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

Optimal. Leaf size=386 \[ \frac{1}{9} d^2 x^5 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{4}{63} d^2 x^5 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{16 b d^2 x^4 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{1575 c}+\frac{64 b d^2 x^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{4725 c^3}-\frac{2 b d^2 \left (c^2 x^2+1\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{81 c^5}+\frac{20 b d^2 \left (c^2 x^2+1\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{441 c^5}+\frac{2 b d^2 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{315 c^5}-\frac{8 b d^2 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{189 c^5}-\frac{128 b d^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{4725 c^5}+\frac{8}{315} d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{2}{729} b^2 c^4 d^2 x^9+\frac{212 b^2 c^2 d^2 x^7}{27783}-\frac{2104 b^2 d^2 x^3}{297675 c^2}+\frac{4208 b^2 d^2 x}{99225 c^4}+\frac{526 b^2 d^2 x^5}{165375} \]

[Out]

(4208*b^2*d^2*x)/(99225*c^4) - (2104*b^2*d^2*x^3)/(297675*c^2) + (526*b^2*d^2*x^5)/165375 + (212*b^2*c^2*d^2*x
^7)/27783 + (2*b^2*c^4*d^2*x^9)/729 - (128*b*d^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(4725*c^5) + (64*b*d^
2*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(4725*c^3) - (16*b*d^2*x^4*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]
))/(1575*c) - (8*b*d^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(189*c^5) + (2*b*d^2*(1 + c^2*x^2)^(5/2)*(a +
b*ArcSinh[c*x]))/(315*c^5) + (20*b*d^2*(1 + c^2*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(441*c^5) - (2*b*d^2*(1 + c^
2*x^2)^(9/2)*(a + b*ArcSinh[c*x]))/(81*c^5) + (8*d^2*x^5*(a + b*ArcSinh[c*x])^2)/315 + (4*d^2*x^5*(1 + c^2*x^2
)*(a + b*ArcSinh[c*x])^2)/63 + (d^2*x^5*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/9

________________________________________________________________________________________

Rubi [A]  time = 0.738931, antiderivative size = 386, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 11, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.423, Rules used = {5744, 5661, 5758, 5717, 8, 30, 266, 43, 5732, 12, 1153} \[ \frac{1}{9} d^2 x^5 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{4}{63} d^2 x^5 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{16 b d^2 x^4 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{1575 c}+\frac{64 b d^2 x^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{4725 c^3}-\frac{2 b d^2 \left (c^2 x^2+1\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{81 c^5}+\frac{20 b d^2 \left (c^2 x^2+1\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{441 c^5}+\frac{2 b d^2 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{315 c^5}-\frac{8 b d^2 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{189 c^5}-\frac{128 b d^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{4725 c^5}+\frac{8}{315} d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{2}{729} b^2 c^4 d^2 x^9+\frac{212 b^2 c^2 d^2 x^7}{27783}-\frac{2104 b^2 d^2 x^3}{297675 c^2}+\frac{4208 b^2 d^2 x}{99225 c^4}+\frac{526 b^2 d^2 x^5}{165375} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4208*b^2*d^2*x)/(99225*c^4) - (2104*b^2*d^2*x^3)/(297675*c^2) + (526*b^2*d^2*x^5)/165375 + (212*b^2*c^2*d^2*x
^7)/27783 + (2*b^2*c^4*d^2*x^9)/729 - (128*b*d^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(4725*c^5) + (64*b*d^
2*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(4725*c^3) - (16*b*d^2*x^4*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]
))/(1575*c) - (8*b*d^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(189*c^5) + (2*b*d^2*(1 + c^2*x^2)^(5/2)*(a +
b*ArcSinh[c*x]))/(315*c^5) + (20*b*d^2*(1 + c^2*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(441*c^5) - (2*b*d^2*(1 + c^
2*x^2)^(9/2)*(a + b*ArcSinh[c*x]))/(81*c^5) + (8*d^2*x^5*(a + b*ArcSinh[c*x])^2)/315 + (4*d^2*x^5*(1 + c^2*x^2
)*(a + b*ArcSinh[c*x])^2)/63 + (d^2*x^5*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/9

Rule 5744

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 + 2*p + 1)), x] + (Dist[(2*d*p)/(m + 2*p + 1), Int
[(f*x)^m*(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 + 2*p + 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, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m, -1]
&& (RationalQ[m] || EqQ[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 5758

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp
[(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^n)/(e*m), x] + (-Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)
^(m - 2)*(a + b*ArcSinh[c*x])^n)/Sqrt[d + e*x^2], x], x] - Dist[(b*f*n*Sqrt[1 + c^2*x^2])/(c*m*Sqrt[d + e*x^2]
), Int[(f*x)^(m - 1)*(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[m, 1] && IntegerQ[m]

Rule 5717

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 5732

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x
^m*(1 + c^2*x^2)^p, x]}, Dist[d^p*(a + b*ArcSinh[c*x]), u, x] - Dist[b*c*d^p, Int[SimplifyIntegrand[u/Sqrt[1 +
c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegerQ[p - 1/2] && (IGtQ[(m + 1)/2,
0] || ILtQ[(m + 2*p + 3)/2, 0]) && NeQ[p, -2^(-1)] && GtQ[d, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1153

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rubi steps

\begin{align*} \int x^4 \left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac{1}{9} d^2 x^5 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{9} (4 d) \int x^4 \left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac{1}{9} \left (2 b c d^2\right ) \int x^5 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx\\ &=-\frac{2 b d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{45 c^5}+\frac{4 b d^2 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{63 c^5}-\frac{2 b d^2 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{81 c^5}+\frac{4}{63} d^2 x^5 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{9} d^2 x^5 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{63} \left (8 d^2\right ) \int x^4 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac{1}{63} \left (8 b c d^2\right ) \int x^5 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx+\frac{1}{9} \left (2 b^2 c^2 d^2\right ) \int \frac{\left (1+c^2 x^2\right )^2 \left (8-20 c^2 x^2+35 c^4 x^4\right )}{315 c^6} \, dx\\ &=-\frac{8 b d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{189 c^5}+\frac{2 b d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{315 c^5}+\frac{20 b d^2 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{441 c^5}-\frac{2 b d^2 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{81 c^5}+\frac{8}{315} d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{4}{63} d^2 x^5 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{9} d^2 x^5 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{\left (2 b^2 d^2\right ) \int \left (1+c^2 x^2\right )^2 \left (8-20 c^2 x^2+35 c^4 x^4\right ) \, dx}{2835 c^4}-\frac{1}{315} \left (16 b c d^2\right ) \int \frac{x^5 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx+\frac{1}{63} \left (8 b^2 c^2 d^2\right ) \int \frac{8-4 c^2 x^2+3 c^4 x^4+15 c^6 x^6}{105 c^6} \, dx\\ &=-\frac{16 b d^2 x^4 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{1575 c}-\frac{8 b d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{189 c^5}+\frac{2 b d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{315 c^5}+\frac{20 b d^2 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{441 c^5}-\frac{2 b d^2 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{81 c^5}+\frac{8}{315} d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{4}{63} d^2 x^5 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{9} d^2 x^5 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{\left (16 b^2 d^2\right ) \int x^4 \, dx}{1575}+\frac{\left (2 b^2 d^2\right ) \int \left (8-4 c^2 x^2+3 c^4 x^4+50 c^6 x^6+35 c^8 x^8\right ) \, dx}{2835 c^4}+\frac{\left (8 b^2 d^2\right ) \int \left (8-4 c^2 x^2+3 c^4 x^4+15 c^6 x^6\right ) \, dx}{6615 c^4}+\frac{\left (64 b d^2\right ) \int \frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx}{1575 c}\\ &=\frac{304 b^2 d^2 x}{19845 c^4}-\frac{152 b^2 d^2 x^3}{59535 c^2}+\frac{526 b^2 d^2 x^5}{165375}+\frac{212 b^2 c^2 d^2 x^7}{27783}+\frac{2}{729} b^2 c^4 d^2 x^9+\frac{64 b d^2 x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4725 c^3}-\frac{16 b d^2 x^4 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{1575 c}-\frac{8 b d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{189 c^5}+\frac{2 b d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{315 c^5}+\frac{20 b d^2 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{441 c^5}-\frac{2 b d^2 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{81 c^5}+\frac{8}{315} d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{4}{63} d^2 x^5 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{9} d^2 x^5 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{\left (128 b d^2\right ) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx}{4725 c^3}-\frac{\left (64 b^2 d^2\right ) \int x^2 \, dx}{4725 c^2}\\ &=\frac{304 b^2 d^2 x}{19845 c^4}-\frac{2104 b^2 d^2 x^3}{297675 c^2}+\frac{526 b^2 d^2 x^5}{165375}+\frac{212 b^2 c^2 d^2 x^7}{27783}+\frac{2}{729} b^2 c^4 d^2 x^9-\frac{128 b d^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4725 c^5}+\frac{64 b d^2 x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4725 c^3}-\frac{16 b d^2 x^4 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{1575 c}-\frac{8 b d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{189 c^5}+\frac{2 b d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{315 c^5}+\frac{20 b d^2 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{441 c^5}-\frac{2 b d^2 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{81 c^5}+\frac{8}{315} d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{4}{63} d^2 x^5 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{9} d^2 x^5 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{\left (128 b^2 d^2\right ) \int 1 \, dx}{4725 c^4}\\ &=\frac{4208 b^2 d^2 x}{99225 c^4}-\frac{2104 b^2 d^2 x^3}{297675 c^2}+\frac{526 b^2 d^2 x^5}{165375}+\frac{212 b^2 c^2 d^2 x^7}{27783}+\frac{2}{729} b^2 c^4 d^2 x^9-\frac{128 b d^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4725 c^5}+\frac{64 b d^2 x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4725 c^3}-\frac{16 b d^2 x^4 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{1575 c}-\frac{8 b d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{189 c^5}+\frac{2 b d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{315 c^5}+\frac{20 b d^2 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{441 c^5}-\frac{2 b d^2 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{81 c^5}+\frac{8}{315} d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{4}{63} d^2 x^5 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{9} d^2 x^5 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2\\ \end{align*}

Mathematica [A]  time = 0.3879, size = 251, normalized size = 0.65 \[ \frac{d^2 \left (99225 a^2 c^5 x^5 \left (35 c^4 x^4+90 c^2 x^2+63\right )-630 a b \sqrt{c^2 x^2+1} \left (1225 c^8 x^8+2650 c^6 x^6+789 c^4 x^4-1052 c^2 x^2+2104\right )-630 b \sinh ^{-1}(c x) \left (b \sqrt{c^2 x^2+1} \left (1225 c^8 x^8+2650 c^6 x^6+789 c^4 x^4-1052 c^2 x^2+2104\right )-315 a c^5 x^5 \left (35 c^4 x^4+90 c^2 x^2+63\right )\right )+2 b^2 c x \left (42875 c^8 x^8+119250 c^6 x^6+49707 c^4 x^4-110460 c^2 x^2+662760\right )+99225 b^2 c^5 x^5 \left (35 c^4 x^4+90 c^2 x^2+63\right ) \sinh ^{-1}(c x)^2\right )}{31255875 c^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*(99225*a^2*c^5*x^5*(63 + 90*c^2*x^2 + 35*c^4*x^4) - 630*a*b*Sqrt[1 + c^2*x^2]*(2104 - 1052*c^2*x^2 + 789*
c^4*x^4 + 2650*c^6*x^6 + 1225*c^8*x^8) + 2*b^2*c*x*(662760 - 110460*c^2*x^2 + 49707*c^4*x^4 + 119250*c^6*x^6 +
42875*c^8*x^8) - 630*b*(-315*a*c^5*x^5*(63 + 90*c^2*x^2 + 35*c^4*x^4) + b*Sqrt[1 + c^2*x^2]*(2104 - 1052*c^2*
x^2 + 789*c^4*x^4 + 2650*c^6*x^6 + 1225*c^8*x^8))*ArcSinh[c*x] + 99225*b^2*c^5*x^5*(63 + 90*c^2*x^2 + 35*c^4*x
^4)*ArcSinh[c*x]^2))/(31255875*c^5)

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Maple [A]  time = 0.046, size = 446, normalized size = 1.2 \begin{align*}{\frac{1}{{c}^{5}} \left ({d}^{2}{a}^{2} \left ({\frac{{c}^{9}{x}^{9}}{9}}+{\frac{2\,{c}^{7}{x}^{7}}{7}}+{\frac{{c}^{5}{x}^{5}}{5}} \right ) +{d}^{2}{b}^{2} \left ({\frac{ \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}{c}^{3}{x}^{3} \left ({c}^{2}{x}^{2}+1 \right ) ^{3}}{9}}-{\frac{ \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}cx \left ({c}^{2}{x}^{2}+1 \right ) ^{3}}{21}}+{\frac{8\, \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}cx}{315}}+{\frac{ \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}cx \left ({c}^{2}{x}^{2}+1 \right ) ^{2}}{105}}+{\frac{4\, \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}cx \left ({c}^{2}{x}^{2}+1 \right ) }{315}}-{\frac{2\,{\it Arcsinh} \left ( cx \right ){c}^{2}{x}^{2}}{81} \left ({c}^{2}{x}^{2}+1 \right ) ^{{\frac{7}{2}}}}+{\frac{82\,{\it Arcsinh} \left ( cx \right ){c}^{2}{x}^{2}}{3969} \left ({c}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{2}}}}+{\frac{1672\,{\it Arcsinh} \left ( cx \right ){c}^{2}{x}^{2}}{99225} \left ({c}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{832\,{\it Arcsinh} \left ( cx \right ){c}^{2}{x}^{2}}{99225}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{4208\,{\it Arcsinh} \left ( cx \right ) }{99225}\sqrt{{c}^{2}{x}^{2}+1}}+{\frac{2\,cx \left ({c}^{2}{x}^{2}+1 \right ) ^{4}}{729}}+{\frac{1493104\,cx}{31255875}}-{\frac{836\,cx \left ({c}^{2}{x}^{2}+1 \right ) ^{3}}{250047}}-{\frac{33862\,cx \left ({c}^{2}{x}^{2}+1 \right ) ^{2}}{10418625}}-{\frac{47248\,cx \left ({c}^{2}{x}^{2}+1 \right ) }{31255875}} \right ) +2\,{d}^{2}ab \left ( 1/9\,{\it Arcsinh} \left ( cx \right ){c}^{9}{x}^{9}+2/7\,{\it Arcsinh} \left ( cx \right ){c}^{7}{x}^{7}+1/5\,{\it Arcsinh} \left ( cx \right ){c}^{5}{x}^{5}-{\frac{{c}^{8}{x}^{8}\sqrt{{c}^{2}{x}^{2}+1}}{81}}-{\frac{106\,{c}^{6}{x}^{6}\sqrt{{c}^{2}{x}^{2}+1}}{3969}}-{\frac{263\,{c}^{4}{x}^{4}\sqrt{{c}^{2}{x}^{2}+1}}{33075}}+{\frac{1052\,{c}^{2}{x}^{2}\sqrt{{c}^{2}{x}^{2}+1}}{99225}}-{\frac{2104\,\sqrt{{c}^{2}{x}^{2}+1}}{99225}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^5*(d^2*a^2*(1/9*c^9*x^9+2/7*c^7*x^7+1/5*c^5*x^5)+d^2*b^2*(1/9*arcsinh(c*x)^2*c^3*x^3*(c^2*x^2+1)^3-1/21*ar
csinh(c*x)^2*c*x*(c^2*x^2+1)^3+8/315*arcsinh(c*x)^2*c*x+1/105*arcsinh(c*x)^2*c*x*(c^2*x^2+1)^2+4/315*arcsinh(c
*x)^2*c*x*(c^2*x^2+1)-2/81*arcsinh(c*x)*c^2*x^2*(c^2*x^2+1)^(7/2)+82/3969*arcsinh(c*x)*c^2*x^2*(c^2*x^2+1)^(5/
2)+1672/99225*arcsinh(c*x)*c^2*x^2*(c^2*x^2+1)^(3/2)+832/99225*arcsinh(c*x)*c^2*x^2*(c^2*x^2+1)^(1/2)-4208/992
25*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+2/729*c*x*(c^2*x^2+1)^4+1493104/31255875*c*x-836/250047*c*x*(c^2*x^2+1)^3-33
862/10418625*c*x*(c^2*x^2+1)^2-47248/31255875*c*x*(c^2*x^2+1))+2*d^2*a*b*(1/9*arcsinh(c*x)*c^9*x^9+2/7*arcsinh
(c*x)*c^7*x^7+1/5*arcsinh(c*x)*c^5*x^5-1/81*c^8*x^8*(c^2*x^2+1)^(1/2)-106/3969*c^6*x^6*(c^2*x^2+1)^(1/2)-263/3
3075*c^4*x^4*(c^2*x^2+1)^(1/2)+1052/99225*c^2*x^2*(c^2*x^2+1)^(1/2)-2104/99225*(c^2*x^2+1)^(1/2)))

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Maxima [B]  time = 1.35261, size = 1026, normalized size = 2.66 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b^2*c^4*d^2*x^9*arcsinh(c*x)^2 + 1/9*a^2*c^4*d^2*x^9 + 2/7*b^2*c^2*d^2*x^7*arcsinh(c*x)^2 + 2/7*a^2*c^2*d^
2*x^7 + 1/5*b^2*d^2*x^5*arcsinh(c*x)^2 + 2/2835*(315*x^9*arcsinh(c*x) - (35*sqrt(c^2*x^2 + 1)*x^8/c^2 - 40*sqr
t(c^2*x^2 + 1)*x^6/c^4 + 48*sqrt(c^2*x^2 + 1)*x^4/c^6 - 64*sqrt(c^2*x^2 + 1)*x^2/c^8 + 128*sqrt(c^2*x^2 + 1)/c
^10)*c)*a*b*c^4*d^2 - 2/893025*(315*(35*sqrt(c^2*x^2 + 1)*x^8/c^2 - 40*sqrt(c^2*x^2 + 1)*x^6/c^4 + 48*sqrt(c^2
*x^2 + 1)*x^4/c^6 - 64*sqrt(c^2*x^2 + 1)*x^2/c^8 + 128*sqrt(c^2*x^2 + 1)/c^10)*c*arcsinh(c*x) - (1225*c^8*x^9
- 1800*c^6*x^7 + 3024*c^4*x^5 - 6720*c^2*x^3 + 40320*x)/c^8)*b^2*c^4*d^2 + 1/5*a^2*d^2*x^5 + 4/245*(35*x^7*arc
sinh(c*x) - (5*sqrt(c^2*x^2 + 1)*x^6/c^2 - 6*sqrt(c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(c^2*x^2 + 1)*x^2/c^6 - 16*sqrt
(c^2*x^2 + 1)/c^8)*c)*a*b*c^2*d^2 - 4/25725*(105*(5*sqrt(c^2*x^2 + 1)*x^6/c^2 - 6*sqrt(c^2*x^2 + 1)*x^4/c^4 +
8*sqrt(c^2*x^2 + 1)*x^2/c^6 - 16*sqrt(c^2*x^2 + 1)/c^8)*c*arcsinh(c*x) - (75*c^6*x^7 - 126*c^4*x^5 + 280*c^2*x
^3 - 1680*x)/c^6)*b^2*c^2*d^2 + 2/75*(15*x^5*arcsinh(c*x) - (3*sqrt(c^2*x^2 + 1)*x^4/c^2 - 4*sqrt(c^2*x^2 + 1)
*x^2/c^4 + 8*sqrt(c^2*x^2 + 1)/c^6)*c)*a*b*d^2 - 2/1125*(15*(3*sqrt(c^2*x^2 + 1)*x^4/c^2 - 4*sqrt(c^2*x^2 + 1)
*x^2/c^4 + 8*sqrt(c^2*x^2 + 1)/c^6)*c*arcsinh(c*x) - (9*c^4*x^5 - 20*c^2*x^3 + 120*x)/c^4)*b^2*d^2

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Fricas [A]  time = 3.1998, size = 871, normalized size = 2.26 \begin{align*} \frac{42875 \,{\left (81 \, a^{2} + 2 \, b^{2}\right )} c^{9} d^{2} x^{9} + 2250 \,{\left (3969 \, a^{2} + 106 \, b^{2}\right )} c^{7} d^{2} x^{7} + 189 \,{\left (33075 \, a^{2} + 526 \, b^{2}\right )} c^{5} d^{2} x^{5} - 220920 \, b^{2} c^{3} d^{2} x^{3} + 1325520 \, b^{2} c d^{2} x + 99225 \,{\left (35 \, b^{2} c^{9} d^{2} x^{9} + 90 \, b^{2} c^{7} d^{2} x^{7} + 63 \, b^{2} c^{5} d^{2} x^{5}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 630 \,{\left (11025 \, a b c^{9} d^{2} x^{9} + 28350 \, a b c^{7} d^{2} x^{7} + 19845 \, a b c^{5} d^{2} x^{5} -{\left (1225 \, b^{2} c^{8} d^{2} x^{8} + 2650 \, b^{2} c^{6} d^{2} x^{6} + 789 \, b^{2} c^{4} d^{2} x^{4} - 1052 \, b^{2} c^{2} d^{2} x^{2} + 2104 \, b^{2} d^{2}\right )} \sqrt{c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - 630 \,{\left (1225 \, a b c^{8} d^{2} x^{8} + 2650 \, a b c^{6} d^{2} x^{6} + 789 \, a b c^{4} d^{2} x^{4} - 1052 \, a b c^{2} d^{2} x^{2} + 2104 \, a b d^{2}\right )} \sqrt{c^{2} x^{2} + 1}}{31255875 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/31255875*(42875*(81*a^2 + 2*b^2)*c^9*d^2*x^9 + 2250*(3969*a^2 + 106*b^2)*c^7*d^2*x^7 + 189*(33075*a^2 + 526*
b^2)*c^5*d^2*x^5 - 220920*b^2*c^3*d^2*x^3 + 1325520*b^2*c*d^2*x + 99225*(35*b^2*c^9*d^2*x^9 + 90*b^2*c^7*d^2*x
^7 + 63*b^2*c^5*d^2*x^5)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 630*(11025*a*b*c^9*d^2*x^9 + 28350*a*b*c^7*d^2*x^7 +
19845*a*b*c^5*d^2*x^5 - (1225*b^2*c^8*d^2*x^8 + 2650*b^2*c^6*d^2*x^6 + 789*b^2*c^4*d^2*x^4 - 1052*b^2*c^2*d^2
*x^2 + 2104*b^2*d^2)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) - 630*(1225*a*b*c^8*d^2*x^8 + 2650*a*b*c^
6*d^2*x^6 + 789*a*b*c^4*d^2*x^4 - 1052*a*b*c^2*d^2*x^2 + 2104*a*b*d^2)*sqrt(c^2*x^2 + 1))/c^5

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Sympy [A]  time = 44.9501, size = 563, normalized size = 1.46 \begin{align*} \begin{cases} \frac{a^{2} c^{4} d^{2} x^{9}}{9} + \frac{2 a^{2} c^{2} d^{2} x^{7}}{7} + \frac{a^{2} d^{2} x^{5}}{5} + \frac{2 a b c^{4} d^{2} x^{9} \operatorname{asinh}{\left (c x \right )}}{9} - \frac{2 a b c^{3} d^{2} x^{8} \sqrt{c^{2} x^{2} + 1}}{81} + \frac{4 a b c^{2} d^{2} x^{7} \operatorname{asinh}{\left (c x \right )}}{7} - \frac{212 a b c d^{2} x^{6} \sqrt{c^{2} x^{2} + 1}}{3969} + \frac{2 a b d^{2} x^{5} \operatorname{asinh}{\left (c x \right )}}{5} - \frac{526 a b d^{2} x^{4} \sqrt{c^{2} x^{2} + 1}}{33075 c} + \frac{2104 a b d^{2} x^{2} \sqrt{c^{2} x^{2} + 1}}{99225 c^{3}} - \frac{4208 a b d^{2} \sqrt{c^{2} x^{2} + 1}}{99225 c^{5}} + \frac{b^{2} c^{4} d^{2} x^{9} \operatorname{asinh}^{2}{\left (c x \right )}}{9} + \frac{2 b^{2} c^{4} d^{2} x^{9}}{729} - \frac{2 b^{2} c^{3} d^{2} x^{8} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{81} + \frac{2 b^{2} c^{2} d^{2} x^{7} \operatorname{asinh}^{2}{\left (c x \right )}}{7} + \frac{212 b^{2} c^{2} d^{2} x^{7}}{27783} - \frac{212 b^{2} c d^{2} x^{6} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{3969} + \frac{b^{2} d^{2} x^{5} \operatorname{asinh}^{2}{\left (c x \right )}}{5} + \frac{526 b^{2} d^{2} x^{5}}{165375} - \frac{526 b^{2} d^{2} x^{4} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{33075 c} - \frac{2104 b^{2} d^{2} x^{3}}{297675 c^{2}} + \frac{2104 b^{2} d^{2} x^{2} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{99225 c^{3}} + \frac{4208 b^{2} d^{2} x}{99225 c^{4}} - \frac{4208 b^{2} d^{2} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{99225 c^{5}} & \text{for}\: c \neq 0 \\\frac{a^{2} d^{2} x^{5}}{5} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*c**4*d**2*x**9/9 + 2*a**2*c**2*d**2*x**7/7 + a**2*d**2*x**5/5 + 2*a*b*c**4*d**2*x**9*asinh(c*x
)/9 - 2*a*b*c**3*d**2*x**8*sqrt(c**2*x**2 + 1)/81 + 4*a*b*c**2*d**2*x**7*asinh(c*x)/7 - 212*a*b*c*d**2*x**6*sq
rt(c**2*x**2 + 1)/3969 + 2*a*b*d**2*x**5*asinh(c*x)/5 - 526*a*b*d**2*x**4*sqrt(c**2*x**2 + 1)/(33075*c) + 2104
*a*b*d**2*x**2*sqrt(c**2*x**2 + 1)/(99225*c**3) - 4208*a*b*d**2*sqrt(c**2*x**2 + 1)/(99225*c**5) + b**2*c**4*d
**2*x**9*asinh(c*x)**2/9 + 2*b**2*c**4*d**2*x**9/729 - 2*b**2*c**3*d**2*x**8*sqrt(c**2*x**2 + 1)*asinh(c*x)/81
+ 2*b**2*c**2*d**2*x**7*asinh(c*x)**2/7 + 212*b**2*c**2*d**2*x**7/27783 - 212*b**2*c*d**2*x**6*sqrt(c**2*x**2
+ 1)*asinh(c*x)/3969 + b**2*d**2*x**5*asinh(c*x)**2/5 + 526*b**2*d**2*x**5/165375 - 526*b**2*d**2*x**4*sqrt(c
**2*x**2 + 1)*asinh(c*x)/(33075*c) - 2104*b**2*d**2*x**3/(297675*c**2) + 2104*b**2*d**2*x**2*sqrt(c**2*x**2 +
1)*asinh(c*x)/(99225*c**3) + 4208*b**2*d**2*x/(99225*c**4) - 4208*b**2*d**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/(99
225*c**5), Ne(c, 0)), (a**2*d**2*x**5/5, True))

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Giac [B]  time = 2.73084, size = 998, normalized size = 2.59 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*a^2*c^4*d^2*x^9 + 2/7*a^2*c^2*d^2*x^7 + 2/2835*(315*x^9*log(c*x + sqrt(c^2*x^2 + 1)) - (35*(c^2*x^2 + 1)^(
9/2) - 180*(c^2*x^2 + 1)^(7/2) + 378*(c^2*x^2 + 1)^(5/2) - 420*(c^2*x^2 + 1)^(3/2) + 315*sqrt(c^2*x^2 + 1))/c^
9)*a*b*c^4*d^2 + 1/893025*(99225*x^9*log(c*x + sqrt(c^2*x^2 + 1))^2 + 2*c*((1225*c^8*x^9 - 1800*c^6*x^7 + 3024
*c^4*x^5 - 6720*c^2*x^3 + 40320*x)/c^9 - 315*(35*(c^2*x^2 + 1)^(9/2) - 180*(c^2*x^2 + 1)^(7/2) + 378*(c^2*x^2
+ 1)^(5/2) - 420*(c^2*x^2 + 1)^(3/2) + 315*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1))/c^10))*b^2*c^4*d^2
+ 1/5*a^2*d^2*x^5 + 4/245*(35*x^7*log(c*x + sqrt(c^2*x^2 + 1)) - (5*(c^2*x^2 + 1)^(7/2) - 21*(c^2*x^2 + 1)^(5/
2) + 35*(c^2*x^2 + 1)^(3/2) - 35*sqrt(c^2*x^2 + 1))/c^7)*a*b*c^2*d^2 + 2/25725*(3675*x^7*log(c*x + sqrt(c^2*x^
2 + 1))^2 + 2*c*((75*c^6*x^7 - 126*c^4*x^5 + 280*c^2*x^3 - 1680*x)/c^7 - 105*(5*(c^2*x^2 + 1)^(7/2) - 21*(c^2*
x^2 + 1)^(5/2) + 35*(c^2*x^2 + 1)^(3/2) - 35*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1))/c^8))*b^2*c^2*d^2
+ 2/75*(15*x^5*log(c*x + sqrt(c^2*x^2 + 1)) - (3*(c^2*x^2 + 1)^(5/2) - 10*(c^2*x^2 + 1)^(3/2) + 15*sqrt(c^2*x
^2 + 1))/c^5)*a*b*d^2 + 1/1125*(225*x^5*log(c*x + sqrt(c^2*x^2 + 1))^2 + 2*c*((9*c^4*x^5 - 20*c^2*x^3 + 120*x)
/c^5 - 15*(3*(c^2*x^2 + 1)^(5/2) - 10*(c^2*x^2 + 1)^(3/2) + 15*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1))
/c^6))*b^2*d^2