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

Optimal. Leaf size=303 \[ \frac{1}{7} d^2 x^3 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{4}{35} d^2 x^3 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{16 b d^2 x^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{315 c}-\frac{2 b d^2 \left (c^2 x^2+1\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c^3}+\frac{2 b d^2 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{175 c^3}+\frac{8 b d^2 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{105 c^3}+\frac{32 b d^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{315 c^3}+\frac{8}{105} d^2 x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{2}{343} b^2 c^4 d^2 x^7+\frac{136 b^2 c^2 d^2 x^5}{6125}-\frac{1636 b^2 d^2 x}{11025 c^2}+\frac{818 b^2 d^2 x^3}{33075} \]

[Out]

(-1636*b^2*d^2*x)/(11025*c^2) + (818*b^2*d^2*x^3)/33075 + (136*b^2*c^2*d^2*x^5)/6125 + (2*b^2*c^4*d^2*x^7)/343
+ (32*b*d^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(315*c^3) - (16*b*d^2*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSin
h[c*x]))/(315*c) + (8*b*d^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(105*c^3) + (2*b*d^2*(1 + c^2*x^2)^(5/2)
*(a + b*ArcSinh[c*x]))/(175*c^3) - (2*b*d^2*(1 + c^2*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(49*c^3) + (8*d^2*x^3*(a
+ b*ArcSinh[c*x])^2)/105 + (4*d^2*x^3*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/35 + (d^2*x^3*(1 + c^2*x^2)^2*(a
+ b*ArcSinh[c*x])^2)/7

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Rubi [A]  time = 0.594148, antiderivative size = 303, normalized size of antiderivative = 1., number of steps used = 14, 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, 373} \[ \frac{1}{7} d^2 x^3 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{4}{35} d^2 x^3 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{16 b d^2 x^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{315 c}-\frac{2 b d^2 \left (c^2 x^2+1\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c^3}+\frac{2 b d^2 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{175 c^3}+\frac{8 b d^2 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{105 c^3}+\frac{32 b d^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{315 c^3}+\frac{8}{105} d^2 x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{2}{343} b^2 c^4 d^2 x^7+\frac{136 b^2 c^2 d^2 x^5}{6125}-\frac{1636 b^2 d^2 x}{11025 c^2}+\frac{818 b^2 d^2 x^3}{33075} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1636*b^2*d^2*x)/(11025*c^2) + (818*b^2*d^2*x^3)/33075 + (136*b^2*c^2*d^2*x^5)/6125 + (2*b^2*c^4*d^2*x^7)/343
+ (32*b*d^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(315*c^3) - (16*b*d^2*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSin
h[c*x]))/(315*c) + (8*b*d^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(105*c^3) + (2*b*d^2*(1 + c^2*x^2)^(5/2)
*(a + b*ArcSinh[c*x]))/(175*c^3) - (2*b*d^2*(1 + c^2*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(49*c^3) + (8*d^2*x^3*(a
+ b*ArcSinh[c*x])^2)/105 + (4*d^2*x^3*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/35 + (d^2*x^3*(1 + c^2*x^2)^2*(a
+ b*ArcSinh[c*x])^2)/7

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 373

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]

Rubi steps

\begin{align*} \int x^2 \left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac{1}{7} d^2 x^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{7} (4 d) \int x^2 \left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac{1}{7} \left (2 b c d^2\right ) \int x^3 \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 )}{35 c^3}-\frac{2 b d^2 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c^3}+\frac{4}{35} d^2 x^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{7} d^2 x^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{35} \left (8 d^2\right ) \int x^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac{1}{35} \left (8 b c d^2\right ) \int x^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx+\frac{1}{7} \left (2 b^2 c^2 d^2\right ) \int \frac{\left (1+c^2 x^2\right )^2 \left (-2+5 c^2 x^2\right )}{35 c^4} \, dx\\ &=\frac{8 b d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{105 c^3}+\frac{2 b d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{175 c^3}-\frac{2 b d^2 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c^3}+\frac{8}{105} d^2 x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{4}{35} d^2 x^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{7} d^2 x^3 \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 (-2+5 c^2 x^2\right ) \, dx}{245 c^2}-\frac{1}{105} \left (16 b c d^2\right ) \int \frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx+\frac{1}{35} \left (8 b^2 c^2 d^2\right ) \int \frac{-2+c^2 x^2+3 c^4 x^4}{15 c^4} \, dx\\ &=-\frac{16 b d^2 x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{315 c}+\frac{8 b d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{105 c^3}+\frac{2 b d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{175 c^3}-\frac{2 b d^2 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c^3}+\frac{8}{105} d^2 x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{4}{35} d^2 x^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{7} d^2 x^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{315} \left (16 b^2 d^2\right ) \int x^2 \, dx+\frac{\left (2 b^2 d^2\right ) \int \left (-2+c^2 x^2+8 c^4 x^4+5 c^6 x^6\right ) \, dx}{245 c^2}+\frac{\left (8 b^2 d^2\right ) \int \left (-2+c^2 x^2+3 c^4 x^4\right ) \, dx}{525 c^2}+\frac{\left (32 b d^2\right ) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx}{315 c}\\ &=-\frac{172 b^2 d^2 x}{3675 c^2}+\frac{818 b^2 d^2 x^3}{33075}+\frac{136 b^2 c^2 d^2 x^5}{6125}+\frac{2}{343} b^2 c^4 d^2 x^7+\frac{32 b d^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{315 c^3}-\frac{16 b d^2 x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{315 c}+\frac{8 b d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{105 c^3}+\frac{2 b d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{175 c^3}-\frac{2 b d^2 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c^3}+\frac{8}{105} d^2 x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{4}{35} d^2 x^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{7} d^2 x^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{\left (32 b^2 d^2\right ) \int 1 \, dx}{315 c^2}\\ &=-\frac{1636 b^2 d^2 x}{11025 c^2}+\frac{818 b^2 d^2 x^3}{33075}+\frac{136 b^2 c^2 d^2 x^5}{6125}+\frac{2}{343} b^2 c^4 d^2 x^7+\frac{32 b d^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{315 c^3}-\frac{16 b d^2 x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{315 c}+\frac{8 b d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{105 c^3}+\frac{2 b d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{175 c^3}-\frac{2 b d^2 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c^3}+\frac{8}{105} d^2 x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{4}{35} d^2 x^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{7} d^2 x^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2\\ \end{align*}

Mathematica [A]  time = 0.396393, size = 227, normalized size = 0.75 \[ \frac{d^2 \left (11025 a^2 c^3 x^3 \left (15 c^4 x^4+42 c^2 x^2+35\right )-210 a b \sqrt{c^2 x^2+1} \left (225 c^6 x^6+612 c^4 x^4+409 c^2 x^2-818\right )-210 b \sinh ^{-1}(c x) \left (b \sqrt{c^2 x^2+1} \left (225 c^6 x^6+612 c^4 x^4+409 c^2 x^2-818\right )-105 a c^3 x^3 \left (15 c^4 x^4+42 c^2 x^2+35\right )\right )+2 b^2 c x \left (3375 c^6 x^6+12852 c^4 x^4+14315 c^2 x^2-85890\right )+11025 b^2 c^3 x^3 \left (15 c^4 x^4+42 c^2 x^2+35\right ) \sinh ^{-1}(c x)^2\right )}{1157625 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*(11025*a^2*c^3*x^3*(35 + 42*c^2*x^2 + 15*c^4*x^4) - 210*a*b*Sqrt[1 + c^2*x^2]*(-818 + 409*c^2*x^2 + 612*c
^4*x^4 + 225*c^6*x^6) + 2*b^2*c*x*(-85890 + 14315*c^2*x^2 + 12852*c^4*x^4 + 3375*c^6*x^6) - 210*b*(-105*a*c^3*
x^3*(35 + 42*c^2*x^2 + 15*c^4*x^4) + b*Sqrt[1 + c^2*x^2]*(-818 + 409*c^2*x^2 + 612*c^4*x^4 + 225*c^6*x^6))*Arc
Sinh[c*x] + 11025*b^2*c^3*x^3*(35 + 42*c^2*x^2 + 15*c^4*x^4)*ArcSinh[c*x]^2))/(1157625*c^3)

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Maple [A]  time = 0.046, size = 364, normalized size = 1.2 \begin{align*}{\frac{1}{{c}^{3}} \left ({d}^{2}{a}^{2} \left ({\frac{{c}^{7}{x}^{7}}{7}}+{\frac{2\,{c}^{5}{x}^{5}}{5}}+{\frac{{c}^{3}{x}^{3}}{3}} \right ) +{d}^{2}{b}^{2} \left ({\frac{ \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}cx \left ({c}^{2}{x}^{2}+1 \right ) ^{3}}{7}}-{\frac{8\, \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}cx}{105}}-{\frac{ \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}cx \left ({c}^{2}{x}^{2}+1 \right ) ^{2}}{35}}-{\frac{4\, \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}cx \left ({c}^{2}{x}^{2}+1 \right ) }{105}}-{\frac{2\,{\it Arcsinh} \left ( cx \right ){c}^{2}{x}^{2}}{49} \left ({c}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{2}}}}-{\frac{36\,{\it Arcsinh} \left ( cx \right ){c}^{2}{x}^{2}}{1225} \left ({c}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{44\,{\it Arcsinh} \left ( cx \right ){c}^{2}{x}^{2}}{11025}\sqrt{{c}^{2}{x}^{2}+1}}+{\frac{1636\,{\it Arcsinh} \left ( cx \right ) }{11025}\sqrt{{c}^{2}{x}^{2}+1}}+{\frac{2\,cx \left ({c}^{2}{x}^{2}+1 \right ) ^{3}}{343}}-{\frac{181456\,cx}{1157625}}+{\frac{202\,cx \left ({c}^{2}{x}^{2}+1 \right ) ^{2}}{42875}}-{\frac{2528\,cx \left ({c}^{2}{x}^{2}+1 \right ) }{1157625}} \right ) +2\,{d}^{2}ab \left ( 1/7\,{\it Arcsinh} \left ( cx \right ){c}^{7}{x}^{7}+2/5\,{\it Arcsinh} \left ( cx \right ){c}^{5}{x}^{5}+1/3\,{\it Arcsinh} \left ( cx \right ){c}^{3}{x}^{3}-1/49\,{c}^{6}{x}^{6}\sqrt{{c}^{2}{x}^{2}+1}-{\frac{68\,{c}^{4}{x}^{4}\sqrt{{c}^{2}{x}^{2}+1}}{1225}}-{\frac{409\,{c}^{2}{x}^{2}\sqrt{{c}^{2}{x}^{2}+1}}{11025}}+{\frac{818\,\sqrt{{c}^{2}{x}^{2}+1}}{11025}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^3*(d^2*a^2*(1/7*c^7*x^7+2/5*c^5*x^5+1/3*c^3*x^3)+d^2*b^2*(1/7*arcsinh(c*x)^2*c*x*(c^2*x^2+1)^3-8/105*arcsi
nh(c*x)^2*c*x-1/35*arcsinh(c*x)^2*c*x*(c^2*x^2+1)^2-4/105*arcsinh(c*x)^2*c*x*(c^2*x^2+1)-2/49*arcsinh(c*x)*c^2
*x^2*(c^2*x^2+1)^(5/2)-36/1225*arcsinh(c*x)*c^2*x^2*(c^2*x^2+1)^(3/2)-44/11025*arcsinh(c*x)*c^2*x^2*(c^2*x^2+1
)^(1/2)+1636/11025*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+2/343*c*x*(c^2*x^2+1)^3-181456/1157625*c*x+202/42875*c*x*(c^
2*x^2+1)^2-2528/1157625*c*x*(c^2*x^2+1))+2*d^2*a*b*(1/7*arcsinh(c*x)*c^7*x^7+2/5*arcsinh(c*x)*c^5*x^5+1/3*arcs
inh(c*x)*c^3*x^3-1/49*c^6*x^6*(c^2*x^2+1)^(1/2)-68/1225*c^4*x^4*(c^2*x^2+1)^(1/2)-409/11025*c^2*x^2*(c^2*x^2+1
)^(1/2)+818/11025*(c^2*x^2+1)^(1/2)))

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Maxima [B]  time = 1.22513, size = 836, normalized size = 2.76 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^2*c^4*d^2*x^7*arcsinh(c*x)^2 + 1/7*a^2*c^4*d^2*x^7 + 2/5*b^2*c^2*d^2*x^5*arcsinh(c*x)^2 + 2/5*a^2*c^2*d^
2*x^5 + 2/245*(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^2 - 2/25725*(105*(5*sqrt(c^2*x^2 + 1)*x^6/c^2 - 6*sqr
t(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^4*d^2 + 1/3*b^2*d^2*x^3*arcsinh(c*x)^2 + 4/75*(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*c^2*d^2 -
4/1125*(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*c^2*d^2 + 1/3*a^2*d^2*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^2 - 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^2

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Fricas [A]  time = 2.70777, size = 755, normalized size = 2.49 \begin{align*} \frac{3375 \,{\left (49 \, a^{2} + 2 \, b^{2}\right )} c^{7} d^{2} x^{7} + 378 \,{\left (1225 \, a^{2} + 68 \, b^{2}\right )} c^{5} d^{2} x^{5} + 35 \,{\left (11025 \, a^{2} + 818 \, b^{2}\right )} c^{3} d^{2} x^{3} - 171780 \, b^{2} c d^{2} x + 11025 \,{\left (15 \, b^{2} c^{7} d^{2} x^{7} + 42 \, b^{2} c^{5} d^{2} x^{5} + 35 \, b^{2} c^{3} d^{2} x^{3}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 210 \,{\left (1575 \, a b c^{7} d^{2} x^{7} + 4410 \, a b c^{5} d^{2} x^{5} + 3675 \, a b c^{3} d^{2} x^{3} -{\left (225 \, b^{2} c^{6} d^{2} x^{6} + 612 \, b^{2} c^{4} d^{2} x^{4} + 409 \, b^{2} c^{2} d^{2} x^{2} - 818 \, b^{2} d^{2}\right )} \sqrt{c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - 210 \,{\left (225 \, a b c^{6} d^{2} x^{6} + 612 \, a b c^{4} d^{2} x^{4} + 409 \, a b c^{2} d^{2} x^{2} - 818 \, a b d^{2}\right )} \sqrt{c^{2} x^{2} + 1}}{1157625 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1157625*(3375*(49*a^2 + 2*b^2)*c^7*d^2*x^7 + 378*(1225*a^2 + 68*b^2)*c^5*d^2*x^5 + 35*(11025*a^2 + 818*b^2)*
c^3*d^2*x^3 - 171780*b^2*c*d^2*x + 11025*(15*b^2*c^7*d^2*x^7 + 42*b^2*c^5*d^2*x^5 + 35*b^2*c^3*d^2*x^3)*log(c*
x + sqrt(c^2*x^2 + 1))^2 + 210*(1575*a*b*c^7*d^2*x^7 + 4410*a*b*c^5*d^2*x^5 + 3675*a*b*c^3*d^2*x^3 - (225*b^2*
c^6*d^2*x^6 + 612*b^2*c^4*d^2*x^4 + 409*b^2*c^2*d^2*x^2 - 818*b^2*d^2)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x
^2 + 1)) - 210*(225*a*b*c^6*d^2*x^6 + 612*a*b*c^4*d^2*x^4 + 409*a*b*c^2*d^2*x^2 - 818*a*b*d^2)*sqrt(c^2*x^2 +
1))/c^3

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Sympy [A]  time = 17.3361, size = 483, normalized size = 1.59 \begin{align*} \begin{cases} \frac{a^{2} c^{4} d^{2} x^{7}}{7} + \frac{2 a^{2} c^{2} d^{2} x^{5}}{5} + \frac{a^{2} d^{2} x^{3}}{3} + \frac{2 a b c^{4} d^{2} x^{7} \operatorname{asinh}{\left (c x \right )}}{7} - \frac{2 a b c^{3} d^{2} x^{6} \sqrt{c^{2} x^{2} + 1}}{49} + \frac{4 a b c^{2} d^{2} x^{5} \operatorname{asinh}{\left (c x \right )}}{5} - \frac{136 a b c d^{2} x^{4} \sqrt{c^{2} x^{2} + 1}}{1225} + \frac{2 a b d^{2} x^{3} \operatorname{asinh}{\left (c x \right )}}{3} - \frac{818 a b d^{2} x^{2} \sqrt{c^{2} x^{2} + 1}}{11025 c} + \frac{1636 a b d^{2} \sqrt{c^{2} x^{2} + 1}}{11025 c^{3}} + \frac{b^{2} c^{4} d^{2} x^{7} \operatorname{asinh}^{2}{\left (c x \right )}}{7} + \frac{2 b^{2} c^{4} d^{2} x^{7}}{343} - \frac{2 b^{2} c^{3} d^{2} x^{6} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{49} + \frac{2 b^{2} c^{2} d^{2} x^{5} \operatorname{asinh}^{2}{\left (c x \right )}}{5} + \frac{136 b^{2} c^{2} d^{2} x^{5}}{6125} - \frac{136 b^{2} c d^{2} x^{4} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{1225} + \frac{b^{2} d^{2} x^{3} \operatorname{asinh}^{2}{\left (c x \right )}}{3} + \frac{818 b^{2} d^{2} x^{3}}{33075} - \frac{818 b^{2} d^{2} x^{2} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{11025 c} - \frac{1636 b^{2} d^{2} x}{11025 c^{2}} + \frac{1636 b^{2} d^{2} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{11025 c^{3}} & \text{for}\: c \neq 0 \\\frac{a^{2} d^{2} x^{3}}{3} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*c**4*d**2*x**7/7 + 2*a**2*c**2*d**2*x**5/5 + a**2*d**2*x**3/3 + 2*a*b*c**4*d**2*x**7*asinh(c*x
)/7 - 2*a*b*c**3*d**2*x**6*sqrt(c**2*x**2 + 1)/49 + 4*a*b*c**2*d**2*x**5*asinh(c*x)/5 - 136*a*b*c*d**2*x**4*sq
rt(c**2*x**2 + 1)/1225 + 2*a*b*d**2*x**3*asinh(c*x)/3 - 818*a*b*d**2*x**2*sqrt(c**2*x**2 + 1)/(11025*c) + 1636
*a*b*d**2*sqrt(c**2*x**2 + 1)/(11025*c**3) + b**2*c**4*d**2*x**7*asinh(c*x)**2/7 + 2*b**2*c**4*d**2*x**7/343 -
2*b**2*c**3*d**2*x**6*sqrt(c**2*x**2 + 1)*asinh(c*x)/49 + 2*b**2*c**2*d**2*x**5*asinh(c*x)**2/5 + 136*b**2*c*
*2*d**2*x**5/6125 - 136*b**2*c*d**2*x**4*sqrt(c**2*x**2 + 1)*asinh(c*x)/1225 + b**2*d**2*x**3*asinh(c*x)**2/3
+ 818*b**2*d**2*x**3/33075 - 818*b**2*d**2*x**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/(11025*c) - 1636*b**2*d**2*x/(1
1025*c**2) + 1636*b**2*d**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/(11025*c**3), Ne(c, 0)), (a**2*d**2*x**3/3, True))

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Giac [B]  time = 2.76466, size = 853, normalized size = 2.82 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*a^2*c^4*d^2*x^7 + 2/5*a^2*c^2*d^2*x^5 + 2/245*(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^4*d^2 + 1/25725*(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^4*d^2 + 4/75*(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*c^2*d^2 + 2/1125*(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*c^2*d^2 + 1/3*a^2*d^2*x^3 + 2/9*(3*x^3*log(c*x + sqrt(c^2*x^2 + 1)) - (
(c^2*x^2 + 1)^(3/2) - 3*sqrt(c^2*x^2 + 1))/c^3)*a*b*d^2 + 1/27*(9*x^3*log(c*x + sqrt(c^2*x^2 + 1))^2 + 2*c*((c
^2*x^3 - 6*x)/c^3 - 3*((c^2*x^2 + 1)^(3/2) - 3*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1))/c^4))*b^2*d^2