Integrand size = 27, antiderivative size = 243 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {b x \sqrt {d-c^2 d x^2}}{c^5 d^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b x \sqrt {d-c^2 d x^2}}{6 c^5 d^3 \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )}+\frac {a+b \text {arccosh}(c x)}{3 c^6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 (a+b \text {arccosh}(c x))}{c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{c^6 d^3}+\frac {11 b \sqrt {d-c^2 d x^2} \text {arctanh}(c x)}{6 c^6 d^3 \sqrt {-1+c x} \sqrt {1+c x}} \]
1/3*(a+b*arccosh(c*x))/c^6/d/(-c^2*d*x^2+d)^(3/2)-2*(a+b*arccosh(c*x))/c^6 /d^2/(-c^2*d*x^2+d)^(1/2)-(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c^6/d^3+ b*x*(-c^2*d*x^2+d)^(1/2)/c^5/d^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/6*b*x*(-c^2 *d*x^2+d)^(1/2)/c^5/d^3/(-c^2*x^2+1)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+11/6*b*ar ctanh(c*x)*(-c^2*d*x^2+d)^(1/2)/c^6/d^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 0.13 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.69 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {16 a-24 a c^2 x^2+6 a c^4 x^4+5 b c x \sqrt {-1+c x} \sqrt {1+c x}-6 b c^3 x^3 \sqrt {-1+c x} \sqrt {1+c x}+2 b \left (8-12 c^2 x^2+3 c^4 x^4\right ) \text {arccosh}(c x)-11 b \sqrt {-1+c x} \sqrt {1+c x} \left (-1+c^2 x^2\right ) \text {arctanh}(c x)}{6 c^6 d^2 \left (-1+c^2 x^2\right ) \sqrt {d-c^2 d x^2}} \]
(16*a - 24*a*c^2*x^2 + 6*a*c^4*x^4 + 5*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 6*b*c^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 2*b*(8 - 12*c^2*x^2 + 3*c^4*x ^4)*ArcCosh[c*x] - 11*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-1 + c^2*x^2)*ArcTan h[c*x])/(6*c^6*d^2*(-1 + c^2*x^2)*Sqrt[d - c^2*d*x^2])
Time = 0.49 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.72, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6337, 27, 1471, 25, 299, 219}
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.
| \(\displaystyle \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6337 |
| \(\displaystyle -\frac {b c \sqrt {d-c^2 d x^2} \int -\frac {3 c^4 x^4-12 c^2 x^2+8}{3 c^6 d^3 \left (1-c^2 x^2\right )^2}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{c^6 d^3}-\frac {2 (a+b \text {arccosh}(c x))}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{3 c^6 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \sqrt {d-c^2 d x^2} \int \frac {3 c^4 x^4-12 c^2 x^2+8}{\left (1-c^2 x^2\right )^2}dx}{3 c^5 d^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{c^6 d^3}-\frac {2 (a+b \text {arccosh}(c x))}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{3 c^6 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1471 |
| \(\displaystyle \frac {b \sqrt {d-c^2 d x^2} \left (-\frac {1}{2} \int -\frac {17-6 c^2 x^2}{1-c^2 x^2}dx-\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c^5 d^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{c^6 d^3}-\frac {2 (a+b \text {arccosh}(c x))}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{3 c^6 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \sqrt {d-c^2 d x^2} \left (\frac {1}{2} \int \frac {17-6 c^2 x^2}{1-c^2 x^2}dx-\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c^5 d^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{c^6 d^3}-\frac {2 (a+b \text {arccosh}(c x))}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{3 c^6 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {b \sqrt {d-c^2 d x^2} \left (\frac {1}{2} \left (11 \int \frac {1}{1-c^2 x^2}dx+6 x\right )-\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c^5 d^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{c^6 d^3}-\frac {2 (a+b \text {arccosh}(c x))}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{3 c^6 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{c^6 d^3}-\frac {2 (a+b \text {arccosh}(c x))}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{3 c^6 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b \left (\frac {1}{2} \left (\frac {11 \text {arctanh}(c x)}{c}+6 x\right )-\frac {x}{2 \left (1-c^2 x^2\right )}\right ) \sqrt {d-c^2 d x^2}}{3 c^5 d^3 \sqrt {c x-1} \sqrt {c x+1}}\) |
(a + b*ArcCosh[c*x])/(3*c^6*d*(d - c^2*d*x^2)^(3/2)) - (2*(a + b*ArcCosh[c *x]))/(c^6*d^2*Sqrt[d - c^2*d*x^2]) - (Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[ c*x]))/(c^6*d^3) + (b*Sqrt[d - c^2*d*x^2]*(-1/2*x/(1 - c^2*x^2) + (6*x + ( 11*ArcTanh[c*x])/c)/2))/(3*c^5*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
3.2.24.3.1 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_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2 , x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x , 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Simp[1/(2*d*(q + 1)) Int[(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^ 2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]
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])
Time = 1.30 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.64
| method | result | size |
| default | \(a \left (-\frac {x^{4}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {\frac {4 x^{2}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {8}{3 d \,c^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}}{c^{2}}\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (6 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}-6 c^{5} x^{5}-11 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{4} x^{4}+11 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) c^{4} x^{4}-24 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}+11 c^{3} x^{3}+22 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}-22 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) x^{2} c^{2}+16 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}-5 c x -11 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+11 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )\right )}{6 c^{6} \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) d^{3}}\) | \(398\) |
| parts | \(a \left (-\frac {x^{4}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {\frac {4 x^{2}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {8}{3 d \,c^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}}{c^{2}}\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (6 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}-6 c^{5} x^{5}-11 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{4} x^{4}+11 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) c^{4} x^{4}-24 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}+11 c^{3} x^{3}+22 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}-22 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) x^{2} c^{2}+16 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}-5 c x -11 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+11 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )\right )}{6 c^{6} \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) d^{3}}\) | \(398\) |
a*(-x^4/c^2/d/(-c^2*d*x^2+d)^(3/2)+4/c^2*(x^2/c^2/d/(-c^2*d*x^2+d)^(3/2)-2 /3/d/c^4/(-c^2*d*x^2+d)^(3/2)))-1/6*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2) *(c*x+1)^(1/2)*(6*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)*c^4*x^4-6*c^5*x ^5-11*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*c^4*x^4+11*ln((c*x-1)^(1/2)*(c *x+1)^(1/2)+c*x-1)*c^4*x^4-24*(c*x+1)^(1/2)*arccosh(c*x)*(c*x-1)^(1/2)*c^2 *x^2+11*c^3*x^3+22*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*x^2*c^2-22*ln((c* x-1)^(1/2)*(c*x+1)^(1/2)+c*x-1)*x^2*c^2+16*arccosh(c*x)*(c*x-1)^(1/2)*(c*x +1)^(1/2)-5*c*x-11*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+11*ln((c*x-1)^(1/ 2)*(c*x+1)^(1/2)+c*x-1))/c^6/(c^6*x^6-3*c^4*x^4+3*c^2*x^2-1)/d^3
Time = 0.32 (sec) , antiderivative size = 529, normalized size of antiderivative = 2.18 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\left [-\frac {8 \, {\left (3 \, b c^{4} x^{4} - 12 \, b c^{2} x^{2} + 8 \, b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 11 \, {\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \sqrt {-d} \log \left (-\frac {c^{6} d x^{6} + 5 \, c^{4} d x^{4} - 5 \, c^{2} d x^{2} + 4 \, {\left (c^{3} x^{3} + c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} \sqrt {-d} - d}{c^{6} x^{6} - 3 \, c^{4} x^{4} + 3 \, c^{2} x^{2} - 1}\right ) - 4 \, {\left (6 \, b c^{3} x^{3} - 5 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 8 \, {\left (3 \, a c^{4} x^{4} - 12 \, a c^{2} x^{2} + 8 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{24 \, {\left (c^{10} d^{3} x^{4} - 2 \, c^{8} d^{3} x^{2} + c^{6} d^{3}\right )}}, \frac {11 \, {\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \sqrt {d} \arctan \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} c \sqrt {d} x}{c^{4} d x^{4} - d}\right ) - 4 \, {\left (3 \, b c^{4} x^{4} - 12 \, b c^{2} x^{2} + 8 \, b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, {\left (6 \, b c^{3} x^{3} - 5 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 4 \, {\left (3 \, a c^{4} x^{4} - 12 \, a c^{2} x^{2} + 8 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{12 \, {\left (c^{10} d^{3} x^{4} - 2 \, c^{8} d^{3} x^{2} + c^{6} d^{3}\right )}}\right ] \]
[-1/24*(8*(3*b*c^4*x^4 - 12*b*c^2*x^2 + 8*b)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) + 11*(b*c^4*x^4 - 2*b*c^2*x^2 + b)*sqrt(-d)*log(-(c^6 *d*x^6 + 5*c^4*d*x^4 - 5*c^2*d*x^2 + 4*(c^3*x^3 + c*x)*sqrt(-c^2*d*x^2 + d )*sqrt(c^2*x^2 - 1)*sqrt(-d) - d)/(c^6*x^6 - 3*c^4*x^4 + 3*c^2*x^2 - 1)) - 4*(6*b*c^3*x^3 - 5*b*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 8*(3*a *c^4*x^4 - 12*a*c^2*x^2 + 8*a)*sqrt(-c^2*d*x^2 + d))/(c^10*d^3*x^4 - 2*c^8 *d^3*x^2 + c^6*d^3), 1/12*(11*(b*c^4*x^4 - 2*b*c^2*x^2 + b)*sqrt(d)*arctan (2*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1)*c*sqrt(d)*x/(c^4*d*x^4 - d)) - 4 *(3*b*c^4*x^4 - 12*b*c^2*x^2 + 8*b)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^ 2*x^2 - 1)) + 2*(6*b*c^3*x^3 - 5*b*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 4*(3*a*c^4*x^4 - 12*a*c^2*x^2 + 8*a)*sqrt(-c^2*d*x^2 + d))/(c^10*d^ 3*x^4 - 2*c^8*d^3*x^2 + c^6*d^3)]
\[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^{5} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{5}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
-1/9*b*(((9*c^4*sqrt(d)*x^4 - 8*sqrt(d))*sqrt(c*x + 1)*sqrt(c*x - 1)/sqrt( -c*x + 1) - 3*(3*c^5*sqrt(d)*x^5 - 12*c^3*sqrt(d)*x^3 + 8*c*sqrt(d)*x + (3 *c^4*sqrt(d)*x^4 - 12*c^2*sqrt(d)*x^2 + 8*sqrt(d))*sqrt(c*x + 1)*sqrt(c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/sqrt(-c*x + 1))/((c^8*d^3*x^2 - c^6*d^3)*(c*x + 1)*sqrt(c*x - 1) + (c^9*d^3*x^3 - c^7*d^3*x)*sqrt(c*x + 1)) + 9*integrate(1/9*(9*c^7*sqrt(d)*x^7 - 45*c^5*sqrt(d)*x^5 + 60*c^3*sq rt(d)*x^3 - 24*c*sqrt(d)*x + (9*c^6*sqrt(d)*x^6 - 54*c^4*sqrt(d)*x^4 + 60* c^2*sqrt(d)*x^2 - 16*sqrt(d))*e^(1/2*log(c*x + 1) + 1/2*log(c*x - 1)))/(sq rt(-c*x + 1)*((c^9*d^3*x^4 - 2*c^7*d^3*x^2 + c^5*d^3)*e^(3/2*log(c*x + 1) + log(c*x - 1)) + 2*(c^10*d^3*x^5 - 2*c^8*d^3*x^3 + c^6*d^3*x)*e^(log(c*x + 1) + 1/2*log(c*x - 1)) + (c^11*d^3*x^6 - 2*c^9*d^3*x^4 + c^7*d^3*x^2)*sq rt(c*x + 1))), x)) - 1/3*a*(3*x^4/((-c^2*d*x^2 + d)^(3/2)*c^2*d) - 12*x^2/ ((-c^2*d*x^2 + d)^(3/2)*c^4*d) + 8/((-c^2*d*x^2 + d)^(3/2)*c^6*d))
Exception generated. \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^5\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]