Integrand size = 23, antiderivative size = 134 \[ \int e^{\coth ^{-1}(a x)} (e x)^m (c-a c x)^{3/2} \, dx=\frac {2 \left (1+\frac {1}{a x}\right )^{3/2} x (e x)^m (c-a c x)^{3/2}}{(5+2 m) \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {2 (7+4 m) (e x)^m (c-a c x)^{3/2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {3}{2}-m,-\frac {1}{2}-m,-\frac {1}{a x}\right )}{a (3+2 m) (5+2 m) \left (1-\frac {1}{a x}\right )^{3/2}} \] Output:
2*(1+1/a/x)^(3/2)*x*(e*x)^m*(-a*c*x+c)^(3/2)/(5+2*m)/(1-1/a/x)^(3/2)-2*(7+ 4*m)*(e*x)^m*(-a*c*x+c)^(3/2)*hypergeom([-1/2, -3/2-m],[-1/2-m],-1/a/x)/a/ (3+2*m)/(5+2*m)/(1-1/a/x)^(3/2)
Time = 0.12 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.79 \[ \int e^{\coth ^{-1}(a x)} (e x)^m (c-a c x)^{3/2} \, dx=-\frac {2 c x (e x)^m \sqrt {c-a c x} \left ((3+2 m) \sqrt {1+\frac {1}{a x}} (1+a x)-(7+4 m) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {3}{2}-m,-\frac {1}{2}-m,-\frac {1}{a x}\right )\right )}{(3+2 m) (5+2 m) \sqrt {1-\frac {1}{a x}}} \] Input:
Integrate[E^ArcCoth[a*x]*(e*x)^m*(c - a*c*x)^(3/2),x]
Output:
(-2*c*x*(e*x)^m*Sqrt[c - a*c*x]*((3 + 2*m)*Sqrt[1 + 1/(a*x)]*(1 + a*x) - ( 7 + 4*m)*Hypergeometric2F1[-1/2, -3/2 - m, -1/2 - m, -(1/(a*x))]))/((3 + 2 *m)*(5 + 2*m)*Sqrt[1 - 1/(a*x)])
Time = 0.59 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6730, 27, 88, 74}
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.
\(\displaystyle \int (c-a c x)^{3/2} e^{\coth ^{-1}(a x)} (e x)^m \, dx\) |
\(\Big \downarrow \) 6730 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{m+\frac {3}{2}} (c-a c x)^{3/2} (e x)^m \int \frac {\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{-m-\frac {7}{2}}}{a}d\frac {1}{x}}{\left (1-\frac {1}{a x}\right )^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{m+\frac {3}{2}} (c-a c x)^{3/2} (e x)^m \int \left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{-m-\frac {7}{2}}d\frac {1}{x}}{a \left (1-\frac {1}{a x}\right )^{3/2}}\) |
\(\Big \downarrow \) 88 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{m+\frac {3}{2}} (c-a c x)^{3/2} (e x)^m \left (-\frac {(4 m+7) \int \sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{-m-\frac {5}{2}}d\frac {1}{x}}{2 m+5}-\frac {2 a \left (\frac {1}{a x}+1\right )^{3/2} \left (\frac {1}{x}\right )^{-m-\frac {5}{2}}}{2 m+5}\right )}{a \left (1-\frac {1}{a x}\right )^{3/2}}\) |
\(\Big \downarrow \) 74 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{m+\frac {3}{2}} (c-a c x)^{3/2} (e x)^m \left (\frac {2 (4 m+7) \left (\frac {1}{x}\right )^{-m-\frac {3}{2}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-m-\frac {3}{2},-m-\frac {1}{2},-\frac {1}{a x}\right )}{(2 m+3) (2 m+5)}-\frac {2 a \left (\frac {1}{a x}+1\right )^{3/2} \left (\frac {1}{x}\right )^{-m-\frac {5}{2}}}{2 m+5}\right )}{a \left (1-\frac {1}{a x}\right )^{3/2}}\) |
Input:
Int[E^ArcCoth[a*x]*(e*x)^m*(c - a*c*x)^(3/2),x]
Output:
-(((x^(-1))^(3/2 + m)*(e*x)^m*(c - a*c*x)^(3/2)*((-2*a*(1 + 1/(a*x))^(3/2) *(x^(-1))^(-5/2 - m))/(5 + 2*m) + (2*(7 + 4*m)*(x^(-1))^(-3/2 - m)*Hyperge ometric2F1[-1/2, -3/2 - m, -1/2 - m, -(1/(a*x))])/((3 + 2*m)*(5 + 2*m))))/ (a*(1 - 1/(a*x))^(3/2)))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x )^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] && !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && !RationalQ[p] && SumSimpl erQ[p, 1]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(p _), x_Symbol] :> Simp[(-(e*x)^m)*(1/x)^(m + p)*((c + d*x)^p/(1 + c/(d*x))^p ) Subst[Int[((1 + c*(x/d))^p*((1 + x/a)^(n/2)/x^(m + p + 2)))/(1 - x/a)^( n/2), x], x, 1/x], x] /; FreeQ[{a, c, d, e, m, n, p}, x] && EqQ[a^2*c^2 - d ^2, 0] && !IntegerQ[p]
\[\int \frac {\left (e x \right )^{m} \left (-a c x +c \right )^{\frac {3}{2}}}{\sqrt {\frac {a x -1}{a x +1}}}d x\]
Input:
int(1/((a*x-1)/(a*x+1))^(1/2)*(e*x)^m*(-a*c*x+c)^(3/2),x)
Output:
int(1/((a*x-1)/(a*x+1))^(1/2)*(e*x)^m*(-a*c*x+c)^(3/2),x)
\[ \int e^{\coth ^{-1}(a x)} (e x)^m (c-a c x)^{3/2} \, dx=\int { \frac {{\left (-a c x + c\right )}^{\frac {3}{2}} \left (e x\right )^{m}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)*(e*x)^m*(-a*c*x+c)^(3/2),x, algorithm= "fricas")
Output:
integral(-(a*c*x + c)*sqrt(-a*c*x + c)*(e*x)^m*sqrt((a*x - 1)/(a*x + 1)), x)
Timed out. \[ \int e^{\coth ^{-1}(a x)} (e x)^m (c-a c x)^{3/2} \, dx=\text {Timed out} \] Input:
integrate(1/((a*x-1)/(a*x+1))**(1/2)*(e*x)**m*(-a*c*x+c)**(3/2),x)
Output:
Timed out
\[ \int e^{\coth ^{-1}(a x)} (e x)^m (c-a c x)^{3/2} \, dx=\int { \frac {{\left (-a c x + c\right )}^{\frac {3}{2}} \left (e x\right )^{m}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)*(e*x)^m*(-a*c*x+c)^(3/2),x, algorithm= "maxima")
Output:
integrate((-a*c*x + c)^(3/2)*(e*x)^m/sqrt((a*x - 1)/(a*x + 1)), x)
Exception generated. \[ \int e^{\coth ^{-1}(a x)} (e x)^m (c-a c x)^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)*(e*x)^m*(-a*c*x+c)^(3/2),x, algorithm= "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. \[ \int e^{\coth ^{-1}(a x)} (e x)^m (c-a c x)^{3/2} \, dx=\int \frac {{\left (e\,x\right )}^m\,{\left (c-a\,c\,x\right )}^{3/2}}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \] Input:
int(((e*x)^m*(c - a*c*x)^(3/2))/((a*x - 1)/(a*x + 1))^(1/2),x)
Output:
int(((e*x)^m*(c - a*c*x)^(3/2))/((a*x - 1)/(a*x + 1))^(1/2), x)
\[ \int e^{\coth ^{-1}(a x)} (e x)^m (c-a c x)^{3/2} \, dx=e^{m} \sqrt {c}\, c \left (-\left (\int \frac {x^{m} \sqrt {a x +1}\, \sqrt {-a x +1}\, x}{\sqrt {a x -1}}d x \right ) a +\int \frac {x^{m} \sqrt {a x +1}\, \sqrt {-a x +1}}{\sqrt {a x -1}}d x \right ) \] Input:
int(1/((a*x-1)/(a*x+1))^(1/2)*(e*x)^m*(-a*c*x+c)^(3/2),x)
Output:
e**m*sqrt(c)*c*( - int((x**m*sqrt(a*x + 1)*sqrt( - a*x + 1)*x)/sqrt(a*x - 1),x)*a + int((x**m*sqrt(a*x + 1)*sqrt( - a*x + 1))/sqrt(a*x - 1),x))