3.272 \(\int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx\)

Optimal. Leaf size=254 \[ -\frac{256 (c-a c x)^{5/2}}{7 a^3 x^2 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{\frac{1}{a x}+1}}-\frac{2944 (c-a c x)^{5/2}}{35 a^4 x^3 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{\frac{1}{a x}+1}}+\frac{2 x \left (a-\frac{1}{x}\right )^4 (c-a c x)^{5/2}}{7 a^4 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{\frac{1}{a x}+1}}-\frac{32 \left (a-\frac{1}{x}\right )^3 (c-a c x)^{5/2}}{35 a^4 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{\frac{1}{a x}+1}}+\frac{128 (c-a c x)^{5/2}}{35 a^2 x \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{\frac{1}{a x}+1}} \]

[Out]

(-32*(a - x^(-1))^3*(c - a*c*x)^(5/2))/(35*a^4*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)]) - (2944*(c - a*c*x)^(5/2
))/(35*a^4*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)]*x^3) - (256*(c - a*c*x)^(5/2))/(7*a^3*(1 - 1/(a*x))^(5/2)*Sqr
t[1 + 1/(a*x)]*x^2) + (128*(c - a*c*x)^(5/2))/(35*a^2*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)]*x) + (2*(a - x^(-1
))^4*x*(c - a*c*x)^(5/2))/(7*a^4*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)])

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Rubi [A]  time = 0.213266, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6176, 6181, 94, 89, 78, 37} \[ -\frac{256 (c-a c x)^{5/2}}{7 a^3 x^2 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{\frac{1}{a x}+1}}-\frac{2944 (c-a c x)^{5/2}}{35 a^4 x^3 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{\frac{1}{a x}+1}}+\frac{2 x \left (a-\frac{1}{x}\right )^4 (c-a c x)^{5/2}}{7 a^4 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{\frac{1}{a x}+1}}-\frac{32 \left (a-\frac{1}{x}\right )^3 (c-a c x)^{5/2}}{35 a^4 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{\frac{1}{a x}+1}}+\frac{128 (c-a c x)^{5/2}}{35 a^2 x \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{\frac{1}{a x}+1}} \]

Antiderivative was successfully verified.

[In]

Int[(c - a*c*x)^(5/2)/E^(3*ArcCoth[a*x]),x]

[Out]

(-32*(a - x^(-1))^3*(c - a*c*x)^(5/2))/(35*a^4*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)]) - (2944*(c - a*c*x)^(5/2
))/(35*a^4*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)]*x^3) - (256*(c - a*c*x)^(5/2))/(7*a^3*(1 - 1/(a*x))^(5/2)*Sqr
t[1 + 1/(a*x)]*x^2) + (128*(c - a*c*x)^(5/2))/(35*a^2*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)]*x) + (2*(a - x^(-1
))^4*x*(c - a*c*x)^(5/2))/(7*a^4*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)])

Rule 6176

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 6181

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_), x_Symbol] :> -Dist[c^p*x^m*(1/x)^m, Sub
st[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]

Rule 94

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[p, 1] &&  !SumSimplerQ[m, 1])

Rule 89

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 78

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] || LtQ
[p, n]))))

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]

Rubi steps

\begin{align*} \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx &=\frac{(c-a c x)^{5/2} \int e^{-3 \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 ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^4}{x^{9/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{5/2}}\\ &=\frac{2 \left (a-\frac{1}{x}\right )^4 x (c-a c x)^{5/2}}{7 a^4 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}}}+\frac{\left (16 \left (\frac{1}{x}\right )^{5/2} (c-a c x)^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^3}{x^{7/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{7 a \left (1-\frac{1}{a x}\right )^{5/2}}\\ &=-\frac{32 \left (a-\frac{1}{x}\right )^3 (c-a c x)^{5/2}}{35 a^4 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}}}+\frac{2 \left (a-\frac{1}{x}\right )^4 x (c-a c x)^{5/2}}{7 a^4 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}}}-\frac{\left (192 \left (\frac{1}{x}\right )^{5/2} (c-a c x)^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^2}{x^{5/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{35 a^2 \left (1-\frac{1}{a x}\right )^{5/2}}\\ &=-\frac{32 \left (a-\frac{1}{x}\right )^3 (c-a c x)^{5/2}}{35 a^4 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}}}+\frac{128 (c-a c x)^{5/2}}{35 a^2 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}} x}+\frac{2 \left (a-\frac{1}{x}\right )^4 x (c-a c x)^{5/2}}{7 a^4 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}}}-\frac{\left (128 \left (\frac{1}{x}\right )^{5/2} (c-a c x)^{5/2}\right ) \operatorname{Subst}\left (\int \frac{-\frac{5}{a}+\frac{3 x}{2 a^2}}{x^{3/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{35 a^2 \left (1-\frac{1}{a x}\right )^{5/2}}\\ &=-\frac{32 \left (a-\frac{1}{x}\right )^3 (c-a c x)^{5/2}}{35 a^4 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}}}-\frac{256 (c-a c x)^{5/2}}{7 a^3 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}} x^2}+\frac{128 (c-a c x)^{5/2}}{35 a^2 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}} x}+\frac{2 \left (a-\frac{1}{x}\right )^4 x (c-a c x)^{5/2}}{7 a^4 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}}}-\frac{\left (1472 \left (\frac{1}{x}\right )^{5/2} (c-a c x)^{5/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{35 a^4 \left (1-\frac{1}{a x}\right )^{5/2}}\\ &=-\frac{32 \left (a-\frac{1}{x}\right )^3 (c-a c x)^{5/2}}{35 a^4 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}}}-\frac{2944 (c-a c x)^{5/2}}{35 a^4 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}} x^3}-\frac{256 (c-a c x)^{5/2}}{7 a^3 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}} x^2}+\frac{128 (c-a c x)^{5/2}}{35 a^2 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}} x}+\frac{2 \left (a-\frac{1}{x}\right )^4 x (c-a c x)^{5/2}}{7 a^4 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}}}\\ \end{align*}

Mathematica [A]  time = 0.0381353, size = 68, normalized size = 0.27 \[ \frac{2 c^2 \left (5 a^4 x^4-36 a^3 x^3+142 a^2 x^2-708 a x-1451\right ) \sqrt{c-a c x}}{35 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c - a*c*x)^(5/2)/E^(3*ArcCoth[a*x]),x]

[Out]

(2*c^2*Sqrt[c - a*c*x]*(-1451 - 708*a*x + 142*a^2*x^2 - 36*a^3*x^3 + 5*a^4*x^4))/(35*a^2*Sqrt[1 - 1/(a^2*x^2)]
*x)

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Maple [A]  time = 0.052, size = 72, normalized size = 0.3 \begin{align*}{\frac{ \left ( 2\,ax+2 \right ) \left ( 5\,{x}^{4}{a}^{4}-36\,{x}^{3}{a}^{3}+142\,{a}^{2}{x}^{2}-708\,ax-1451 \right ) }{35\,a \left ( ax-1 \right ) ^{4}} \left ( -acx+c \right ) ^{{\frac{5}{2}}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a*c*x+c)^(5/2)*((a*x-1)/(a*x+1))^(3/2),x)

[Out]

2/35*(a*x+1)*(5*a^4*x^4-36*a^3*x^3+142*a^2*x^2-708*a*x-1451)*(-a*c*x+c)^(5/2)*((a*x-1)/(a*x+1))^(3/2)/a/(a*x-1
)^4

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Maxima [A]  time = 1.11333, size = 162, normalized size = 0.64 \begin{align*} \frac{2 \,{\left (5 \, a^{5} \sqrt{-c} c^{2} x^{5} - 31 \, a^{4} \sqrt{-c} c^{2} x^{4} + 106 \, a^{3} \sqrt{-c} c^{2} x^{3} - 566 \, a^{2} \sqrt{-c} c^{2} x^{2} - 2159 \, a \sqrt{-c} c^{2} x - 1451 \, \sqrt{-c} c^{2}\right )}{\left (a x - 1\right )}^{2}}{35 \,{\left (a^{3} x^{2} - 2 \, a^{2} x + a\right )}{\left (a x + 1\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^(5/2)*((a*x-1)/(a*x+1))^(3/2),x, algorithm="maxima")

[Out]

2/35*(5*a^5*sqrt(-c)*c^2*x^5 - 31*a^4*sqrt(-c)*c^2*x^4 + 106*a^3*sqrt(-c)*c^2*x^3 - 566*a^2*sqrt(-c)*c^2*x^2 -
 2159*a*sqrt(-c)*c^2*x - 1451*sqrt(-c)*c^2)*(a*x - 1)^2/((a^3*x^2 - 2*a^2*x + a)*(a*x + 1)^(3/2))

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Fricas [A]  time = 1.50044, size = 185, normalized size = 0.73 \begin{align*} \frac{2 \,{\left (5 \, a^{4} c^{2} x^{4} - 36 \, a^{3} c^{2} x^{3} + 142 \, a^{2} c^{2} x^{2} - 708 \, a c^{2} x - 1451 \, c^{2}\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{35 \,{\left (a^{2} x - a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^(5/2)*((a*x-1)/(a*x+1))^(3/2),x, algorithm="fricas")

[Out]

2/35*(5*a^4*c^2*x^4 - 36*a^3*c^2*x^3 + 142*a^2*c^2*x^2 - 708*a*c^2*x - 1451*c^2)*sqrt(-a*c*x + c)*sqrt((a*x -
1)/(a*x + 1))/(a^2*x - a)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)**(5/2)*((a*x-1)/(a*x+1))**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.29361, size = 138, normalized size = 0.54 \begin{align*} -\frac{2 \,{\left (5 \,{\left (a c x + c\right )}^{3} \sqrt{-a c x - c} - 56 \,{\left (a c x + c\right )}^{2} \sqrt{-a c x - c} c - 280 \,{\left (-a c x - c\right )}^{\frac{3}{2}} c^{2} - 1120 \, \sqrt{-a c x - c} c^{3} + \frac{560 \, c^{4}}{\sqrt{-a c x - c}}\right )}{\left | c \right |}}{35 \, a c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^(5/2)*((a*x-1)/(a*x+1))^(3/2),x, algorithm="giac")

[Out]

-2/35*(5*(a*c*x + c)^3*sqrt(-a*c*x - c) - 56*(a*c*x + c)^2*sqrt(-a*c*x - c)*c - 280*(-a*c*x - c)^(3/2)*c^2 - 1
120*sqrt(-a*c*x - c)*c^3 + 560*c^4/sqrt(-a*c*x - c))*abs(c)/(a*c^2)