3.3.18 \(\int x^2 (d+c^2 d x^2)^3 (a+b \sinh ^{-1}(c x))^2 \, dx\) [218]

Optimal. Leaf size=382 \[ -\frac {10516 b^2 d^3 x}{99225 c^2}+\frac {5258 b^2 d^3 x^3}{297675}+\frac {4198 b^2 c^2 d^3 x^5}{165375}+\frac {374 b^2 c^4 d^3 x^7}{27783}+\frac {2}{729} b^2 c^6 d^3 x^9+\frac {64 b d^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{945 c^3}-\frac {32 b d^3 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{945 c}+\frac {16 b d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{315 c^3}+\frac {4 b d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{525 c^3}+\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{441 c^3}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{81 c^3}+\frac {16}{315} d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {8}{105} d^3 x^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{21} d^3 x^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{9} d^3 x^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \]

[Out]

-10516/99225*b^2*d^3*x/c^2+5258/297675*b^2*d^3*x^3+4198/165375*b^2*c^2*d^3*x^5+374/27783*b^2*c^4*d^3*x^7+2/729
*b^2*c^6*d^3*x^9+16/315*b*d^3*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))/c^3+4/525*b*d^3*(c^2*x^2+1)^(5/2)*(a+b*arcs
inh(c*x))/c^3+2/441*b*d^3*(c^2*x^2+1)^(7/2)*(a+b*arcsinh(c*x))/c^3-2/81*b*d^3*(c^2*x^2+1)^(9/2)*(a+b*arcsinh(c
*x))/c^3+16/315*d^3*x^3*(a+b*arcsinh(c*x))^2+8/105*d^3*x^3*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2+2/21*d^3*x^3*(c^2*
x^2+1)^2*(a+b*arcsinh(c*x))^2+1/9*d^3*x^3*(c^2*x^2+1)^3*(a+b*arcsinh(c*x))^2+64/945*b*d^3*(a+b*arcsinh(c*x))*(
c^2*x^2+1)^(1/2)/c^3-32/945*b*d^3*x^2*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c

________________________________________________________________________________________

Rubi [A]
time = 0.59, antiderivative size = 382, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 11, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {5808, 5776, 5812, 5798, 8, 30, 272, 45, 5804, 12, 380} \begin {gather*} -\frac {32 b d^3 x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{945 c}+\frac {1}{9} d^3 x^3 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{21} d^3 x^3 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {8}{105} d^3 x^3 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {2 b d^3 \left (c^2 x^2+1\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{81 c^3}+\frac {2 b d^3 \left (c^2 x^2+1\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{441 c^3}+\frac {4 b d^3 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{525 c^3}+\frac {16 b d^3 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{315 c^3}+\frac {64 b d^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{945 c^3}+\frac {16}{315} d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{729} b^2 c^6 d^3 x^9+\frac {374 b^2 c^4 d^3 x^7}{27783}+\frac {4198 b^2 c^2 d^3 x^5}{165375}-\frac {10516 b^2 d^3 x}{99225 c^2}+\frac {5258 b^2 d^3 x^3}{297675} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-10516*b^2*d^3*x)/(99225*c^2) + (5258*b^2*d^3*x^3)/297675 + (4198*b^2*c^2*d^3*x^5)/165375 + (374*b^2*c^4*d^3*
x^7)/27783 + (2*b^2*c^6*d^3*x^9)/729 + (64*b*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(945*c^3) - (32*b*d^3
*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(945*c) + (16*b*d^3*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(31
5*c^3) + (4*b*d^3*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(525*c^3) + (2*b*d^3*(1 + c^2*x^2)^(7/2)*(a + b*Ar
cSinh[c*x]))/(441*c^3) - (2*b*d^3*(1 + c^2*x^2)^(9/2)*(a + b*ArcSinh[c*x]))/(81*c^3) + (16*d^3*x^3*(a + b*ArcS
inh[c*x])^2)/315 + (8*d^3*x^3*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/105 + (2*d^3*x^3*(1 + c^2*x^2)^2*(a + b*Ar
cSinh[c*x])^2)/21 + (d^3*x^3*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x])^2)/9

Rule 8

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

Rule 12

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

Rule 30

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

Rule 45

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 272

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 380

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

Rule 5776

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcS
inh[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[
1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5798

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/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)
^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 5804

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

Rule 5808

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

Rule 5812

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

Rubi steps

\begin {align*} \int x^2 \left (d+c^2 d x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac {1}{9} d^3 x^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{3} (2 d) \int x^2 \left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac {1}{9} \left (2 b c d^3\right ) \int x^3 \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 )}{63 c^3}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{81 c^3}+\frac {2}{21} d^3 x^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{9} d^3 x^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{21} \left (8 d^2\right ) \int x^2 \left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac {1}{21} \left (4 b c d^3\right ) \int x^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx+\frac {1}{9} \left (2 b^2 c^2 d^3\right ) \int \frac {\left (1+c^2 x^2\right )^3 \left (-2+7 c^2 x^2\right )}{63 c^4} \, dx\\ &=\frac {4 b d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{105 c^3}+\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{441 c^3}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{81 c^3}+\frac {8}{105} d^3 x^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{21} d^3 x^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{9} d^3 x^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{105} \left (16 d^3\right ) \int x^2 \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 (-2+7 c^2 x^2\right ) \, dx}{567 c^2}-\frac {1}{105} \left (16 b c d^3\right ) \int x^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx+\frac {1}{21} \left (4 b^2 c^2 d^3\right ) \int \frac {\left (1+c^2 x^2\right )^2 \left (-2+5 c^2 x^2\right )}{35 c^4} \, dx\\ &=\frac {16 b d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{315 c^3}+\frac {4 b d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{525 c^3}+\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{441 c^3}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{81 c^3}+\frac {16}{315} d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {8}{105} d^3 x^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{21} d^3 x^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{9} d^3 x^3 \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 (-2+c^2 x^2+15 c^4 x^4+19 c^6 x^6+7 c^8 x^8\right ) \, dx}{567 c^2}+\frac {\left (4 b^2 d^3\right ) \int \left (1+c^2 x^2\right )^2 \left (-2+5 c^2 x^2\right ) \, dx}{735 c^2}-\frac {1}{315} \left (32 b c d^3\right ) \int \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx+\frac {1}{105} \left (16 b^2 c^2 d^3\right ) \int \frac {-2+c^2 x^2+3 c^4 x^4}{15 c^4} \, dx\\ &=-\frac {4 b^2 d^3 x}{567 c^2}+\frac {2 b^2 d^3 x^3}{1701}+\frac {2}{189} b^2 c^2 d^3 x^5+\frac {38 b^2 c^4 d^3 x^7}{3969}+\frac {2}{729} b^2 c^6 d^3 x^9-\frac {32 b d^3 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{945 c}+\frac {16 b d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{315 c^3}+\frac {4 b d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{525 c^3}+\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{441 c^3}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{81 c^3}+\frac {16}{315} d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {8}{105} d^3 x^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{21} d^3 x^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{9} d^3 x^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{945} \left (32 b^2 d^3\right ) \int x^2 \, dx+\frac {\left (4 b^2 d^3\right ) \int \left (-2+c^2 x^2+8 c^4 x^4+5 c^6 x^6\right ) \, dx}{735 c^2}+\frac {\left (16 b^2 d^3\right ) \int \left (-2+c^2 x^2+3 c^4 x^4\right ) \, dx}{1575 c^2}+\frac {\left (64 b d^3\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{945 c}\\ &=-\frac {3796 b^2 d^3 x}{99225 c^2}+\frac {5258 b^2 d^3 x^3}{297675}+\frac {4198 b^2 c^2 d^3 x^5}{165375}+\frac {374 b^2 c^4 d^3 x^7}{27783}+\frac {2}{729} b^2 c^6 d^3 x^9+\frac {64 b d^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{945 c^3}-\frac {32 b d^3 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{945 c}+\frac {16 b d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{315 c^3}+\frac {4 b d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{525 c^3}+\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{441 c^3}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{81 c^3}+\frac {16}{315} d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {8}{105} d^3 x^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{21} d^3 x^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{9} d^3 x^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {\left (64 b^2 d^3\right ) \int 1 \, dx}{945 c^2}\\ &=-\frac {10516 b^2 d^3 x}{99225 c^2}+\frac {5258 b^2 d^3 x^3}{297675}+\frac {4198 b^2 c^2 d^3 x^5}{165375}+\frac {374 b^2 c^4 d^3 x^7}{27783}+\frac {2}{729} b^2 c^6 d^3 x^9+\frac {64 b d^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{945 c^3}-\frac {32 b d^3 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{945 c}+\frac {16 b d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{315 c^3}+\frac {4 b d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{525 c^3}+\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{441 c^3}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{81 c^3}+\frac {16}{315} d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {8}{105} d^3 x^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{21} d^3 x^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{9} d^3 x^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2\\ \end {align*}

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Mathematica [A]
time = 0.28, size = 275, normalized size = 0.72 \begin {gather*} \frac {d^3 \left (99225 a^2 c^3 x^3 \left (105+189 c^2 x^2+135 c^4 x^4+35 c^6 x^6\right )-630 a b \sqrt {1+c^2 x^2} \left (-5258+2629 c^2 x^2+6297 c^4 x^4+4675 c^6 x^6+1225 c^8 x^8\right )+b^2 \left (-3312540 c x+552090 c^3 x^3+793422 c^5 x^5+420750 c^7 x^7+85750 c^9 x^9\right )-630 b \left (-315 a c^3 x^3 \left (105+189 c^2 x^2+135 c^4 x^4+35 c^6 x^6\right )+b \sqrt {1+c^2 x^2} \left (-5258+2629 c^2 x^2+6297 c^4 x^4+4675 c^6 x^6+1225 c^8 x^8\right )\right ) \sinh ^{-1}(c x)+99225 b^2 c^3 x^3 \left (105+189 c^2 x^2+135 c^4 x^4+35 c^6 x^6\right ) \sinh ^{-1}(c x)^2\right )}{31255875 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d^3*(99225*a^2*c^3*x^3*(105 + 189*c^2*x^2 + 135*c^4*x^4 + 35*c^6*x^6) - 630*a*b*Sqrt[1 + c^2*x^2]*(-5258 + 26
29*c^2*x^2 + 6297*c^4*x^4 + 4675*c^6*x^6 + 1225*c^8*x^8) + b^2*(-3312540*c*x + 552090*c^3*x^3 + 793422*c^5*x^5
+ 420750*c^7*x^7 + 85750*c^9*x^9) - 630*b*(-315*a*c^3*x^3*(105 + 189*c^2*x^2 + 135*c^4*x^4 + 35*c^6*x^6) + b*
Sqrt[1 + c^2*x^2]*(-5258 + 2629*c^2*x^2 + 6297*c^4*x^4 + 4675*c^6*x^6 + 1225*c^8*x^8))*ArcSinh[c*x] + 99225*b^
2*c^3*x^3*(105 + 189*c^2*x^2 + 135*c^4*x^4 + 35*c^6*x^6)*ArcSinh[c*x]^2))/(31255875*c^3)

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int x^{2} \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arcsinh \left (c x \right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 922 vs. \(2 (340) = 680\).
time = 0.34, size = 922, normalized size = 2.41 \begin {gather*} \frac {1}{9} \, b^{2} c^{6} d^{3} x^{9} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{9} \, a^{2} c^{6} d^{3} x^{9} + \frac {3}{7} \, b^{2} c^{4} d^{3} x^{7} \operatorname {arsinh}\left (c x\right )^{2} + \frac {3}{7} \, a^{2} c^{4} d^{3} x^{7} + \frac {3}{5} \, b^{2} c^{2} d^{3} x^{5} \operatorname {arsinh}\left (c x\right )^{2} + \frac {2}{2835} \, {\left (315 \, x^{9} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {35 \, \sqrt {c^{2} x^{2} + 1} x^{8}}{c^{2}} - \frac {40 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{6}} - \frac {64 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} + 1}}{c^{10}}\right )} c\right )} a b c^{6} d^{3} - \frac {2}{893025} \, {\left (315 \, {\left (\frac {35 \, \sqrt {c^{2} x^{2} + 1} x^{8}}{c^{2}} - \frac {40 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{6}} - \frac {64 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} + 1}}{c^{10}}\right )} c \operatorname {arsinh}\left (c x\right ) - \frac {1225 \, c^{8} x^{9} - 1800 \, c^{6} x^{7} + 3024 \, c^{4} x^{5} - 6720 \, c^{2} x^{3} + 40320 \, x}{c^{8}}\right )} b^{2} c^{6} d^{3} + \frac {3}{5} \, a^{2} c^{2} d^{3} x^{5} + \frac {6}{245} \, {\left (35 \, x^{7} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{2}} - \frac {6 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{6}} - \frac {16 \, \sqrt {c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} a b c^{4} d^{3} - \frac {2}{8575} \, {\left (105 \, {\left (\frac {5 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{2}} - \frac {6 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{6}} - \frac {16 \, \sqrt {c^{2} x^{2} + 1}}{c^{8}}\right )} c \operatorname {arsinh}\left (c x\right ) - \frac {75 \, c^{6} x^{7} - 126 \, c^{4} x^{5} + 280 \, c^{2} x^{3} - 1680 \, x}{c^{6}}\right )} b^{2} c^{4} d^{3} + \frac {1}{3} \, b^{2} d^{3} x^{3} \operatorname {arsinh}\left (c x\right )^{2} + \frac {2}{25} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b c^{2} d^{3} - \frac {2}{375} \, {\left (15 \, {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c \operatorname {arsinh}\left (c x\right ) - \frac {9 \, c^{4} x^{5} - 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} c^{2} d^{3} + \frac {1}{3} \, a^{2} d^{3} x^{3} + \frac {2}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b d^{3} - \frac {2}{27} \, {\left (3 \, c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )} \operatorname {arsinh}\left (c x\right ) - \frac {c^{2} x^{3} - 6 \, x}{c^{2}}\right )} b^{2} d^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b^2*c^6*d^3*x^9*arcsinh(c*x)^2 + 1/9*a^2*c^6*d^3*x^9 + 3/7*b^2*c^4*d^3*x^7*arcsinh(c*x)^2 + 3/7*a^2*c^4*d^
3*x^7 + 3/5*b^2*c^2*d^3*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
*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^6*d^3 - 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^6*d^3 + 3/5*a^2*c^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^4*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 + 28
0*c^2*x^3 - 1680*x)/c^6)*b^2*c^4*d^3 + 1/3*b^2*d^3*x^3*arcsinh(c*x)^2 + 2/25*(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*c^2*d^3 - 2/375*(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 - 2
0*c^2*x^3 + 120*x)/c^4)*b^2*c^2*d^3 + 1/3*a^2*d^3*x^3 + 2/9*(3*x^3*arcsinh(c*x) - c*(sqrt(c^2*x^2 + 1)*x^2/c^2
- 2*sqrt(c^2*x^2 + 1)/c^4))*a*b*d^3 - 2/27*(3*c*(sqrt(c^2*x^2 + 1)*x^2/c^2 - 2*sqrt(c^2*x^2 + 1)/c^4)*arcsinh
(c*x) - (c^2*x^3 - 6*x)/c^2)*b^2*d^3

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Fricas [A]
time = 0.43, size = 403, normalized size = 1.05 \begin {gather*} \frac {42875 \, {\left (81 \, a^{2} + 2 \, b^{2}\right )} c^{9} d^{3} x^{9} + 1125 \, {\left (11907 \, a^{2} + 374 \, b^{2}\right )} c^{7} d^{3} x^{7} + 189 \, {\left (99225 \, a^{2} + 4198 \, b^{2}\right )} c^{5} d^{3} x^{5} + 105 \, {\left (99225 \, a^{2} + 5258 \, b^{2}\right )} c^{3} d^{3} x^{3} - 3312540 \, b^{2} c d^{3} x + 99225 \, {\left (35 \, b^{2} c^{9} d^{3} x^{9} + 135 \, b^{2} c^{7} d^{3} x^{7} + 189 \, b^{2} c^{5} d^{3} x^{5} + 105 \, b^{2} c^{3} d^{3} x^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 630 \, {\left (11025 \, a b c^{9} d^{3} x^{9} + 42525 \, a b c^{7} d^{3} x^{7} + 59535 \, a b c^{5} d^{3} x^{5} + 33075 \, a b c^{3} d^{3} x^{3} - {\left (1225 \, b^{2} c^{8} d^{3} x^{8} + 4675 \, b^{2} c^{6} d^{3} x^{6} + 6297 \, b^{2} c^{4} d^{3} x^{4} + 2629 \, b^{2} c^{2} d^{3} x^{2} - 5258 \, b^{2} d^{3}\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^{3} x^{8} + 4675 \, a b c^{6} d^{3} x^{6} + 6297 \, a b c^{4} d^{3} x^{4} + 2629 \, a b c^{2} d^{3} x^{2} - 5258 \, a b d^{3}\right )} \sqrt {c^{2} x^{2} + 1}}{31255875 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/31255875*(42875*(81*a^2 + 2*b^2)*c^9*d^3*x^9 + 1125*(11907*a^2 + 374*b^2)*c^7*d^3*x^7 + 189*(99225*a^2 + 419
8*b^2)*c^5*d^3*x^5 + 105*(99225*a^2 + 5258*b^2)*c^3*d^3*x^3 - 3312540*b^2*c*d^3*x + 99225*(35*b^2*c^9*d^3*x^9
+ 135*b^2*c^7*d^3*x^7 + 189*b^2*c^5*d^3*x^5 + 105*b^2*c^3*d^3*x^3)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 630*(11025
*a*b*c^9*d^3*x^9 + 42525*a*b*c^7*d^3*x^7 + 59535*a*b*c^5*d^3*x^5 + 33075*a*b*c^3*d^3*x^3 - (1225*b^2*c^8*d^3*x
^8 + 4675*b^2*c^6*d^3*x^6 + 6297*b^2*c^4*d^3*x^4 + 2629*b^2*c^2*d^3*x^2 - 5258*b^2*d^3)*sqrt(c^2*x^2 + 1))*log
(c*x + sqrt(c^2*x^2 + 1)) - 630*(1225*a*b*c^8*d^3*x^8 + 4675*a*b*c^6*d^3*x^6 + 6297*a*b*c^4*d^3*x^4 + 2629*a*b
*c^2*d^3*x^2 - 5258*a*b*d^3)*sqrt(c^2*x^2 + 1))/c^3

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Sympy [A]
time = 2.17, size = 626, normalized size = 1.64 \begin {gather*} \begin {cases} \frac {a^{2} c^{6} d^{3} x^{9}}{9} + \frac {3 a^{2} c^{4} d^{3} x^{7}}{7} + \frac {3 a^{2} c^{2} d^{3} x^{5}}{5} + \frac {a^{2} d^{3} x^{3}}{3} + \frac {2 a b c^{6} d^{3} x^{9} \operatorname {asinh}{\left (c x \right )}}{9} - \frac {2 a b c^{5} d^{3} x^{8} \sqrt {c^{2} x^{2} + 1}}{81} + \frac {6 a b c^{4} d^{3} x^{7} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {374 a b c^{3} d^{3} x^{6} \sqrt {c^{2} x^{2} + 1}}{3969} + \frac {6 a b c^{2} d^{3} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {4198 a b c d^{3} x^{4} \sqrt {c^{2} x^{2} + 1}}{33075} + \frac {2 a b d^{3} x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {5258 a b d^{3} x^{2} \sqrt {c^{2} x^{2} + 1}}{99225 c} + \frac {10516 a b d^{3} \sqrt {c^{2} x^{2} + 1}}{99225 c^{3}} + \frac {b^{2} c^{6} d^{3} x^{9} \operatorname {asinh}^{2}{\left (c x \right )}}{9} + \frac {2 b^{2} c^{6} d^{3} x^{9}}{729} - \frac {2 b^{2} c^{5} d^{3} x^{8} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{81} + \frac {3 b^{2} c^{4} d^{3} x^{7} \operatorname {asinh}^{2}{\left (c x \right )}}{7} + \frac {374 b^{2} c^{4} d^{3} x^{7}}{27783} - \frac {374 b^{2} c^{3} d^{3} x^{6} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3969} + \frac {3 b^{2} c^{2} d^{3} x^{5} \operatorname {asinh}^{2}{\left (c x \right )}}{5} + \frac {4198 b^{2} c^{2} d^{3} x^{5}}{165375} - \frac {4198 b^{2} c d^{3} x^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{33075} + \frac {b^{2} d^{3} x^{3} \operatorname {asinh}^{2}{\left (c x \right )}}{3} + \frac {5258 b^{2} d^{3} x^{3}}{297675} - \frac {5258 b^{2} d^{3} x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{99225 c} - \frac {10516 b^{2} d^{3} x}{99225 c^{2}} + \frac {10516 b^{2} d^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{99225 c^{3}} & \text {for}\: c \neq 0 \\\frac {a^{2} d^{3} x^{3}}{3} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*c**6*d**3*x**9/9 + 3*a**2*c**4*d**3*x**7/7 + 3*a**2*c**2*d**3*x**5/5 + a**2*d**3*x**3/3 + 2*a*
b*c**6*d**3*x**9*asinh(c*x)/9 - 2*a*b*c**5*d**3*x**8*sqrt(c**2*x**2 + 1)/81 + 6*a*b*c**4*d**3*x**7*asinh(c*x)/
7 - 374*a*b*c**3*d**3*x**6*sqrt(c**2*x**2 + 1)/3969 + 6*a*b*c**2*d**3*x**5*asinh(c*x)/5 - 4198*a*b*c*d**3*x**4
*sqrt(c**2*x**2 + 1)/33075 + 2*a*b*d**3*x**3*asinh(c*x)/3 - 5258*a*b*d**3*x**2*sqrt(c**2*x**2 + 1)/(99225*c) +
10516*a*b*d**3*sqrt(c**2*x**2 + 1)/(99225*c**3) + b**2*c**6*d**3*x**9*asinh(c*x)**2/9 + 2*b**2*c**6*d**3*x**9
/729 - 2*b**2*c**5*d**3*x**8*sqrt(c**2*x**2 + 1)*asinh(c*x)/81 + 3*b**2*c**4*d**3*x**7*asinh(c*x)**2/7 + 374*b
**2*c**4*d**3*x**7/27783 - 374*b**2*c**3*d**3*x**6*sqrt(c**2*x**2 + 1)*asinh(c*x)/3969 + 3*b**2*c**2*d**3*x**5
*asinh(c*x)**2/5 + 4198*b**2*c**2*d**3*x**5/165375 - 4198*b**2*c*d**3*x**4*sqrt(c**2*x**2 + 1)*asinh(c*x)/3307
5 + b**2*d**3*x**3*asinh(c*x)**2/3 + 5258*b**2*d**3*x**3/297675 - 5258*b**2*d**3*x**2*sqrt(c**2*x**2 + 1)*asin
h(c*x)/(99225*c) - 10516*b**2*d**3*x/(99225*c**2) + 10516*b**2*d**3*sqrt(c**2*x**2 + 1)*asinh(c*x)/(99225*c**3
), Ne(c, 0)), (a**2*d**3*x**3/3, True))

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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