\(\int x^3 (d-c^2 d x^2)^{3/2} (a+b \text {arccosh}(c x)) \, dx\) [82]

Optimal result

Integrand size = 27, antiderivative size = 243 \[ \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\frac {2 b d x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d x^3 \sqrt {d-c^2 d x^2}}{105 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {8 b c d x^5 \sqrt {d-c^2 d x^2}}{175 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 c^4 d}+\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^4 d^2} \]

[Out]

-1/5*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/c^4/d+1/7*(-c^2*d*x^2+d)^(7/2)*(a+b*arccosh(c*x))/c^4/d^2+2/35*b*
d*x*(-c^2*d*x^2+d)^(1/2)/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/105*b*d*x^3*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c
*x+1)^(1/2)-8/175*b*c*d*x^5*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/49*b*c^3*d*x^7*(-c^2*d*x^2+d)^(
1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {272, 45, 5922, 12, 380} \[ \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^4 d^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 c^4 d}-\frac {8 b c d x^5 \sqrt {d-c^2 d x^2}}{175 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d x^3 \sqrt {d-c^2 d x^2}}{105 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b d x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

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

[Out]

(2*b*d*x*Sqrt[d - c^2*d*x^2])/(35*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d*x^3*Sqrt[d - c^2*d*x^2])/(105*c*Sqr
t[-1 + c*x]*Sqrt[1 + c*x]) - (8*b*c*d*x^5*Sqrt[d - c^2*d*x^2])/(175*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*x
^7*Sqrt[d - c^2*d*x^2])/(49*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(5*c^
4*d) + ((d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(7*c^4*d^2)

Rule 12

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

Rule 45

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 272

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 380

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]

Rule 5922

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x
^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 +
c*x])], Int[SimplifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0
] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 c^4 d}+\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^4 d^2}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \frac {d \left (-2-5 c^2 x^2\right ) \left (1-c^2 x^2\right )^2}{35 c^4} \, dx}{\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))}{5 c^4 d}+\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^4 d^2}-\frac {\left (b d \sqrt {d-c^2 d x^2}\right ) \int \left (-2-5 c^2 x^2\right ) \left (1-c^2 x^2\right )^2 \, dx}{35 c^3 \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))}{5 c^4 d}+\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^4 d^2}-\frac {\left (b d \sqrt {d-c^2 d x^2}\right ) \int \left (-2-c^2 x^2+8 c^4 x^4-5 c^6 x^6\right ) \, dx}{35 c^3 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {2 b d x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d x^3 \sqrt {d-c^2 d x^2}}{105 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {8 b c d x^5 \sqrt {d-c^2 d x^2}}{175 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d x^7 \sqrt {d-c^2 d x^2}}{49 \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))}{5 c^4 d}+\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^4 d^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.56 \[ \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=-\frac {d \sqrt {d-c^2 d x^2} \left (-b c x \left (210+35 c^2 x^2-168 c^4 x^4+75 c^6 x^6\right )+210 (-1+c x)^{5/2} (1+c x)^{5/2} (a+b \text {arccosh}(c x))+525 c^2 x^2 (-1+c x)^{5/2} (1+c x)^{5/2} (a+b \text {arccosh}(c x))\right )}{3675 c^4 \sqrt {-1+c x} \sqrt {1+c x}} \]

[In]

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

[Out]

-1/3675*(d*Sqrt[d - c^2*d*x^2]*(-(b*c*x*(210 + 35*c^2*x^2 - 168*c^4*x^4 + 75*c^6*x^6)) + 210*(-1 + c*x)^(5/2)*
(1 + c*x)^(5/2)*(a + b*ArcCosh[c*x]) + 525*c^2*x^2*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcCosh[c*x])))/(c^
4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(965\) vs. \(2(203)=406\).

Time = 0.53 (sec) , antiderivative size = 966, normalized size of antiderivative = 3.98

method	result	size
default	\(a \left (-\frac {x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{7 c^{2} d}-\frac {2 \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{35 d \,c^{4}}\right )+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (64 c^{8} x^{8}-144 c^{6} x^{6}+64 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{7} c^{7}+104 c^{4} x^{4}-112 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} x^{5}-25 c^{2} x^{2}+56 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-7 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) \left (-1+7 \,\operatorname {arccosh}\left (c x \right )\right ) d}{6272 \left (c x +1\right ) c^{4} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (16 c^{6} x^{6}-28 c^{4} x^{4}+16 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} x^{5}+13 c^{2} x^{2}-20 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+5 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -1\right ) \left (-1+5 \,\operatorname {arccosh}\left (c x \right )\right ) d}{3200 \left (c x +1\right ) c^{4} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) \left (-1+3 \,\operatorname {arccosh}\left (c x \right )\right ) d}{384 \left (c x +1\right ) c^{4} \left (c x -1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (-1+\operatorname {arccosh}\left (c x \right )\right ) d}{128 \left (c x +1\right ) c^{4} \left (c x -1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (1+\operatorname {arccosh}\left (c x \right )\right ) d}{128 \left (c x +1\right ) c^{4} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+4 c^{4} x^{4}+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -5 c^{2} x^{2}+1\right ) \left (1+3 \,\operatorname {arccosh}\left (c x \right )\right ) d}{384 \left (c x +1\right ) c^{4} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-16 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} x^{5}+16 c^{6} x^{6}+20 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-28 c^{4} x^{4}-5 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +13 c^{2} x^{2}-1\right ) \left (1+5 \,\operatorname {arccosh}\left (c x \right )\right ) d}{3200 \left (c x +1\right ) c^{4} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-64 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{7} c^{7}+64 c^{8} x^{8}+112 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} x^{5}-144 c^{6} x^{6}-56 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+104 c^{4} x^{4}+7 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -25 c^{2} x^{2}+1\right ) \left (1+7 \,\operatorname {arccosh}\left (c x \right )\right ) d}{6272 \left (c x +1\right ) c^{4} \left (c x -1\right )}\right )\)	\(966\)
parts	\(a \left (-\frac {x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{7 c^{2} d}-\frac {2 \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{35 d \,c^{4}}\right )+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (64 c^{8} x^{8}-144 c^{6} x^{6}+64 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{7} c^{7}+104 c^{4} x^{4}-112 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} x^{5}-25 c^{2} x^{2}+56 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-7 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) \left (-1+7 \,\operatorname {arccosh}\left (c x \right )\right ) d}{6272 \left (c x +1\right ) c^{4} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (16 c^{6} x^{6}-28 c^{4} x^{4}+16 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} x^{5}+13 c^{2} x^{2}-20 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+5 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -1\right ) \left (-1+5 \,\operatorname {arccosh}\left (c x \right )\right ) d}{3200 \left (c x +1\right ) c^{4} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) \left (-1+3 \,\operatorname {arccosh}\left (c x \right )\right ) d}{384 \left (c x +1\right ) c^{4} \left (c x -1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (-1+\operatorname {arccosh}\left (c x \right )\right ) d}{128 \left (c x +1\right ) c^{4} \left (c x -1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (1+\operatorname {arccosh}\left (c x \right )\right ) d}{128 \left (c x +1\right ) c^{4} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+4 c^{4} x^{4}+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -5 c^{2} x^{2}+1\right ) \left (1+3 \,\operatorname {arccosh}\left (c x \right )\right ) d}{384 \left (c x +1\right ) c^{4} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-16 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} x^{5}+16 c^{6} x^{6}+20 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-28 c^{4} x^{4}-5 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +13 c^{2} x^{2}-1\right ) \left (1+5 \,\operatorname {arccosh}\left (c x \right )\right ) d}{3200 \left (c x +1\right ) c^{4} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-64 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{7} c^{7}+64 c^{8} x^{8}+112 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} x^{5}-144 c^{6} x^{6}-56 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+104 c^{4} x^{4}+7 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -25 c^{2} x^{2}+1\right ) \left (1+7 \,\operatorname {arccosh}\left (c x \right )\right ) d}{6272 \left (c x +1\right ) c^{4} \left (c x -1\right )}\right )\)	\(966\)

[In]

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

[Out]

a*(-1/7*x^2*(-c^2*d*x^2+d)^(5/2)/c^2/d-2/35/d/c^4*(-c^2*d*x^2+d)^(5/2))+b*(-1/6272*(-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)*c^5*x^5
-25*c^2*x^2+56*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3-7*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+1)*(-1+7*arccosh(c*x))*d/
(c*x+1)/c^4/(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)*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^4/(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)*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^4/(c*x-1)-3/128*(-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^4/(c*x-1)-3/128*(-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^4/(c*x-1)+1/384*(-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*arcco
sh(c*x))*d/(c*x+1)/c^4/(c*x-1)+1/3200*(-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*arcc
osh(c*x))*d/(c*x+1)/c^4/(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)*c^5*x^5-144*c^6*x^6-56*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3+104*c^4*x^4+7*(
c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-25*c^2*x^2+1)*(1+7*arccosh(c*x))*d/(c*x+1)/c^4/(c*x-1))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.88 \[ \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=-\frac {105 \, {\left (5 \, b c^{8} d x^{8} - 13 \, b c^{6} d x^{6} + 9 \, b c^{4} d x^{4} + b c^{2} d x^{2} - 2 \, b d\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (75 \, b c^{7} d x^{7} - 168 \, b c^{5} d x^{5} + 35 \, b c^{3} d x^{3} + 210 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 105 \, {\left (5 \, a c^{8} d x^{8} - 13 \, a c^{6} d x^{6} + 9 \, a c^{4} d x^{4} + a c^{2} d x^{2} - 2 \, a d\right )} \sqrt {-c^{2} d x^{2} + d}}{3675 \, {\left (c^{6} x^{2} - c^{4}\right )}} \]

[In]

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

[Out]

-1/3675*(105*(5*b*c^8*d*x^8 - 13*b*c^6*d*x^6 + 9*b*c^4*d*x^4 + b*c^2*d*x^2 - 2*b*d)*sqrt(-c^2*d*x^2 + d)*log(c
*x + sqrt(c^2*x^2 - 1)) - (75*b*c^7*d*x^7 - 168*b*c^5*d*x^5 + 35*b*c^3*d*x^3 + 210*b*c*d*x)*sqrt(-c^2*d*x^2 +
d)*sqrt(c^2*x^2 - 1) + 105*(5*a*c^8*d*x^8 - 13*a*c^6*d*x^6 + 9*a*c^4*d*x^4 + a*c^2*d*x^2 - 2*a*d)*sqrt(-c^2*d*
x^2 + d))/(c^6*x^2 - c^4)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.66 \[ \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=-\frac {1}{35} \, {\left (\frac {5 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} b \operatorname {arcosh}\left (c x\right ) - \frac {1}{35} \, {\left (\frac {5 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} a + \frac {{\left (75 \, c^{6} \sqrt {-d} d x^{7} - 168 \, c^{4} \sqrt {-d} d x^{5} + 35 \, c^{2} \sqrt {-d} d x^{3} + 210 \, \sqrt {-d} d x\right )} b}{3675 \, c^{3}} \]

[In]

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

[Out]

-1/35*(5*(-c^2*d*x^2 + d)^(5/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(5/2)/(c^4*d))*b*arccosh(c*x) - 1/35*(5*(-c^2
*d*x^2 + d)^(5/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(5/2)/(c^4*d))*a + 1/3675*(75*c^6*sqrt(-d)*d*x^7 - 168*c^4*
sqrt(-d)*d*x^5 + 35*c^2*sqrt(-d)*d*x^3 + 210*sqrt(-d)*d*x)*b/c^3

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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