3.295 \(\int e^{\coth ^{-1}(a x)} x^m \sqrt {c-a c x} \, dx\)

Optimal. Leaf size=65 \[ \frac {2 x^{m+1} \sqrt {c-a c x} \, _2F_1\left (-\frac {1}{2},-m-\frac {3}{2};-m-\frac {1}{2};-\frac {1}{a x}\right )}{(2 m+3) \sqrt {1-\frac {1}{a x}}} \]

[Out]

2*x^(1+m)*hypergeom([-1/2, -3/2-m],[-1/2-m],-1/a/x)*(-a*c*x+c)^(1/2)/(3+2*m)/(1-1/a/x)^(1/2)

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Rubi [A]  time = 0.19, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6176, 6181, 64} \[ \frac {2 x^{m+1} \sqrt {c-a c x} \, _2F_1\left (-\frac {1}{2},-m-\frac {3}{2};-m-\frac {1}{2};-\frac {1}{a x}\right )}{(2 m+3) \sqrt {1-\frac {1}{a x}}} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcCoth[a*x]*x^m*Sqrt[c - a*c*x],x]

[Out]

(2*x^(1 + m)*Sqrt[c - a*c*x]*Hypergeometric2F1[-1/2, -3/2 - m, -1/2 - m, -(1/(a*x))])/((3 + 2*m)*Sqrt[1 - 1/(a
*x)])

Rule 64

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c^n*(b*x)^(m + 1)*Hypergeometric2F1[-n, m +
1, m + 2, -((d*x)/c)])/(b*(m + 1)), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[
c, 0] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-(d/(b*c)), 0])))

Rule 6176

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_), x_Symbol] :> Dist[(c + d*x)^p/(x^p*(1 + c/(d
*x))^p), Int[u*x^p*(1 + c/(d*x))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^
2, 0] &&  !IntegerQ[n/2] &&  !IntegerQ[p]

Rule 6181

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_), x_Symbol] :> -Dist[c^p*x^m*(1/x)^m, Sub
st[Int[((1 + (d*x)/c)^p*(1 + x/a)^(n/2))/(x^(m + 2)*(1 - x/a)^(n/2)), x], x, 1/x], x] /; FreeQ[{a, c, d, m, n,
p}, x] && EqQ[c^2 - a^2*d^2, 0] &&  !IntegerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) &&  !IntegerQ[m]

Rubi steps

\begin {align*} \int e^{\coth ^{-1}(a x)} x^m \sqrt {c-a c x} \, dx &=\frac {\sqrt {c-a c x} \int e^{\coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}} x^{\frac {1}{2}+m} \, dx}{\sqrt {1-\frac {1}{a x}} \sqrt {x}}\\ &=-\frac {\left (\left (\frac {1}{x}\right )^{\frac {1}{2}+m} x^m \sqrt {c-a c x}\right ) \operatorname {Subst}\left (\int x^{-\frac {5}{2}-m} \sqrt {1+\frac {x}{a}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=\frac {2 x^{1+m} \sqrt {c-a c x} \, _2F_1\left (-\frac {1}{2},-\frac {3}{2}-m;-\frac {1}{2}-m;-\frac {1}{a x}\right )}{(3+2 m) \sqrt {1-\frac {1}{a x}}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 67, normalized size = 1.03 \[ -\frac {x^{m+1} \sqrt {c-a c x} \, _2F_1\left (-\frac {1}{2},-m-\frac {3}{2};-m-\frac {1}{2};-\frac {1}{a x}\right )}{\left (-m-\frac {3}{2}\right ) \sqrt {1-\frac {1}{a x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcCoth[a*x]*x^m*Sqrt[c - a*c*x],x]

[Out]

-((x^(1 + m)*Sqrt[c - a*c*x]*Hypergeometric2F1[-1/2, -3/2 - m, -1/2 - m, -(1/(a*x))])/((-3/2 - m)*Sqrt[1 - 1/(
a*x)]))

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fricas [F]  time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a c x + c} {\left (a x + 1\right )} x^{m} \sqrt {\frac {a x - 1}{a x + 1}}}{a x - 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/2)*x^m*(-a*c*x+c)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(-a*c*x + c)*(a*x + 1)*x^m*sqrt((a*x - 1)/(a*x + 1))/(a*x - 1), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-a c x + c} x^{m}}{\sqrt {\frac {a x - 1}{a x + 1}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/2)*x^m*(-a*c*x+c)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(-a*c*x + c)*x^m/sqrt((a*x - 1)/(a*x + 1)), x)

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maple [F]  time = 0.36, size = 0, normalized size = 0.00 \[ \int \frac {x^{m} \sqrt {-a c x +c}}{\sqrt {\frac {a x -1}{a x +1}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*x-1)/(a*x+1))^(1/2)*x^m*(-a*c*x+c)^(1/2),x)

[Out]

int(1/((a*x-1)/(a*x+1))^(1/2)*x^m*(-a*c*x+c)^(1/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-a c x + c} x^{m}}{\sqrt {\frac {a x - 1}{a x + 1}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/2)*x^m*(-a*c*x+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-a*c*x + c)*x^m/sqrt((a*x - 1)/(a*x + 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^m\,\sqrt {c-a\,c\,x}}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^m*(c - a*c*x)^(1/2))/((a*x - 1)/(a*x + 1))^(1/2),x)

[Out]

int((x^m*(c - a*c*x)^(1/2))/((a*x - 1)/(a*x + 1))^(1/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))**(1/2)*x**m*(-a*c*x+c)**(1/2),x)

[Out]

Timed out

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