Integrand size = 31, antiderivative size = 1385 \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {3 b d^2 f^2 g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b d^2 g^3 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {25 b c d^2 f^3 x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {15 b d^2 f g^2 x^2 \sqrt {d-c^2 d x^2}}{256 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c d^2 f^2 g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 g^3 x^3 \sqrt {d-c^2 d x^2}}{189 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b c^3 d^2 f^3 x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {59 b c d^2 f g^2 x^4 \sqrt {d-c^2 d x^2}}{256 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {9 b c^3 d^2 f^2 g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 g^3 x^5 \sqrt {d-c^2 d x^2}}{21 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {17 b c^3 d^2 f g^2 x^6 \sqrt {d-c^2 d x^2}}{96 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c^5 d^2 f^2 g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {19 b c^3 d^2 g^3 x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c^5 d^2 f g^2 x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 g^3 x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 f^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{36 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{16} d^2 f^3 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {15 d^2 f g^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{128 c^2}+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {5}{24} d^2 f^3 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {5}{16} d^2 f g^2 x^3 (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^3 x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {3}{8} d^2 f g^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {3 d^2 f^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 {2 d^2 g^3 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{63 c^4}-\frac {d^2 g^3 x^2 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{9 c^2}-\frac {5 d^2 f^3 \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}}-\frac {15 d^2 f g^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{256 b c^3 \sqrt {-1+c x} \sqrt {1+c x}} \]
2/63*b*d^2*g^3*x*(-c^2*d*x^2+d)^(1/2)/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)-25/9 6*b*c*d^2*f^3*x^2*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/189*b *d^2*g^3*x^3*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)+5/96*b*c^3 *d^2*f^3*x^4*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/21*b*c*d^2 *g^3*x^5*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+19/441*b*c^3*d^2 *g^3*x^7*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/81*b*c^5*d^2*g ^3*x^9*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/36*b*d^2*f^3*(-c ^2*x^2+1)^3*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-5/32*d^2*f^ 3*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/c/(c*x-1)^(1/2)/(c*x+1)^(1/2 )+5/16*d^2*f^3*x*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)-15/128*d^2*f*g^2* x*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+5/24*d^2*f^3*x*(-c*x+1)*(c*x +1)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)+1/6*d^2*f^3*x*(-c*x+1)^2*(c*x+ 1)^2*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)-2/63*d^2*g^3*(-c*x+1)^3*(c*x+ 1)^3*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c^4+5/16*d^2*f*g^2*x^3*(-c*x+ 1)*(c*x+1)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)+3/8*d^2*f*g^2*x^3*(-c*x +1)^2*(c*x+1)^2*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)-3/7*d^2*f^2*g*(-c* x+1)^3*(c*x+1)^3*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2-1/9*d^2*g^3*x ^2*(-c*x+1)^3*(c*x+1)^3*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+3/7*b* d^2*f^2*g*x*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)+15/256*b*d^ 2*f*g^2*x^2*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-3/7*b*c*...
Time = 7.85 (sec) , antiderivative size = 1802, normalized size of antiderivative = 1.30 \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx =\text {Too large to display} \]
Sqrt[-(d*(-1 + c^2*x^2))]*(-1/63*(a*d^2*g*(27*c^2*f^2 + 2*g^2))/c^4 + (a*d ^2*f*(88*c^2*f^2 - 15*g^2)*x)/(128*c^2) - (a*d^2*g*(-81*c^2*f^2 + g^2)*x^2 )/(63*c^2) - (a*d^2*f*(104*c^2*f^2 - 177*g^2)*x^3)/192 + (a*d^2*g*(-27*c^2 *f^2 + 5*g^2)*x^4)/21 + (a*c^2*d^2*f*(8*c^2*f^2 - 51*g^2)*x^5)/48 - (a*c^2 *d^2*g*(-27*c^2*f^2 + 19*g^2)*x^6)/63 + (3*a*c^4*d^2*f*g^2*x^7)/8 + (a*c^4 *d^2*g^3*x^8)/9) - (5*a*d^(5/2)*f*(8*c^2*f^2 + 3*g^2)*ArcTan[(c*x*Sqrt[-(d *(-1 + c^2*x^2))])/(Sqrt[d]*(-1 + c^2*x^2))])/(128*c^3) - (b*d^2*f^2*g*Sqr t[-(d*(-1 + c*x)*(1 + c*x))]*(-9*c*x - 12*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*ArcCosh[c*x] + Cosh[3*ArcCosh[c*x]]))/(12*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) - (b*d^2*f^3*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*(Cosh[2*Ar cCosh[c*x]] + 2*ArcCosh[c*x]*(ArcCosh[c*x] - Sinh[2*ArcCosh[c*x]])))/(8*c* Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + (b*d^2*f^3*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*Sinh[4* ArcCosh[c*x]]))/(64*c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) - (3*b*d^2*f*g ^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x] ] - 4*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]))/(128*c^3*Sqrt[(-1 + c*x)/(1 + c* x)]*(1 + c*x)) + (b*d^2*f^2*g*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*(-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*ArcCosh[c*x]] - 75*ArcCosh[c*x]*Sinh[3*ArcCosh[c*x]] - 45 *ArcCosh[c*x]*Sinh[5*ArcCosh[c*x]]))/(600*c^2*Sqrt[(-1 + c*x)/(1 + c*x)...
Time = 2.70 (sec) , antiderivative size = 677, normalized size of antiderivative = 0.49, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, 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.
| \(\displaystyle \int \left (d-c^2 d x^2\right )^{5/2} (f+g x)^3 (a+b \text {arccosh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6387 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int (c x-1)^{5/2} (c x+1)^{5/2} (f+g x)^3 (a+b \text {arccosh}(c x))dx}{\sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6390 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int \left ((c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x)) f^3+3 g x (c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x)) f^2+3 g^2 x^2 (c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x)) f+g^3 x^3 (c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))\right )dx}{\sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \left (\frac {2 g^3 (c x-1)^{7/2} (c x+1)^{7/2} (a+b \text {arccosh}(c x))}{63 c^4}-\frac {15 f g^2 (a+b \text {arccosh}(c x))^2}{256 b c^3}+\frac {3 f^2 g (c x-1)^{7/2} (c x+1)^{7/2} (a+b \text {arccosh}(c x))}{7 c^2}-\frac {15 f g^2 x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{128 c^2}+\frac {g^3 x^2 (c x-1)^{7/2} (c x+1)^{7/2} (a+b \text {arccosh}(c x))}{9 c^2}+\frac {1}{6} f^3 x (c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))-\frac {5}{24} f^3 x (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))+\frac {5}{16} f^3 x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))-\frac {5 f^3 (a+b \text {arccosh}(c x))^2}{32 b c}+\frac {3}{8} f g^2 x^3 (c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))-\frac {5}{16} f g^2 x^3 (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))+\frac {15}{64} f g^2 x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))-\frac {3}{49} b c^5 f^2 g x^7-\frac {3}{64} b c^5 f g^2 x^8-\frac {1}{81} b c^5 g^3 x^9+\frac {5}{96} b c^3 f^3 x^4+\frac {9}{35} b c^3 f^2 g x^5+\frac {17}{96} b c^3 f g^2 x^6+\frac {19}{441} b c^3 g^3 x^7+\frac {2 b g^3 x}{63 c^3}+\frac {b f^3 \left (1-c^2 x^2\right )^3}{36 c}-\frac {25}{96} b c f^3 x^2-\frac {3}{7} b c f^2 g x^3+\frac {3 b f^2 g x}{7 c}-\frac {59}{256} b c f g^2 x^4+\frac {15 b f g^2 x^2}{256 c}-\frac {1}{21} b c g^3 x^5+\frac {b g^3 x^3}{189 c}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\) |
(d^2*Sqrt[d - c^2*d*x^2]*((3*b*f^2*g*x)/(7*c) + (2*b*g^3*x)/(63*c^3) - (25 *b*c*f^3*x^2)/96 + (15*b*f*g^2*x^2)/(256*c) - (3*b*c*f^2*g*x^3)/7 + (b*g^3 *x^3)/(189*c) + (5*b*c^3*f^3*x^4)/96 - (59*b*c*f*g^2*x^4)/256 + (9*b*c^3*f ^2*g*x^5)/35 - (b*c*g^3*x^5)/21 + (17*b*c^3*f*g^2*x^6)/96 - (3*b*c^5*f^2*g *x^7)/49 + (19*b*c^3*g^3*x^7)/441 - (3*b*c^5*f*g^2*x^8)/64 - (b*c^5*g^3*x^ 9)/81 + (b*f^3*(1 - c^2*x^2)^3)/(36*c) + (5*f^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/16 - (15*f*g^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*( a + b*ArcCosh[c*x]))/(128*c^2) + (15*f*g^2*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x ]*(a + b*ArcCosh[c*x]))/64 - (5*f^3*x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]))/24 - (5*f*g^2*x^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]))/16 + (f^3*x*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcC osh[c*x]))/6 + (3*f*g^2*x^3*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcCos h[c*x]))/8 + (3*f^2*g*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(a + b*ArcCosh[c*x] ))/(7*c^2) + (2*g^3*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(a + b*ArcCosh[c*x])) /(63*c^4) + (g^3*x^2*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(a + b*ArcCosh[c*x]) )/(9*c^2) - (5*f^3*(a + b*ArcCosh[c*x])^2)/(32*b*c) - (15*f*g^2*(a + b*Arc Cosh[c*x])^2)/(256*b*c^3)))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])
3.1.62.3.1 Defintions of rubi rules used
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. \(3137\) vs. \(2(1201)=2402\).
Time = 1.47 (sec) , antiderivative size = 3138, normalized size of antiderivative = 2.27
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(3138\) |
| parts | \(\text {Expression too large to display}\) | \(3138\) |
a*(f^3*(1/6*x*(-c^2*d*x^2+d)^(5/2)+5/6*d*(1/4*x*(-c^2*d*x^2+d)^(3/2)+3/4*d *(1/2*x*(-c^2*d*x^2+d)^(1/2)+1/2*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(- c^2*d*x^2+d)^(1/2)))))+g^3*(-1/9*x^2*(-c^2*d*x^2+d)^(7/2)/c^2/d-2/63/d/c^4 *(-c^2*d*x^2+d)^(7/2))+3*f*g^2*(-1/8*x*(-c^2*d*x^2+d)^(7/2)/c^2/d+1/8/c^2* (1/6*x*(-c^2*d*x^2+d)^(5/2)+5/6*d*(1/4*x*(-c^2*d*x^2+d)^(3/2)+3/4*d*(1/2*x *(-c^2*d*x^2+d)^(1/2)+1/2*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x ^2+d)^(1/2))))))-3/7*f^2*g/c^2/d*(-c^2*d*x^2+d)^(7/2))+b*(-5/256*(-d*(c^2* x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c^3*f*arccosh(c*x)^2*(8*c^2*f^2+ 3*g^2)*d^2+1/41472*(-d*(c^2*x^2-1))^(1/2)*(256*c^10*x^10-704*c^8*x^8+256*( c*x+1)^(1/2)*(c*x-1)^(1/2)*x^9*c^9+688*c^6*x^6-576*(c*x+1)^(1/2)*(c*x-1)^( 1/2)*x^7*c^7-280*c^4*x^4+432*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^5*x^5+41*c^2*x^ 2-120*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3+9*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c* x-1)*g^3*(-1+9*arccosh(c*x))*d^2/(c*x+1)/c^4/(c*x-1)+3/16384*(-d*(c^2*x^2- 1))^(1/2)*(128*c^9*x^9-320*c^7*x^7+128*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^8*x^8 +272*c^5*x^5-256*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^6*x^6-88*c^3*x^3+160*(c*x+1 )^(1/2)*(c*x-1)^(1/2)*c^4*x^4+8*c*x-32*(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*g^2*(-1+8*arccosh(c*x))*d^2/(c*x+1)/c^3/(c *x-1)+3/25088*(-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)*(...
\[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \,d x } \]
integral((a*c^4*d^2*g^3*x^7 + 3*a*c^4*d^2*f*g^2*x^6 + 3*a*d^2*f^2*g*x + a* d^2*f^3 + (3*a*c^4*d^2*f^2*g - 2*a*c^2*d^2*g^3)*x^5 + (a*c^4*d^2*f^3 - 6*a *c^2*d^2*f*g^2)*x^4 - (6*a*c^2*d^2*f^2*g - a*d^2*g^3)*x^3 - (2*a*c^2*d^2*f ^3 - 3*a*d^2*f*g^2)*x^2 + (b*c^4*d^2*g^3*x^7 + 3*b*c^4*d^2*f*g^2*x^6 + 3*b *d^2*f^2*g*x + b*d^2*f^3 + (3*b*c^4*d^2*f^2*g - 2*b*c^2*d^2*g^3)*x^5 + (b* c^4*d^2*f^3 - 6*b*c^2*d^2*f*g^2)*x^4 - (6*b*c^2*d^2*f^2*g - b*d^2*g^3)*x^3 - (2*b*c^2*d^2*f^3 - 3*b*d^2*f*g^2)*x^2)*arccosh(c*x))*sqrt(-c^2*d*x^2 + d), x)
Timed out. \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\text {Timed out} \]
\[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \,d x } \]
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^3 + 1/128*(8*(-c^2* d*x^2 + d)^(5/2)*x/c^2 - 48*(-c^2*d*x^2 + d)^(7/2)*x/(c^2*d) + 10*(-c^2*d* x^2 + d)^(3/2)*d*x/c^2 + 15*sqrt(-c^2*d*x^2 + d)*d^2*x/c^2 + 15*d^(5/2)*ar csin(c*x)/c^3)*a*f*g^2 - 1/63*(7*(-c^2*d*x^2 + d)^(7/2)*x^2/(c^2*d) + 2*(- c^2*d*x^2 + d)^(7/2)/(c^4*d))*a*g^3 - 3/7*(-c^2*d*x^2 + d)^(7/2)*a*f^2*g/( c^2*d) + integrate((-c^2*d*x^2 + d)^(5/2)*b*g^3*x^3*log(c*x + sqrt(c*x + 1 )*sqrt(c*x - 1)) + 3*(-c^2*d*x^2 + d)^(5/2)*b*f*g^2*x^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + 3*(-c^2*d*x^2 + d)^(5/2)*b*f^2*g*x*log(c*x + sqrt(c *x + 1)*sqrt(c*x - 1)) + (-c^2*d*x^2 + d)^(5/2)*b*f^3*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)), x)
Exception generated. \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\int {\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]