3.229 \(\int (c e+d e x)^m (a+b \cosh ^{-1}(c+d x)) \, dx\)

Optimal. Leaf size=118 \[ \frac{(e (c+d x))^{m+1} \left (a+b \cosh ^{-1}(c+d x)\right )}{d e (m+1)}-\frac{b \left (1-(c+d x)^2\right ) (e (c+d x))^{m+2} \text{Hypergeometric2F1}\left (1,\frac{m+3}{2},\frac{m+4}{2},(c+d x)^2\right )}{d e^2 (m+1) (m+2) \sqrt{c+d x-1} \sqrt{c+d x+1}} \]

[Out]

((e*(c + d*x))^(1 + m)*(a + b*ArcCosh[c + d*x]))/(d*e*(1 + m)) - (b*(e*(c + d*x))^(2 + m)*(1 - (c + d*x)^2)*Hy
pergeometric2F1[1, (3 + m)/2, (4 + m)/2, (c + d*x)^2])/(d*e^2*(1 + m)*(2 + m)*Sqrt[-1 + c + d*x]*Sqrt[1 + c +
d*x])

________________________________________________________________________________________

Rubi [A]  time = 0.0989388, antiderivative size = 124, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {5866, 5662, 126, 365, 364} \[ \frac{(e (c+d x))^{m+1} \left (a+b \cosh ^{-1}(c+d x)\right )}{d e (m+1)}-\frac{b \sqrt{1-(c+d x)^2} (e (c+d x))^{m+2} \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};(c+d x)^2\right )}{d e^2 (m+1) (m+2) \sqrt{c+d x-1} \sqrt{c+d x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((e*(c + d*x))^(1 + m)*(a + b*ArcCosh[c + d*x]))/(d*e*(1 + m)) - (b*(e*(c + d*x))^(2 + m)*Sqrt[1 - (c + d*x)^2
]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, (c + d*x)^2])/(d*e^2*(1 + m)*(2 + m)*Sqrt[-1 + c + d*x]*Sqrt[1
+ c + d*x])

Rule 5866

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
]

Rule 5662

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

Rule 126

Int[((f_.)*(x_))^(p_.)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Dist[((a + b*x)^Fra
cPart[m]*(c + d*x)^FracPart[m])/(a*c + b*d*x^2)^FracPart[m], Int[(a*c + b*d*x^2)^m*(f*x)^p, x], x] /; FreeQ[{a
, b, c, d, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[m - n, 0]

Rule 365

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

Rule 364

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

Rubi steps

\begin{align*} \int (c e+d e x)^m \left (a+b \cosh ^{-1}(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int (e x)^m \left (a+b \cosh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{(e (c+d x))^{1+m} \left (a+b \cosh ^{-1}(c+d x)\right )}{d e (1+m)}-\frac{b \operatorname{Subst}\left (\int \frac{(e x)^{1+m}}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{d e (1+m)}\\ &=\frac{(e (c+d x))^{1+m} \left (a+b \cosh ^{-1}(c+d x)\right )}{d e (1+m)}-\frac{\left (b \sqrt{-1+(c+d x)^2}\right ) \operatorname{Subst}\left (\int \frac{(e x)^{1+m}}{\sqrt{-1+x^2}} \, dx,x,c+d x\right )}{d e (1+m) \sqrt{-1+c+d x} \sqrt{1+c+d x}}\\ &=\frac{(e (c+d x))^{1+m} \left (a+b \cosh ^{-1}(c+d x)\right )}{d e (1+m)}-\frac{\left (b \sqrt{1-(c+d x)^2}\right ) \operatorname{Subst}\left (\int \frac{(e x)^{1+m}}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{d e (1+m) \sqrt{-1+c+d x} \sqrt{1+c+d x}}\\ &=\frac{(e (c+d x))^{1+m} \left (a+b \cosh ^{-1}(c+d x)\right )}{d e (1+m)}-\frac{b (e (c+d x))^{2+m} \sqrt{1-(c+d x)^2} \, _2F_1\left (\frac{1}{2},\frac{2+m}{2};\frac{4+m}{2};(c+d x)^2\right )}{d e^2 (1+m) (2+m) \sqrt{-1+c+d x} \sqrt{1+c+d x}}\\ \end{align*}

Mathematica [A]  time = 0.192028, size = 106, normalized size = 0.9 \[ \frac{(c+d x) (e (c+d x))^m \left (-\frac{b (c+d x) \sqrt{1-(c+d x)^2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},(c+d x)^2\right )}{(m+2) \sqrt{c+d x-1} \sqrt{c+d x+1}}+a+b \cosh ^{-1}(c+d x)\right )}{d (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((c + d*x)*(e*(c + d*x))^m*(a + b*ArcCosh[c + d*x] - (b*(c + d*x)*Sqrt[1 - (c + d*x)^2]*Hypergeometric2F1[1/2,
(2 + m)/2, (4 + m)/2, (c + d*x)^2])/((2 + m)*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])))/(d*(1 + m))

________________________________________________________________________________________

Maple [F]  time = 1.967, size = 0, normalized size = 0. \begin{align*} \int \left ( dex+ce \right ) ^{m} \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \operatorname{arcosh}\left (d x + c\right ) + a\right )}{\left (d e x + c e\right )}^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e \left (c + d x\right )\right )^{m} \left (a + b \operatorname{acosh}{\left (c + d x \right )}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arcosh}\left (d x + c\right ) + a\right )}{\left (d e x + c e\right )}^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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