\(\int e^{2 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx\) [731]

Optimal result

Integrand size = 27, antiderivative size = 172 \[ \int e^{2 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\frac {x^{1+m} \sqrt {c-a^2 c x^2}}{2+m}-\frac {c (3+2 m) x^{1+m} \sqrt {1-a^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},a^2 x^2\right )}{(1+m) (2+m) \sqrt {c-a^2 c x^2}}-\frac {2 a c x^{2+m} \sqrt {1-a^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},a^2 x^2\right )}{(2+m) \sqrt {c-a^2 c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6302, 6286, 1823, 822, 372, 371} \[ \int e^{2 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=-\frac {c (2 m+3) \sqrt {1-a^2 x^2} x^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},a^2 x^2\right )}{(m+1) (m+2) \sqrt {c-a^2 c x^2}}-\frac {2 a c \sqrt {1-a^2 x^2} x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},a^2 x^2\right )}{(m+2) \sqrt {c-a^2 c x^2}}+\frac {x^{m+1} \sqrt {c-a^2 c x^2}}{m+2} \]

[In]

Int[E^(2*ArcCoth[a*x])*x^m*Sqrt[c - a^2*c*x^2],x]

[Out]

(x^(1 + m)*Sqrt[c - a^2*c*x^2])/(2 + m) - (c*(3 + 2*m)*x^(1 + m)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1/2, (1 +
m)/2, (3 + m)/2, a^2*x^2])/((1 + m)*(2 + m)*Sqrt[c - a^2*c*x^2]) - (2*a*c*x^(2 + m)*Sqrt[1 - a^2*x^2]*Hyperge
ometric2F1[1/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/((2 + m)*Sqrt[c - a^2*c*x^2])

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^FracPart[p]/
(1 + b*(x^n/a))^FracPart[p]), Int[(c*x)^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 822

Int[((e_.)*(x_))^(m_)*((f_) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[f, Int[(e*x)^m*(a + c*
x^2)^p, x], x] + Dist[g/e, Int[(e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, p}, x] &&  !Ration
alQ[m] &&  !IGtQ[p, 0]

Rule 1823

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rule 6286

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

Rule 6302

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rubi steps \begin{align*} \text {integral}& = -\int e^{2 \text {arctanh}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx \\ & = -\left (c \int \frac {x^m (1+a x)^2}{\sqrt {c-a^2 c x^2}} \, dx\right ) \\ & = \frac {x^{1+m} \sqrt {c-a^2 c x^2}}{2+m}+\frac {\int \frac {x^m \left (-a^2 c (3+2 m)-2 a^3 c (2+m) x\right )}{\sqrt {c-a^2 c x^2}} \, dx}{a^2 (2+m)} \\ & = \frac {x^{1+m} \sqrt {c-a^2 c x^2}}{2+m}-(2 a c) \int \frac {x^{1+m}}{\sqrt {c-a^2 c x^2}} \, dx-\frac {(c (3+2 m)) \int \frac {x^m}{\sqrt {c-a^2 c x^2}} \, dx}{2+m} \\ & = \frac {x^{1+m} \sqrt {c-a^2 c x^2}}{2+m}-\frac {\left (2 a c \sqrt {1-a^2 x^2}\right ) \int \frac {x^{1+m}}{\sqrt {1-a^2 x^2}} \, dx}{\sqrt {c-a^2 c x^2}}-\frac {\left (c (3+2 m) \sqrt {1-a^2 x^2}\right ) \int \frac {x^m}{\sqrt {1-a^2 x^2}} \, dx}{(2+m) \sqrt {c-a^2 c x^2}} \\ & = \frac {x^{1+m} \sqrt {c-a^2 c x^2}}{2+m}-\frac {c (3+2 m) x^{1+m} \sqrt {1-a^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},a^2 x^2\right )}{(1+m) (2+m) \sqrt {c-a^2 c x^2}}-\frac {2 a c x^{2+m} \sqrt {1-a^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},a^2 x^2\right )}{(2+m) \sqrt {c-a^2 c x^2}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 0.30 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.75 \[ \int e^{2 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\frac {x^{1+m} \left (\frac {2 \sqrt {1-a x} \sqrt {-c (1+a x)} \operatorname {AppellF1}\left (1+m,\frac {1}{2},-\frac {1}{2},2+m,a x,-a x\right )}{\sqrt {-1+a x} \sqrt {1+a x}}+\frac {\sqrt {c-a^2 c x^2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},a^2 x^2\right )}{\sqrt {1-a^2 x^2}}\right )}{1+m} \]

[In]

Integrate[E^(2*ArcCoth[a*x])*x^m*Sqrt[c - a^2*c*x^2],x]

[Out]

(x^(1 + m)*((2*Sqrt[1 - a*x]*Sqrt[-(c*(1 + a*x))]*AppellF1[1 + m, 1/2, -1/2, 2 + m, a*x, -(a*x)])/(Sqrt[-1 + a
*x]*Sqrt[1 + a*x]) + (Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[-1/2, (1 + m)/2, (3 + m)/2, a^2*x^2])/Sqrt[1 - a^2
*x^2]))/(1 + m)

Maple [F]

\[\int \frac {\left (a x +1\right ) x^{m} \sqrt {-a^{2} c \,x^{2}+c}}{a x -1}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int e^{2 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x + 1\right )} x^{m}}{a x - 1} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int e^{2 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\int \frac {x^{m} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}{a x - 1}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int e^{2 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x + 1\right )} x^{m}}{a x - 1} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int e^{2 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\text {Exception raised: TypeError} \]

[In]

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

Mupad [F(-1)]

Timed out. \[ \int e^{2 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\int \frac {x^m\,\sqrt {c-a^2\,c\,x^2}\,\left (a\,x+1\right )}{a\,x-1} \,d x \]

[In]

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

[Out]

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