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]

(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)

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Rubi [A]  time = 0.08, antiderivative size = 125, normalized size of antiderivative = 1.00, 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 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])

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 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]

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*}

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Mathematica [A]  time = 0.23, 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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fricas [F]  time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} {\left (e x + d\right )}^{m}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} {\left (e x + d\right )}^{m}\,{d x} \]

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)

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maple [F]  time = 4.84, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right )^{m} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )\, dx \]

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]  time = 0.00, size = 0, normalized size = 0.00 \[ b {\left (\frac {{\left (e x + d\right )} {\left (e x + d\right )}^{m} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{e {\left (m + 1\right )}} - \int \frac {{\left (c^{2} e x^{2} + c^{2} d x\right )} {\left (e x + d\right )}^{m}}{c^{2} e {\left (m + 1\right )} x^{2} - e {\left (m + 1\right )}}\,{d x} + \int \frac {{\left (c e x + c d\right )} {\left (e x + d\right )}^{m}}{c^{3} e {\left (m + 1\right )} x^{3} - c e {\left (m + 1\right )} x + {\left (c^{2} e {\left (m + 1\right )} x^{2} - e {\left (m + 1\right )}\right )} \sqrt {c x + 1} \sqrt {c x - 1}}\,{d x}\right )} + \frac {{\left (e x + d\right )}^{m + 1} a}{e {\left (m + 1\right )}} \]

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]

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
))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d+e\,x\right )}^m \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(c*x))*(d + e*x)^m,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x\right )^{m}\, dx \]

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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