\(\int e^{n \text {arctanh}(a x)} (c-a c x)^{3/2} \, dx\) [287]

Optimal result

Integrand size = 20, antiderivative size = 83 \[ \int e^{n \text {arctanh}(a x)} (c-a c x)^{3/2} \, dx=-\frac {2^{1+\frac {n}{2}} c (1-a x)^{\frac {4-n}{2}} \sqrt {c-a c x} \operatorname {Hypergeometric2F1}\left (\frac {5-n}{2},-\frac {n}{2},\frac {7-n}{2},\frac {1}{2} (1-a x)\right )}{a (5-n)} \] Output:

-2^(1+1/2*n)*c*(-a*x+1)^(2-1/2*n)*(-a*c*x+c)^(1/2)*hypergeom([-1/2*n, 5/2- 
1/2*n],[7/2-1/2*n],-1/2*a*x+1/2)/a/(5-n)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.94 \[ \int e^{n \text {arctanh}(a x)} (c-a c x)^{3/2} \, dx=\frac {2^{1+\frac {n}{2}} c (1-a x)^{2-\frac {n}{2}} \sqrt {c-a c x} \operatorname {Hypergeometric2F1}\left (\frac {5}{2}-\frac {n}{2},-\frac {n}{2},\frac {7}{2}-\frac {n}{2},\frac {1}{2}-\frac {a x}{2}\right )}{a (-5+n)} \] Input:

Integrate[E^(n*ArcTanh[a*x])*(c - a*c*x)^(3/2),x]
 

Output:

(2^(1 + n/2)*c*(1 - a*x)^(2 - n/2)*Sqrt[c - a*c*x]*Hypergeometric2F1[5/2 - 
 n/2, -1/2*n, 7/2 - n/2, 1/2 - (a*x)/2])/(a*(-5 + n))
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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.

\(\displaystyle \int (c-a c x)^{3/2} e^{n \text {arctanh}(a x)} \, dx\)

\(\Big \downarrow \) 6680

\(\displaystyle \int (c-a c x)^{3/2} (1-a x)^{-n/2} (a x+1)^{n/2}dx\)

\(\Big \downarrow \) 37

\(\displaystyle \frac {(c-a c x)^{3/2} \int (1-a x)^{\frac {3-n}{2}} (a x+1)^{n/2}dx}{(1-a x)^{3/2}}\)

\(\Big \downarrow \) 79

\(\displaystyle -\frac {2^{\frac {n}{2}+1} (c-a c x)^{3/2} (1-a x)^{\frac {5-n}{2}-\frac {3}{2}} \operatorname {Hypergeometric2F1}\left (\frac {5-n}{2},-\frac {n}{2},\frac {7-n}{2},\frac {1}{2} (1-a x)\right )}{a (5-n)}\)

Input:

Int[E^(n*ArcTanh[a*x])*(c - a*c*x)^(3/2),x]
 

Output:

-((2^(1 + n/2)*(1 - a*x)^(-3/2 + (5 - n)/2)*(c - a*c*x)^(3/2)*Hypergeometr 
ic2F1[(5 - n)/2, -1/2*n, (7 - n)/2, (1 - a*x)/2])/(a*(5 - n)))
 

Defintions of rubi rules used

rule 37

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]
 

rule 79

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]))
 

rule 6680

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])
 

Maple [F]

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

Input:

int(exp(n*arctanh(a*x))*(-a*c*x+c)^(3/2),x)
 

Output:

int(exp(n*arctanh(a*x))*(-a*c*x+c)^(3/2),x)
 

Fricas [F]

\[ \int e^{n \text {arctanh}(a x)} (c-a c x)^{3/2} \, dx=\int { {\left (-a c x + c\right )}^{\frac {3}{2}} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \] Input:

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

Output:

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

Sympy [F]

\[ \int e^{n \text {arctanh}(a x)} (c-a c x)^{3/2} \, dx=\int \left (- c \left (a x - 1\right )\right )^{\frac {3}{2}} e^{n \operatorname {atanh}{\left (a x \right )}}\, dx \] Input:

integrate(exp(n*atanh(a*x))*(-a*c*x+c)**(3/2),x)
 

Output:

Integral((-c*(a*x - 1))**(3/2)*exp(n*atanh(a*x)), x)
 

Maxima [F]

\[ \int e^{n \text {arctanh}(a x)} (c-a c x)^{3/2} \, dx=\int { {\left (-a c x + c\right )}^{\frac {3}{2}} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \] Input:

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

Output:

integrate((-a*c*x + c)^(3/2)*(-(a*x + 1)/(a*x - 1))^(1/2*n), x)
 

Giac [F]

\[ \int e^{n \text {arctanh}(a x)} (c-a c x)^{3/2} \, dx=\int { {\left (-a c x + c\right )}^{\frac {3}{2}} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \] Input:

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

Output:

integrate((-a*c*x + c)^(3/2)*(-(a*x + 1)/(a*x - 1))^(1/2*n), x)
 

Mupad [F(-1)]

Timed out. \[ \int e^{n \text {arctanh}(a x)} (c-a c x)^{3/2} \, dx=\int {\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,{\left (c-a\,c\,x\right )}^{3/2} \,d x \] Input:

int(exp(n*atanh(a*x))*(c - a*c*x)^(3/2),x)
 

Output:

int(exp(n*atanh(a*x))*(c - a*c*x)^(3/2), x)
 

Reduce [F]

\[ \int e^{n \text {arctanh}(a x)} (c-a c x)^{3/2} \, dx=\sqrt {c}\, c \left (-\left (\int e^{\mathit {atanh} \left (a x \right ) n} \sqrt {-a x +1}\, x d x \right ) a +\int e^{\mathit {atanh} \left (a x \right ) n} \sqrt {-a x +1}d x \right ) \] Input:

int(exp(n*atanh(a*x))*(-a*c*x+c)^(3/2),x)
 

Output:

sqrt(c)*c*( - int(e**(atanh(a*x)*n)*sqrt( - a*x + 1)*x,x)*a + int(e**(atan 
h(a*x)*n)*sqrt( - a*x + 1),x))