\(\int \frac {e^{n \arctan (a x)} x^m}{(c+a^2 c x^2)^{3/2}} \, dx\) [369]

Optimal result

Integrand size = 26, antiderivative size = 82 \[ \int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {x^{1+m} \sqrt {1+a^2 x^2} \operatorname {AppellF1}\left (1+m,\frac {1}{2} (3-i n),\frac {1}{2} (3+i n),2+m,i a x,-i a x\right )}{c (1+m) \sqrt {c+a^2 c x^2}} \]

[Out]

x^(1+m)*AppellF1(1+m,3/2+1/2*I*n,3/2-1/2*I*n,2+m,-I*a*x,I*a*x)*(a^2*x^2+1)^(1/2)/c/(1+m)/(a^2*c*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {5193, 5190, 138} \[ \int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {a^2 x^2+1} x^{m+1} \operatorname {AppellF1}\left (m+1,\frac {1}{2} (3-i n),\frac {1}{2} (i n+3),m+2,i a x,-i a x\right )}{c (m+1) \sqrt {a^2 c x^2+c}} \]

[In]

Int[(E^(n*ArcTan[a*x])*x^m)/(c + a^2*c*x^2)^(3/2),x]

[Out]

(x^(1 + m)*Sqrt[1 + a^2*x^2]*AppellF1[1 + m, (3 - I*n)/2, (3 + I*n)/2, 2 + m, I*a*x, (-I)*a*x])/(c*(1 + m)*Sqr
t[c + a^2*c*x^2])

Rule 138

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[c^n*e^p*((b*x)^(m +
1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p},
x] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])

Rule 5190

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

Rule 5193

Int[E^(ArcTan[(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*ArcTan[a*x]), x], x] /; FreeQ[{a, c
, d, m, n, p}, x] && EqQ[d, a^2*c] &&  !(IntegerQ[p] || GtQ[c, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+a^2 x^2} \int \frac {e^{n \arctan (a x)} x^m}{\left (1+a^2 x^2\right )^{3/2}} \, dx}{c \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {1+a^2 x^2} \int x^m (1-i a x)^{-\frac {3}{2}+\frac {i n}{2}} (1+i a x)^{-\frac {3}{2}-\frac {i n}{2}} \, dx}{c \sqrt {c+a^2 c x^2}} \\ & = \frac {x^{1+m} \sqrt {1+a^2 x^2} \operatorname {AppellF1}\left (1+m,\frac {1}{2} (3-i n),\frac {1}{2} (3+i n),2+m,i a x,-i a x\right )}{c (1+m) \sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [F]

\[ \int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^{3/2}} \, dx \]

[In]

Integrate[(E^(n*ArcTan[a*x])*x^m)/(c + a^2*c*x^2)^(3/2),x]

[Out]

Integrate[(E^(n*ArcTan[a*x])*x^m)/(c + a^2*c*x^2)^(3/2), x]

Maple [F]

\[\int \frac {{\mathrm e}^{n \arctan \left (a x \right )} x^{m}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}d x\]

[In]

int(exp(n*arctan(a*x))*x^m/(a^2*c*x^2+c)^(3/2),x)

[Out]

int(exp(n*arctan(a*x))*x^m/(a^2*c*x^2+c)^(3/2),x)

Fricas [F]

\[ \int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {x^{m} e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(exp(n*arctan(a*x))*x^m/(a^2*c*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(a^2*c*x^2 + c)*x^m*e^(n*arctan(a*x))/(a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2), x)

Sympy [F]

\[ \int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x^{m} e^{n \operatorname {atan}{\left (a x \right )}}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(exp(n*atan(a*x))*x**m/(a**2*c*x**2+c)**(3/2),x)

[Out]

Integral(x**m*exp(n*atan(a*x))/(c*(a**2*x**2 + 1))**(3/2), x)

Maxima [F]

\[ \int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {x^{m} e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(exp(n*arctan(a*x))*x^m/(a^2*c*x^2+c)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^m*e^(n*arctan(a*x))/(a^2*c*x^2 + c)^(3/2), x)

Giac [F]

\[ \int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {x^{m} e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(exp(n*arctan(a*x))*x^m/(a^2*c*x^2+c)^(3/2),x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x^m\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]

[In]

int((x^m*exp(n*atan(a*x)))/(c + a^2*c*x^2)^(3/2),x)

[Out]

int((x^m*exp(n*atan(a*x)))/(c + a^2*c*x^2)^(3/2), x)