\(\int (d+e x)^m (a+b \text {arccosh}(c x)) \, dx\) [38]

Optimal result

Integrand size = 16, antiderivative size = 125 \[ \int (d+e x)^m (a+b \text {arccosh}(c x)) \, dx=-\frac {\sqrt {2} b (c d+e) \sqrt {-1+c x} (d+e x)^m \left (\frac {c (d+e x)}{c d+e}\right )^{-m} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-1-m,\frac {3}{2},\frac {1}{2} (1-c x),\frac {e (1-c x)}{c d+e}\right )}{c e (1+m)}+\frac {(d+e x)^{1+m} (a+b \text {arccosh}(c x))}{e (1+m)} \] Output:

-2^(1/2)*b*(c*d+e)*(c*x-1)^(1/2)*(e*x+d)^m*AppellF1(1/2,-1-m,1/2,3/2,e*(-c 
*x+1)/(c*d+e),-1/2*c*x+1/2)/c/e/(1+m)/((c*(e*x+d)/(c*d+e))^m)+(e*x+d)^(1+m 
)*(a+b*arccosh(c*x))/e/(1+m)
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.42 \[ \int (d+e x)^m (a+b \text {arccosh}(c x)) \, dx=\frac {(d+e x)^m \left (\frac {c (d+e x)}{c d+e}\right )^{-m} \left (-2 b e \sqrt {-2+2 c x} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},-m,\frac {3}{2},\frac {1}{2}-\frac {c x}{2},\frac {e-c e x}{c d+e}\right )+b (-c d+e) \sqrt {-2+2 c x} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-m,\frac {3}{2},\frac {1}{2}-\frac {c x}{2},\frac {e-c e x}{c d+e}\right )+c (d+e x) \left (\frac {c (d+e x)}{c d+e}\right )^m (a+b \text {arccosh}(c x))\right )}{c e (1+m)} \] Input:

Integrate[(d + e*x)^m*(a + b*ArcCosh[c*x]),x]
 

Output:

((d + e*x)^m*(-2*b*e*Sqrt[-2 + 2*c*x]*AppellF1[1/2, -1/2, -m, 3/2, 1/2 - ( 
c*x)/2, (e - c*e*x)/(c*d + e)] + b*(-(c*d) + e)*Sqrt[-2 + 2*c*x]*AppellF1[ 
1/2, 1/2, -m, 3/2, 1/2 - (c*x)/2, (e - c*e*x)/(c*d + e)] + c*(d + e*x)*((c 
*(d + e*x))/(c*d + e))^m*(a + b*ArcCosh[c*x])))/(c*e*(1 + m)*((c*(d + e*x) 
)/(c*d + e))^m)
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6378, 156, 155}

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.

\(\displaystyle \int (d+e x)^m (a+b \text {arccosh}(c x)) \, dx\)

\(\Big \downarrow \) 6378

\(\displaystyle \frac {(d+e x)^{m+1} (a+b \text {arccosh}(c x))}{e (m+1)}-\frac {b c \int \frac {(d+e x)^{m+1}}{\sqrt {c x-1} \sqrt {c x+1}}dx}{e (m+1)}\)

\(\Big \downarrow \) 156

\(\displaystyle \frac {(d+e x)^{m+1} (a+b \text {arccosh}(c x))}{e (m+1)}-\frac {b (c d+e) (d+e x)^m \left (\frac {c (d+e x)}{c d+e}\right )^{-m} \int \frac {\left (\frac {c d}{c d+e}+\frac {c e x}{c d+e}\right )^{m+1}}{\sqrt {c x-1} \sqrt {c x+1}}dx}{e (m+1)}\)

\(\Big \downarrow \) 155

\(\displaystyle \frac {(d+e x)^{m+1} (a+b \text {arccosh}(c x))}{e (m+1)}-\frac {\sqrt {2} b \sqrt {c x-1} (c d+e) (d+e x)^m \left (\frac {c (d+e x)}{c d+e}\right )^{-m} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-m-1,\frac {3}{2},\frac {1}{2} (1-c x),\frac {e (1-c x)}{c d+e}\right )}{c e (m+1)}\)

Input:

Int[(d + e*x)^m*(a + b*ArcCosh[c*x]),x]
 

Output:

-((Sqrt[2]*b*(c*d + e)*Sqrt[-1 + c*x]*(d + e*x)^m*AppellF1[1/2, 1/2, -1 - 
m, 3/2, (1 - c*x)/2, (e*(1 - c*x))/(c*d + e)])/(c*e*(1 + m)*((c*(d + e*x)) 
/(c*d + e))^m)) + ((d + e*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(e*(1 + m))
 

Defintions of rubi rules used

rule 155

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) 
^(p_), x_] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*Simplify[b/(b*c - a*d)]^n* 
Simplify[b/(b*e - a*f)]^p))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/ 
(b*c - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, 
 m, n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !IntegerQ[p] && GtQ[Sim 
plify[b/(b*c - a*d)], 0] && GtQ[Simplify[b/(b*e - a*f)], 0] &&  !(GtQ[Simpl 
ify[d/(d*a - c*b)], 0] && GtQ[Simplify[d/(d*e - c*f)], 0] && SimplerQ[c + d 
*x, a + b*x]) &&  !(GtQ[Simplify[f/(f*a - e*b)], 0] && GtQ[Simplify[f/(f*c 
- e*d)], 0] && SimplerQ[e + f*x, a + b*x])
 

rule 156

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) 
^(p_), x_] :> Simp[(e + f*x)^FracPart[p]/(Simplify[b/(b*e - a*f)]^IntPart[p 
]*(b*((e + f*x)/(b*e - a*f)))^FracPart[p])   Int[(a + b*x)^m*(c + d*x)^n*Si 
mp[b*(e/(b*e - a*f)) + b*f*(x/(b*e - a*f)), x]^p, x], x] /; FreeQ[{a, b, c, 
 d, e, f, m, n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !IntegerQ[p] & 
& GtQ[Simplify[b/(b*c - a*d)], 0] &&  !GtQ[Simplify[b/(b*e - a*f)], 0]
 

rule 6378

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

Maple [F]

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

Input:

int((e*x+d)^m*(a+b*arccosh(c*x)),x)
 

Output:

int((e*x+d)^m*(a+b*arccosh(c*x)),x)
 

Fricas [F]

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

integrate((e*x+d)^m*(a+b*arccosh(c*x)),x, algorithm="fricas")
 

Output:

integral((b*arccosh(c*x) + a)*(e*x + d)^m, x)
 

Sympy [F]

\[ \int (d+e x)^m (a+b \text {arccosh}(c x)) \, dx=\int \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x\right )^{m}\, dx \] Input:

integrate((e*x+d)**m*(a+b*acosh(c*x)),x)
 

Output:

Integral((a + b*acosh(c*x))*(d + e*x)**m, x)
 

Maxima [F]

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

integrate((e*x+d)^m*(a+b*arccosh(c*x)),x, algorithm="maxima")
 

Output:

b*((e*x + d)*(e*x + d)^m*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(e*(m + 1) 
) - integrate((c^2*e*x^2 + c^2*d*x)*(e*x + d)^m/(c^2*e*(m + 1)*x^2 - e*(m 
+ 1)), x) + integrate((c*e*x + c*d)*(e*x + d)^m/(c^3*e*(m + 1)*x^3 - c*e*( 
m + 1)*x + (c^2*e*(m + 1)*x^2 - e*(m + 1))*sqrt(c*x + 1)*sqrt(c*x - 1)), x 
)) + (e*x + d)^(m + 1)*a/(e*(m + 1))
 

Giac [F]

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

integrate((e*x+d)^m*(a+b*arccosh(c*x)),x, algorithm="giac")
 

Output:

integrate((b*arccosh(c*x) + a)*(e*x + d)^m, x)
 

Mupad [F(-1)]

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

int((a + b*acosh(c*x))*(d + e*x)^m,x)
 

Output:

int((a + b*acosh(c*x))*(d + e*x)^m, x)
 

Reduce [F]

\[ \int (d+e x)^m (a+b \text {arccosh}(c x)) \, dx=\frac {\left (e x +d \right )^{m} a d +\left (e x +d \right )^{m} a e x +\left (\int \left (e x +d \right )^{m} \mathit {acosh} \left (c x \right )d x \right ) b e m +\left (\int \left (e x +d \right )^{m} \mathit {acosh} \left (c x \right )d x \right ) b e}{e \left (m +1\right )} \] Input:

int((e*x+d)^m*(a+b*acosh(c*x)),x)
 

Output:

((d + e*x)**m*a*d + (d + e*x)**m*a*e*x + int((d + e*x)**m*acosh(c*x),x)*b* 
e*m + int((d + e*x)**m*acosh(c*x),x)*b*e)/(e*(m + 1))