3.3.81 \(\int e^{n \tanh ^{-1}(a x)} \sqrt {c-a c x} \, dx\) [281]

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

[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 = 0.05, 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 = {6265, 23, 71} \begin {gather*} -\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)} \end {gather*}

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, -1/2*n, (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 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], 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 6265

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 \begin {gather*} \frac {2^{1+\frac {n}{2}} (1-a x)^{1-\frac {n}{2}} \sqrt {c-a c x} \, _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 (-3+n)} \end {gather*}

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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Maple [F]
time = 0.39, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{n \arctanh \left (a x \right )} \sqrt {-c x a +c}\, dx\]

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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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