Integrand size = 27, antiderivative size = 149 \[ \int \frac {e^{\coth ^{-1}(a x)} (e x)^m}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\frac {a^3 \sqrt {1-\frac {1}{a^2 x^2}} (e x)^{4+m} \operatorname {Hypergeometric2F1}\left (2,\frac {4+m}{2},\frac {6+m}{2},a^2 x^2\right )}{c e^4 (4+m) \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {a^4 \sqrt {1-\frac {1}{a^2 x^2}} (e x)^{5+m} \operatorname {Hypergeometric2F1}\left (2,\frac {5+m}{2},\frac {7+m}{2},a^2 x^2\right )}{c e^5 (5+m) \sqrt {c-\frac {c}{a^2 x^2}}} \] Output:
a^3*(1-1/a^2/x^2)^(1/2)*(e*x)^(4+m)*hypergeom([2, 2+1/2*m],[3+1/2*m],a^2*x ^2)/c/e^4/(4+m)/(c-c/a^2/x^2)^(1/2)+a^4*(1-1/a^2/x^2)^(1/2)*(e*x)^(5+m)*hy pergeom([2, 5/2+1/2*m],[7/2+1/2*m],a^2*x^2)/c/e^5/(5+m)/(c-c/a^2/x^2)^(1/2 )
Time = 0.10 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.74 \[ \int \frac {e^{\coth ^{-1}(a x)} (e x)^m}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\frac {a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^4 (e x)^m \left ((5+m) \operatorname {Hypergeometric2F1}\left (2,2+\frac {m}{2},3+\frac {m}{2},a^2 x^2\right )+a (4+m) x \operatorname {Hypergeometric2F1}\left (2,\frac {5+m}{2},\frac {7+m}{2},a^2 x^2\right )\right )}{c (4+m) (5+m) \sqrt {c-\frac {c}{a^2 x^2}}} \] Input:
Integrate[(E^ArcCoth[a*x]*(e*x)^m)/(c - c/(a^2*x^2))^(3/2),x]
Output:
(a^3*Sqrt[1 - 1/(a^2*x^2)]*x^4*(e*x)^m*((5 + m)*Hypergeometric2F1[2, 2 + m /2, 3 + m/2, a^2*x^2] + a*(4 + m)*x*Hypergeometric2F1[2, (5 + m)/2, (7 + m )/2, a^2*x^2]))/(c*(4 + m)*(5 + m)*Sqrt[c - c/(a^2*x^2)])
Time = 0.97 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.80, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6751, 6747, 8, 92, 82, 278}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {e^{\coth ^{-1}(a x)} (e x)^m}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 6751 |
\(\displaystyle \frac {\sqrt {1-\frac {1}{a^2 x^2}} \int \frac {e^{\coth ^{-1}(a x)} (e x)^m}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}dx}{c \sqrt {c-\frac {c}{a^2 x^2}}}\) |
\(\Big \downarrow \) 6747 |
\(\displaystyle \frac {a^3 \sqrt {1-\frac {1}{a^2 x^2}} \int \frac {x^3 (e x)^m}{(1-a x)^2 (a x+1)}dx}{c \sqrt {c-\frac {c}{a^2 x^2}}}\) |
\(\Big \downarrow \) 8 |
\(\displaystyle \frac {a^3 \sqrt {1-\frac {1}{a^2 x^2}} \int \frac {(e x)^{m+3}}{(1-a x)^2 (a x+1)}dx}{c e^3 \sqrt {c-\frac {c}{a^2 x^2}}}\) |
\(\Big \downarrow \) 92 |
\(\displaystyle \frac {a^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (\int \frac {(e x)^{m+3}}{(1-a x)^2 (a x+1)^2}dx+\frac {a \int \frac {(e x)^{m+4}}{(1-a x)^2 (a x+1)^2}dx}{e}\right )}{c e^3 \sqrt {c-\frac {c}{a^2 x^2}}}\) |
\(\Big \downarrow \) 82 |
\(\displaystyle \frac {a^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (\int \frac {(e x)^{m+3}}{\left (1-a^2 x^2\right )^2}dx+\frac {a \int \frac {(e x)^{m+4}}{\left (1-a^2 x^2\right )^2}dx}{e}\right )}{c e^3 \sqrt {c-\frac {c}{a^2 x^2}}}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \frac {a^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (\frac {a (e x)^{m+5} \operatorname {Hypergeometric2F1}\left (2,\frac {m+5}{2},\frac {m+7}{2},a^2 x^2\right )}{e^2 (m+5)}+\frac {(e x)^{m+4} \operatorname {Hypergeometric2F1}\left (2,\frac {m+4}{2},\frac {m+6}{2},a^2 x^2\right )}{e (m+4)}\right )}{c e^3 \sqrt {c-\frac {c}{a^2 x^2}}}\) |
Input:
Int[(E^ArcCoth[a*x]*(e*x)^m)/(c - c/(a^2*x^2))^(3/2),x]
Output:
(a^3*Sqrt[1 - 1/(a^2*x^2)]*(((e*x)^(4 + m)*Hypergeometric2F1[2, (4 + m)/2, (6 + m)/2, a^2*x^2])/(e*(4 + m)) + (a*(e*x)^(5 + m)*Hypergeometric2F1[2, (5 + m)/2, (7 + m)/2, a^2*x^2])/(e^2*(5 + m))))/(c*e^3*Sqrt[c - c/(a^2*x^2 )])
Int[(u_.)*(x_)^(m_.)*((a_.)*(x_))^(p_), x_Symbol] :> Simp[1/a^m Int[u*(a* x)^(m + p), x], x] /; FreeQ[{a, m, p}, x] && IntegerQ[m]
Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_) )^(p_.), x_] :> Int[(a*c + b*d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && IntegerQ[m]
Int[((f_.)*(x_))^(p_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_] :> Simp[a Int[(a + b*x)^n*(c + d*x)^n*(f*x)^p, x], x] + Simp[b/f In t[(a + b*x)^n*(c + d*x)^n*(f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[m - n - 1, 0] && !RationalQ[p] && !IGtQ[m, 0] && NeQ[m + n + p + 2, 0]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symb ol] :> Simp[c^p/a^(2*p) Int[(u/x^(2*p))*(-1 + a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && !Inte gerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) && IntegersQ[2*p, p + n/2]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbo l] :> Simp[c^IntPart[p]*((c + d/x^2)^FracPart[p]/(1 - 1/(a^2*x^2))^FracPart [p]) Int[u*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[n/2] && !(IntegerQ[p] || GtQ[c, 0])
\[\int \frac {\left (e x \right )^{m}}{\sqrt {\frac {a x -1}{a x +1}}\, \left (c -\frac {c}{a^{2} x^{2}}\right )^{\frac {3}{2}}}d x\]
Input:
int(1/((a*x-1)/(a*x+1))^(1/2)*(e*x)^m/(c-c/a^2/x^2)^(3/2),x)
Output:
int(1/((a*x-1)/(a*x+1))^(1/2)*(e*x)^m/(c-c/a^2/x^2)^(3/2),x)
\[ \int \frac {e^{\coth ^{-1}(a x)} (e x)^m}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\int { \frac {\left (e x\right )^{m}}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {3}{2}} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)*(e*x)^m/(c-c/a^2/x^2)^(3/2),x, algorit hm="fricas")
Output:
integral((e*x)^m*a^4*x^4*sqrt((a*x - 1)/(a*x + 1))*sqrt((a^2*c*x^2 - c)/(a ^2*x^2))/(a^4*c^2*x^4 - 2*a^3*c^2*x^3 + 2*a*c^2*x - c^2), x)
Timed out. \[ \int \frac {e^{\coth ^{-1}(a x)} (e x)^m}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/((a*x-1)/(a*x+1))**(1/2)*(e*x)**m/(c-c/a**2/x**2)**(3/2),x)
Output:
Timed out
\[ \int \frac {e^{\coth ^{-1}(a x)} (e x)^m}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\int { \frac {\left (e x\right )^{m}}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {3}{2}} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)*(e*x)^m/(c-c/a^2/x^2)^(3/2),x, algorit hm="maxima")
Output:
integrate((e*x)^m/((c - c/(a^2*x^2))^(3/2)*sqrt((a*x - 1)/(a*x + 1))), x)
Exception generated. \[ \int \frac {e^{\coth ^{-1}(a x)} (e x)^m}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)*(e*x)^m/(c-c/a^2/x^2)^(3/2),x, algorit hm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {e^{\coth ^{-1}(a x)} (e x)^m}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\int \frac {{\left (e\,x\right )}^m}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{3/2}\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \] Input:
int((e*x)^m/((c - c/(a^2*x^2))^(3/2)*((a*x - 1)/(a*x + 1))^(1/2)),x)
Output:
int((e*x)^m/((c - c/(a^2*x^2))^(3/2)*((a*x - 1)/(a*x + 1))^(1/2)), x)
\[ \int \frac {e^{\coth ^{-1}(a x)} (e x)^m}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\frac {e^{m} \sqrt {c}\, \left (x^{m} a m x +x^{m} m +x^{m}-\left (\int \frac {x^{m}}{a^{3} x^{4}-a^{2} x^{3}-a \,x^{2}+x}d x \right ) m^{2}-\left (\int \frac {x^{m}}{a^{3} x^{4}-a^{2} x^{3}-a \,x^{2}+x}d x \right ) m +2 \left (\int \frac {x^{m} x}{a^{3} x^{3}-a^{2} x^{2}-a x +1}d x \right ) a^{2} m^{2}+2 \left (\int \frac {x^{m} x}{a^{3} x^{3}-a^{2} x^{2}-a x +1}d x \right ) a^{2} m \right )}{a \,c^{2} m \left (m +1\right )} \] Input:
int(1/((a*x-1)/(a*x+1))^(1/2)*(e*x)^m/(c-c/a^2/x^2)^(3/2),x)
Output:
(e**m*sqrt(c)*(x**m*a*m*x + x**m*m + x**m - int(x**m/(a**3*x**4 - a**2*x** 3 - a*x**2 + x),x)*m**2 - int(x**m/(a**3*x**4 - a**2*x**3 - a*x**2 + x),x) *m + 2*int((x**m*x)/(a**3*x**3 - a**2*x**2 - a*x + 1),x)*a**2*m**2 + 2*int ((x**m*x)/(a**3*x**3 - a**2*x**2 - a*x + 1),x)*a**2*m))/(a*c**2*m*(m + 1))