3.59 \(\int (f+g x)^2 (d-c^2 d x^2)^{3/2} (a+b \cosh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=725 \[ \frac{3}{8} d f^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} d f^2 x (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{3 d f^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b c \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 d f g (1-c x)^2 (c x+1)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2}+\frac{1}{8} d g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} d g^2 x^3 (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{d g^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 c^2}-\frac{d g^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 d f^2 x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 b c d f^2 x^2 \sqrt{d-c^2 d x^2}}{16 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b c^3 d f g x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{c x-1} \sqrt{c x+1}}-\frac{4 b c d f g x^3 \sqrt{d-c^2 d x^2}}{15 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b d f g x \sqrt{d-c^2 d x^2}}{5 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 d g^2 x^6 \sqrt{d-c^2 d x^2}}{36 \sqrt{c x-1} \sqrt{c x+1}}-\frac{7 b c d g^2 x^4 \sqrt{d-c^2 d x^2}}{96 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d g^2 x^2 \sqrt{d-c^2 d x^2}}{32 c \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

(2*b*d*f*g*x*Sqrt[d - c^2*d*x^2])/(5*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (5*b*c*d*f^2*x^2*Sqrt[d - c^2*d*x^2])/(
16*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d*g^2*x^2*Sqrt[d - c^2*d*x^2])/(32*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (4*
b*c*d*f*g*x^3*Sqrt[d - c^2*d*x^2])/(15*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*f^2*x^4*Sqrt[d - c^2*d*x^2])/(
16*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (7*b*c*d*g^2*x^4*Sqrt[d - c^2*d*x^2])/(96*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (
2*b*c^3*d*f*g*x^5*Sqrt[d - c^2*d*x^2])/(25*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*g^2*x^6*Sqrt[d - c^2*d*x^2
])/(36*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*d*f^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/8 - (d*g^2*x*Sqrt[
d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(16*c^2) + (d*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/8 + (d*f^
2*x*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/4 + (d*g^2*x^3*(1 - c*x)*(1 + c*x)*Sqrt[d -
c^2*d*x^2]*(a + b*ArcCosh[c*x]))/6 - (2*d*f*g*(1 - c*x)^2*(1 + c*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])
)/(5*c^2) - (3*d*f^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(16*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d*g^
2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(32*b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

________________________________________________________________________________________

Rubi [A]  time = 1.6929, antiderivative size = 725, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 12, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.387, Rules used = {5836, 5822, 5685, 5683, 5676, 30, 14, 5718, 194, 5745, 5743, 5759} \[ \frac{3}{8} d f^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} d f^2 x (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{3 d f^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b c \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 d f g (1-c x)^2 (c x+1)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2}+\frac{1}{8} d g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} d g^2 x^3 (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{d g^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 c^2}-\frac{d g^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 d f^2 x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 b c d f^2 x^2 \sqrt{d-c^2 d x^2}}{16 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b c^3 d f g x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{c x-1} \sqrt{c x+1}}-\frac{4 b c d f g x^3 \sqrt{d-c^2 d x^2}}{15 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b d f g x \sqrt{d-c^2 d x^2}}{5 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 d g^2 x^6 \sqrt{d-c^2 d x^2}}{36 \sqrt{c x-1} \sqrt{c x+1}}-\frac{7 b c d g^2 x^4 \sqrt{d-c^2 d x^2}}{96 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d g^2 x^2 \sqrt{d-c^2 d x^2}}{32 c \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

Int[(f + g*x)^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]),x]

[Out]

(2*b*d*f*g*x*Sqrt[d - c^2*d*x^2])/(5*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (5*b*c*d*f^2*x^2*Sqrt[d - c^2*d*x^2])/(
16*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d*g^2*x^2*Sqrt[d - c^2*d*x^2])/(32*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (4*
b*c*d*f*g*x^3*Sqrt[d - c^2*d*x^2])/(15*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*f^2*x^4*Sqrt[d - c^2*d*x^2])/(
16*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (7*b*c*d*g^2*x^4*Sqrt[d - c^2*d*x^2])/(96*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (
2*b*c^3*d*f*g*x^5*Sqrt[d - c^2*d*x^2])/(25*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*g^2*x^6*Sqrt[d - c^2*d*x^2
])/(36*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*d*f^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/8 - (d*g^2*x*Sqrt[
d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(16*c^2) + (d*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/8 + (d*f^
2*x*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/4 + (d*g^2*x^3*(1 - c*x)*(1 + c*x)*Sqrt[d -
c^2*d*x^2]*(a + b*ArcCosh[c*x]))/6 - (2*d*f*g*(1 - c*x)^2*(1 + c*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])
)/(5*c^2) - (3*d*f^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(16*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d*g^
2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(32*b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 5836

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(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 + g*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, g, n}, x] && EqQ[c^2*d
+ e, 0] && IntegerQ[m] && IntegerQ[p - 1/2]

Rule 5822

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_)*((f_) + (g
_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, (f + g*x
)^m, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[m,
0] && IntegerQ[p + 1/2] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[n, 0] && ((EqQ[n, 1] && GtQ[p, -1]) || GtQ[p, 0] |
| EqQ[m, 1] || (EqQ[m, 2] && LtQ[p, -2]))

Rule 5685

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

Rule 5683

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

Rule 5676

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

Rule 30

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

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 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 5745

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_
))^(p_), x_Symbol] :> Simp[((f*x)^(m + 1)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n)/(f*(m + 2*p + 1)
), x] + (Dist[(2*d1*d2*p)/(m + 2*p + 1), Int[(f*x)^m*(d1 + e1*x)^(p - 1)*(d2 + e2*x)^(p - 1)*(a + b*ArcCosh[c*
x])^n, x], x] - Dist[(b*c*n*(-(d1*d2))^(p - 1/2)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(f*(m + 2*p + 1)*Sqrt[1 + c*
x]*Sqrt[-1 + c*x]), Int[(f*x)^(m + 1)*(-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, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[p, 0] &&  !L
tQ[m, -1] && IntegerQ[p - 1/2] && (RationalQ[m] || EqQ[n, 1])

Rule 5743

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*
(x_)], x_Symbol] :> Simp[((f*x)^(m + 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(f*(m + 2)), x
] + (-Dist[(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/((m + 2)*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[((f*x)^m*(a + b*ArcCo
sh[c*x])^n)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Dist[(b*c*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(f*(m + 2)*S
qrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e
1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] &&  !LtQ[m, -1] && (RationalQ[m] |
| EqQ[n, 1])

Rule 5759

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[(f*(f*x)^(m - 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(e1*e2*m
), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b*ArcCosh[c*x])^n)/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*
x]), x], x] + Dist[(b*f*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(c*d1*d2*m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)
^(m - 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0]
&& EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rubi steps

\begin{align*} \int (f+g x)^2 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=-\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} (f+g x)^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \int \left (f^2 (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )+2 f g x (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )+g^2 x^2 (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )\right ) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\left (d f^2 \sqrt{d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 d f g \sqrt{d-c^2 d x^2}\right ) \int x (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (d g^2 \sqrt{d-c^2 d x^2}\right ) \int x^2 (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{4} d f^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} d g^2 x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{2 d f g (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2}+\frac{\left (3 d f^2 \sqrt{d-c^2 d x^2}\right ) \int \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b c d f^2 \sqrt{d-c^2 d x^2}\right ) \int x \left (-1+c^2 x^2\right ) \, dx}{4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 b d f g \sqrt{d-c^2 d x^2}\right ) \int \left (-1+c^2 x^2\right )^2 \, dx}{5 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (d g^2 \sqrt{d-c^2 d x^2}\right ) \int x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b c d g^2 \sqrt{d-c^2 d x^2}\right ) \int x^3 \left (-1+c^2 x^2\right ) \, dx}{6 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{3}{8} d f^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} d g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} d f^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} d g^2 x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{2 d f g (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2}-\frac{\left (3 d f^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{8 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b c d f^2 \sqrt{d-c^2 d x^2}\right ) \int \left (-x+c^2 x^3\right ) \, dx}{4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 b c d f^2 \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{8 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 b d f g \sqrt{d-c^2 d x^2}\right ) \int \left (1-2 c^2 x^2+c^4 x^4\right ) \, dx}{5 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (d g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{8 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b c d g^2 \sqrt{d-c^2 d x^2}\right ) \int x^3 \, dx}{8 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b c d g^2 \sqrt{d-c^2 d x^2}\right ) \int \left (-x^3+c^2 x^5\right ) \, dx}{6 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{2 b d f g x \sqrt{d-c^2 d x^2}}{5 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{5 b c d f^2 x^2 \sqrt{d-c^2 d x^2}}{16 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{4 b c d f g x^3 \sqrt{d-c^2 d x^2}}{15 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d f^2 x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{7 b c d g^2 x^4 \sqrt{d-c^2 d x^2}}{96 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b c^3 d f g x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d g^2 x^6 \sqrt{d-c^2 d x^2}}{36 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3}{8} d f^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{d g^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 c^2}+\frac{1}{8} d g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} d f^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} d g^2 x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{2 d f g (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2}-\frac{3 d f^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (d g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{16 c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b d g^2 \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{16 c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{2 b d f g x \sqrt{d-c^2 d x^2}}{5 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{5 b c d f^2 x^2 \sqrt{d-c^2 d x^2}}{16 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d g^2 x^2 \sqrt{d-c^2 d x^2}}{32 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{4 b c d f g x^3 \sqrt{d-c^2 d x^2}}{15 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d f^2 x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{7 b c d g^2 x^4 \sqrt{d-c^2 d x^2}}{96 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b c^3 d f g x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d g^2 x^6 \sqrt{d-c^2 d x^2}}{36 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3}{8} d f^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{d g^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 c^2}+\frac{1}{8} d g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} d f^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} d g^2 x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{2 d f g (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2}-\frac{3 d f^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d g^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}

Mathematica [A]  time = 2.86543, size = 623, normalized size = 0.86 \[ \frac{-3600 a d^{3/2} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (6 c^2 f^2+g^2\right ) \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )-240 a c d \sqrt{\frac{c x-1}{c x+1}} (c x+1) \sqrt{d-c^2 d x^2} \left (30 c^2 f^2 x \left (2 c^2 x^2-5\right )+96 f g \left (c^2 x^2-1\right )^2+5 g^2 x \left (8 c^4 x^4-14 c^2 x^2+3\right )\right )-7200 b c^2 d f^2 \sqrt{d-c^2 d x^2} \left (\cosh \left (2 \cosh ^{-1}(c x)\right )+2 \cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)-\sinh \left (2 \cosh ^{-1}(c x)\right )\right )\right )+450 b c^2 d f^2 \sqrt{d-c^2 d x^2} \left (8 \cosh ^{-1}(c x)^2+\cosh \left (4 \cosh ^{-1}(c x)\right )-4 \cosh ^{-1}(c x) \sinh \left (4 \cosh ^{-1}(c x)\right )\right )+3200 b c d f g \sqrt{d-c^2 d x^2} \left (9 c x+12 \left (\frac{c x-1}{c x+1}\right )^{3/2} (c x+1)^3 \cosh ^{-1}(c x)-\cosh \left (3 \cosh ^{-1}(c x)\right )\right )-32 b c d f g \sqrt{d-c^2 d x^2} \left (450 c x-450 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)-25 \cosh \left (3 \cosh ^{-1}(c x)\right )-9 \cosh \left (5 \cosh ^{-1}(c x)\right )+75 \cosh ^{-1}(c x) \sinh \left (3 \cosh ^{-1}(c x)\right )+45 \cosh ^{-1}(c x) \sinh \left (5 \cosh ^{-1}(c x)\right )\right )-450 b d g^2 \sqrt{d-c^2 d x^2} \left (8 \cosh ^{-1}(c x)^2+\cosh \left (4 \cosh ^{-1}(c x)\right )-4 \cosh ^{-1}(c x) \sinh \left (4 \cosh ^{-1}(c x)\right )\right )-25 b d g^2 \sqrt{d-c^2 d x^2} \left (-72 \cosh ^{-1}(c x)^2+18 \cosh \left (2 \cosh ^{-1}(c x)\right )-9 \cosh \left (4 \cosh ^{-1}(c x)\right )-2 \cosh \left (6 \cosh ^{-1}(c x)\right )+12 \cosh ^{-1}(c x) \left (-3 \sinh \left (2 \cosh ^{-1}(c x)\right )+3 \sinh \left (4 \cosh ^{-1}(c x)\right )+\sinh \left (6 \cosh ^{-1}(c x)\right )\right )\right )}{57600 c^3 \sqrt{\frac{c x-1}{c x+1}} (c x+1)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(f + g*x)^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]),x]

[Out]

(-240*a*c*d*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(96*f*g*(-1 + c^2*x^2)^2 + 30*c^2*f^2*x*(
-5 + 2*c^2*x^2) + 5*g^2*x*(3 - 14*c^2*x^2 + 8*c^4*x^4)) - 3600*a*d^(3/2)*(6*c^2*f^2 + g^2)*Sqrt[(-1 + c*x)/(1
+ c*x)]*(1 + c*x)*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 3200*b*c*d*f*g*Sqrt[d - c^2*d*x
^2]*(9*c*x + 12*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*ArcCosh[c*x] - Cosh[3*ArcCosh[c*x]]) - 7200*b*c^2*d*f
^2*Sqrt[d - c^2*d*x^2]*(Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]*(ArcCosh[c*x] - Sinh[2*ArcCosh[c*x]])) + 450*b*c
^2*d*f^2*Sqrt[d - c^2*d*x^2]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]) -
450*b*d*g^2*Sqrt[d - c^2*d*x^2]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]
]) - 32*b*c*d*f*g*Sqrt[d - c^2*d*x^2]*(450*c*x - 450*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] - 25*Co
sh[3*ArcCosh[c*x]] - 9*Cosh[5*ArcCosh[c*x]] + 75*ArcCosh[c*x]*Sinh[3*ArcCosh[c*x]] + 45*ArcCosh[c*x]*Sinh[5*Ar
cCosh[c*x]]) - 25*b*d*g^2*Sqrt[d - c^2*d*x^2]*(-72*ArcCosh[c*x]^2 + 18*Cosh[2*ArcCosh[c*x]] - 9*Cosh[4*ArcCosh
[c*x]] - 2*Cosh[6*ArcCosh[c*x]] + 12*ArcCosh[c*x]*(-3*Sinh[2*ArcCosh[c*x]] + 3*Sinh[4*ArcCosh[c*x]] + Sinh[6*A
rcCosh[c*x]])))/(57600*c^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))

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Maple [A]  time = 0.44, size = 1177, normalized size = 1.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x)),x)

[Out]

-3/16*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c*arccosh(c*x)^2*d*f^2-1/32*b*(-d*(c^2*x^2-1))^(1/2
)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c^3*arccosh(c*x)^2*d*g^2-17/48*b*(-d*(c^2*x^2-1))^(1/2)*g^2*d/(c*x+1)/(c*x-1)*ar
ccosh(c*x)*x^3-5/8*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)/(c*x-1)*arccosh(c*x)*x*f^2+1/16*b*(-d*(c^2*x^2-1))^(1/2)
*d/(c*x+1)^(1/2)*c^3/(c*x-1)^(1/2)*x^4*f^2-5/16*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)^(1/2)*c/(c*x-1)^(1/2)*x^2*f
^2+1/36*b*(-d*(c^2*x^2-1))^(1/2)*g^2*d/(c*x+1)^(1/2)*c^3/(c*x-1)^(1/2)*x^6-7/96*b*(-d*(c^2*x^2-1))^(1/2)*g^2*d
/(c*x+1)^(1/2)*c/(c*x-1)^(1/2)*x^4+1/32*b*(-d*(c^2*x^2-1))^(1/2)*g^2*d/(c*x+1)^(1/2)/c/(c*x-1)^(1/2)*x^2+7/230
4*b*(-d*(c^2*x^2-1))^(1/2)*g^2*d/(c*x+1)^(1/2)/c^3/(c*x-1)^(1/2)+17/128*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)^(1/
2)/c/(c*x-1)^(1/2)*f^2+1/24*a*g^2/c^2*x*(-c^2*d*x^2+d)^(3/2)+3/8*a*f^2*d*x*(-c^2*d*x^2+d)^(1/2)+3/8*a*f^2*d^2/
(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-2/5*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d/(c*x+1)*c^4/(c*x
-1)*arccosh(c*x)*x^6+6/5*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d/(c*x+1)*c^2/(c*x-1)*arccosh(c*x)*x^4+1/4*a*f^2*x*(-c^2
*d*x^2+d)^(3/2)+1/16*b*(-d*(c^2*x^2-1))^(1/2)*g^2*d/(c*x+1)/c^2/(c*x-1)*arccosh(c*x)*x+2/5*b*(-d*(c^2*x^2-1))^
(1/2)*f*g*d/(c*x+1)/c^2/(c*x-1)*arccosh(c*x)-1/4*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)*c^4/(c*x-1)*arccosh(c*x)*x
^5*f^2+7/8*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)*c^2/(c*x-1)*arccosh(c*x)*x^3*f^2-6/5*b*(-d*(c^2*x^2-1))^(1/2)*f*
g*d/(c*x+1)/(c*x-1)*arccosh(c*x)*x^2+2/25*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d/(c*x+1)^(1/2)*c^3/(c*x-1)^(1/2)*x^5-4
/15*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d/(c*x+1)^(1/2)*c/(c*x-1)^(1/2)*x^3+2/5*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d/(c*x+1
)^(1/2)/c/(c*x-1)^(1/2)*x-1/6*b*(-d*(c^2*x^2-1))^(1/2)*g^2*d/(c*x+1)*c^4/(c*x-1)*arccosh(c*x)*x^7+11/24*b*(-d*
(c^2*x^2-1))^(1/2)*g^2*d/(c*x+1)*c^2/(c*x-1)*arccosh(c*x)*x^5-1/6*a*g^2*x*(-c^2*d*x^2+d)^(5/2)/c^2/d+1/16*a*g^
2/c^2*d*x*(-c^2*d*x^2+d)^(1/2)+1/16*a*g^2/c^2*d^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-2
/5*a*f*g/c^2/d*(-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((g*x+f)^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x)),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 c^{2} d g^{2} x^{4} + 2 \, a c^{2} d f g x^{3} - 2 \, a d f g x - a d f^{2} +{\left (a c^{2} d f^{2} - a d g^{2}\right )} x^{2} +{\left (b c^{2} d g^{2} x^{4} + 2 \, b c^{2} d f g x^{3} - 2 \, b d f g x - b d f^{2} +{\left (b c^{2} d f^{2} - b d g^{2}\right )} x^{2}\right )} \operatorname{arcosh}\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((g*x+f)^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x)),x, algorithm="fricas")

[Out]

integral(-(a*c^2*d*g^2*x^4 + 2*a*c^2*d*f*g*x^3 - 2*a*d*f*g*x - a*d*f^2 + (a*c^2*d*f^2 - a*d*g^2)*x^2 + (b*c^2*
d*g^2*x^4 + 2*b*c^2*d*f*g*x^3 - 2*b*d*f*g*x - b*d*f^2 + (b*c^2*d*f^2 - b*d*g^2)*x^2)*arccosh(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((g*x+f)**2*(-c**2*d*x**2+d)**(3/2)*(a+b*acosh(c*x)),x)

[Out]

Timed out

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Giac [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((g*x+f)^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x)),x, algorithm="giac")

[Out]

Timed out