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

Optimal. Leaf size=470 \[ -\frac{2 b c^5 d^2 x^7 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{49 \sqrt{c x-1} \sqrt{c x+1}}+\frac{6 b c^3 d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b c d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{\left (d-c^2 d x^2\right )^{7/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{7 c^2 d}-\frac{2 b^2 d^2 \left (1-c^2 x^2\right )^4 \sqrt{d-c^2 d x^2}}{343 c^2 (1-c x) (c x+1)}-\frac{12 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{1225 c^2 (1-c x) (c x+1)}-\frac{16 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{735 c^2 (1-c x) (c x+1)}-\frac{32 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{245 c^2 (1-c x) (c x+1)} \]

[Out]

(-32*b^2*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(245*c^2*(1 - c*x)*(1 + c*x)) - (16*b^2*d^2*(1 - c^2*x^2)^2*Sq
rt[d - c^2*d*x^2])/(735*c^2*(1 - c*x)*(1 + c*x)) - (12*b^2*d^2*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(1225*c^2*
(1 - c*x)*(1 + c*x)) - (2*b^2*d^2*(1 - c^2*x^2)^4*Sqrt[d - c^2*d*x^2])/(343*c^2*(1 - c*x)*(1 + c*x)) + (2*b*d^
2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(7*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c*d^2*x^3*Sqrt[d - c^2
*d*x^2]*(a + b*ArcCosh[c*x]))/(7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (6*b*c^3*d^2*x^5*Sqrt[d - c^2*d*x^2]*(a + b*A
rcCosh[c*x]))/(35*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c^5*d^2*x^7*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(
49*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x])^2)/(7*c^2*d)

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Rubi [A]  time = 0.680231, antiderivative size = 485, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {5798, 5718, 194, 5680, 12, 1610, 1799, 1850} \[ -\frac{2 b c^5 d^2 x^7 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{49 \sqrt{c x-1} \sqrt{c x+1}}+\frac{6 b c^3 d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b c d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{d^2 (1-c x)^3 (c x+1)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{7 c^2}-\frac{2 b^2 d^2 \left (1-c^2 x^2\right )^4 \sqrt{d-c^2 d x^2}}{343 c^2 (1-c x) (c x+1)}-\frac{12 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{1225 c^2 (1-c x) (c x+1)}-\frac{16 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{735 c^2 (1-c x) (c x+1)}-\frac{32 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{245 c^2 (1-c x) (c x+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-32*b^2*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(245*c^2*(1 - c*x)*(1 + c*x)) - (16*b^2*d^2*(1 - c^2*x^2)^2*Sq
rt[d - c^2*d*x^2])/(735*c^2*(1 - c*x)*(1 + c*x)) - (12*b^2*d^2*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(1225*c^2*
(1 - c*x)*(1 + c*x)) - (2*b^2*d^2*(1 - c^2*x^2)^4*Sqrt[d - c^2*d*x^2])/(343*c^2*(1 - c*x)*(1 + c*x)) + (2*b*d^
2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(7*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c*d^2*x^3*Sqrt[d - c^2
*d*x^2]*(a + b*ArcCosh[c*x]))/(7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (6*b*c^3*d^2*x^5*Sqrt[d - c^2*d*x^2]*(a + b*A
rcCosh[c*x]))/(35*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c^5*d^2*x^7*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(
49*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d^2*(1 - c*x)^3*(1 + c*x)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(7
*c^2)

Rule 5798

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Dist
[((-d)^IntPart[p]*(d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^m*(1 + c*
x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[c^2*d + e, 0]
&&  !IntegerQ[p]

Rule 5718

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_
Symbol] :> Simp[((d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e1*e2*(p + 1)), x] - Dist[
(b*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*c*(p + 1)*(1 + c*x)^FracPart[p]
*(-1 + c*x)^FracPart[p]), Int[(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c,
d1, e1, d2, e2, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p + 1
/2]

Rule 194

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

Rule 5680

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2
)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x
], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && 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 1610

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Dist[((
a + b*x)^FracPart[m]*(c + d*x)^FracPart[m])/(a*c + b*d*x^2)^FracPart[m], Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] && EqQ[m, n] &&  !Intege
rQ[m]

Rule 1799

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin{align*} \int x \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx &=\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int x (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{d^2 (1-c x)^3 (1+c x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{7 c^2}-\frac{\left (2 b d^2 \sqrt{d-c^2 d x^2}\right ) \int \left (-1+c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{7 c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{2 b d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{6 b c^3 d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c^5 d^2 x^7 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{49 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^2 (1-c x)^3 (1+c x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{7 c^2}+\frac{\left (2 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right )}{35 \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{7 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{2 b d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{6 b c^3 d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c^5 d^2 x^7 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{49 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^2 (1-c x)^3 (1+c x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{7 c^2}+\frac{\left (2 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{245 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{2 b d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{6 b c^3 d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c^5 d^2 x^7 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{49 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^2 (1-c x)^3 (1+c x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{7 c^2}+\frac{\left (2 b^2 d^2 \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2}\right ) \int \frac{x \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right )}{\sqrt{-1+c^2 x^2}} \, dx}{245 (-1+c x) (1+c x)}\\ &=\frac{2 b d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{6 b c^3 d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c^5 d^2 x^7 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{49 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^2 (1-c x)^3 (1+c x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{7 c^2}+\frac{\left (b^2 d^2 \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{-35+35 c^2 x-21 c^4 x^2+5 c^6 x^3}{\sqrt{-1+c^2 x}} \, dx,x,x^2\right )}{245 (-1+c x) (1+c x)}\\ &=\frac{2 b d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{6 b c^3 d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c^5 d^2 x^7 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{49 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^2 (1-c x)^3 (1+c x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{7 c^2}+\frac{\left (b^2 d^2 \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (-\frac{16}{\sqrt{-1+c^2 x}}+8 \sqrt{-1+c^2 x}-6 \left (-1+c^2 x\right )^{3/2}+5 \left (-1+c^2 x\right )^{5/2}\right ) \, dx,x,x^2\right )}{245 (-1+c x) (1+c x)}\\ &=-\frac{32 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{245 c^2 (1-c x) (1+c x)}-\frac{16 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{735 c^2 (1-c x) (1+c x)}-\frac{12 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{1225 c^2 (1-c x) (1+c x)}-\frac{2 b^2 d^2 \left (1-c^2 x^2\right )^4 \sqrt{d-c^2 d x^2}}{343 c^2 (1-c x) (1+c x)}+\frac{2 b d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{6 b c^3 d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c^5 d^2 x^7 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{49 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^2 (1-c x)^3 (1+c x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{7 c^2}\\ \end{align*}

Mathematica [A]  time = 0.59218, size = 234, normalized size = 0.5 \[ \frac{d^2 \sqrt{d-c^2 d x^2} \left (3675 a^2 \left (c^2 x^2-1\right )^4-210 a b c x \sqrt{c x-1} \sqrt{c x+1} \left (5 c^6 x^6-21 c^4 x^4+35 c^2 x^2-35\right )+210 b \cosh ^{-1}(c x) \left (35 a \left (c^2 x^2-1\right )^4+b c x \sqrt{c x-1} \sqrt{c x+1} \left (-5 c^6 x^6+21 c^4 x^4-35 c^2 x^2+35\right )\right )+2 b^2 \left (75 c^8 x^8-426 c^6 x^6+1108 c^4 x^4-2918 c^2 x^2+2161\right )+3675 b^2 \left (c^2 x^2-1\right )^4 \cosh ^{-1}(c x)^2\right )}{25725 c^2 \left (c^2 x^2-1\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*Sqrt[d - c^2*d*x^2]*(3675*a^2*(-1 + c^2*x^2)^4 - 210*a*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-35 + 35*c^2*x
^2 - 21*c^4*x^4 + 5*c^6*x^6) + 2*b^2*(2161 - 2918*c^2*x^2 + 1108*c^4*x^4 - 426*c^6*x^6 + 75*c^8*x^8) + 210*b*(
35*a*(-1 + c^2*x^2)^4 + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(35 - 35*c^2*x^2 + 21*c^4*x^4 - 5*c^6*x^6))*ArcCosh
[c*x] + 3675*b^2*(-1 + c^2*x^2)^4*ArcCosh[c*x]^2))/(25725*c^2*(-1 + c^2*x^2))

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Maple [B]  time = 0.468, size = 1958, normalized size = 4.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*a^2/c^2/d*(-c^2*d*x^2+d)^(7/2)+b^2*(1/43904*(-d*(c^2*x^2-1))^(1/2)*(64*c^8*x^8-144*c^6*x^6+64*(c*x+1)^(1/
2)*(c*x-1)^(1/2)*x^7*c^7+104*c^4*x^4-112*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5-25*c^2*x^2+56*(c*x+1)^(1/2)*(c*x-
1)^(1/2)*x^3*c^3-7*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+1)*(49*arccosh(c*x)^2-14*arccosh(c*x)+2)*d^2/(c*x+1)/c^2/(c
*x-1)-1/3200*(-d*(c^2*x^2-1))^(1/2)*(16*c^6*x^6-28*c^4*x^4+16*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5+13*c^2*x^2-2
0*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+5*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-1)*(25*arccosh(c*x)^2-10*arccosh(c*x)+
2)*d^2/(c*x+1)/c^2/(c*x-1)+1/384*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2+4*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3
*c^3-3*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+1)*(9*arccosh(c*x)^2-6*arccosh(c*x)+2)*d^2/(c*x+1)/c^2/(c*x-1)-5/128*(-
d*(c^2*x^2-1))^(1/2)*((c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*(arccosh(c*x)^2-2*arccosh(c*x)+2)*d^2/(c*x+1)
/c^2/(c*x-1)-5/128*(-d*(c^2*x^2-1))^(1/2)*(-(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*(arccosh(c*x)^2+2*arcco
sh(c*x)+2)*d^2/(c*x+1)/c^2/(c*x-1)+1/384*(-d*(c^2*x^2-1))^(1/2)*(-4*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+4*c^4*
x^4+3*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-5*c^2*x^2+1)*(9*arccosh(c*x)^2+6*arccosh(c*x)+2)*d^2/(c*x+1)/c^2/(c*x-1)
-1/3200*(-d*(c^2*x^2-1))^(1/2)*(-16*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5+16*c^6*x^6+20*(c*x+1)^(1/2)*(c*x-1)^(1
/2)*x^3*c^3-28*c^4*x^4-5*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+13*c^2*x^2-1)*(25*arccosh(c*x)^2+10*arccosh(c*x)+2)*d
^2/(c*x+1)/c^2/(c*x-1)+1/43904*(-d*(c^2*x^2-1))^(1/2)*(-64*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^7*c^7+64*c^8*x^8+112*
(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5-144*c^6*x^6-56*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+104*c^4*x^4+7*(c*x+1)^(
1/2)*(c*x-1)^(1/2)*x*c-25*c^2*x^2+1)*(49*arccosh(c*x)^2+14*arccosh(c*x)+2)*d^2/(c*x+1)/c^2/(c*x-1))+2*a*b*(1/6
272*(-d*(c^2*x^2-1))^(1/2)*(64*c^8*x^8-144*c^6*x^6+64*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^7*c^7+104*c^4*x^4-112*(c*x
+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5-25*c^2*x^2+56*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3-7*(c*x+1)^(1/2)*(c*x-1)^(1/2
)*x*c+1)*(-1+7*arccosh(c*x))*d^2/(c*x+1)/c^2/(c*x-1)-1/640*(-d*(c^2*x^2-1))^(1/2)*(16*c^6*x^6-28*c^4*x^4+16*(c
*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5+13*c^2*x^2-20*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+5*(c*x+1)^(1/2)*(c*x-1)^(1
/2)*x*c-1)*(-1+5*arccosh(c*x))*d^2/(c*x+1)/c^2/(c*x-1)+1/128*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2+4*(c*
x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3-3*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+1)*(-1+3*arccosh(c*x))*d^2/(c*x+1)/c^2/(c*x
-1)-5/128*(-d*(c^2*x^2-1))^(1/2)*((c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*(-1+arccosh(c*x))*d^2/(c*x+1)/c^2
/(c*x-1)-5/128*(-d*(c^2*x^2-1))^(1/2)*(-(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*(1+arccosh(c*x))*d^2/(c*x+1
)/c^2/(c*x-1)+1/128*(-d*(c^2*x^2-1))^(1/2)*(-4*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+4*c^4*x^4+3*(c*x+1)^(1/2)*(
c*x-1)^(1/2)*x*c-5*c^2*x^2+1)*(1+3*arccosh(c*x))*d^2/(c*x+1)/c^2/(c*x-1)-1/640*(-d*(c^2*x^2-1))^(1/2)*(-16*(c*
x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5+16*c^6*x^6+20*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3-28*c^4*x^4-5*(c*x+1)^(1/2)*
(c*x-1)^(1/2)*x*c+13*c^2*x^2-1)*(1+5*arccosh(c*x))*d^2/(c*x+1)/c^2/(c*x-1)+1/6272*(-d*(c^2*x^2-1))^(1/2)*(-64*
(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^7*c^7+64*c^8*x^8+112*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5-144*c^6*x^6-56*(c*x+1)^
(1/2)*(c*x-1)^(1/2)*x^3*c^3+104*c^4*x^4+7*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-25*c^2*x^2+1)*(1+7*arccosh(c*x))*d^2
/(c*x+1)/c^2/(c*x-1))

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Maxima [A]  time = 1.3219, size = 455, normalized size = 0.97 \begin{align*} -\frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{7}{2}} b^{2} \operatorname{arcosh}\left (c x\right )^{2}}{7 \, c^{2} d} - \frac{2 \,{\left (-c^{2} d x^{2} + d\right )}^{\frac{7}{2}} a b \operatorname{arcosh}\left (c x\right )}{7 \, c^{2} d} + \frac{2}{25725} \, b^{2}{\left (\frac{75 \, \sqrt{c^{2} x^{2} - 1} c^{4} \sqrt{-d} d^{3} x^{6} - 351 \, \sqrt{c^{2} x^{2} - 1} c^{2} \sqrt{-d} d^{3} x^{4} + 757 \, \sqrt{c^{2} x^{2} - 1} \sqrt{-d} d^{3} x^{2} - \frac{2161 \, \sqrt{c^{2} x^{2} - 1} \sqrt{-d} d^{3}}{c^{2}}}{d} - \frac{105 \,{\left (5 \, c^{6} \sqrt{-d} d^{3} x^{7} - 21 \, c^{4} \sqrt{-d} d^{3} x^{5} + 35 \, c^{2} \sqrt{-d} d^{3} x^{3} - 35 \, \sqrt{-d} d^{3} x\right )} \operatorname{arcosh}\left (c x\right )}{c d}\right )} - \frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{7}{2}} a^{2}}{7 \, c^{2} d} - \frac{2 \,{\left (5 \, c^{6} \sqrt{-d} d^{3} x^{7} - 21 \, c^{4} \sqrt{-d} d^{3} x^{5} + 35 \, c^{2} \sqrt{-d} d^{3} x^{3} - 35 \, \sqrt{-d} d^{3} x\right )} a b}{245 \, c d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2,x, algorithm="maxima")

[Out]

-1/7*(-c^2*d*x^2 + d)^(7/2)*b^2*arccosh(c*x)^2/(c^2*d) - 2/7*(-c^2*d*x^2 + d)^(7/2)*a*b*arccosh(c*x)/(c^2*d) +
2/25725*b^2*((75*sqrt(c^2*x^2 - 1)*c^4*sqrt(-d)*d^3*x^6 - 351*sqrt(c^2*x^2 - 1)*c^2*sqrt(-d)*d^3*x^4 + 757*sq
rt(c^2*x^2 - 1)*sqrt(-d)*d^3*x^2 - 2161*sqrt(c^2*x^2 - 1)*sqrt(-d)*d^3/c^2)/d - 105*(5*c^6*sqrt(-d)*d^3*x^7 -
21*c^4*sqrt(-d)*d^3*x^5 + 35*c^2*sqrt(-d)*d^3*x^3 - 35*sqrt(-d)*d^3*x)*arccosh(c*x)/(c*d)) - 1/7*(-c^2*d*x^2 +
d)^(7/2)*a^2/(c^2*d) - 2/245*(5*c^6*sqrt(-d)*d^3*x^7 - 21*c^4*sqrt(-d)*d^3*x^5 + 35*c^2*sqrt(-d)*d^3*x^3 - 35
*sqrt(-d)*d^3*x)*a*b/(c*d)

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Fricas [A]  time = 2.56769, size = 1031, normalized size = 2.19 \begin{align*} \frac{3675 \,{\left (b^{2} c^{8} d^{2} x^{8} - 4 \, b^{2} c^{6} d^{2} x^{6} + 6 \, b^{2} c^{4} d^{2} x^{4} - 4 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \sqrt{-c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )^{2} - 210 \,{\left (5 \, a b c^{7} d^{2} x^{7} - 21 \, a b c^{5} d^{2} x^{5} + 35 \, a b c^{3} d^{2} x^{3} - 35 \, a b c d^{2} x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} - 210 \,{\left ({\left (5 \, b^{2} c^{7} d^{2} x^{7} - 21 \, b^{2} c^{5} d^{2} x^{5} + 35 \, b^{2} c^{3} d^{2} x^{3} - 35 \, b^{2} c d^{2} x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} - 35 \,{\left (a b c^{8} d^{2} x^{8} - 4 \, a b c^{6} d^{2} x^{6} + 6 \, a b c^{4} d^{2} x^{4} - 4 \, a b c^{2} d^{2} x^{2} + a b d^{2}\right )} \sqrt{-c^{2} d x^{2} + d}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) +{\left (75 \,{\left (49 \, a^{2} + 2 \, b^{2}\right )} c^{8} d^{2} x^{8} - 12 \,{\left (1225 \, a^{2} + 71 \, b^{2}\right )} c^{6} d^{2} x^{6} + 2 \,{\left (11025 \, a^{2} + 1108 \, b^{2}\right )} c^{4} d^{2} x^{4} - 4 \,{\left (3675 \, a^{2} + 1459 \, b^{2}\right )} c^{2} d^{2} x^{2} +{\left (3675 \, a^{2} + 4322 \, b^{2}\right )} d^{2}\right )} \sqrt{-c^{2} d x^{2} + d}}{25725 \,{\left (c^{4} x^{2} - c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25725*(3675*(b^2*c^8*d^2*x^8 - 4*b^2*c^6*d^2*x^6 + 6*b^2*c^4*d^2*x^4 - 4*b^2*c^2*d^2*x^2 + b^2*d^2)*sqrt(-c^
2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1))^2 - 210*(5*a*b*c^7*d^2*x^7 - 21*a*b*c^5*d^2*x^5 + 35*a*b*c^3*d^2*x^3
- 35*a*b*c*d^2*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 210*((5*b^2*c^7*d^2*x^7 - 21*b^2*c^5*d^2*x^5 + 35*
b^2*c^3*d^2*x^3 - 35*b^2*c*d^2*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 35*(a*b*c^8*d^2*x^8 - 4*a*b*c^6*d^2
*x^6 + 6*a*b*c^4*d^2*x^4 - 4*a*b*c^2*d^2*x^2 + a*b*d^2)*sqrt(-c^2*d*x^2 + d))*log(c*x + sqrt(c^2*x^2 - 1)) + (
75*(49*a^2 + 2*b^2)*c^8*d^2*x^8 - 12*(1225*a^2 + 71*b^2)*c^6*d^2*x^6 + 2*(11025*a^2 + 1108*b^2)*c^4*d^2*x^4 -
4*(3675*a^2 + 1459*b^2)*c^2*d^2*x^2 + (3675*a^2 + 4322*b^2)*d^2)*sqrt(-c^2*d*x^2 + d))/(c^4*x^2 - c^2)

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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*(-c**2*d*x**2+d)**(5/2)*(a+b*acosh(c*x))**2,x)

[Out]

Timed out

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

[Out]

Exception raised: NotImplementedError