\(\int e^{-\coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx\) [254]

Optimal result

Integrand size = 20, antiderivative size = 128 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\frac {256 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x}{35 \sqrt {c-a c x}}+\frac {64}{35} c^2 \sqrt {1-\frac {1}{a^2 x^2}} x \sqrt {c-a c x}+\frac {24}{35} c \sqrt {1-\frac {1}{a^2 x^2}} x (c-a c x)^{3/2}+\frac {2}{7} \sqrt {1-\frac {1}{a^2 x^2}} x (c-a c x)^{5/2} \]

[Out]

24/35*c*x*(-a*c*x+c)^(3/2)*(1-1/a^2/x^2)^(1/2)+2/7*x*(-a*c*x+c)^(5/2)*(1-1/a^2/x^2)^(1/2)+256/35*c^3*x*(1-1/a^
2/x^2)^(1/2)/(-a*c*x+c)^(1/2)+64/35*c^2*x*(1-1/a^2/x^2)^(1/2)*(-a*c*x+c)^(1/2)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.54, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6311, 6316, 96, 91, 79, 37} \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=-\frac {344 \sqrt {\frac {1}{a x}+1} (c-a c x)^{5/2}}{35 a^3 x^2 \left (1-\frac {1}{a x}\right )^{5/2}}+\frac {2 x \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^3 (c-a c x)^{5/2}}{7 a^3 \left (1-\frac {1}{a x}\right )^{5/2}}+\frac {16 \sqrt {\frac {1}{a x}+1} (c-a c x)^{5/2}}{5 a^2 x \left (1-\frac {1}{a x}\right )^{5/2}}-\frac {24 \sqrt {\frac {1}{a x}+1} (c-a c x)^{5/2}}{35 a \left (1-\frac {1}{a x}\right )^{5/2}} \]

[In]

Int[(c - a*c*x)^(5/2)/E^ArcCoth[a*x],x]

[Out]

(-24*Sqrt[1 + 1/(a*x)]*(c - a*c*x)^(5/2))/(35*a*(1 - 1/(a*x))^(5/2)) - (344*Sqrt[1 + 1/(a*x)]*(c - a*c*x)^(5/2
))/(35*a^3*(1 - 1/(a*x))^(5/2)*x^2) + (16*Sqrt[1 + 1/(a*x)]*(c - a*c*x)^(5/2))/(5*a^2*(1 - 1/(a*x))^(5/2)*x) +
(2*(a - x^(-1))^3*Sqrt[1 + 1/(a*x)]*x*(c - a*c*x)^(5/2))/(7*a^3*(1 - 1/(a*x))^(5/2))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(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)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 91

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c - a*d
)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d*e - c*f)*(n + 1))), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 96

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

Rule 6311

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_), x_Symbol] :> Dist[(c + d*x)^p/(x^p*(1 + c/(d
*x))^p), Int[u*x^p*(1 + c/(d*x))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^
2, 0] &&  !IntegerQ[n/2] &&  !IntegerQ[p]

Rule 6316

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

Rubi steps \begin{align*} \text {integral}& = \frac {(c-a c x)^{5/2} \int e^{-\coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{5/2} x^{5/2} \, dx}{\left (1-\frac {1}{a x}\right )^{5/2} x^{5/2}} \\ & = -\frac {\left (\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^3}{x^{9/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{5/2}} \\ & = \frac {2 \left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}} x (c-a c x)^{5/2}}{7 a^3 \left (1-\frac {1}{a x}\right )^{5/2}}+\frac {\left (12 \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2}{x^{7/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{7 a \left (1-\frac {1}{a x}\right )^{5/2}} \\ & = -\frac {24 \sqrt {1+\frac {1}{a x}} (c-a c x)^{5/2}}{35 a \left (1-\frac {1}{a x}\right )^{5/2}}+\frac {2 \left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}} x (c-a c x)^{5/2}}{7 a^3 \left (1-\frac {1}{a x}\right )^{5/2}}+\frac {\left (24 \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}\right ) \text {Subst}\left (\int \frac {-\frac {7}{a}+\frac {5 x}{2 a^2}}{x^{5/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{35 a \left (1-\frac {1}{a x}\right )^{5/2}} \\ & = -\frac {24 \sqrt {1+\frac {1}{a x}} (c-a c x)^{5/2}}{35 a \left (1-\frac {1}{a x}\right )^{5/2}}+\frac {16 \sqrt {1+\frac {1}{a x}} (c-a c x)^{5/2}}{5 a^2 \left (1-\frac {1}{a x}\right )^{5/2} x}+\frac {2 \left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}} x (c-a c x)^{5/2}}{7 a^3 \left (1-\frac {1}{a x}\right )^{5/2}}+\frac {\left (172 \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}\right ) \text {Subst}\left (\int \frac {1}{x^{3/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{35 a^3 \left (1-\frac {1}{a x}\right )^{5/2}} \\ & = -\frac {24 \sqrt {1+\frac {1}{a x}} (c-a c x)^{5/2}}{35 a \left (1-\frac {1}{a x}\right )^{5/2}}-\frac {344 \sqrt {1+\frac {1}{a x}} (c-a c x)^{5/2}}{35 a^3 \left (1-\frac {1}{a x}\right )^{5/2} x^2}+\frac {16 \sqrt {1+\frac {1}{a x}} (c-a c x)^{5/2}}{5 a^2 \left (1-\frac {1}{a x}\right )^{5/2} x}+\frac {2 \left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}} x (c-a c x)^{5/2}}{7 a^3 \left (1-\frac {1}{a x}\right )^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.55 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\frac {2 c^2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x} \left (-177+71 a x-27 a^2 x^2+5 a^3 x^3\right )}{35 a \sqrt {1-\frac {1}{a x}}} \]

[In]

Integrate[(c - a*c*x)^(5/2)/E^ArcCoth[a*x],x]

[Out]

(2*c^2*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x]*(-177 + 71*a*x - 27*a^2*x^2 + 5*a^3*x^3))/(35*a*Sqrt[1 - 1/(a*x)])

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.48

method	result	size
risch	\(-\frac {2 c^{3} \sqrt {\frac {a x -1}{a x +1}}\, \left (5 a^{3} x^{3}-27 a^{2} x^{2}+71 a x -177\right ) \left (a x +1\right )}{35 \sqrt {-c \left (a x -1\right )}\, a}\)	\(61\)
gosper	\(\frac {2 \left (a x +1\right ) \left (5 a^{3} x^{3}-27 a^{2} x^{2}+71 a x -177\right ) \left (-a c x +c \right )^{\frac {5}{2}} \sqrt {\frac {a x -1}{a x +1}}}{35 a \left (a x -1\right )^{3}}\)	\(64\)
default	\(\frac {2 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, c^{2} \left (5 a^{3} x^{3}-27 a^{2} x^{2}+71 a x -177\right )}{35 \left (a x -1\right ) a}\)	\(68\)

[In]

int((-a*c*x+c)^(5/2)*((a*x-1)/(a*x+1))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/35*c^3*((a*x-1)/(a*x+1))^(1/2)/(-c*(a*x-1))^(1/2)*(5*a^3*x^3-27*a^2*x^2+71*a*x-177)/a*(a*x+1)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.65 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\frac {2 \, {\left (5 \, a^{4} c^{2} x^{4} - 22 \, a^{3} c^{2} x^{3} + 44 \, a^{2} c^{2} x^{2} - 106 \, a c^{2} x - 177 \, c^{2}\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{35 \, {\left (a^{2} x - a\right )}} \]

[In]

integrate((-a*c*x+c)^(5/2)*((a*x-1)/(a*x+1))^(1/2),x, algorithm="fricas")

[Out]

2/35*(5*a^4*c^2*x^4 - 22*a^3*c^2*x^3 + 44*a^2*c^2*x^2 - 106*a*c^2*x - 177*c^2)*sqrt(-a*c*x + c)*sqrt((a*x - 1)
/(a*x + 1))/(a^2*x - a)

Sympy [F(-1)]

Timed out. \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\text {Timed out} \]

[In]

integrate((-a*c*x+c)**(5/2)*((a*x-1)/(a*x+1))**(1/2),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.75 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\frac {2 \, {\left (5 \, a^{4} \sqrt {-c} c^{2} x^{4} - 22 \, a^{3} \sqrt {-c} c^{2} x^{3} + 44 \, a^{2} \sqrt {-c} c^{2} x^{2} - 106 \, a \sqrt {-c} c^{2} x - 177 \, \sqrt {-c} c^{2}\right )} {\left (a x - 1\right )}}{35 \, {\left (a^{2} x - a\right )} \sqrt {a x + 1}} \]

[In]

integrate((-a*c*x+c)^(5/2)*((a*x-1)/(a*x+1))^(1/2),x, algorithm="maxima")

[Out]

2/35*(5*a^4*sqrt(-c)*c^2*x^4 - 22*a^3*sqrt(-c)*c^2*x^3 + 44*a^2*sqrt(-c)*c^2*x^2 - 106*a*sqrt(-c)*c^2*x - 177*
sqrt(-c)*c^2)*(a*x - 1)/((a^2*x - a)*sqrt(a*x + 1))

Giac [F(-2)]

Exception generated. \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((-a*c*x+c)^(5/2)*((a*x-1)/(a*x+1))^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [B] (verification not implemented)

Time = 4.30 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.73 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\frac {2\,c^2\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (5\,a^3\,x^3-17\,a^2\,x^2+27\,a\,x-79\right )}{35\,a}-\frac {512\,c^2\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{35\,a\,\left (a\,x-1\right )} \]

[In]

int((c - a*c*x)^(5/2)*((a*x - 1)/(a*x + 1))^(1/2),x)

[Out]

(2*c^2*(c - a*c*x)^(1/2)*((a*x - 1)/(a*x + 1))^(1/2)*(27*a*x - 17*a^2*x^2 + 5*a^3*x^3 - 79))/(35*a) - (512*c^2
*(c - a*c*x)^(1/2)*((a*x - 1)/(a*x + 1))^(1/2))/(35*a*(a*x - 1))