Integrand size = 29, antiderivative size = 553 \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {b d^2 g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 b c d^2 f x^2 \sqrt {d-c^2 d x^2}}{32 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b c^3 d^2 g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 f (-1+c x)^{3/2} (1+c x)^{3/2} \sqrt {d-c^2 d x^2}}{96 c}-\frac {b d^2 f (-1+c x)^{5/2} (1+c x)^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {5}{24} d^2 f x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{6} d^2 f x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {d^2 g (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{7 c^2}-\frac {5 d^2 f \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{32 b c \sqrt {-1+c x} \sqrt {1+c x}} \] Output:
1/7*b*d^2*g*x*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-5/32*b*c* d^2*f*x^2*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/7*b*c*d^2*g*x ^3*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+3/35*b*c^3*d^2*g*x^5*( -c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/49*b*c^5*d^2*g*x^7*(-c^2 *d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+5/96*b*d^2*f*(c*x-1)^(3/2)*(c* x+1)^(3/2)*(-c^2*d*x^2+d)^(1/2)/c-1/36*b*d^2*f*(c*x-1)^(5/2)*(c*x+1)^(5/2) *(-c^2*d*x^2+d)^(1/2)/c+5/16*d^2*f*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x ))+5/24*d^2*f*x*(-c*x+1)*(c*x+1)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))+1 /6*d^2*f*x*(-c*x+1)^2*(c*x+1)^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))-1/ 7*d^2*g*(-c*x+1)^3*(c*x+1)^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/c^2-5 /32*d^2*f*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2/b/c/(c*x-1)^(1/2)/(c*x +1)^(1/2)
Time = 4.54 (sec) , antiderivative size = 644, normalized size of antiderivative = 1.16 \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {d^2 \left (8400 a \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2} \left (48 g \left (-1+c^2 x^2\right )^3+7 c^2 f x \left (33-26 c^2 x^2+8 c^4 x^4\right )\right )-882000 a c \sqrt {d} f \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+78400 b g \sqrt {d-c^2 d x^2} \left (9 c x+12 \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3 \text {arccosh}(c x)-\cosh (3 \text {arccosh}(c x))\right )-352800 b c f \sqrt {d-c^2 d x^2} (\cosh (2 \text {arccosh}(c x))+2 \text {arccosh}(c x) (\text {arccosh}(c x)-\sinh (2 \text {arccosh}(c x))))+44100 b c f \sqrt {d-c^2 d x^2} \left (8 \text {arccosh}(c x)^2+\cosh (4 \text {arccosh}(c x))-4 \text {arccosh}(c x) \sinh (4 \text {arccosh}(c x))\right )-1568 b g \sqrt {d-c^2 d x^2} \left (450 c x-450 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)-25 \cosh (3 \text {arccosh}(c x))-9 \cosh (5 \text {arccosh}(c x))+75 \text {arccosh}(c x) \sinh (3 \text {arccosh}(c x))+45 \text {arccosh}(c x) \sinh (5 \text {arccosh}(c x))\right )+1225 b c f \sqrt {d-c^2 d x^2} \left (-72 \text {arccosh}(c x)^2+18 \cosh (2 \text {arccosh}(c x))-9 \cosh (4 \text {arccosh}(c x))-2 \cosh (6 \text {arccosh}(c x))+12 \text {arccosh}(c x) (-3 \sinh (2 \text {arccosh}(c x))+3 \sinh (4 \text {arccosh}(c x))+\sinh (6 \text {arccosh}(c x)))\right )+4 b g \sqrt {d-c^2 d x^2} \left (55125 c x-55125 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)-1225 \cosh (3 \text {arccosh}(c x))-1323 \cosh (5 \text {arccosh}(c x))-225 \cosh (7 \text {arccosh}(c x))+3675 \text {arccosh}(c x) \sinh (3 \text {arccosh}(c x))+6615 \text {arccosh}(c x) \sinh (5 \text {arccosh}(c x))+1575 \text {arccosh}(c x) \sinh (7 \text {arccosh}(c x))\right )\right )}{2822400 c^2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \] Input:
Integrate[(f + g*x)*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]),x]
Output:
(d^2*(8400*a*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(48* g*(-1 + c^2*x^2)^3 + 7*c^2*f*x*(33 - 26*c^2*x^2 + 8*c^4*x^4)) - 882000*a*c *Sqrt[d]*f*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))] + 78400*b*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]]) - 352800*b*c*f*Sqrt[d - c^2*d*x^2]*(Cosh[2*ArcCosh[c*x]] + 2*ArcCos h[c*x]*(ArcCosh[c*x] - Sinh[2*ArcCosh[c*x]])) + 44100*b*c*f*Sqrt[d - c^2*d *x^2]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*Sinh[4*Arc Cosh[c*x]]) - 1568*b*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*Cosh[3*ArcCosh[c*x]] - 9*Cosh[5*Arc Cosh[c*x]] + 75*ArcCosh[c*x]*Sinh[3*ArcCosh[c*x]] + 45*ArcCosh[c*x]*Sinh[5 *ArcCosh[c*x]]) + 1225*b*c*f*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]] + 1 2*ArcCosh[c*x]*(-3*Sinh[2*ArcCosh[c*x]] + 3*Sinh[4*ArcCosh[c*x]] + Sinh[6* ArcCosh[c*x]])) + 4*b*g*Sqrt[d - c^2*d*x^2]*(55125*c*x - 55125*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] - 1225*Cosh[3*ArcCosh[c*x]] - 1323* Cosh[5*ArcCosh[c*x]] - 225*Cosh[7*ArcCosh[c*x]] + 3675*ArcCosh[c*x]*Sinh[3 *ArcCosh[c*x]] + 6615*ArcCosh[c*x]*Sinh[5*ArcCosh[c*x]] + 1575*ArcCosh[c*x ]*Sinh[7*ArcCosh[c*x]])))/(2822400*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x ))
Time = 1.11 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.50, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {6387, 6390, 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.
| \(\displaystyle \int \left (d-c^2 d x^2\right )^{5/2} (f+g x) (a+b \text {arccosh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6387 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int (c x-1)^{5/2} (c x+1)^{5/2} (f+g x) (a+b \text {arccosh}(c x))dx}{\sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6390 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int \left (f (c x-1)^{5/2} (a+b \text {arccosh}(c x)) (c x+1)^{5/2}+g x (c x-1)^{5/2} (a+b \text {arccosh}(c x)) (c x+1)^{5/2}\right )dx}{\sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \left (\frac {g (c x-1)^{7/2} (c x+1)^{7/2} (a+b \text {arccosh}(c x))}{7 c^2}+\frac {1}{6} f x (c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))-\frac {5}{24} f x (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))+\frac {5}{16} f x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))-\frac {5 f (a+b \text {arccosh}(c x))^2}{32 b c}-\frac {1}{49} b c^5 g x^7+\frac {5}{96} b c^3 f x^4+\frac {3}{35} b c^3 g x^5+\frac {b f \left (1-c^2 x^2\right )^3}{36 c}-\frac {25}{96} b c f x^2-\frac {1}{7} b c g x^3+\frac {b g x}{7 c}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\) |
Input:
Int[(f + g*x)*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]),x]
Output:
(d^2*Sqrt[d - c^2*d*x^2]*((b*g*x)/(7*c) - (25*b*c*f*x^2)/96 - (b*c*g*x^3)/ 7 + (5*b*c^3*f*x^4)/96 + (3*b*c^3*g*x^5)/35 - (b*c^5*g*x^7)/49 + (b*f*(1 - c^2*x^2)^3)/(36*c) + (5*f*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c *x]))/16 - (5*f*x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]))/2 4 + (f*x*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcCosh[c*x]))/6 + (g*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(a + b*ArcCosh[c*x]))/(7*c^2) - (5*f*(a + b* ArcCosh[c*x])^2)/(32*b*c)))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d _) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-d)^IntPart[p]*((d + e*x^2)^Fra cPart[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[p - 1/2] && IntegerQ[m]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_)*(( d2_) + (e2_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(m_.), x_Symbol] :> Int[Expand Integrand[(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]))
Leaf count of result is larger than twice the leaf count of optimal. \(1855\) vs. \(2(473)=946\).
Time = 0.88 (sec) , antiderivative size = 1856, normalized size of antiderivative = 3.36
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1856\) |
| parts | \(\text {Expression too large to display}\) | \(1856\) |
Input:
int((g*x+f)*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x)),x,method=_RETURNVERBOS E)
Output:
1/6*a*f*x*(-c^2*d*x^2+d)^(5/2)+5/24*a*f*d*x*(-c^2*d*x^2+d)^(3/2)+5/16*a*f* d^2*x*(-c^2*d*x^2+d)^(1/2)+5/16*a*f*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2) *x/(-c^2*d*x^2+d)^(1/2))-1/7*a*g*(-c^2*d*x^2+d)^(7/2)/c^2/d+b*(-5/32*(-d*( c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c*arccosh(c*x)^2*f*d^2+1/627 2*(-d*(c^2*x^2-1))^(1/2)*(64*c^8*x^8-144*c^6*x^6+64*c^7*x^7*(c*x-1)^(1/2)* (c*x+1)^(1/2)+104*c^4*x^4-112*c^5*x^5*(c*x-1)^(1/2)*(c*x+1)^(1/2)-25*c^2*x ^2+56*c^3*x^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)-7*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c* x+1)*g*(-1+7*arccosh(c*x))*d^2/(c*x+1)/c^2/(c*x-1)+1/2304*(-d*(c^2*x^2-1)) ^(1/2)*(32*c^7*x^7-64*c^5*x^5+32*c^6*x^6*(c*x-1)^(1/2)*(c*x+1)^(1/2)+38*c^ 3*x^3-48*c^4*x^4*(c*x-1)^(1/2)*(c*x+1)^(1/2)-6*c*x+18*(c*x-1)^(1/2)*(c*x+1 )^(1/2)*c^2*x^2-(c*x-1)^(1/2)*(c*x+1)^(1/2))*f*(-1+6*arccosh(c*x))*d^2/(c* x-1)/(c*x+1)/c-1/640*(-d*(c^2*x^2-1))^(1/2)*(16*c^6*x^6-28*c^4*x^4+16*c^5* x^5*(c*x-1)^(1/2)*(c*x+1)^(1/2)+13*c^2*x^2-20*c^3*x^3*(c*x-1)^(1/2)*(c*x+1 )^(1/2)+5*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-1)*g*(-1+5*arccosh(c*x))*d^2/(c* x+1)/c^2/(c*x-1)-3/512*(-d*(c^2*x^2-1))^(1/2)*(8*c^5*x^5-12*c^3*x^3+8*c^4* x^4*(c*x-1)^(1/2)*(c*x+1)^(1/2)+4*c*x-8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^ 2+(c*x-1)^(1/2)*(c*x+1)^(1/2))*f*(-1+4*arccosh(c*x))*d^2/(c*x-1)/(c*x+1)/c +1/128*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2+4*c^3*x^3*(c*x-1)^(1/2) *(c*x+1)^(1/2)-3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+1)*g*(-1+3*arccosh(c*x))* d^2/(c*x+1)/c^2/(c*x-1)+15/256*(-d*(c^2*x^2-1))^(1/2)*(2*c^3*x^3-2*c*x+...
\[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \,d x } \] Input:
integrate((g*x+f)*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x)),x, algorithm="fr icas")
Output:
integral((a*c^4*d^2*g*x^5 + a*c^4*d^2*f*x^4 - 2*a*c^2*d^2*g*x^3 - 2*a*c^2* d^2*f*x^2 + a*d^2*g*x + a*d^2*f + (b*c^4*d^2*g*x^5 + b*c^4*d^2*f*x^4 - 2*b *c^2*d^2*g*x^3 - 2*b*c^2*d^2*f*x^2 + b*d^2*g*x + b*d^2*f)*arccosh(c*x))*sq rt(-c^2*d*x^2 + d), x)
Timed out. \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\text {Timed out} \] Input:
integrate((g*x+f)*(-c**2*d*x**2+d)**(5/2)*(a+b*acosh(c*x)),x)
Output:
Timed out
\[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \,d x } \] Input:
integrate((g*x+f)*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x)),x, algorithm="ma xima")
Output:
1/48*(8*(-c^2*d*x^2 + d)^(5/2)*x + 10*(-c^2*d*x^2 + d)^(3/2)*d*x + 15*sqrt (-c^2*d*x^2 + d)*d^2*x + 15*d^(5/2)*arcsin(c*x)/c)*a*f - 1/7*(-c^2*d*x^2 + d)^(7/2)*a*g/(c^2*d) + integrate((-c^2*d*x^2 + d)^(5/2)*b*g*x*log(c*x + s qrt(c*x + 1)*sqrt(c*x - 1)) + (-c^2*d*x^2 + d)^(5/2)*b*f*log(c*x + sqrt(c* x + 1)*sqrt(c*x - 1)), x)
Exception generated. \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((g*x+f)*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x)),x, algorithm="gi ac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\int \left (f+g\,x\right )\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \] Input:
int((f + g*x)*(a + b*acosh(c*x))*(d - c^2*d*x^2)^(5/2),x)
Output:
int((f + g*x)*(a + b*acosh(c*x))*(d - c^2*d*x^2)^(5/2), x)
\[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {\sqrt {d}\, d^{2} \left (105 \mathit {asin} \left (c x \right ) a c f +56 \sqrt {-c^{2} x^{2}+1}\, a \,c^{6} f \,x^{5}+48 \sqrt {-c^{2} x^{2}+1}\, a \,c^{6} g \,x^{6}-182 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} f \,x^{3}-144 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} g \,x^{4}+231 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} f x +144 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} g \,x^{2}-48 \sqrt {-c^{2} x^{2}+1}\, a g +336 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{5}d x \right ) b \,c^{6} g +336 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{4}d x \right ) b \,c^{6} f -672 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{3}d x \right ) b \,c^{4} g -672 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{2}d x \right ) b \,c^{4} f +336 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x d x \right ) b \,c^{2} g +336 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )d x \right ) b \,c^{2} f +48 a g \right )}{336 c^{2}} \] Input:
int((g*x+f)*(-c^2*d*x^2+d)^(5/2)*(a+b*acosh(c*x)),x)
Output:
(sqrt(d)*d**2*(105*asin(c*x)*a*c*f + 56*sqrt( - c**2*x**2 + 1)*a*c**6*f*x* *5 + 48*sqrt( - c**2*x**2 + 1)*a*c**6*g*x**6 - 182*sqrt( - c**2*x**2 + 1)* a*c**4*f*x**3 - 144*sqrt( - c**2*x**2 + 1)*a*c**4*g*x**4 + 231*sqrt( - c** 2*x**2 + 1)*a*c**2*f*x + 144*sqrt( - c**2*x**2 + 1)*a*c**2*g*x**2 - 48*sqr t( - c**2*x**2 + 1)*a*g + 336*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**5,x )*b*c**6*g + 336*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**4,x)*b*c**6*f - 672*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**3,x)*b*c**4*g - 672*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**2,x)*b*c**4*f + 336*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x,x)*b*c**2*g + 336*int(sqrt( - c**2*x**2 + 1)*acosh(c*x), x)*b*c**2*f + 48*a*g))/(336*c**2)