\(\int (f x)^m (d-c^2 d x^2)^3 (a+b \text {arccosh}(c x)) \, dx\) [140]

Optimal result

Integrand size = 27, antiderivative size = 385 \[ \int (f x)^m \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {b c d^3 \left (2271+1329 m+284 m^2+27 m^3+m^4\right ) (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x}}{f^2 (3+m)^2 (5+m)^2 (7+m)^2}-\frac {b c^3 d^3 (9+m) (13+2 m) (f x)^{4+m} \sqrt {-1+c x} \sqrt {1+c x}}{f^4 (5+m)^2 (7+m)^2}+\frac {b c^5 d^3 (f x)^{6+m} \sqrt {-1+c x} \sqrt {1+c x}}{f^6 (7+m)^2}+\frac {d^3 (f x)^{1+m} (a+b \text {arccosh}(c x))}{f (1+m)}-\frac {3 c^2 d^3 (f x)^{3+m} (a+b \text {arccosh}(c x))}{f^3 (3+m)}+\frac {3 c^4 d^3 (f x)^{5+m} (a+b \text {arccosh}(c x))}{f^5 (5+m)}-\frac {c^6 d^3 (f x)^{7+m} (a+b \text {arccosh}(c x))}{f^7 (7+m)}-\frac {3 b c d^3 \left (2161+1813 m+455 m^2+35 m^3\right ) (f x)^{2+m} \sqrt {1-c x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{f^2 (1+m) (2+m) (3+m)^2 (5+m)^2 (7+m)^2 \sqrt {-1+c x}} \] Output:

b*c*d^3*(m^4+27*m^3+284*m^2+1329*m+2271)*(f*x)^(2+m)*(c*x-1)^(1/2)*(c*x+1) 
^(1/2)/f^2/(3+m)^2/(5+m)^2/(7+m)^2-b*c^3*d^3*(9+m)*(13+2*m)*(f*x)^(4+m)*(c 
*x-1)^(1/2)*(c*x+1)^(1/2)/f^4/(5+m)^2/(7+m)^2+b*c^5*d^3*(f*x)^(6+m)*(c*x-1 
)^(1/2)*(c*x+1)^(1/2)/f^6/(7+m)^2+d^3*(f*x)^(1+m)*(a+b*arccosh(c*x))/f/(1+ 
m)-3*c^2*d^3*(f*x)^(3+m)*(a+b*arccosh(c*x))/f^3/(3+m)+3*c^4*d^3*(f*x)^(5+m 
)*(a+b*arccosh(c*x))/f^5/(5+m)-c^6*d^3*(f*x)^(7+m)*(a+b*arccosh(c*x))/f^7/ 
(7+m)-3*b*c*d^3*(35*m^3+455*m^2+1813*m+2161)*(f*x)^(2+m)*(-c*x+1)^(1/2)*hy 
pergeom([1/2, 1+1/2*m],[2+1/2*m],c^2*x^2)/f^2/(1+m)/(2+m)/(3+m)^2/(5+m)^2/ 
(7+m)^2/(c*x-1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.01 \[ \int (f x)^m \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=d^3 x (f x)^m \left (\frac {a+b \text {arccosh}(c x)}{1+m}-\frac {3 c^2 x^2 (a+b \text {arccosh}(c x))}{3+m}+\frac {3 c^4 x^4 (a+b \text {arccosh}(c x))}{5+m}-\frac {c^6 x^6 (a+b \text {arccosh}(c x))}{7+m}+\frac {b c^7 x^7 \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},4+\frac {m}{2},5+\frac {m}{2},c^2 x^2\right )}{(7+m) (8+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c x \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{\left (2+3 m+m^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b c^3 x^3 \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4+m}{2},\frac {6+m}{2},c^2 x^2\right )}{\left (12+7 m+m^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c^5 x^5 \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {6+m}{2},\frac {8+m}{2},c^2 x^2\right )}{(5+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}\right ) \] Input:

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

Output:

d^3*x*(f*x)^m*((a + b*ArcCosh[c*x])/(1 + m) - (3*c^2*x^2*(a + b*ArcCosh[c* 
x]))/(3 + m) + (3*c^4*x^4*(a + b*ArcCosh[c*x]))/(5 + m) - (c^6*x^6*(a + b* 
ArcCosh[c*x]))/(7 + m) + (b*c^7*x^7*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/ 
2, 4 + m/2, 5 + m/2, c^2*x^2])/((7 + m)*(8 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c* 
x]) - (b*c*x*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2 
, c^2*x^2])/((2 + 3*m + m^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*b*c^3*x^3* 
Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (4 + m)/2, (6 + m)/2, c^2*x^2])/( 
(12 + 7*m + m^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*b*c^5*x^5*Sqrt[1 - c^2 
*x^2]*Hypergeometric2F1[1/2, (6 + m)/2, (8 + m)/2, c^2*x^2])/((5 + m)*(6 + 
 m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))
 

Rubi [A] (verified)

Time = 2.25 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6336, 27, 2113, 2340, 1590, 27, 363, 279, 278}

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.

\(\displaystyle \int \left (d-c^2 d x^2\right )^3 (f x)^m (a+b \text {arccosh}(c x)) \, dx\)

\(\Big \downarrow \) 6336

\(\displaystyle -b c \int \frac {d^3 (f x)^{m+1} \left (-\frac {c^6 x^6}{m+7}+\frac {3 c^4 x^4}{m+5}-\frac {3 c^2 x^2}{m+3}+\frac {1}{m+1}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {c^6 d^3 (f x)^{m+7} (a+b \text {arccosh}(c x))}{f^7 (m+7)}+\frac {3 c^4 d^3 (f x)^{m+5} (a+b \text {arccosh}(c x))}{f^5 (m+5)}-\frac {3 c^2 d^3 (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {d^3 (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b c d^3 \int \frac {(f x)^{m+1} \left (-\frac {c^6 x^6}{m+7}+\frac {3 c^4 x^4}{m+5}-\frac {3 c^2 x^2}{m+3}+\frac {1}{m+1}\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx}{f}-\frac {c^6 d^3 (f x)^{m+7} (a+b \text {arccosh}(c x))}{f^7 (m+7)}+\frac {3 c^4 d^3 (f x)^{m+5} (a+b \text {arccosh}(c x))}{f^5 (m+5)}-\frac {3 c^2 d^3 (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {d^3 (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}\)

\(\Big \downarrow \) 2113

\(\displaystyle -\frac {b c d^3 \sqrt {c^2 x^2-1} \int \frac {(f x)^{m+1} \left (-\frac {c^6 x^6}{m+7}+\frac {3 c^4 x^4}{m+5}-\frac {3 c^2 x^2}{m+3}+\frac {1}{m+1}\right )}{\sqrt {c^2 x^2-1}}dx}{f \sqrt {c x-1} \sqrt {c x+1}}-\frac {c^6 d^3 (f x)^{m+7} (a+b \text {arccosh}(c x))}{f^7 (m+7)}+\frac {3 c^4 d^3 (f x)^{m+5} (a+b \text {arccosh}(c x))}{f^5 (m+5)}-\frac {3 c^2 d^3 (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {d^3 (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}\)

\(\Big \downarrow \) 2340

\(\displaystyle -\frac {b c d^3 \sqrt {c^2 x^2-1} \left (\frac {\int \frac {(f x)^{m+1} \left (\frac {(m+9) (2 m+13) x^4 c^6}{(m+5) (m+7)}-\frac {3 (m+7) x^2 c^4}{m+3}+\frac {(m+7) c^2}{m+1}\right )}{\sqrt {c^2 x^2-1}}dx}{c^2 (m+7)}-\frac {c^4 \sqrt {c^2 x^2-1} (f x)^{m+6}}{f^5 (m+7)^2}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}-\frac {c^6 d^3 (f x)^{m+7} (a+b \text {arccosh}(c x))}{f^7 (m+7)}+\frac {3 c^4 d^3 (f x)^{m+5} (a+b \text {arccosh}(c x))}{f^5 (m+5)}-\frac {3 c^2 d^3 (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {d^3 (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}\)

\(\Big \downarrow \) 1590

\(\displaystyle -\frac {b c d^3 \sqrt {c^2 x^2-1} \left (\frac {\frac {\int \frac {c^4 (f x)^{m+1} \left (\frac {(m+5) (m+7)}{m+1}-\frac {c^2 \left (m^4+27 m^3+284 m^2+1329 m+2271\right ) x^2}{(m+3) (m+5) (m+7)}\right )}{\sqrt {c^2 x^2-1}}dx}{c^2 (m+5)}+\frac {c^4 (m+9) (2 m+13) \sqrt {c^2 x^2-1} (f x)^{m+4}}{f^3 (m+5)^2 (m+7)}}{c^2 (m+7)}-\frac {c^4 \sqrt {c^2 x^2-1} (f x)^{m+6}}{f^5 (m+7)^2}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}-\frac {c^6 d^3 (f x)^{m+7} (a+b \text {arccosh}(c x))}{f^7 (m+7)}+\frac {3 c^4 d^3 (f x)^{m+5} (a+b \text {arccosh}(c x))}{f^5 (m+5)}-\frac {3 c^2 d^3 (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {d^3 (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b c d^3 \sqrt {c^2 x^2-1} \left (\frac {\frac {c^2 \int \frac {(f x)^{m+1} \left (\frac {(m+5) (m+7)}{m+1}-\frac {c^2 \left (m^4+27 m^3+284 m^2+1329 m+2271\right ) x^2}{(m+3) (m+5) (m+7)}\right )}{\sqrt {c^2 x^2-1}}dx}{m+5}+\frac {c^4 (m+9) (2 m+13) \sqrt {c^2 x^2-1} (f x)^{m+4}}{f^3 (m+5)^2 (m+7)}}{c^2 (m+7)}-\frac {c^4 \sqrt {c^2 x^2-1} (f x)^{m+6}}{f^5 (m+7)^2}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}-\frac {c^6 d^3 (f x)^{m+7} (a+b \text {arccosh}(c x))}{f^7 (m+7)}+\frac {3 c^4 d^3 (f x)^{m+5} (a+b \text {arccosh}(c x))}{f^5 (m+5)}-\frac {3 c^2 d^3 (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {d^3 (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}\)

\(\Big \downarrow \) 363

\(\displaystyle -\frac {b c d^3 \sqrt {c^2 x^2-1} \left (\frac {\frac {c^2 \left (\frac {3 \left (35 m^3+455 m^2+1813 m+2161\right ) \int \frac {(f x)^{m+1}}{\sqrt {c^2 x^2-1}}dx}{(m+1) (m+3)^2 (m+5) (m+7)}-\frac {\left (m^4+27 m^3+284 m^2+1329 m+2271\right ) \sqrt {c^2 x^2-1} (f x)^{m+2}}{f (m+3)^2 (m+5) (m+7)}\right )}{m+5}+\frac {c^4 (m+9) (2 m+13) \sqrt {c^2 x^2-1} (f x)^{m+4}}{f^3 (m+5)^2 (m+7)}}{c^2 (m+7)}-\frac {c^4 \sqrt {c^2 x^2-1} (f x)^{m+6}}{f^5 (m+7)^2}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}-\frac {c^6 d^3 (f x)^{m+7} (a+b \text {arccosh}(c x))}{f^7 (m+7)}+\frac {3 c^4 d^3 (f x)^{m+5} (a+b \text {arccosh}(c x))}{f^5 (m+5)}-\frac {3 c^2 d^3 (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {d^3 (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}\)

\(\Big \downarrow \) 279

\(\displaystyle -\frac {b c d^3 \sqrt {c^2 x^2-1} \left (\frac {\frac {c^2 \left (\frac {3 \left (35 m^3+455 m^2+1813 m+2161\right ) \sqrt {1-c^2 x^2} \int \frac {(f x)^{m+1}}{\sqrt {1-c^2 x^2}}dx}{(m+1) (m+3)^2 (m+5) (m+7) \sqrt {c^2 x^2-1}}-\frac {\left (m^4+27 m^3+284 m^2+1329 m+2271\right ) \sqrt {c^2 x^2-1} (f x)^{m+2}}{f (m+3)^2 (m+5) (m+7)}\right )}{m+5}+\frac {c^4 (m+9) (2 m+13) \sqrt {c^2 x^2-1} (f x)^{m+4}}{f^3 (m+5)^2 (m+7)}}{c^2 (m+7)}-\frac {c^4 \sqrt {c^2 x^2-1} (f x)^{m+6}}{f^5 (m+7)^2}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}-\frac {c^6 d^3 (f x)^{m+7} (a+b \text {arccosh}(c x))}{f^7 (m+7)}+\frac {3 c^4 d^3 (f x)^{m+5} (a+b \text {arccosh}(c x))}{f^5 (m+5)}-\frac {3 c^2 d^3 (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {d^3 (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}\)

\(\Big \downarrow \) 278

\(\displaystyle -\frac {c^6 d^3 (f x)^{m+7} (a+b \text {arccosh}(c x))}{f^7 (m+7)}+\frac {3 c^4 d^3 (f x)^{m+5} (a+b \text {arccosh}(c x))}{f^5 (m+5)}-\frac {3 c^2 d^3 (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {d^3 (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}-\frac {b c d^3 \sqrt {c^2 x^2-1} \left (\frac {\frac {c^2 \left (\frac {3 \left (35 m^3+455 m^2+1813 m+2161\right ) \sqrt {1-c^2 x^2} (f x)^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right )}{f (m+1) (m+2) (m+3)^2 (m+5) (m+7) \sqrt {c^2 x^2-1}}-\frac {\left (m^4+27 m^3+284 m^2+1329 m+2271\right ) \sqrt {c^2 x^2-1} (f x)^{m+2}}{f (m+3)^2 (m+5) (m+7)}\right )}{m+5}+\frac {c^4 (m+9) (2 m+13) \sqrt {c^2 x^2-1} (f x)^{m+4}}{f^3 (m+5)^2 (m+7)}}{c^2 (m+7)}-\frac {c^4 \sqrt {c^2 x^2-1} (f x)^{m+6}}{f^5 (m+7)^2}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

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

Output:

(d^3*(f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(f*(1 + m)) - (3*c^2*d^3*(f*x)^(3 
 + m)*(a + b*ArcCosh[c*x]))/(f^3*(3 + m)) + (3*c^4*d^3*(f*x)^(5 + m)*(a + 
b*ArcCosh[c*x]))/(f^5*(5 + m)) - (c^6*d^3*(f*x)^(7 + m)*(a + b*ArcCosh[c*x 
]))/(f^7*(7 + m)) - (b*c*d^3*Sqrt[-1 + c^2*x^2]*(-((c^4*(f*x)^(6 + m)*Sqrt 
[-1 + c^2*x^2])/(f^5*(7 + m)^2)) + ((c^4*(9 + m)*(13 + 2*m)*(f*x)^(4 + m)* 
Sqrt[-1 + c^2*x^2])/(f^3*(5 + m)^2*(7 + m)) + (c^2*(-(((2271 + 1329*m + 28 
4*m^2 + 27*m^3 + m^4)*(f*x)^(2 + m)*Sqrt[-1 + c^2*x^2])/(f*(3 + m)^2*(5 + 
m)*(7 + m))) + (3*(2161 + 1813*m + 455*m^2 + 35*m^3)*(f*x)^(2 + m)*Sqrt[1 
- c^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(f*(1 + 
m)*(2 + m)*(3 + m)^2*(5 + m)*(7 + m)*Sqrt[-1 + c^2*x^2])))/(5 + m))/(c^2*( 
7 + m))))/(f*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP 
art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[(c*x)^m* 
(1 + b*(x^2/a))^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && 
!(ILtQ[p, 0] || GtQ[a, 0])
 

rule 363

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), 
 x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3))   Int[(e*x)^ 
m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d 
, 0] && NeQ[m + 2*p + 3, 0]
 

rule 1590

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( 
c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[c^p*(f*x)^(m + 4*p - 1)*((d + e*x^2)^ 
(q + 1)/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1))), x] + Simp[1/(e*(m + 4*p + 2*q 
 + 1))   Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + 
b*x^2 + c*x^4)^p - c^p*x^(4*p)) - d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], 
x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 
0] &&  !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 0]
 

rule 2113

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_. 
)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*x)^FracPart[m]*((c + d*x)^FracPart[ 
m]/(a*c + b*d*x^2)^FracPart[m])   Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a 
*d, 0] && EqQ[m, n] &&  !IntegerQ[m]
 

rule 2340

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 
)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m 
+ q + 2*p + 1))   Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) 
*Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; 
GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ 
[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])
 

rule 6336

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

Maple [F]

\[\int \left (f x \right )^{m} \left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )d x\]

Input:

int((f*x)^m*(-c^2*d*x^2+d)^3*(a+b*arccosh(c*x)),x)
 

Output:

int((f*x)^m*(-c^2*d*x^2+d)^3*(a+b*arccosh(c*x)),x)
 

Fricas [F]

\[ \int (f x)^m \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\int { -{\left (c^{2} d x^{2} - d\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \] Input:

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

Output:

integral(-(a*c^6*d^3*x^6 - 3*a*c^4*d^3*x^4 + 3*a*c^2*d^3*x^2 - a*d^3 + (b* 
c^6*d^3*x^6 - 3*b*c^4*d^3*x^4 + 3*b*c^2*d^3*x^2 - b*d^3)*arccosh(c*x))*(f* 
x)^m, x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

\[ \int (f x)^m \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\int { -{\left (c^{2} d x^{2} - d\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \] Input:

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

Output:

-a*c^6*d^3*f^m*x^7*x^m/(m + 7) + 3*a*c^4*d^3*f^m*x^5*x^m/(m + 5) - 3*a*c^2 
*d^3*f^m*x^3*x^m/(m + 3) + (f*x)^(m + 1)*a*d^3/(f*(m + 1)) - ((m^3 + 9*m^2 
 + 23*m + 15)*b*c^6*d^3*f^m*x^7 - 3*(m^3 + 11*m^2 + 31*m + 21)*b*c^4*d^3*f 
^m*x^5 + 3*(m^3 + 13*m^2 + 47*m + 35)*b*c^2*d^3*f^m*x^3 - (m^3 + 15*m^2 + 
71*m + 105)*b*d^3*f^m*x)*x^m*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(m^4 + 
 16*m^3 + 86*m^2 + 176*m + 105) - integrate(((m^3 + 9*m^2 + 23*m + 15)*b*c 
^7*d^3*f^m*x^7 - 3*(m^3 + 11*m^2 + 31*m + 21)*b*c^5*d^3*f^m*x^5 + 3*(m^3 + 
 13*m^2 + 47*m + 35)*b*c^3*d^3*f^m*x^3 - (m^3 + 15*m^2 + 71*m + 105)*b*c*d 
^3*f^m*x)*x^m/((m^4 + 16*m^3 + 86*m^2 + 176*m + 105)*c^3*x^3 - (m^4 + 16*m 
^3 + 86*m^2 + 176*m + 105)*c*x + ((m^4 + 16*m^3 + 86*m^2 + 176*m + 105)*c^ 
2*x^2 - m^4 - 16*m^3 - 86*m^2 - 176*m - 105)*sqrt(c*x + 1)*sqrt(c*x - 1)), 
 x) + integrate(((m^3 + 9*m^2 + 23*m + 15)*b*c^8*d^3*f^m*x^8 - 3*(m^3 + 11 
*m^2 + 31*m + 21)*b*c^6*d^3*f^m*x^6 + 3*(m^3 + 13*m^2 + 47*m + 35)*b*c^4*d 
^3*f^m*x^4 - (m^3 + 15*m^2 + 71*m + 105)*b*c^2*d^3*f^m*x^2)*x^m/((m^4 + 16 
*m^3 + 86*m^2 + 176*m + 105)*c^2*x^2 - m^4 - 16*m^3 - 86*m^2 - 176*m - 105 
), x)
                                                                                    
                                                                                    
 

Giac [F(-2)]

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

integrate((f*x)^m*(-c^2*d*x^2+d)^3*(a+b*arccosh(c*x)),x, algorithm="giac")
 

Output:

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
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (f x)^m \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx =\text {Too large to display} \] Input:

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

Output:

(f**m*d**3*( - x**m*a*c**6*m**3*x**7 - 9*x**m*a*c**6*m**2*x**7 - 23*x**m*a 
*c**6*m*x**7 - 15*x**m*a*c**6*x**7 + 3*x**m*a*c**4*m**3*x**5 + 33*x**m*a*c 
**4*m**2*x**5 + 93*x**m*a*c**4*m*x**5 + 63*x**m*a*c**4*x**5 - 3*x**m*a*c** 
2*m**3*x**3 - 39*x**m*a*c**2*m**2*x**3 - 141*x**m*a*c**2*m*x**3 - 105*x**m 
*a*c**2*x**3 + x**m*a*m**3*x + 15*x**m*a*m**2*x + 71*x**m*a*m*x + 105*x**m 
*a*x - int(x**m*acosh(c*x)*x**6,x)*b*c**6*m**4 - 16*int(x**m*acosh(c*x)*x* 
*6,x)*b*c**6*m**3 - 86*int(x**m*acosh(c*x)*x**6,x)*b*c**6*m**2 - 176*int(x 
**m*acosh(c*x)*x**6,x)*b*c**6*m - 105*int(x**m*acosh(c*x)*x**6,x)*b*c**6 + 
 3*int(x**m*acosh(c*x)*x**4,x)*b*c**4*m**4 + 48*int(x**m*acosh(c*x)*x**4,x 
)*b*c**4*m**3 + 258*int(x**m*acosh(c*x)*x**4,x)*b*c**4*m**2 + 528*int(x**m 
*acosh(c*x)*x**4,x)*b*c**4*m + 315*int(x**m*acosh(c*x)*x**4,x)*b*c**4 - 3* 
int(x**m*acosh(c*x)*x**2,x)*b*c**2*m**4 - 48*int(x**m*acosh(c*x)*x**2,x)*b 
*c**2*m**3 - 258*int(x**m*acosh(c*x)*x**2,x)*b*c**2*m**2 - 528*int(x**m*ac 
osh(c*x)*x**2,x)*b*c**2*m - 315*int(x**m*acosh(c*x)*x**2,x)*b*c**2 + int(x 
**m*acosh(c*x),x)*b*m**4 + 16*int(x**m*acosh(c*x),x)*b*m**3 + 86*int(x**m* 
acosh(c*x),x)*b*m**2 + 176*int(x**m*acosh(c*x),x)*b*m + 105*int(x**m*acosh 
(c*x),x)*b))/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105)