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

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.07 (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 = {5963, 144, 143} \[ \int (d+e x)^m (a+b \text {arccosh}(c x)) \, dx=\frac {(d+e x)^{m+1} (a+b \text {arccosh}(c x))}{e (m+1)}-\frac {\sqrt {2} b \sqrt {c x-1} (c d+e) (d+e x)^m \left (\frac {c (d+e x)}{c d+e}\right )^{-m} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-m-1,\frac {3}{2},\frac {1}{2} (1-c x),\frac {e (1-c x)}{c d+e}\right )}{c e (m+1)} \]

[In]

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

[Out]

-((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))

Rule 143

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((a + b*x)
^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n*(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] &&  !Inte
gerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !(GtQ[d/(d*a - c*b), 0] && GtQ[
d/(d*e - c*f), 0] && SimplerQ[c + d*x, a + b*x]) &&  !(GtQ[f/(f*a - e*b), 0] && GtQ[f/(f*c - e*d), 0] && Simpl
erQ[e + f*x, a + b*x])

Rule 144

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

Rule 5963

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] - Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x)^{1+m} (a+b \text {arccosh}(c x))}{e (1+m)}-\frac {(b c) \int \frac {(d+e x)^{1+m}}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{e (1+m)} \\ & = \frac {(d+e x)^{1+m} (a+b \text {arccosh}(c x))}{e (1+m)}-\frac {\left (b (c d+e) (d+e x)^m \left (\frac {c (d+e x)}{c d+e}\right )^{-m}\right ) \int \frac {\left (\frac {c d}{c d+e}+\frac {c e x}{c d+e}\right )^{1+m}}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{e (1+m)} \\ & = -\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)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.42 \[ \int (d+e x)^m (a+b \text {arccosh}(c x)) \, dx=\frac {(d+e x)^m \left (\frac {c (d+e x)}{c d+e}\right )^{-m} \left (-2 b e \sqrt {-2+2 c x} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},-m,\frac {3}{2},\frac {1}{2}-\frac {c x}{2},\frac {e-c e x}{c d+e}\right )+b (-c d+e) \sqrt {-2+2 c x} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-m,\frac {3}{2},\frac {1}{2}-\frac {c x}{2},\frac {e-c e x}{c d+e}\right )+c (d+e x) \left (\frac {c (d+e x)}{c d+e}\right )^m (a+b \text {arccosh}(c x))\right )}{c e (1+m)} \]

[In]

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

[Out]

((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)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((a + b*acosh(c*x))*(d + e*x)^m,x)

[Out]

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