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

Optimal. Leaf size=125 \[ \frac{(d+e x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{e (m+1)}-\frac{\sqrt{2} b \sqrt{c x-1} (c d+e) (d+e x)^m \left (\frac{c (d+e x)}{c d+e}\right )^{-m} F_1\left (\frac{1}{2};\frac{1}{2},-m-1;\frac{3}{2};\frac{1}{2} (1-c x),\frac{e (1-c x)}{c d+e}\right )}{c e (m+1)} \]

[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))

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Rubi [A]  time = 0.0811056, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {5802, 139, 138} \[ \frac{(d+e x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{e (m+1)}-\frac{\sqrt{2} b \sqrt{c x-1} (c d+e) (d+e x)^m \left (\frac{c (d+e x)}{c d+e}\right )^{-m} F_1\left (\frac{1}{2};\frac{1}{2},-m-1;\frac{3}{2};\frac{1}{2} (1-c x),\frac{e (1-c x)}{c d+e}\right )}{c e (m+1)} \]

Antiderivative was successfully verified.

[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 5802

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]

Rule 139

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 138

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((a + b*x)
^(m + 1)*AppellF1[m + 1, -n, -p, m + 2, -((d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f))])/(b*(m + 1
)*(b/(b*c - a*d))^n*(b/(b*e - a*f))^p), 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])

Rubi steps

\begin{align*} \int (d+e x)^m \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac{(d+e x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{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} F_1\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} \left (a+b \cosh ^{-1}(c x)\right )}{e (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.223621, size = 177, normalized size = 1.42 \[ \frac{(d+e x)^m \left (\frac{c (d+e x)}{c d+e}\right )^{-m} \left (c (d+e x) \left (a+b \cosh ^{-1}(c x)\right ) \left (\frac{c (d+e x)}{c d+e}\right )^m-2 b e \sqrt{2 c x-2} F_1\left (\frac{1}{2};-\frac{1}{2},-m;\frac{3}{2};\frac{1}{2}-\frac{c x}{2},\frac{e-c e x}{c d+e}\right )+b \sqrt{2 c x-2} (e-c d) F_1\left (\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2}-\frac{c x}{2},\frac{e-c e x}{c d+e}\right )\right )}{c e (m+1)} \]

Warning: Unable to verify antiderivative.

[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)

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Maple [F]  time = 3.666, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{m} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[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)