3.216 \(\int x^4 (d+c^2 d x^2)^3 (a+b \sinh ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=465 \[ \frac{1}{11} d^3 x^5 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{2}{33} d^3 x^5 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{8}{231} d^3 x^5 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{32 b d^3 x^4 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{5775 c}+\frac{128 b d^3 x^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{17325 c^3}-\frac{2 b d^3 \left (c^2 x^2+1\right )^{11/2} \left (a+b \sinh ^{-1}(c x)\right )}{121 c^5}+\frac{8 b d^3 \left (c^2 x^2+1\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{297 c^5}-\frac{2 b d^3 \left (c^2 x^2+1\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{1617 c^5}+\frac{4 b d^3 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{1155 c^5}-\frac{16 b d^3 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{693 c^5}-\frac{256 b d^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{17325 c^5}+\frac{16 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2}{1155}+\frac{2 b^2 c^6 d^3 x^{11}}{1331}+\frac{182 b^2 c^4 d^3 x^9}{29403}+\frac{9410 b^2 c^2 d^3 x^7}{1120581}-\frac{50488 b^2 d^3 x^3}{12006225 c^2}+\frac{100976 b^2 d^3 x}{4002075 c^4}+\frac{12622 b^2 d^3 x^5}{6670125} \]

[Out]

(100976*b^2*d^3*x)/(4002075*c^4) - (50488*b^2*d^3*x^3)/(12006225*c^2) + (12622*b^2*d^3*x^5)/6670125 + (9410*b^
2*c^2*d^3*x^7)/1120581 + (182*b^2*c^4*d^3*x^9)/29403 + (2*b^2*c^6*d^3*x^11)/1331 - (256*b*d^3*Sqrt[1 + c^2*x^2
]*(a + b*ArcSinh[c*x]))/(17325*c^5) + (128*b*d^3*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(17325*c^3) - (32
*b*d^3*x^4*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(5775*c) - (16*b*d^3*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x
]))/(693*c^5) + (4*b*d^3*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(1155*c^5) - (2*b*d^3*(1 + c^2*x^2)^(7/2)*(
a + b*ArcSinh[c*x]))/(1617*c^5) + (8*b*d^3*(1 + c^2*x^2)^(9/2)*(a + b*ArcSinh[c*x]))/(297*c^5) - (2*b*d^3*(1 +
c^2*x^2)^(11/2)*(a + b*ArcSinh[c*x]))/(121*c^5) + (16*d^3*x^5*(a + b*ArcSinh[c*x])^2)/1155 + (8*d^3*x^5*(1 +
c^2*x^2)*(a + b*ArcSinh[c*x])^2)/231 + (2*d^3*x^5*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/33 + (d^3*x^5*(1 + c
^2*x^2)^3*(a + b*ArcSinh[c*x])^2)/11

________________________________________________________________________________________

Rubi [A]  time = 1.05765, antiderivative size = 465, normalized size of antiderivative = 1., number of steps used = 21, 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} \[ \frac{1}{11} d^3 x^5 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{2}{33} d^3 x^5 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{8}{231} d^3 x^5 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{32 b d^3 x^4 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{5775 c}+\frac{128 b d^3 x^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{17325 c^3}-\frac{2 b d^3 \left (c^2 x^2+1\right )^{11/2} \left (a+b \sinh ^{-1}(c x)\right )}{121 c^5}+\frac{8 b d^3 \left (c^2 x^2+1\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{297 c^5}-\frac{2 b d^3 \left (c^2 x^2+1\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{1617 c^5}+\frac{4 b d^3 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{1155 c^5}-\frac{16 b d^3 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{693 c^5}-\frac{256 b d^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{17325 c^5}+\frac{16 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2}{1155}+\frac{2 b^2 c^6 d^3 x^{11}}{1331}+\frac{182 b^2 c^4 d^3 x^9}{29403}+\frac{9410 b^2 c^2 d^3 x^7}{1120581}-\frac{50488 b^2 d^3 x^3}{12006225 c^2}+\frac{100976 b^2 d^3 x}{4002075 c^4}+\frac{12622 b^2 d^3 x^5}{6670125} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2,x]

[Out]

(100976*b^2*d^3*x)/(4002075*c^4) - (50488*b^2*d^3*x^3)/(12006225*c^2) + (12622*b^2*d^3*x^5)/6670125 + (9410*b^
2*c^2*d^3*x^7)/1120581 + (182*b^2*c^4*d^3*x^9)/29403 + (2*b^2*c^6*d^3*x^11)/1331 - (256*b*d^3*Sqrt[1 + c^2*x^2
]*(a + b*ArcSinh[c*x]))/(17325*c^5) + (128*b*d^3*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(17325*c^3) - (32
*b*d^3*x^4*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(5775*c) - (16*b*d^3*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x
]))/(693*c^5) + (4*b*d^3*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(1155*c^5) - (2*b*d^3*(1 + c^2*x^2)^(7/2)*(
a + b*ArcSinh[c*x]))/(1617*c^5) + (8*b*d^3*(1 + c^2*x^2)^(9/2)*(a + b*ArcSinh[c*x]))/(297*c^5) - (2*b*d^3*(1 +
c^2*x^2)^(11/2)*(a + b*ArcSinh[c*x]))/(121*c^5) + (16*d^3*x^5*(a + b*ArcSinh[c*x])^2)/1155 + (8*d^3*x^5*(1 +
c^2*x^2)*(a + b*ArcSinh[c*x])^2)/231 + (2*d^3*x^5*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/33 + (d^3*x^5*(1 + c
^2*x^2)^3*(a + b*ArcSinh[c*x])^2)/11

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 )^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac{1}{11} d^3 x^5 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{11} (6 d) \int x^4 \left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac{1}{11} \left (2 b c d^3\right ) \int x^5 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx\\ &=-\frac{2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{77 c^5}+\frac{4 b d^3 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{99 c^5}-\frac{2 b d^3 \left (1+c^2 x^2\right )^{11/2} \left (a+b \sinh ^{-1}(c x)\right )}{121 c^5}+\frac{2}{33} d^3 x^5 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{11} d^3 x^5 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{33} \left (8 d^2\right ) \int x^4 \left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac{1}{33} \left (4 b c d^3\right ) \int x^5 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx+\frac{1}{11} \left (2 b^2 c^2 d^3\right ) \int \frac{\left (1+c^2 x^2\right )^3 \left (8-28 c^2 x^2+63 c^4 x^4\right )}{693 c^6} \, dx\\ &=-\frac{4 b d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{165 c^5}+\frac{2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{231 c^5}+\frac{8 b d^3 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{297 c^5}-\frac{2 b d^3 \left (1+c^2 x^2\right )^{11/2} \left (a+b \sinh ^{-1}(c x)\right )}{121 c^5}+\frac{8}{231} d^3 x^5 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{2}{33} d^3 x^5 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{11} d^3 x^5 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{231} \left (16 d^3\right ) \int x^4 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+\frac{\left (2 b^2 d^3\right ) \int \left (1+c^2 x^2\right )^3 \left (8-28 c^2 x^2+63 c^4 x^4\right ) \, dx}{7623 c^4}-\frac{1}{231} \left (16 b c d^3\right ) \int x^5 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx+\frac{1}{33} \left (4 b^2 c^2 d^3\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{16 b d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{693 c^5}+\frac{4 b d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{1155 c^5}-\frac{2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{1617 c^5}+\frac{8 b d^3 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{297 c^5}-\frac{2 b d^3 \left (1+c^2 x^2\right )^{11/2} \left (a+b \sinh ^{-1}(c x)\right )}{121 c^5}+\frac{16 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2}{1155}+\frac{8}{231} d^3 x^5 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{2}{33} d^3 x^5 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{11} d^3 x^5 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{\left (2 b^2 d^3\right ) \int \left (8-4 c^2 x^2+3 c^4 x^4+113 c^6 x^6+161 c^8 x^8+63 c^{10} x^{10}\right ) \, dx}{7623 c^4}+\frac{\left (4 b^2 d^3\right ) \int \left (1+c^2 x^2\right )^2 \left (8-20 c^2 x^2+35 c^4 x^4\right ) \, dx}{10395 c^4}-\frac{\left (32 b c d^3\right ) \int \frac{x^5 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx}{1155}+\frac{1}{231} \left (16 b^2 c^2 d^3\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^2 d^3 x}{7623 c^4}-\frac{8 b^2 d^3 x^3}{22869 c^2}+\frac{2 b^2 d^3 x^5}{12705}+\frac{226 b^2 c^2 d^3 x^7}{53361}+\frac{46 b^2 c^4 d^3 x^9}{9801}+\frac{2 b^2 c^6 d^3 x^{11}}{1331}-\frac{32 b d^3 x^4 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5775 c}-\frac{16 b d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{693 c^5}+\frac{4 b d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{1155 c^5}-\frac{2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{1617 c^5}+\frac{8 b d^3 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{297 c^5}-\frac{2 b d^3 \left (1+c^2 x^2\right )^{11/2} \left (a+b \sinh ^{-1}(c x)\right )}{121 c^5}+\frac{16 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2}{1155}+\frac{8}{231} d^3 x^5 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{2}{33} d^3 x^5 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{11} d^3 x^5 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{\left (32 b^2 d^3\right ) \int x^4 \, dx}{5775}+\frac{\left (4 b^2 d^3\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}{10395 c^4}+\frac{\left (16 b^2 d^3\right ) \int \left (8-4 c^2 x^2+3 c^4 x^4+15 c^6 x^6\right ) \, dx}{24255 c^4}+\frac{\left (128 b d^3\right ) \int \frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx}{5775 c}\\ &=\frac{8368 b^2 d^3 x}{800415 c^4}-\frac{4184 b^2 d^3 x^3}{2401245 c^2}+\frac{12622 b^2 d^3 x^5}{6670125}+\frac{9410 b^2 c^2 d^3 x^7}{1120581}+\frac{182 b^2 c^4 d^3 x^9}{29403}+\frac{2 b^2 c^6 d^3 x^{11}}{1331}+\frac{128 b d^3 x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{17325 c^3}-\frac{32 b d^3 x^4 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5775 c}-\frac{16 b d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{693 c^5}+\frac{4 b d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{1155 c^5}-\frac{2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{1617 c^5}+\frac{8 b d^3 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{297 c^5}-\frac{2 b d^3 \left (1+c^2 x^2\right )^{11/2} \left (a+b \sinh ^{-1}(c x)\right )}{121 c^5}+\frac{16 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2}{1155}+\frac{8}{231} d^3 x^5 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{2}{33} d^3 x^5 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{11} d^3 x^5 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{\left (256 b d^3\right ) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx}{17325 c^3}-\frac{\left (128 b^2 d^3\right ) \int x^2 \, dx}{17325 c^2}\\ &=\frac{8368 b^2 d^3 x}{800415 c^4}-\frac{50488 b^2 d^3 x^3}{12006225 c^2}+\frac{12622 b^2 d^3 x^5}{6670125}+\frac{9410 b^2 c^2 d^3 x^7}{1120581}+\frac{182 b^2 c^4 d^3 x^9}{29403}+\frac{2 b^2 c^6 d^3 x^{11}}{1331}-\frac{256 b d^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{17325 c^5}+\frac{128 b d^3 x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{17325 c^3}-\frac{32 b d^3 x^4 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5775 c}-\frac{16 b d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{693 c^5}+\frac{4 b d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{1155 c^5}-\frac{2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{1617 c^5}+\frac{8 b d^3 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{297 c^5}-\frac{2 b d^3 \left (1+c^2 x^2\right )^{11/2} \left (a+b \sinh ^{-1}(c x)\right )}{121 c^5}+\frac{16 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2}{1155}+\frac{8}{231} d^3 x^5 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{2}{33} d^3 x^5 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{11} d^3 x^5 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{\left (256 b^2 d^3\right ) \int 1 \, dx}{17325 c^4}\\ &=\frac{100976 b^2 d^3 x}{4002075 c^4}-\frac{50488 b^2 d^3 x^3}{12006225 c^2}+\frac{12622 b^2 d^3 x^5}{6670125}+\frac{9410 b^2 c^2 d^3 x^7}{1120581}+\frac{182 b^2 c^4 d^3 x^9}{29403}+\frac{2 b^2 c^6 d^3 x^{11}}{1331}-\frac{256 b d^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{17325 c^5}+\frac{128 b d^3 x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{17325 c^3}-\frac{32 b d^3 x^4 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5775 c}-\frac{16 b d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{693 c^5}+\frac{4 b d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{1155 c^5}-\frac{2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{1617 c^5}+\frac{8 b d^3 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{297 c^5}-\frac{2 b d^3 \left (1+c^2 x^2\right )^{11/2} \left (a+b \sinh ^{-1}(c x)\right )}{121 c^5}+\frac{16 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2}{1155}+\frac{8}{231} d^3 x^5 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{2}{33} d^3 x^5 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{11} d^3 x^5 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2\\ \end{align*}

Mathematica [A]  time = 0.461173, size = 299, normalized size = 0.64 \[ \frac{d^3 \left (12006225 a^2 c^5 x^5 \left (105 c^6 x^6+385 c^4 x^4+495 c^2 x^2+231\right )-6930 a b \sqrt{c^2 x^2+1} \left (33075 c^{10} x^{10}+111475 c^8 x^8+117625 c^6 x^6+18933 c^4 x^4-25244 c^2 x^2+50488\right )-6930 b \sinh ^{-1}(c x) \left (b \sqrt{c^2 x^2+1} \left (33075 c^{10} x^{10}+111475 c^8 x^8+117625 c^6 x^6+18933 c^4 x^4-25244 c^2 x^2+50488\right )-3465 a c^5 x^5 \left (105 c^6 x^6+385 c^4 x^4+495 c^2 x^2+231\right )\right )+2 b^2 c x \left (10418625 c^{10} x^{10}[phone] c^8 x^8+58224375 c^6 x^6+13120569 c^4 x^4-29156820 c^2 x^2+174940920\right )[phone] b^2 c^5 x^5 \left (105 c^6 x^6+385 c^4 x^4+495 c^2 x^2+231\right ) \sinh ^{-1}(c x)^2\right )}{13867189875 c^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2,x]

[Out]

(d^3*(12006225*a^2*c^5*x^5*(231 + 495*c^2*x^2 + 385*c^4*x^4 + 105*c^6*x^6) - 6930*a*b*Sqrt[1 + c^2*x^2]*(50488
- 25244*c^2*x^2 + 18933*c^4*x^4 + 117625*c^6*x^6 + 111475*c^8*x^8 + 33075*c^10*x^10) + 2*b^2*c*x*(174940920 -
29156820*c^2*x^2 + 13120569*c^4*x^4 + 58224375*c^6*x^6 + 42917875*c^8*x^8 + 10418625*c^10*x^10) - 6930*b*(-34
65*a*c^5*x^5*(231 + 495*c^2*x^2 + 385*c^4*x^4 + 105*c^6*x^6) + b*Sqrt[1 + c^2*x^2]*(50488 - 25244*c^2*x^2 + 18
933*c^4*x^4 + 117625*c^6*x^6 + 111475*c^8*x^8 + 33075*c^10*x^10))*ArcSinh[c*x] + 12006225*b^2*c^5*x^5*(231 + 4
95*c^2*x^2 + 385*c^4*x^4 + 105*c^6*x^6)*ArcSinh[c*x]^2))/(13867189875*c^5)

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Maple [A]  time = 0.092, size = 544, normalized size = 1.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2,x)

[Out]

1/c^5*(d^3*a^2*(1/11*c^11*x^11+1/3*c^9*x^9+3/7*c^7*x^7+1/5*c^5*x^5)+d^3*b^2*(1/11*arcsinh(c*x)^2*c^3*x^3*(c^2*
x^2+1)^4-1/33*arcsinh(c*x)^2*c*x*(c^2*x^2+1)^4+16/1155*arcsinh(c*x)^2*c*x+1/231*arcsinh(c*x)^2*c*x*(c^2*x^2+1)
^3+2/385*arcsinh(c*x)^2*c*x*(c^2*x^2+1)^2+8/1155*arcsinh(c*x)^2*c*x*(c^2*x^2+1)-5487704/4622396625*c*x*(c^2*x^
2+1)^2-100976/4002075*arcsinh(c*x)*(c^2*x^2+1)^(1/2)-606416/13867189875*c*x*(c^2*x^2+1)-428/323433*c*x*(c^2*x^
2+1)^4-148174/110937519*c*x*(c^2*x^2+1)^3+382986368/13867189875*c*x+34/3267*arcsinh(c*x)*c^2*x^2*(c^2*x^2+1)^(
7/2)+1468/160083*arcsinh(c*x)*c^2*x^2*(c^2*x^2+1)^(5/2)+2/1331*c*x*(c^2*x^2+1)^5-2/121*arcsinh(c*x)*c^2*x^2*(c
^2*x^2+1)^(9/2)+28384/4002075*arcsinh(c*x)*c^2*x^2*(c^2*x^2+1)^(3/2)+9904/4002075*arcsinh(c*x)*c^2*x^2*(c^2*x^
2+1)^(1/2))+2*d^3*a*b*(1/11*arcsinh(c*x)*c^11*x^11+1/3*arcsinh(c*x)*c^9*x^9+3/7*arcsinh(c*x)*c^7*x^7+1/5*arcsi
nh(c*x)*c^5*x^5-1/121*c^10*x^10*(c^2*x^2+1)^(1/2)-91/3267*c^8*x^8*(c^2*x^2+1)^(1/2)-4705/160083*c^6*x^6*(c^2*x
^2+1)^(1/2)-6311/1334025*c^4*x^4*(c^2*x^2+1)^(1/2)+25244/4002075*c^2*x^2*(c^2*x^2+1)^(1/2)-50488/4002075*(c^2*
x^2+1)^(1/2)))

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Maxima [B]  time = 1.24128, size = 1497, normalized size = 3.22 \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)^3*(a+b*arcsinh(c*x))^2,x, algorithm="maxima")

[Out]

1/11*b^2*c^6*d^3*x^11*arcsinh(c*x)^2 + 1/11*a^2*c^6*d^3*x^11 + 1/3*b^2*c^4*d^3*x^9*arcsinh(c*x)^2 + 1/3*a^2*c^
4*d^3*x^9 + 3/7*b^2*c^2*d^3*x^7*arcsinh(c*x)^2 + 3/7*a^2*c^2*d^3*x^7 + 2/7623*(693*x^11*arcsinh(c*x) - (63*sqr
t(c^2*x^2 + 1)*x^10/c^2 - 70*sqrt(c^2*x^2 + 1)*x^8/c^4 + 80*sqrt(c^2*x^2 + 1)*x^6/c^6 - 96*sqrt(c^2*x^2 + 1)*x
^4/c^8 + 128*sqrt(c^2*x^2 + 1)*x^2/c^10 - 256*sqrt(c^2*x^2 + 1)/c^12)*c)*a*b*c^6*d^3 - 2/26413695*(3465*(63*sq
rt(c^2*x^2 + 1)*x^10/c^2 - 70*sqrt(c^2*x^2 + 1)*x^8/c^4 + 80*sqrt(c^2*x^2 + 1)*x^6/c^6 - 96*sqrt(c^2*x^2 + 1)*
x^4/c^8 + 128*sqrt(c^2*x^2 + 1)*x^2/c^10 - 256*sqrt(c^2*x^2 + 1)/c^12)*c*arcsinh(c*x) - (19845*c^10*x^11 - 269
50*c^8*x^9 + 39600*c^6*x^7 - 66528*c^4*x^5 + 147840*c^2*x^3 - 887040*x)/c^10)*b^2*c^6*d^3 + 1/5*b^2*d^3*x^5*ar
csinh(c*x)^2 + 2/945*(315*x^9*arcsinh(c*x) - (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)*a*b*c^4*d^3 - 2/297
675*(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^3 + 1/5*a^2*d^3*x^5 + 6/245*(35*x^7*arcsinh(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^3 - 2/8575*(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^
3 + 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^3 - 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^3

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Fricas [A]  time = 2.87466, size = 1135, normalized size = 2.44 \begin{align*} \frac{10418625 \,{\left (121 \, a^{2} + 2 \, b^{2}\right )} c^{11} d^{3} x^{11} + 471625 \,{\left (9801 \, a^{2} + 182 \, b^{2}\right )} c^{9} d^{3} x^{9} + 12375 \,{\left (480249 \, a^{2} + 9410 \, b^{2}\right )} c^{7} d^{3} x^{7} + 2079 \,{\left (1334025 \, a^{2} + 12622 \, b^{2}\right )} c^{5} d^{3} x^{5} - 58313640 \, b^{2} c^{3} d^{3} x^{3} + 349881840 \, b^{2} c d^{3} x + 12006225 \,{\left (105 \, b^{2} c^{11} d^{3} x^{11} + 385 \, b^{2} c^{9} d^{3} x^{9} + 495 \, b^{2} c^{7} d^{3} x^{7} + 231 \, b^{2} c^{5} d^{3} x^{5}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 6930 \,{\left (363825 \, a b c^{11} d^{3} x^{11} + 1334025 \, a b c^{9} d^{3} x^{9} + 1715175 \, a b c^{7} d^{3} x^{7} + 800415 \, a b c^{5} d^{3} x^{5} -{\left (33075 \, b^{2} c^{10} d^{3} x^{10} + 111475 \, b^{2} c^{8} d^{3} x^{8} + 117625 \, b^{2} c^{6} d^{3} x^{6} + 18933 \, b^{2} c^{4} d^{3} x^{4} - 25244 \, b^{2} c^{2} d^{3} x^{2} + 50488 \, b^{2} d^{3}\right )} \sqrt{c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - 6930 \,{\left (33075 \, a b c^{10} d^{3} x^{10} + 111475 \, a b c^{8} d^{3} x^{8} + 117625 \, a b c^{6} d^{3} x^{6} + 18933 \, a b c^{4} d^{3} x^{4} - 25244 \, a b c^{2} d^{3} x^{2} + 50488 \, a b d^{3}\right )} \sqrt{c^{2} x^{2} + 1}}{13867189875 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2,x, algorithm="fricas")

[Out]

1/13867189875*(10418625*(121*a^2 + 2*b^2)*c^11*d^3*x^11 + 471625*(9801*a^2 + 182*b^2)*c^9*d^3*x^9 + 12375*(480
249*a^2 + 9410*b^2)*c^7*d^3*x^7 + 2079*(1334025*a^2 + 12622*b^2)*c^5*d^3*x^5 - 58313640*b^2*c^3*d^3*x^3 + 3498
81840*b^2*c*d^3*x + 12006225*(105*b^2*c^11*d^3*x^11 + 385*b^2*c^9*d^3*x^9 + 495*b^2*c^7*d^3*x^7 + 231*b^2*c^5*
d^3*x^5)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 6930*(363825*a*b*c^11*d^3*x^11 + 1334025*a*b*c^9*d^3*x^9 + 1715175*a
*b*c^7*d^3*x^7 + 800415*a*b*c^5*d^3*x^5 - (33075*b^2*c^10*d^3*x^10 + 111475*b^2*c^8*d^3*x^8 + 117625*b^2*c^6*d
^3*x^6 + 18933*b^2*c^4*d^3*x^4 - 25244*b^2*c^2*d^3*x^2 + 50488*b^2*d^3)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*
x^2 + 1)) - 6930*(33075*a*b*c^10*d^3*x^10 + 111475*a*b*c^8*d^3*x^8 + 117625*a*b*c^6*d^3*x^6 + 18933*a*b*c^4*d^
3*x^4 - 25244*a*b*c^2*d^3*x^2 + 50488*a*b*d^3)*sqrt(c^2*x^2 + 1))/c^5

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Sympy [A]  time = 115.345, size = 702, normalized size = 1.51 \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)**3*(a+b*asinh(c*x))**2,x)

[Out]

Piecewise((a**2*c**6*d**3*x**11/11 + a**2*c**4*d**3*x**9/3 + 3*a**2*c**2*d**3*x**7/7 + a**2*d**3*x**5/5 + 2*a*
b*c**6*d**3*x**11*asinh(c*x)/11 - 2*a*b*c**5*d**3*x**10*sqrt(c**2*x**2 + 1)/121 + 2*a*b*c**4*d**3*x**9*asinh(c
*x)/3 - 182*a*b*c**3*d**3*x**8*sqrt(c**2*x**2 + 1)/3267 + 6*a*b*c**2*d**3*x**7*asinh(c*x)/7 - 9410*a*b*c*d**3*
x**6*sqrt(c**2*x**2 + 1)/160083 + 2*a*b*d**3*x**5*asinh(c*x)/5 - 12622*a*b*d**3*x**4*sqrt(c**2*x**2 + 1)/(1334
025*c) + 50488*a*b*d**3*x**2*sqrt(c**2*x**2 + 1)/(4002075*c**3) - 100976*a*b*d**3*sqrt(c**2*x**2 + 1)/(4002075
*c**5) + b**2*c**6*d**3*x**11*asinh(c*x)**2/11 + 2*b**2*c**6*d**3*x**11/1331 - 2*b**2*c**5*d**3*x**10*sqrt(c**
2*x**2 + 1)*asinh(c*x)/121 + b**2*c**4*d**3*x**9*asinh(c*x)**2/3 + 182*b**2*c**4*d**3*x**9/29403 - 182*b**2*c*
*3*d**3*x**8*sqrt(c**2*x**2 + 1)*asinh(c*x)/3267 + 3*b**2*c**2*d**3*x**7*asinh(c*x)**2/7 + 9410*b**2*c**2*d**3
*x**7/1120581 - 9410*b**2*c*d**3*x**6*sqrt(c**2*x**2 + 1)*asinh(c*x)/160083 + b**2*d**3*x**5*asinh(c*x)**2/5 +
12622*b**2*d**3*x**5/6670125 - 12622*b**2*d**3*x**4*sqrt(c**2*x**2 + 1)*asinh(c*x)/(1334025*c) - 50488*b**2*d
**3*x**3/(12006225*c**2) + 50488*b**2*d**3*x**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/(4002075*c**3) + 100976*b**2*d*
*3*x/(4002075*c**4) - 100976*b**2*d**3*sqrt(c**2*x**2 + 1)*asinh(c*x)/(4002075*c**5), Ne(c, 0)), (a**2*d**3*x*
*5/5, True))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError