\(\int x (d-c^2 d x^2)^{3/2} (a+b \text {arccosh}(c x))^2 \, dx\) [180]

Optimal result

Integrand size = 27, antiderivative size = 348 \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {16 b^2 d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{75 c^2 (1-c x) (1+c x)}-\frac {8 b^2 d \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{225 c^2 (1-c x) (1+c x)}-\frac {2 b^2 d \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{125 c^2 (1-c x) (1+c x)}+\frac {2 b d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{5 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4 b c d x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{25 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{5 c^2 d} \]

[Out]

-1/5*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2/c^2/d-16/75*b^2*d*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/c^2/(-c*x+1
)/(c*x+1)-8/225*b^2*d*(-c^2*x^2+1)^2*(-c^2*d*x^2+d)^(1/2)/c^2/(-c*x+1)/(c*x+1)-2/125*b^2*d*(-c^2*x^2+1)^3*(-c^
2*d*x^2+d)^(1/2)/c^2/(-c*x+1)/(c*x+1)+2/5*b*d*x*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1
)^(1/2)-4/15*b*c*d*x^3*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+2/25*b*c^3*d*x^5*(a
+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {5914, 5889, 200, 5894, 12, 534, 1261, 712} \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {2 b d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{5 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{5 c^2 d}-\frac {4 b c d x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b c^3 d x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{25 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b^2 d \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{125 c^2 (1-c x) (c x+1)}-\frac {8 b^2 d \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{225 c^2 (1-c x) (c x+1)}-\frac {16 b^2 d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{75 c^2 (1-c x) (c x+1)} \]

[In]

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

[Out]

(-16*b^2*d*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(75*c^2*(1 - c*x)*(1 + c*x)) - (8*b^2*d*(1 - c^2*x^2)^2*Sqrt[d -
c^2*d*x^2])/(225*c^2*(1 - c*x)*(1 + c*x)) - (2*b^2*d*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(125*c^2*(1 - c*x)*
(1 + c*x)) + (2*b*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(5*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (4*b*c*d*
x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(15*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*c^3*d*x^5*Sqrt[d - c^2*
d*x^2]*(a + b*ArcCosh[c*x]))/(25*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2
)/(5*c^2*d)

Rule 12

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

Rule 200

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 534

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.) + (e_.)*(x_)^(n2_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b
2_.)*(x_)^(non2_.))^(p_.), x_Symbol] :> Dist[(a1 + b1*x^(n/2))^FracPart[p]*((a2 + b2*x^(n/2))^FracPart[p]/(a1*
a2 + b1*b2*x^n)^FracPart[p]), Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n + e*x^(2*n))^q, x], x] /; FreeQ[{a1, b1,
a2, b2, c, d, e, n, p, q}, x] && EqQ[non2, n/2] && EqQ[n2, 2*n] && EqQ[a2*b1 + a1*b2, 0]

Rule 712

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rule 1261

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 5889

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

Rule 5894

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 5914

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^
(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e*(p + 1))), x] - Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*
(-1 + c*x)^p)], Int[(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a,
b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{5 c^2 d}+\frac {\left (2 b d \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^2 (1+c x)^2 (a+b \text {arccosh}(c x)) \, dx}{5 c \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{5 c^2 d}+\frac {\left (2 b d \sqrt {d-c^2 d x^2}\right ) \int \left (-1+c^2 x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx}{5 c \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {2 b d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{5 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4 b c d x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{25 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{5 c^2 d}-\frac {\left (2 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{15 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{5 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {2 b d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{5 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4 b c d x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{25 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{5 c^2 d}-\frac {\left (2 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{75 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {2 b d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{5 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4 b c d x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{25 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{5 c^2 d}-\frac {\left (2 b^2 d \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{\sqrt {-1+c^2 x^2}} \, dx}{75 (-1+c x) (1+c x)} \\ & = \frac {2 b d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{5 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4 b c d x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{25 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{5 c^2 d}-\frac {\left (b^2 d \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {15-10 c^2 x+3 c^4 x^2}{\sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{75 (-1+c x) (1+c x)} \\ & = \frac {2 b d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{5 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4 b c d x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{25 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{5 c^2 d}-\frac {\left (b^2 d \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {8}{\sqrt {-1+c^2 x}}-4 \sqrt {-1+c^2 x}+3 \left (-1+c^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )}{75 (-1+c x) (1+c x)} \\ & = -\frac {16 b^2 d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{75 c^2 (1-c x) (1+c x)}-\frac {8 b^2 d \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{225 c^2 (1-c x) (1+c x)}-\frac {2 b^2 d \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{125 c^2 (1-c x) (1+c x)}+\frac {2 b d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{5 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4 b c d x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{25 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{5 c^2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.48 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.60 \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {d \sqrt {d-c^2 d x^2} \left (225 a^2 \left (-1+c^2 x^2\right )^3-30 a b c x \sqrt {-1+c x} \sqrt {1+c x} \left (15-10 c^2 x^2+3 c^4 x^4\right )+2 b^2 \left (-149+187 c^2 x^2-47 c^4 x^4+9 c^6 x^6\right )-30 b \left (-15 a \left (-1+c^2 x^2\right )^3+b c x \sqrt {-1+c x} \sqrt {1+c x} \left (15-10 c^2 x^2+3 c^4 x^4\right )\right ) \text {arccosh}(c x)+225 b^2 \left (-1+c^2 x^2\right )^3 \text {arccosh}(c x)^2\right )}{1125 c^2 \left (-1+c^2 x^2\right )} \]

[In]

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

[Out]

-1/1125*(d*Sqrt[d - c^2*d*x^2]*(225*a^2*(-1 + c^2*x^2)^3 - 30*a*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(15 - 10*c^
2*x^2 + 3*c^4*x^4) + 2*b^2*(-149 + 187*c^2*x^2 - 47*c^4*x^4 + 9*c^6*x^6) - 30*b*(-15*a*(-1 + c^2*x^2)^3 + b*c*
x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(15 - 10*c^2*x^2 + 3*c^4*x^4))*ArcCosh[c*x] + 225*b^2*(-1 + c^2*x^2)^3*ArcCosh[
c*x]^2))/(c^2*(-1 + c^2*x^2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1269\) vs. \(2(308)=616\).

Time = 1.18 (sec) , antiderivative size = 1270, normalized size of antiderivative = 3.65

method	result	size
default	\(\text {Expression too large to display}\)	\(1270\)
parts	\(\text {Expression too large to display}\)	\(1270\)

[In]

int(x*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2,x,method=_RETURNVERBOSE)

[Out]

-1/5*a^2*(-c^2*d*x^2+d)^(5/2)/c^2/d+b^2*(-1/4000*(-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)*c^5*x^5+13*c^2*x^2-20*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3+5*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-1)
*(25*arccosh(c*x)^2-10*arccosh(c*x)+2)*d/(c*x+1)/c^2/(c*x-1)+1/288*(-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)*c^3*x^3-3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+1)*(9*arccosh(c*x)^2-6*arccosh(c*x)+2
)*d/(c*x+1)/c^2/(c*x-1)-1/16*(-d*(c^2*x^2-1))^(1/2)*((c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(arccosh(c*x)^
2-2*arccosh(c*x)+2)*d/(c*x+1)/c^2/(c*x-1)-1/16*(-d*(c^2*x^2-1))^(1/2)*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^
2-1)*(arccosh(c*x)^2+2*arccosh(c*x)+2)*d/(c*x+1)/c^2/(c*x-1)+1/288*(-d*(c^2*x^2-1))^(1/2)*(-4*(c*x-1)^(1/2)*(c
*x+1)^(1/2)*c^3*x^3+4*c^4*x^4+3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-5*c^2*x^2+1)*(9*arccosh(c*x)^2+6*arccosh(c*x)+
2)*d/(c*x+1)/c^2/(c*x-1)-1/4000*(-d*(c^2*x^2-1))^(1/2)*(-16*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^5*x^5+16*c^6*x^6+20*
(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3-28*c^4*x^4-5*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+13*c^2*x^2-1)*(25*arccosh(c*x
)^2+10*arccosh(c*x)+2)*d/(c*x+1)/c^2/(c*x-1))+2*a*b*(-1/800*(-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)*c^5*x^5+13*c^2*x^2-20*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3+5*(c*x-1)^(1/2)*(c*x+1)^(
1/2)*c*x-1)*(-1+5*arccosh(c*x))*d/(c*x+1)/c^2/(c*x-1)+1/96*(-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)*c^3*x^3-3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+1)*(-1+3*arccosh(c*x))*d/(c*x+1)/c^2/(c*x-1)-
1/16*(-d*(c^2*x^2-1))^(1/2)*((c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(-1+arccosh(c*x))*d/(c*x+1)/c^2/(c*x-1
)-1/16*(-d*(c^2*x^2-1))^(1/2)*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(1+arccosh(c*x))*d/(c*x+1)/c^2/(c*x
-1)+1/96*(-d*(c^2*x^2-1))^(1/2)*(-4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3+4*c^4*x^4+3*(c*x-1)^(1/2)*(c*x+1)^(1/2
)*c*x-5*c^2*x^2+1)*(1+3*arccosh(c*x))*d/(c*x+1)/c^2/(c*x-1)-1/800*(-d*(c^2*x^2-1))^(1/2)*(-16*(c*x+1)^(1/2)*(c
*x-1)^(1/2)*c^5*x^5+16*c^6*x^6+20*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3-28*c^4*x^4-5*(c*x-1)^(1/2)*(c*x+1)^(1/2)
*c*x+13*c^2*x^2-1)*(1+5*arccosh(c*x))*d/(c*x+1)/c^2/(c*x-1))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.05 \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {225 \, {\left (b^{2} c^{6} d x^{6} - 3 \, b^{2} c^{4} d x^{4} + 3 \, b^{2} c^{2} d x^{2} - b^{2} d\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} - 30 \, {\left (3 \, a b c^{5} d x^{5} - 10 \, a b c^{3} d x^{3} + 15 \, a b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 30 \, {\left ({\left (3 \, b^{2} c^{5} d x^{5} - 10 \, b^{2} c^{3} d x^{3} + 15 \, b^{2} c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 15 \, {\left (a b c^{6} d x^{6} - 3 \, a b c^{4} d x^{4} + 3 \, a b c^{2} d x^{2} - a b d\right )} \sqrt {-c^{2} d x^{2} + d}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (9 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{6} d x^{6} - {\left (675 \, a^{2} + 94 \, b^{2}\right )} c^{4} d x^{4} + {\left (675 \, a^{2} + 374 \, b^{2}\right )} c^{2} d x^{2} - {\left (225 \, a^{2} + 298 \, b^{2}\right )} d\right )} \sqrt {-c^{2} d x^{2} + d}}{1125 \, {\left (c^{4} x^{2} - c^{2}\right )}} \]

[In]

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

[Out]

-1/1125*(225*(b^2*c^6*d*x^6 - 3*b^2*c^4*d*x^4 + 3*b^2*c^2*d*x^2 - b^2*d)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c
^2*x^2 - 1))^2 - 30*(3*a*b*c^5*d*x^5 - 10*a*b*c^3*d*x^3 + 15*a*b*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1)
- 30*((3*b^2*c^5*d*x^5 - 10*b^2*c^3*d*x^3 + 15*b^2*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 15*(a*b*c^
6*d*x^6 - 3*a*b*c^4*d*x^4 + 3*a*b*c^2*d*x^2 - a*b*d)*sqrt(-c^2*d*x^2 + d))*log(c*x + sqrt(c^2*x^2 - 1)) + (9*(
25*a^2 + 2*b^2)*c^6*d*x^6 - (675*a^2 + 94*b^2)*c^4*d*x^4 + (675*a^2 + 374*b^2)*c^2*d*x^2 - (225*a^2 + 298*b^2)
*d)*sqrt(-c^2*d*x^2 + d))/(c^4*x^2 - c^2)

Sympy [F]

\[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=\int x \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}\, dx \]

[In]

integrate(x*(-c**2*d*x**2+d)**(3/2)*(a+b*acosh(c*x))**2,x)

[Out]

Integral(x*(-d*(c*x - 1)*(c*x + 1))**(3/2)*(a + b*acosh(c*x))**2, x)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.80 \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} b^{2} \operatorname {arcosh}\left (c x\right )^{2}}{5 \, c^{2} d} - \frac {2}{1125} \, b^{2} {\left (\frac {9 \, \sqrt {c^{2} x^{2} - 1} c^{2} \sqrt {-d} d^{2} x^{4} - 38 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d} d^{2} x^{2} + \frac {149 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d} d^{2}}{c^{2}}}{d} - \frac {15 \, {\left (3 \, c^{4} \sqrt {-d} d^{2} x^{5} - 10 \, c^{2} \sqrt {-d} d^{2} x^{3} + 15 \, \sqrt {-d} d^{2} x\right )} \operatorname {arcosh}\left (c x\right )}{c d}\right )} - \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} a b \operatorname {arcosh}\left (c x\right )}{5 \, c^{2} d} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} a^{2}}{5 \, c^{2} d} + \frac {2 \, {\left (3 \, c^{4} \sqrt {-d} d^{2} x^{5} - 10 \, c^{2} \sqrt {-d} d^{2} x^{3} + 15 \, \sqrt {-d} d^{2} x\right )} a b}{75 \, c d} \]

[In]

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

[Out]

-1/5*(-c^2*d*x^2 + d)^(5/2)*b^2*arccosh(c*x)^2/(c^2*d) - 2/1125*b^2*((9*sqrt(c^2*x^2 - 1)*c^2*sqrt(-d)*d^2*x^4
- 38*sqrt(c^2*x^2 - 1)*sqrt(-d)*d^2*x^2 + 149*sqrt(c^2*x^2 - 1)*sqrt(-d)*d^2/c^2)/d - 15*(3*c^4*sqrt(-d)*d^2*
x^5 - 10*c^2*sqrt(-d)*d^2*x^3 + 15*sqrt(-d)*d^2*x)*arccosh(c*x)/(c*d)) - 2/5*(-c^2*d*x^2 + d)^(5/2)*a*b*arccos
h(c*x)/(c^2*d) - 1/5*(-c^2*d*x^2 + d)^(5/2)*a^2/(c^2*d) + 2/75*(3*c^4*sqrt(-d)*d^2*x^5 - 10*c^2*sqrt(-d)*d^2*x
^3 + 15*sqrt(-d)*d^2*x)*a*b/(c*d)

Giac [F(-2)]

Exception generated. \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=\int x\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]

[In]

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

[Out]

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