3.281 \(\int e^{n \tanh ^{-1}(a x)} \sqrt {c-a c x} \, dx\)

Optimal. Leaf size=81 \[ -\frac {2^{\frac {n}{2}+1} (c-a c x)^{3/2} (1-a x)^{-n/2} \, _2F_1\left (\frac {3-n}{2},-\frac {n}{2};\frac {5-n}{2};\frac {1}{2} (1-a x)\right )}{a c (3-n)} \]

[Out]

-2^(1+1/2*n)*(-a*c*x+c)^(3/2)*hypergeom([-1/2*n, 3/2-1/2*n],[5/2-1/2*n],-1/2*a*x+1/2)/a/c/(3-n)/((-a*x+1)^(1/2
*n))

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Rubi [A]  time = 0.06, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6130, 23, 69} \[ -\frac {2^{\frac {n}{2}+1} (c-a c x)^{3/2} (1-a x)^{-n/2} \, _2F_1\left (\frac {3-n}{2},-\frac {n}{2};\frac {5-n}{2};\frac {1}{2} (1-a x)\right )}{a c (3-n)} \]

Antiderivative was successfully verified.

[In]

Int[E^(n*ArcTanh[a*x])*Sqrt[c - a*c*x],x]

[Out]

-((2^(1 + n/2)*(c - a*c*x)^(3/2)*Hypergeometric2F1[(3 - n)/2, -n/2, (5 - n)/2, (1 - a*x)/2])/(a*c*(3 - n)*(1 -
a*x)^(n/2)))

Rule 23

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((c_) + (d_.)*(v_))^(n_), x_Symbol] :> Dist[(a + b*v)^m/(c + d*v)^m, Int[u*
(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] &&  !(IntegerQ[m] || IntegerQ[n
] || GtQ[b/d, 0])

Rule 69

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

Rule 6130

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

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 77, normalized size = 0.95 \[ \frac {2^{\frac {n}{2}+1} \sqrt {c-a c x} (1-a x)^{1-\frac {n}{2}} \, _2F_1\left (\frac {3}{2}-\frac {n}{2},-\frac {n}{2};\frac {5}{2}-\frac {n}{2};\frac {1}{2}-\frac {a x}{2}\right )}{a (n-3)} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(n*ArcTanh[a*x])*Sqrt[c - a*c*x],x]

[Out]

(2^(1 + n/2)*(1 - a*x)^(1 - n/2)*Sqrt[c - a*c*x]*Hypergeometric2F1[3/2 - n/2, -1/2*n, 5/2 - n/2, 1/2 - (a*x)/2
])/(a*(-3 + n))

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fricas [F]  time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-a c x + c} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(-a*c*x + c)*((a*x + 1)/(a*x - 1))^(1/2*n), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {-a c x + c} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-a*c*x + c)*((a*x + 1)/(a*x - 1))^(1/2*n), x)

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maple [F]  time = 0.21, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \arctanh \left (a x \right )} \sqrt {-a c x +c}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arctanh(a*x))*(-a*c*x+c)^(1/2),x)

[Out]

int(exp(n*arctanh(a*x))*(-a*c*x+c)^(1/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {-a c x + c} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-a*c*x + c)*((a*x + 1)/(a*x - 1))^(1/2*n), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*atanh(a*x))*(c - a*c*x)^(1/2),x)

[Out]

int(exp(n*atanh(a*x))*(c - a*c*x)^(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(exp(n*atanh(a*x))*(-a*c*x+c)**(1/2),x)

[Out]

Timed out

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