3.14.29 \(\int e^{n \tanh ^{-1}(a x)} x \sqrt {c-a^2 c x^2} \, dx\) [1329]

Optimal. Leaf size=173 \[ -\frac {(1-a x)^{\frac {3-n}{2}} (1+a x)^{\frac {3+n}{2}} \sqrt {c-a^2 c x^2}}{3 a^2 \sqrt {1-a^2 x^2}}-\frac {2^{\frac {3+n}{2}} n (1-a x)^{\frac {3-n}{2}} \sqrt {c-a^2 c x^2} \, _2F_1\left (\frac {1}{2} (-1-n),\frac {3-n}{2};\frac {5-n}{2};\frac {1}{2} (1-a x)\right )}{3 a^2 (3-n) \sqrt {1-a^2 x^2}} \]

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6288, 6285, 81, 71} \begin {gather*} -\frac {2^{\frac {n+3}{2}} n \sqrt {c-a^2 c x^2} (1-a x)^{\frac {3-n}{2}} \, _2F_1\left (\frac {1}{2} (-n-1),\frac {3-n}{2};\frac {5-n}{2};\frac {1}{2} (1-a x)\right )}{3 a^2 (3-n) \sqrt {1-a^2 x^2}}-\frac {\sqrt {c-a^2 c x^2} (a x+1)^{\frac {n+3}{2}} (1-a x)^{\frac {3-n}{2}}}{3 a^2 \sqrt {1-a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(n*ArcTanh[a*x])*x*Sqrt[c - a^2*c*x^2],x]

[Out]

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

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 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 6285

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

Rule 6288

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

Rubi steps

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

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Mathematica [A]
time = 0.18, size = 125, normalized size = 0.72 \begin {gather*} \frac {(1-a x)^{\frac {3}{2}-\frac {n}{2}} \sqrt {c-a^2 c x^2} \left (-\left ((-3+n) (1+a x)^{\frac {3+n}{2}}\right )+2^{\frac {3+n}{2}} n \, _2F_1\left (-\frac {1}{2}-\frac {n}{2},\frac {3}{2}-\frac {n}{2};\frac {5}{2}-\frac {n}{2};\frac {1}{2}-\frac {a x}{2}\right )\right )}{3 a^2 (-3+n) \sqrt {1-a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(n*ArcTanh[a*x])*x*Sqrt[c - a^2*c*x^2],x]

[Out]

((1 - a*x)^(3/2 - n/2)*Sqrt[c - a^2*c*x^2]*(-((-3 + n)*(1 + a*x)^((3 + n)/2)) + 2^((3 + n)/2)*n*Hypergeometric
2F1[-1/2 - n/2, 3/2 - n/2, 5/2 - n/2, 1/2 - (a*x)/2]))/(3*a^2*(-3 + n)*Sqrt[1 - a^2*x^2])

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{n \arctanh \left (a x \right )} x \sqrt {-a^{2} c \,x^{2}+c}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} e^{n \operatorname {atanh}{\left (a x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*sqrt(-c*(a*x - 1)*(a*x + 1))*exp(n*atanh(a*x)), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x\,{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,\sqrt {c-a^2\,c\,x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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