Integrand size = 27, antiderivative size = 272 \[ \int x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {8 b x \sqrt {d-c^2 d x^2}}{105 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b x^3 \sqrt {d-c^2 d x^2}}{315 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x^5 \sqrt {d-c^2 d x^2}}{175 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 c^6 d}+\frac {2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^6 d^3} \]
-1/3*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/c^6/d+2/5*(-c^2*d*x^2+d)^(5/2 )*(a+b*arccosh(c*x))/c^6/d^2-1/7*(-c^2*d*x^2+d)^(7/2)*(a+b*arccosh(c*x))/c ^6/d^3+8/105*b*x*(-c^2*d*x^2+d)^(1/2)/c^5/(c*x-1)^(1/2)/(c*x+1)^(1/2)+4/31 5*b*x^3*(-c^2*d*x^2+d)^(1/2)/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/175*b*x^5*( -c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/49*b*c*x^7*(-c^2*d*x^2 +d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 0.17 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.62 \[ \int x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {\sqrt {d-c^2 d x^2} \left (b c x \left (840+140 c^2 x^2+63 c^4 x^4-225 c^6 x^6\right )+105 a \sqrt {-1+c x} \sqrt {1+c x} \left (-8-4 c^2 x^2-3 c^4 x^4+15 c^6 x^6\right )+105 b \sqrt {-1+c x} \sqrt {1+c x} \left (-8-4 c^2 x^2-3 c^4 x^4+15 c^6 x^6\right ) \text {arccosh}(c x)\right )}{11025 c^6 \sqrt {-1+c x} \sqrt {1+c x}} \]
(Sqrt[d - c^2*d*x^2]*(b*c*x*(840 + 140*c^2*x^2 + 63*c^4*x^4 - 225*c^6*x^6) + 105*a*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-8 - 4*c^2*x^2 - 3*c^4*x^4 + 15*c^6 *x^6) + 105*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-8 - 4*c^2*x^2 - 3*c^4*x^4 + 1 5*c^6*x^6)*ArcCosh[c*x]))/(11025*c^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
Time = 0.46 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.64, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6337, 27, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6337 |
| \(\displaystyle -\frac {b c \sqrt {d-c^2 d x^2} \int -\frac {-15 c^6 x^6+3 c^4 x^4+4 c^2 x^2+8}{105 c^6}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^6 d^3}+\frac {2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 c^6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \sqrt {d-c^2 d x^2} \int \left (-15 c^6 x^6+3 c^4 x^4+4 c^2 x^2+8\right )dx}{105 c^5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^6 d^3}+\frac {2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 c^6 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^6 d^3}+\frac {2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 c^6 d}+\frac {b \left (-\frac {15}{7} c^6 x^7+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right ) \sqrt {d-c^2 d x^2}}{105 c^5 \sqrt {c x-1} \sqrt {c x+1}}\) |
(b*Sqrt[d - c^2*d*x^2]*(8*x + (4*c^2*x^3)/3 + (3*c^4*x^5)/5 - (15*c^6*x^7) /7))/(105*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(3*c^6*d) + (2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]) )/(5*c^6*d^2) - ((d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(7*c^6*d^3)
3.1.65.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]}, Simp[(a + b*ArcCo sh[c*x]) u, x] - Simp[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])
Leaf count of result is larger than twice the leaf count of optimal. \(987\) vs. \(2(228)=456\).
Time = 0.69 (sec) , antiderivative size = 988, normalized size of antiderivative = 3.63
| method | result | size |
| default | \(a \left (-\frac {x^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{7 c^{2} d}+\frac {-\frac {4 x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{35 c^{2} d}-\frac {8 \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{105 d \,c^{4}}}{c^{2}}\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 )}{6272 \left (c x +1\right ) c^{6} \left (c x -1\right )}+\frac {3 \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 )}{3200 \left (c x +1\right ) c^{6} \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 )}{1152 \left (c x +1\right ) c^{6} \left (c x -1\right )}-\frac {5 \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 )}{128 \left (c x +1\right ) c^{6} \left (c x -1\right )}-\frac {5 \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 )}{128 \left (c x +1\right ) c^{6} \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 )}{1152 \left (c x +1\right ) c^{6} \left (c x -1\right )}+\frac {3 \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 )}{3200 \left (c x +1\right ) c^{6} \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 )}{6272 \left (c x +1\right ) c^{6} \left (c x -1\right )}\right )\) | \(988\) |
| parts | \(a \left (-\frac {x^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{7 c^{2} d}+\frac {-\frac {4 x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{35 c^{2} d}-\frac {8 \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{105 d \,c^{4}}}{c^{2}}\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 )}{6272 \left (c x +1\right ) c^{6} \left (c x -1\right )}+\frac {3 \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 )}{3200 \left (c x +1\right ) c^{6} \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 )}{1152 \left (c x +1\right ) c^{6} \left (c x -1\right )}-\frac {5 \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 )}{128 \left (c x +1\right ) c^{6} \left (c x -1\right )}-\frac {5 \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 )}{128 \left (c x +1\right ) c^{6} \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 )}{1152 \left (c x +1\right ) c^{6} \left (c x -1\right )}+\frac {3 \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 )}{3200 \left (c x +1\right ) c^{6} \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 )}{6272 \left (c x +1\right ) c^{6} \left (c x -1\right )}\right )\) | \(988\) |
a*(-1/7*x^4*(-c^2*d*x^2+d)^(3/2)/c^2/d+4/7/c^2*(-1/5*x^2*(-c^2*d*x^2+d)^(3 /2)/c^2/d-2/15/d/c^4*(-c^2*d*x^2+d)^(3/2)))+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 ))/(c*x+1)/c^6/(c*x-1)+3/3200*(-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))/ (c*x+1)/c^6/(c*x-1)+1/1152*(-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))/(c*x+1)/c^6/(c*x-1)-5/128*(-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))/(c*x+1)/c^6/(c*x-1) -5/128*(-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))/(c*x+1)/c^6/(c*x-1)+1/1152*(-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))/(c*x+1)/c^6/(c*x-1)+3/3200*(-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*arccosh(c*x))/(c*x+1)/c^6/(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...
Time = 0.27 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.75 \[ \int x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {105 \, {\left (15 \, b c^{8} x^{8} - 18 \, b c^{6} x^{6} - b c^{4} x^{4} - 4 \, b c^{2} x^{2} + 8 \, b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (225 \, b c^{7} x^{7} - 63 \, b c^{5} x^{5} - 140 \, b c^{3} x^{3} - 840 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 105 \, {\left (15 \, a c^{8} x^{8} - 18 \, a c^{6} x^{6} - a c^{4} x^{4} - 4 \, a c^{2} x^{2} + 8 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{11025 \, {\left (c^{8} x^{2} - c^{6}\right )}} \]
1/11025*(105*(15*b*c^8*x^8 - 18*b*c^6*x^6 - b*c^4*x^4 - 4*b*c^2*x^2 + 8*b) *sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) - (225*b*c^7*x^7 - 63*b *c^5*x^5 - 140*b*c^3*x^3 - 840*b*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 105*(15*a*c^8*x^8 - 18*a*c^6*x^6 - a*c^4*x^4 - 4*a*c^2*x^2 + 8*a)*sqr t(-c^2*d*x^2 + d))/(c^8*x^2 - c^6)
\[ \int x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\int x^{5} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )\, dx \]
Time = 0.30 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.75 \[ \int x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=-\frac {1}{105} \, {\left (\frac {15 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4}}{c^{2} d} + \frac {12 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{4} d} + \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{6} d}\right )} b \operatorname {arcosh}\left (c x\right ) - \frac {1}{105} \, {\left (\frac {15 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4}}{c^{2} d} + \frac {12 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{4} d} + \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{6} d}\right )} a - \frac {{\left (225 \, c^{6} \sqrt {-d} x^{7} - 63 \, c^{4} \sqrt {-d} x^{5} - 140 \, c^{2} \sqrt {-d} x^{3} - 840 \, \sqrt {-d} x\right )} b}{11025 \, c^{5}} \]
-1/105*(15*(-c^2*d*x^2 + d)^(3/2)*x^4/(c^2*d) + 12*(-c^2*d*x^2 + d)^(3/2)* x^2/(c^4*d) + 8*(-c^2*d*x^2 + d)^(3/2)/(c^6*d))*b*arccosh(c*x) - 1/105*(15 *(-c^2*d*x^2 + d)^(3/2)*x^4/(c^2*d) + 12*(-c^2*d*x^2 + d)^(3/2)*x^2/(c^4*d ) + 8*(-c^2*d*x^2 + d)^(3/2)/(c^6*d))*a - 1/11025*(225*c^6*sqrt(-d)*x^7 - 63*c^4*sqrt(-d)*x^5 - 140*c^2*sqrt(-d)*x^3 - 840*sqrt(-d)*x)*b/c^5
Exception generated. \[ \int x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\int x^5\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2} \,d x \]