x^2*Sqrt[1 + c
^2*x^2]*Sqrt[d + c^2*d*x^2] - 8*b^2*c^2*x^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 12*a^2*c^4*x^4*Sqrt[1 + c^
2*x^2]*Sqrt[d + c^2*d*x^2] + 20*b^2*c^3*x^3*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^3 - 6*a*b*c^3*x^3*Sqrt[d + c^2*d*
x^2]*Cosh[2*ArcSinh[c*x]] + 112*a*b*c^3*x^3*Sqrt[d + c^2*d*x^2]*Log[c*x] + 60*a^2*c^3*Sqrt[d]*x^3*Sqrt[1 + c^2
*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 56*b^2*c^3*x^3*Sqrt[d + c^2*d*x^2]*PolyLog[2, E^(-2*ArcSinh[c
*x])] + 3*b^2*c^3*x^3*Sqrt[d + c^2*d*x^2]*Sinh[2*ArcSinh[c*x]] - 2*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*(4*b*c*x
+ 8*a*Sqrt[1 + c^2*x^2] + 56*a*c^2*x^2*Sqrt[1 + c^2*x^2] + 3*b*c^3*x^3*Cosh[2*ArcSinh[c*x]] - 56*b*c^3*x^3*Lo
g[1 - E^(-2*ArcSinh[c*x])] - 6*a*c^3*x^3*Sinh[2*ArcSinh[c*x]]) + 2*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^2*(30*a*
c^3*x^3 - 4*b*(-7*c^3*x^3 + Sqrt[1 + c^2*x^2] + 7*c^2*x^2*Sqrt[1 + c^2*x^2]) + 3*b*c^3*x^3*Sinh[2*ArcSinh[c*x]
])))/(24*x^3*Sqrt[1 + c^2*x^2])

________________________________________________________________________________________

Maple [B]  time = 0.375, size = 3311, normalized size = 5.9 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b^2*(d*(c^2*x^2+1))^(1/2)*c^5*d^2/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*x^2+49/3*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(
63*c^4*x^4+15*c^2*x^2+1)*x^3*arcsinh(c*x)*c^6+7/3*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x*ar
csinh(c*x)*c^4+21*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^4/(c^2*x^2+1)^(1/2)*c^7+14/3*b^2*(
d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*ln(1-c*x-(c^2*x^2+1)^(1/2))*c^3*d^2+5*b^2*(d*(c^2*x^2+1))^
(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^2/(c^2*x^2+1)^(1/2)*c^5+7/3*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15
*c^2*x^2+1)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)^2*c^3-5*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)/(c^
2*x^2+1)^(1/2)*arcsinh(c*x)*c^3-5*a*b*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)/(c^2*x^2+1)^(1/2)*c^
3-1/2*a*b*(d*(c^2*x^2+1))^(1/2)*c^5*d^2/(c^2*x^2+1)^(1/2)*x^2+14/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)
*arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2))*c^3*d^2-56/3*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)
*x^5/(c^2*x^2+1)*c^8-71/3*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^3/(c^2*x^2+1)*c^6-16/3*b^2
*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x/(c^2*x^2+1)*c^4-1/3*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c
^4*x^4+15*c^2*x^2+1)/x/(c^2*x^2+1)*c^2-1/3*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)/x^3/(c^2*x^
2+1)*arcsinh(c*x)^2+1/2*b^2*(d*(c^2*x^2+1))^(1/2)*c^6*d^2/(c^2*x^2+1)*arcsinh(c*x)^2*x^3+1/2*b^2*(d*(c^2*x^2+1
))^(1/2)*c^4*d^2/(c^2*x^2+1)*arcsinh(c*x)^2*x+5/2*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)^2*c
^3*d^2-28/3*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*c^3*d^2+14/3*a*b*(d*(c^2*x^2+1))^(1/2)/(c
^2*x^2+1)^(1/2)*ln((c*x+(c^2*x^2+1)^(1/2))^2-1)*c^3*d^2+5/3*a^2*c^4*d*x*(c^2*d*x^2+d)^(3/2)-294*a*b*(d*(c^2*x^
2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^5/(c^2*x^2+1)*arcsinh(c*x)*c^8-406*a*b*(d*(c^2*x^2+1))^(1/2)*d^2/(
63*c^4*x^4+15*c^2*x^2+1)*x^3/(c^2*x^2+1)*arcsinh(c*x)*c^6-380/3*a*b*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c
^2*x^2+1)*x/(c^2*x^2+1)*arcsinh(c*x)*c^4-46/3*a*b*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)/x/(c^2*x
^2+1)*arcsinh(c*x)*c^2+70*a*b*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^2/(c^2*x^2+1)^(1/2)*arcsin
h(c*x)*c^5+294*a*b*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^4/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*c^7+
5/2*a^2*c^4*d^2*x*(c^2*d*x^2+d)^(1/2)+5/2*a^2*c^4*d^3*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1
/2)-56/3*a*b*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^3/(c^2*x^2+1)*c^6-7/3*a*b*(d*(c^2*x^2+1))^(
1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x/(c^2*x^2+1)*c^4+a*b*(d*(c^2*x^2+1))^(1/2)*c^6*d^2/(c^2*x^2+1)*arcsinh(c*x
)*x^3+a*b*(d*(c^2*x^2+1))^(1/2)*c^4*d^2/(c^2*x^2+1)*arcsinh(c*x)*x-1/3*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x
^4+15*c^2*x^2+1)/x^2/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*c+147*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2
+1)*x^4/(c^2*x^2+1)^(1/2)*arcsinh(c*x)^2*c^7+35*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^2/(c
^2*x^2+1)^(1/2)*arcsinh(c*x)^2*c^5-21*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^2/(c^2*x^2+1)^
(1/2)*arcsinh(c*x)*c^5-2/3*a*b*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)/x^3/(c^2*x^2+1)*arcsinh(c*x
)-49/3*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^5/(c^2*x^2+1)*arcsinh(c*x)*c^8-203*b^2*(d*(c^
2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^3/(c^2*x^2+1)*arcsinh(c*x)^2*c^6-23/3*b^2*(d*(c^2*x^2+1))^(1/2
)*d^2/(63*c^4*x^4+15*c^2*x^2+1)/x/(c^2*x^2+1)*arcsinh(c*x)^2*c^2-190/3*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x
^4+15*c^2*x^2+1)*x/(c^2*x^2+1)*arcsinh(c*x)^2*c^4-7/3*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*
x/(c^2*x^2+1)*arcsinh(c*x)*c^4-56/3*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^3/(c^2*x^2+1)*ar
csinh(c*x)*c^6-49/3*a*b*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^5/(c^2*x^2+1)*c^8+14/3*a*b*(d*(c
^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*c^3-21*a*b*(d*(c^2*x^2+1))^(1/2)
*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^2/(c^2*x^2+1)^(1/2)*c^5-1/3*a*b*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*
x^2+1)/x^2/(c^2*x^2+1)^(1/2)*c-147*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^5/(c^2*x^2+1)*arc
sinh(c*x)^2*c^8+5/6*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)^3*c^3*d^2+1/4*b^2*(d*(c^2*x^2+1))
^(1/2)*c^6*d^2/(c^2*x^2+1)*x^3+1/4*b^2*(d*(c^2*x^2+1))^(1/2)*c^4*d^2/(c^2*x^2+1)*x-1/4*b^2*(d*(c^2*x^2+1))^(1/
2)*c^3*d^2/(c^2*x^2+1)^(1/2)*arcsinh(c*x)-1/4*a*b*(d*(c^2*x^2+1))^(1/2)*c^3*d^2/(c^2*x^2+1)^(1/2)-1/3*a^2/d/x^
3*(c^2*d*x^2+d)^(7/2)+4/3*a^2*c^4*x*(c^2*d*x^2+d)^(5/2)-4/3*a^2*c^2/d/x*(c^2*d*x^2+d)^(7/2)+1/3*b^2*(d*(c^2*x^
2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)/(c^2*x^2+1)^(1/2)*c^3-7/3*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+
15*c^2*x^2+1)*x^3*c^6+14/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*polylog(2,c*x+(c^2*x^2+1)^(1/2))*c^3*d^
2+14/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*polylog(2,-c*x-(c^2*x^2+1)^(1/2))*c^3*d^2-14/3*b^2*(d*(c^2*
x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)^2*c^3*d^2+49/3*a*b*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x
^2+1)*x^3*c^6+7/3*a*b*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x*c^4

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a^{2} c^{4} d^{2} x^{4} + 2 \, a^{2} c^{2} d^{2} x^{2} + a^{2} d^{2} +{\left (b^{2} c^{4} d^{2} x^{4} + 2 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \operatorname{arsinh}\left (c x\right )^{2} + 2 \,{\left (a b c^{4} d^{2} x^{4} + 2 \, a b c^{2} d^{2} x^{2} + a b d^{2}\right )} \operatorname{arsinh}\left (c x\right )\right )} \sqrt{c^{2} d x^{2} + d}}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^2*c^4*d^2*x^4 + 2*a^2*c^2*d^2*x^2 + a^2*d^2 + (b^2*c^4*d^2*x^4 + 2*b^2*c^2*d^2*x^2 + b^2*d^2)*arcs
inh(c*x)^2 + 2*(a*b*c^4*d^2*x^4 + 2*a*b*c^2*d^2*x^2 + a*b*d^2)*arcsinh(c*x))*sqrt(c^2*d*x^2 + d)/x^4, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c^{2} d x^{2} + d\right )}^{\frac{5}{2}}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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