Integrand size = 18, antiderivative size = 65 \[ \int e^{-5 \text {arctanh}(a x)} (c-a c x)^p \, dx=-\frac {2^{\frac {7}{2}+p} (1-a x)^{-p} (c-a c x)^p \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {5}{2}-p,-\frac {1}{2},\frac {1}{2} (1+a x)\right )}{3 a (1+a x)^{3/2}} \] Output:
-1/3*2^(7/2+p)*(-a*c*x+c)^p*hypergeom([-3/2, -5/2-p],[-1/2],1/2*a*x+1/2)/a /((-a*x+1)^p)/(a*x+1)^(3/2)
Time = 0.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.95 \[ \int e^{-5 \text {arctanh}(a x)} (c-a c x)^p \, dx=-\frac {(1-a x)^{7/2} (c-a c x)^p \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {7}{2}+p,\frac {9}{2}+p,\frac {1}{2}-\frac {a x}{2}\right )}{2 \sqrt {2} a (7+2 p)} \] Input:
Integrate[(c - a*c*x)^p/E^(5*ArcTanh[a*x]),x]
Output:
-1/2*((1 - a*x)^(7/2)*(c - a*c*x)^p*Hypergeometric2F1[5/2, 7/2 + p, 9/2 + p, 1/2 - (a*x)/2])/(Sqrt[2]*a*(7 + 2*p))
Time = 0.38 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6680, 37, 79}
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.
\(\displaystyle \int e^{-5 \text {arctanh}(a x)} (c-a c x)^p \, dx\) |
\(\Big \downarrow \) 6680 |
\(\displaystyle \int \frac {(1-a x)^{5/2} (c-a c x)^p}{(a x+1)^{5/2}}dx\) |
\(\Big \downarrow \) 37 |
\(\displaystyle (1-a x)^{-p} (c-a c x)^p \int \frac {(1-a x)^{p+\frac {5}{2}}}{(a x+1)^{5/2}}dx\) |
\(\Big \downarrow \) 79 |
\(\displaystyle -\frac {2^{p+\frac {7}{2}} (1-a x)^{-p} (c-a c x)^p \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-p-\frac {5}{2},-\frac {1}{2},\frac {1}{2} (a x+1)\right )}{3 a (a x+1)^{3/2}}\) |
Input:
Int[(c - a*c*x)^p/E^(5*ArcTanh[a*x]),x]
Output:
-1/3*(2^(7/2 + p)*(c - a*c*x)^p*Hypergeometric2F1[-3/2, -5/2 - p, -1/2, (1 + a*x)/2])/(a*(1 - a*x)^p*(1 + a*x)^(3/2))
Int[(u_.)*((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> S imp[(a + b*x)^m/(c + d*x)^m Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] && !SimplerQ[a + b*x, c + d*x]
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] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol ] :> Int[u*(c + d*x)^p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c , d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && !(IntegerQ[p] || GtQ[c, 0])
\[\int \frac {\left (-a c x +c \right )^{p} \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{\left (a x +1\right )^{5}}d x\]
Input:
int((-a*c*x+c)^p/(a*x+1)^5*(-a^2*x^2+1)^(5/2),x)
Output:
int((-a*c*x+c)^p/(a*x+1)^5*(-a^2*x^2+1)^(5/2),x)
\[ \int e^{-5 \text {arctanh}(a x)} (c-a c x)^p \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} {\left (-a c x + c\right )}^{p}}{{\left (a x + 1\right )}^{5}} \,d x } \] Input:
integrate((-a*c*x+c)^p/(a*x+1)^5*(-a^2*x^2+1)^(5/2),x, algorithm="fricas")
Output:
integral((a^2*x^2 - 2*a*x + 1)*sqrt(-a^2*x^2 + 1)*(-a*c*x + c)^p/(a^3*x^3 + 3*a^2*x^2 + 3*a*x + 1), x)
\[ \int e^{-5 \text {arctanh}(a x)} (c-a c x)^p \, dx=\int \frac {\left (- c \left (a x - 1\right )\right )^{p} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}{\left (a x + 1\right )^{5}}\, dx \] Input:
integrate((-a*c*x+c)**p/(a*x+1)**5*(-a**2*x**2+1)**(5/2),x)
Output:
Integral((-c*(a*x - 1))**p*(-(a*x - 1)*(a*x + 1))**(5/2)/(a*x + 1)**5, x)
\[ \int e^{-5 \text {arctanh}(a x)} (c-a c x)^p \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} {\left (-a c x + c\right )}^{p}}{{\left (a x + 1\right )}^{5}} \,d x } \] Input:
integrate((-a*c*x+c)^p/(a*x+1)^5*(-a^2*x^2+1)^(5/2),x, algorithm="maxima")
Output:
integrate((-a^2*x^2 + 1)^(5/2)*(-a*c*x + c)^p/(a*x + 1)^5, x)
Exception generated. \[ \int e^{-5 \text {arctanh}(a x)} (c-a c x)^p \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-a*c*x+c)^p/(a*x+1)^5*(-a^2*x^2+1)^(5/2),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. \[ \int e^{-5 \text {arctanh}(a x)} (c-a c x)^p \, dx=\int \frac {{\left (1-a^2\,x^2\right )}^{5/2}\,{\left (c-a\,c\,x\right )}^p}{{\left (a\,x+1\right )}^5} \,d x \] Input:
int(((1 - a^2*x^2)^(5/2)*(c - a*c*x)^p)/(a*x + 1)^5,x)
Output:
int(((1 - a^2*x^2)^(5/2)*(c - a*c*x)^p)/(a*x + 1)^5, x)
\[ \int e^{-5 \text {arctanh}(a x)} (c-a c x)^p \, dx=\left (\int \frac {\left (-a c x +c \right )^{p} \sqrt {-a^{2} x^{2}+1}\, x^{2}}{a^{3} x^{3}+3 a^{2} x^{2}+3 a x +1}d x \right ) a^{2}-2 \left (\int \frac {\left (-a c x +c \right )^{p} \sqrt {-a^{2} x^{2}+1}\, x}{a^{3} x^{3}+3 a^{2} x^{2}+3 a x +1}d x \right ) a +\int \frac {\left (-a c x +c \right )^{p} \sqrt {-a^{2} x^{2}+1}}{a^{3} x^{3}+3 a^{2} x^{2}+3 a x +1}d x \] Input:
int((-a*c*x+c)^p/(a*x+1)^5*(-a^2*x^2+1)^(5/2),x)
Output:
int((( - a*c*x + c)**p*sqrt( - a**2*x**2 + 1)*x**2)/(a**3*x**3 + 3*a**2*x* *2 + 3*a*x + 1),x)*a**2 - 2*int((( - a*c*x + c)**p*sqrt( - a**2*x**2 + 1)* x)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1),x)*a + int((( - a*c*x + c)**p*sqr t( - a**2*x**2 + 1))/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1),x)