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

Optimal. Leaf size=536 \[ -\frac{b c^5 d^2 x^8 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{32 \sqrt{c^2 x^2+1}}-\frac{17 b c^3 d^2 x^6 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{144 \sqrt{c^2 x^2+1}}-\frac{59 b c d^2 x^4 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{384 \sqrt{c^2 x^2+1}}+\frac{5}{64} d^2 x^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{5 b d^2 x^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{128 c \sqrt{c^2 x^2+1}}+\frac{5 d^2 x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{128 c^2}-\frac{5 d^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{384 b c^3 \sqrt{c^2 x^2+1}}+\frac{1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5}{48} d x^3 \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{256} b^2 c^4 d^2 x^7 \sqrt{c^2 d x^2+d}+\frac{209 b^2 c^2 d^2 x^5 \sqrt{c^2 d x^2+d}}{13824}+\frac{1079 b^2 d^2 x^3 \sqrt{c^2 d x^2+d}}{55296}-\frac{359 b^2 d^2 x \sqrt{c^2 d x^2+d}}{36864 c^2}+\frac{359 b^2 d^2 \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{36864 c^3 \sqrt{c^2 x^2+1}} \]

[Out]

(-359*b^2*d^2*x*Sqrt[d + c^2*d*x^2])/(36864*c^2) + (1079*b^2*d^2*x^3*Sqrt[d + c^2*d*x^2])/55296 + (209*b^2*c^2
*d^2*x^5*Sqrt[d + c^2*d*x^2])/13824 + (b^2*c^4*d^2*x^7*Sqrt[d + c^2*d*x^2])/256 + (359*b^2*d^2*Sqrt[d + c^2*d*
x^2]*ArcSinh[c*x])/(36864*c^3*Sqrt[1 + c^2*x^2]) - (5*b*d^2*x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(128
*c*Sqrt[1 + c^2*x^2]) - (59*b*c*d^2*x^4*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(384*Sqrt[1 + c^2*x^2]) - (1
7*b*c^3*d^2*x^6*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(144*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*x^8*Sqrt[d + c^
2*d*x^2]*(a + b*ArcSinh[c*x]))/(32*Sqrt[1 + c^2*x^2]) + (5*d^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(
128*c^2) + (5*d^2*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/64 + (5*d*x^3*(d + c^2*d*x^2)^(3/2)*(a + b*A
rcSinh[c*x])^2)/48 + (x^3*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/8 - (5*d^2*Sqrt[d + c^2*d*x^2]*(a + b*
ArcSinh[c*x])^3)/(384*b*c^3*Sqrt[1 + c^2*x^2])

________________________________________________________________________________________

Rubi [A]  time = 1.04717, antiderivative size = 536, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 14, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5744, 5742, 5758, 5675, 5661, 321, 215, 14, 5730, 12, 459, 266, 43, 1267} \[ -\frac{b c^5 d^2 x^8 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{32 \sqrt{c^2 x^2+1}}-\frac{17 b c^3 d^2 x^6 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{144 \sqrt{c^2 x^2+1}}-\frac{59 b c d^2 x^4 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{384 \sqrt{c^2 x^2+1}}+\frac{5}{64} d^2 x^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{5 b d^2 x^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{128 c \sqrt{c^2 x^2+1}}+\frac{5 d^2 x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{128 c^2}-\frac{5 d^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{384 b c^3 \sqrt{c^2 x^2+1}}+\frac{1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5}{48} d x^3 \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{256} b^2 c^4 d^2 x^7 \sqrt{c^2 d x^2+d}+\frac{209 b^2 c^2 d^2 x^5 \sqrt{c^2 d x^2+d}}{13824}+\frac{1079 b^2 d^2 x^3 \sqrt{c^2 d x^2+d}}{55296}-\frac{359 b^2 d^2 x \sqrt{c^2 d x^2+d}}{36864 c^2}+\frac{359 b^2 d^2 \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{36864 c^3 \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-359*b^2*d^2*x*Sqrt[d + c^2*d*x^2])/(36864*c^2) + (1079*b^2*d^2*x^3*Sqrt[d + c^2*d*x^2])/55296 + (209*b^2*c^2
*d^2*x^5*Sqrt[d + c^2*d*x^2])/13824 + (b^2*c^4*d^2*x^7*Sqrt[d + c^2*d*x^2])/256 + (359*b^2*d^2*Sqrt[d + c^2*d*
x^2]*ArcSinh[c*x])/(36864*c^3*Sqrt[1 + c^2*x^2]) - (5*b*d^2*x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(128
*c*Sqrt[1 + c^2*x^2]) - (59*b*c*d^2*x^4*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(384*Sqrt[1 + c^2*x^2]) - (1
7*b*c^3*d^2*x^6*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(144*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*x^8*Sqrt[d + c^
2*d*x^2]*(a + b*ArcSinh[c*x]))/(32*Sqrt[1 + c^2*x^2]) + (5*d^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(
128*c^2) + (5*d^2*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/64 + (5*d*x^3*(d + c^2*d*x^2)^(3/2)*(a + b*A
rcSinh[c*x])^2)/48 + (x^3*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/8 - (5*d^2*Sqrt[d + c^2*d*x^2]*(a + b*
ArcSinh[c*x])^3)/(384*b*c^3*Sqrt[1 + c^2*x^2])

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 5742

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(
(f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^n)/(f*(m + 2)), x] + (Dist[Sqrt[d + e*x^2]/((m + 2)*Sqrt[1
+ c^2*x^2]), Int[((f*x)^m*(a + b*ArcSinh[c*x])^n)/Sqrt[1 + c^2*x^2], x], x] - Dist[(b*c*n*Sqrt[d + e*x^2])/(f*
(m + 2)*Sqrt[1 + c^2*x^2]), Int[(f*x)^(m + 1)*(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] &&  !LtQ[m, -1] && (RationalQ[m] || EqQ[n, 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 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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 5730

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1
+ c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[p, 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 459

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

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 1267

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Si
mp[(c^p*(f*x)^(m + 4*p - 1)*(d + e*x^2)^(q + 1))/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1)), x] + Dist[1/(e*(m + 4*p
+ 2*q + 1)), Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + b*x^2 + c*x^4)^p - c^p*x^(4*p))
- d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] &&
IGtQ[p, 0] &&  !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 0]

Rubi steps

\begin{align*} \int x^2 \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac{1}{8} x^3 \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{8} (5 d) \int x^2 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac{\left (b c d^2 \sqrt{d+c^2 d x^2}\right ) \int x^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{4 \sqrt{1+c^2 x^2}}\\ &=-\frac{b c d^2 x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 \sqrt{1+c^2 x^2}}-\frac{b c^3 d^2 x^6 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{12 \sqrt{1+c^2 x^2}}-\frac{b c^5 d^2 x^8 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{32 \sqrt{1+c^2 x^2}}+\frac{5}{48} d x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{8} x^3 \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{16} \left (5 d^2\right ) \int x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac{\left (5 b c d^2 \sqrt{d+c^2 d x^2}\right ) \int x^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{24 \sqrt{1+c^2 x^2}}+\frac{\left (b^2 c^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{x^4 \left (6+8 c^2 x^2+3 c^4 x^4\right )}{24 \sqrt{1+c^2 x^2}} \, dx}{4 \sqrt{1+c^2 x^2}}\\ &=-\frac{11 b c d^2 x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{96 \sqrt{1+c^2 x^2}}-\frac{17 b c^3 d^2 x^6 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{144 \sqrt{1+c^2 x^2}}-\frac{b c^5 d^2 x^8 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{32 \sqrt{1+c^2 x^2}}+\frac{5}{64} d^2 x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5}{48} d x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{8} x^3 \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{\left (5 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}} \, dx}{64 \sqrt{1+c^2 x^2}}-\frac{\left (5 b c d^2 \sqrt{d+c^2 d x^2}\right ) \int x^3 \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{32 \sqrt{1+c^2 x^2}}+\frac{\left (b^2 c^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{x^4 \left (6+8 c^2 x^2+3 c^4 x^4\right )}{\sqrt{1+c^2 x^2}} \, dx}{96 \sqrt{1+c^2 x^2}}+\frac{\left (5 b^2 c^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{x^4 \left (3+2 c^2 x^2\right )}{12 \sqrt{1+c^2 x^2}} \, dx}{24 \sqrt{1+c^2 x^2}}\\ &=\frac{1}{256} b^2 c^4 d^2 x^7 \sqrt{d+c^2 d x^2}-\frac{59 b c d^2 x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{384 \sqrt{1+c^2 x^2}}-\frac{17 b c^3 d^2 x^6 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{144 \sqrt{1+c^2 x^2}}-\frac{b c^5 d^2 x^8 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{32 \sqrt{1+c^2 x^2}}+\frac{5 d^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{128 c^2}+\frac{5}{64} d^2 x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5}{48} d x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{8} x^3 \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{\left (b^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{x^4 \left (48 c^2+43 c^4 x^2\right )}{\sqrt{1+c^2 x^2}} \, dx}{768 \sqrt{1+c^2 x^2}}-\frac{\left (5 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}{128 c^2 \sqrt{1+c^2 x^2}}-\frac{\left (5 b d^2 \sqrt{d+c^2 d x^2}\right ) \int x \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{64 c \sqrt{1+c^2 x^2}}+\frac{\left (5 b^2 c^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{x^4 \left (3+2 c^2 x^2\right )}{\sqrt{1+c^2 x^2}} \, dx}{288 \sqrt{1+c^2 x^2}}+\frac{\left (5 b^2 c^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{x^4}{\sqrt{1+c^2 x^2}} \, dx}{128 \sqrt{1+c^2 x^2}}\\ &=\frac{5}{512} b^2 d^2 x^3 \sqrt{d+c^2 d x^2}+\frac{209 b^2 c^2 d^2 x^5 \sqrt{d+c^2 d x^2}}{13824}+\frac{1}{256} b^2 c^4 d^2 x^7 \sqrt{d+c^2 d x^2}-\frac{5 b d^2 x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{128 c \sqrt{1+c^2 x^2}}-\frac{59 b c d^2 x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{384 \sqrt{1+c^2 x^2}}-\frac{17 b c^3 d^2 x^6 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{144 \sqrt{1+c^2 x^2}}-\frac{b c^5 d^2 x^8 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{32 \sqrt{1+c^2 x^2}}+\frac{5 d^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{128 c^2}+\frac{5}{64} d^2 x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5}{48} d x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{8} x^3 \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{5 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{384 b c^3 \sqrt{1+c^2 x^2}}-\frac{\left (15 b^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{1+c^2 x^2}} \, dx}{512 \sqrt{1+c^2 x^2}}+\frac{\left (5 b^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{1+c^2 x^2}} \, dx}{128 \sqrt{1+c^2 x^2}}+\frac{\left (73 b^2 c^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{x^4}{\sqrt{1+c^2 x^2}} \, dx}{4608 \sqrt{1+c^2 x^2}}+\frac{\left (5 b^2 c^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{x^4}{\sqrt{1+c^2 x^2}} \, dx}{216 \sqrt{1+c^2 x^2}}\\ &=\frac{5 b^2 d^2 x \sqrt{d+c^2 d x^2}}{1024 c^2}+\frac{1079 b^2 d^2 x^3 \sqrt{d+c^2 d x^2}}{55296}+\frac{209 b^2 c^2 d^2 x^5 \sqrt{d+c^2 d x^2}}{13824}+\frac{1}{256} b^2 c^4 d^2 x^7 \sqrt{d+c^2 d x^2}-\frac{5 b d^2 x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{128 c \sqrt{1+c^2 x^2}}-\frac{59 b c d^2 x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{384 \sqrt{1+c^2 x^2}}-\frac{17 b c^3 d^2 x^6 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{144 \sqrt{1+c^2 x^2}}-\frac{b c^5 d^2 x^8 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{32 \sqrt{1+c^2 x^2}}+\frac{5 d^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{128 c^2}+\frac{5}{64} d^2 x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5}{48} d x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{8} x^3 \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{5 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{384 b c^3 \sqrt{1+c^2 x^2}}-\frac{\left (73 b^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{1+c^2 x^2}} \, dx}{6144 \sqrt{1+c^2 x^2}}-\frac{\left (5 b^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{1+c^2 x^2}} \, dx}{288 \sqrt{1+c^2 x^2}}+\frac{\left (15 b^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx}{1024 c^2 \sqrt{1+c^2 x^2}}-\frac{\left (5 b^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx}{256 c^2 \sqrt{1+c^2 x^2}}\\ &=-\frac{359 b^2 d^2 x \sqrt{d+c^2 d x^2}}{36864 c^2}+\frac{1079 b^2 d^2 x^3 \sqrt{d+c^2 d x^2}}{55296}+\frac{209 b^2 c^2 d^2 x^5 \sqrt{d+c^2 d x^2}}{13824}+\frac{1}{256} b^2 c^4 d^2 x^7 \sqrt{d+c^2 d x^2}-\frac{5 b^2 d^2 \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{1024 c^3 \sqrt{1+c^2 x^2}}-\frac{5 b d^2 x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{128 c \sqrt{1+c^2 x^2}}-\frac{59 b c d^2 x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{384 \sqrt{1+c^2 x^2}}-\frac{17 b c^3 d^2 x^6 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{144 \sqrt{1+c^2 x^2}}-\frac{b c^5 d^2 x^8 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{32 \sqrt{1+c^2 x^2}}+\frac{5 d^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{128 c^2}+\frac{5}{64} d^2 x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5}{48} d x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{8} x^3 \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{5 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{384 b c^3 \sqrt{1+c^2 x^2}}+\frac{\left (73 b^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx}{12288 c^2 \sqrt{1+c^2 x^2}}+\frac{\left (5 b^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx}{576 c^2 \sqrt{1+c^2 x^2}}\\ &=-\frac{359 b^2 d^2 x \sqrt{d+c^2 d x^2}}{36864 c^2}+\frac{1079 b^2 d^2 x^3 \sqrt{d+c^2 d x^2}}{55296}+\frac{209 b^2 c^2 d^2 x^5 \sqrt{d+c^2 d x^2}}{13824}+\frac{1}{256} b^2 c^4 d^2 x^7 \sqrt{d+c^2 d x^2}+\frac{359 b^2 d^2 \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{36864 c^3 \sqrt{1+c^2 x^2}}-\frac{5 b d^2 x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{128 c \sqrt{1+c^2 x^2}}-\frac{59 b c d^2 x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{384 \sqrt{1+c^2 x^2}}-\frac{17 b c^3 d^2 x^6 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{144 \sqrt{1+c^2 x^2}}-\frac{b c^5 d^2 x^8 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{32 \sqrt{1+c^2 x^2}}+\frac{5 d^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{128 c^2}+\frac{5}{64} d^2 x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5}{48} d x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{8} x^3 \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{5 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{384 b c^3 \sqrt{1+c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 2.04668, size = 619, normalized size = 1.15 \[ \frac{d^2 \left (110592 a^2 c^7 x^7 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+313344 a^2 c^5 x^5 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+271872 a^2 c^3 x^3 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+34560 a^2 c x \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}-34560 a^2 \sqrt{d} \sqrt{c^2 x^2+1} \log \left (\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right )+288 b \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)^2 \left (-120 a-48 b \sinh \left (2 \sinh ^{-1}(c x)\right )+24 b \sinh \left (4 \sinh ^{-1}(c x)\right )+16 b \sinh \left (6 \sinh ^{-1}(c x)\right )+3 b \sinh \left (8 \sinh ^{-1}(c x)\right )\right )+13824 a b \sqrt{c^2 d x^2+d} \cosh \left (2 \sinh ^{-1}(c x)\right )-3456 a b \sqrt{c^2 d x^2+d} \cosh \left (4 \sinh ^{-1}(c x)\right )-1536 a b \sqrt{c^2 d x^2+d} \cosh \left (6 \sinh ^{-1}(c x)\right )-216 a b \sqrt{c^2 d x^2+d} \cosh \left (8 \sinh ^{-1}(c x)\right )+24 b \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x) \left (-1152 a \sinh \left (2 \sinh ^{-1}(c x)\right )+576 a \sinh \left (4 \sinh ^{-1}(c x)\right )+384 a \sinh \left (6 \sinh ^{-1}(c x)\right )+72 a \sinh \left (8 \sinh ^{-1}(c x)\right )+576 b \cosh \left (2 \sinh ^{-1}(c x)\right )-144 b \cosh \left (4 \sinh ^{-1}(c x)\right )-64 b \cosh \left (6 \sinh ^{-1}(c x)\right )-9 b \cosh \left (8 \sinh ^{-1}(c x)\right )\right )-11520 b^2 \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)^3-6912 b^2 \sqrt{c^2 d x^2+d} \sinh \left (2 \sinh ^{-1}(c x)\right )+864 b^2 \sqrt{c^2 d x^2+d} \sinh \left (4 \sinh ^{-1}(c x)\right )+256 b^2 \sqrt{c^2 d x^2+d} \sinh \left (6 \sinh ^{-1}(c x)\right )+27 b^2 \sqrt{c^2 d x^2+d} \sinh \left (8 \sinh ^{-1}(c x)\right )\right )}{884736 c^3 \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*(34560*a^2*c*x*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 271872*a^2*c^3*x^3*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*
d*x^2] + 313344*a^2*c^5*x^5*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 110592*a^2*c^7*x^7*Sqrt[1 + c^2*x^2]*Sqrt[
d + c^2*d*x^2] - 11520*b^2*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^3 + 13824*a*b*Sqrt[d + c^2*d*x^2]*Cosh[2*ArcSinh[c
*x]] - 3456*a*b*Sqrt[d + c^2*d*x^2]*Cosh[4*ArcSinh[c*x]] - 1536*a*b*Sqrt[d + c^2*d*x^2]*Cosh[6*ArcSinh[c*x]] -
216*a*b*Sqrt[d + c^2*d*x^2]*Cosh[8*ArcSinh[c*x]] - 34560*a^2*Sqrt[d]*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sq
rt[d + c^2*d*x^2]] - 6912*b^2*Sqrt[d + c^2*d*x^2]*Sinh[2*ArcSinh[c*x]] + 864*b^2*Sqrt[d + c^2*d*x^2]*Sinh[4*Ar
cSinh[c*x]] + 256*b^2*Sqrt[d + c^2*d*x^2]*Sinh[6*ArcSinh[c*x]] + 27*b^2*Sqrt[d + c^2*d*x^2]*Sinh[8*ArcSinh[c*x
]] + 24*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*(576*b*Cosh[2*ArcSinh[c*x]] - 144*b*Cosh[4*ArcSinh[c*x]] - 64*b*Cos
h[6*ArcSinh[c*x]] - 9*b*Cosh[8*ArcSinh[c*x]] - 1152*a*Sinh[2*ArcSinh[c*x]] + 576*a*Sinh[4*ArcSinh[c*x]] + 384*
a*Sinh[6*ArcSinh[c*x]] + 72*a*Sinh[8*ArcSinh[c*x]]) + 288*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^2*(-120*a - 48*b*
Sinh[2*ArcSinh[c*x]] + 24*b*Sinh[4*ArcSinh[c*x]] + 16*b*Sinh[6*ArcSinh[c*x]] + 3*b*Sinh[8*ArcSinh[c*x]])))/(88
4736*c^3*Sqrt[1 + c^2*x^2])

________________________________________________________________________________________

Maple [B]  time = 0.426, size = 1204, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*a^2*x*(c^2*d*x^2+d)^(7/2)/c^2/d-5/192*a^2/c^2*d*x*(c^2*d*x^2+d)^(3/2)-5/128*a^2/c^2*d^2*x*(c^2*d*x^2+d)^(1
/2)+1081/110592*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(c^2*x^2+1)*x^3+5/64*a*b*(d*(c^2*x^2+1))^(1/2)*d^2/c^2/(c^2*x^2+
1)*arcsinh(c*x)*x+127/96*a*b*(d*(c^2*x^2+1))^(1/2)*d^2*c^2/(c^2*x^2+1)*arcsinh(c*x)*x^5+1/4*a*b*(d*(c^2*x^2+1)
)^(1/2)*d^2*c^6/(c^2*x^2+1)*arcsinh(c*x)*x^9+23/24*a*b*(d*(c^2*x^2+1))^(1/2)*d^2*c^4/(c^2*x^2+1)*arcsinh(c*x)*
x^7-5/128*a^2/c^2*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)+359/36864*b^2*(d*(c^2*x^2+1)
)^(1/2)*d^2/c^3/(c^2*x^2+1)^(1/2)*arcsinh(c*x)+1/256*b^2*(d*(c^2*x^2+1))^(1/2)*d^2*c^6/(c^2*x^2+1)*x^9+263/138
24*b^2*(d*(c^2*x^2+1))^(1/2)*d^2*c^4/(c^2*x^2+1)*x^7+1915/55296*b^2*(d*(c^2*x^2+1))^(1/2)*d^2*c^2/(c^2*x^2+1)*
x^5-359/36864*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/c^2/(c^2*x^2+1)*x-5/384*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2
)/c^3*arcsinh(c*x)^3*d^2+133/384*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(c^2*x^2+1)*arcsinh(c*x)^2*x^3+359/36864*a*b*(d
*(c^2*x^2+1))^(1/2)*d^2/c^3/(c^2*x^2+1)^(1/2)-5/128*a*b*(d*(c^2*x^2+1))^(1/2)*d^2/c/(c^2*x^2+1)^(1/2)*x^2-59/3
84*a*b*(d*(c^2*x^2+1))^(1/2)*d^2*c/(c^2*x^2+1)^(1/2)*x^4-17/144*a*b*(d*(c^2*x^2+1))^(1/2)*d^2*c^3/(c^2*x^2+1)^
(1/2)*x^6+1/8*b^2*(d*(c^2*x^2+1))^(1/2)*d^2*c^6/(c^2*x^2+1)*arcsinh(c*x)^2*x^9+23/48*b^2*(d*(c^2*x^2+1))^(1/2)
*d^2*c^4/(c^2*x^2+1)*arcsinh(c*x)^2*x^7+127/192*b^2*(d*(c^2*x^2+1))^(1/2)*d^2*c^2/(c^2*x^2+1)*arcsinh(c*x)^2*x
^5+5/128*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/c^2/(c^2*x^2+1)*arcsinh(c*x)^2*x-1/32*a*b*(d*(c^2*x^2+1))^(1/2)*d^2*c^5
/(c^2*x^2+1)^(1/2)*x^8-5/128*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/c/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*x^2-1/32*b^2*(d*(c
^2*x^2+1))^(1/2)*d^2*c^5/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*x^8-17/144*b^2*(d*(c^2*x^2+1))^(1/2)*d^2*c^3/(c^2*x^2+
1)^(1/2)*arcsinh(c*x)*x^6-59/384*b^2*(d*(c^2*x^2+1))^(1/2)*d^2*c/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*x^4-5/128*a*b*
(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^3*arcsinh(c*x)^2*d^2+133/192*a*b*(d*(c^2*x^2+1))^(1/2)*d^2/(c^2*x^2+
1)*arcsinh(c*x)*x^3-1/48*a^2/c^2*x*(c^2*d*x^2+d)^(5/2)

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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(x^2*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2,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 ({\left (a^{2} c^{4} d^{2} x^{6} + 2 \, a^{2} c^{2} d^{2} x^{4} + a^{2} d^{2} x^{2} +{\left (b^{2} c^{4} d^{2} x^{6} + 2 \, b^{2} c^{2} d^{2} x^{4} + b^{2} d^{2} x^{2}\right )} \operatorname{arsinh}\left (c x\right )^{2} + 2 \,{\left (a b c^{4} d^{2} x^{6} + 2 \, a b c^{2} d^{2} x^{4} + a b d^{2} x^{2}\right )} \operatorname{arsinh}\left (c x\right )\right )} \sqrt{c^{2} d x^{2} + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^2*c^4*d^2*x^6 + 2*a^2*c^2*d^2*x^4 + a^2*d^2*x^2 + (b^2*c^4*d^2*x^6 + 2*b^2*c^2*d^2*x^4 + b^2*d^2*x
^2)*arcsinh(c*x)^2 + 2*(a*b*c^4*d^2*x^6 + 2*a*b*c^2*d^2*x^4 + a*b*d^2*x^2)*arcsinh(c*x))*sqrt(c^2*d*x^2 + d),
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(x**2*(c**2*d*x**2+d)**(5/2)*(a+b*asinh(c*x))**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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