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

Optimal. Leaf size=195 \[ \frac{184 (c-a c x)^{3/2}}{5 a^3 x^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}+\frac{2 x \left (a-\frac{1}{x}\right )^3 (c-a c x)^{3/2}}{5 a^3 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}+\frac{16 (c-a c x)^{3/2}}{a^2 x \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}-\frac{8 (c-a c x)^{3/2}}{5 a \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}} \]

[Out]

(-8*(c - a*c*x)^(3/2))/(5*a*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)]) + (184*(c - a*c*x)^(3/2))/(5*a^3*(1 - 1/(a*
x))^(3/2)*Sqrt[1 + 1/(a*x)]*x^2) + (16*(c - a*c*x)^(3/2))/(a^2*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)]*x) + (2*(
a - x^(-1))^3*x*(c - a*c*x)^(3/2))/(5*a^3*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)])

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Rubi [A]  time = 0.191599, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ \frac{184 (c-a c x)^{3/2}}{5 a^3 x^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}+\frac{2 x \left (a-\frac{1}{x}\right )^3 (c-a c x)^{3/2}}{5 a^3 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}+\frac{16 (c-a c x)^{3/2}}{a^2 x \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}-\frac{8 (c-a c x)^{3/2}}{5 a \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*(c - a*c*x)^(3/2))/(5*a*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)]) + (184*(c - a*c*x)^(3/2))/(5*a^3*(1 - 1/(a*
x))^(3/2)*Sqrt[1 + 1/(a*x)]*x^2) + (16*(c - a*c*x)^(3/2))/(a^2*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)]*x) + (2*(
a - x^(-1))^3*x*(c - a*c*x)^(3/2))/(5*a^3*(1 - 1/(a*x))^(3/2)*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)^{3/2} \, dx &=\frac{(c-a c x)^{3/2} \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right )^{3/2} x^{3/2} \, dx}{\left (1-\frac{1}{a x}\right )^{3/2} x^{3/2}}\\ &=-\frac{\left (\left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}\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 )}{\left (1-\frac{1}{a x}\right )^{3/2}}\\ &=\frac{2 \left (a-\frac{1}{x}\right )^3 x (c-a c x)^{3/2}}{5 a^3 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}+\frac{\left (12 \left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}\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 )}{5 a \left (1-\frac{1}{a x}\right )^{3/2}}\\ &=-\frac{8 (c-a c x)^{3/2}}{5 a \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}+\frac{2 \left (a-\frac{1}{x}\right )^3 x (c-a c x)^{3/2}}{5 a^3 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}+\frac{\left (8 \left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}\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 )}{5 a \left (1-\frac{1}{a x}\right )^{3/2}}\\ &=-\frac{8 (c-a c x)^{3/2}}{5 a \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}+\frac{16 (c-a c x)^{3/2}}{a^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}} x}+\frac{2 \left (a-\frac{1}{x}\right )^3 x (c-a c x)^{3/2}}{5 a^3 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}+\frac{\left (92 \left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{5 a^3 \left (1-\frac{1}{a x}\right )^{3/2}}\\ &=-\frac{8 (c-a c x)^{3/2}}{5 a \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}+\frac{184 (c-a c x)^{3/2}}{5 a^3 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}} x^2}+\frac{16 (c-a c x)^{3/2}}{a^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}} x}+\frac{2 \left (a-\frac{1}{x}\right )^3 x (c-a c x)^{3/2}}{5 a^3 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}\\ \end{align*}

Mathematica [A]  time = 0.0322588, size = 57, normalized size = 0.29 \[ -\frac{2 c \left (a^3 x^3-7 a^2 x^2+43 a x+91\right ) \sqrt{c-a c x}}{5 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*c*Sqrt[c - a*c*x]*(91 + 43*a*x - 7*a^2*x^2 + a^3*x^3))/(5*a^2*Sqrt[1 - 1/(a^2*x^2)]*x)

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Maple [A]  time = 0.043, size = 63, normalized size = 0.3 \begin{align*}{\frac{ \left ( 2\,ax+2 \right ) \left ({x}^{3}{a}^{3}-7\,{a}^{2}{x}^{2}+43\,ax+91 \right ) }{5\,a \left ( ax-1 \right ) ^{3}} \left ( -acx+c \right ) ^{{\frac{3}{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)^(3/2)*((a*x-1)/(a*x+1))^(3/2),x)

[Out]

2/5*(a*x+1)*(a^3*x^3-7*a^2*x^2+43*a*x+91)*(-a*c*x+c)^(3/2)*((a*x-1)/(a*x+1))^(3/2)/a/(a*x-1)^3

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Maxima [A]  time = 1.07883, size = 126, normalized size = 0.65 \begin{align*} -\frac{2 \,{\left (a^{4} \sqrt{-c} c x^{4} - 6 \, a^{3} \sqrt{-c} c x^{3} + 36 \, a^{2} \sqrt{-c} c x^{2} + 134 \, a \sqrt{-c} c x + 91 \, \sqrt{-c} c\right )}{\left (a x - 1\right )}^{2}}{5 \,{\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)^(3/2)*((a*x-1)/(a*x+1))^(3/2),x, algorithm="maxima")

[Out]

-2/5*(a^4*sqrt(-c)*c*x^4 - 6*a^3*sqrt(-c)*c*x^3 + 36*a^2*sqrt(-c)*c*x^2 + 134*a*sqrt(-c)*c*x + 91*sqrt(-c)*c)*
(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 = 1.56261, size = 142, normalized size = 0.73 \begin{align*} -\frac{2 \,{\left (a^{3} c x^{3} - 7 \, a^{2} c x^{2} + 43 \, a c x + 91 \, c\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{5 \,{\left (a^{2} x - a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*(a^3*c*x^3 - 7*a^2*c*x^2 + 43*a*c*x + 91*c)*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)**(3/2)*((a*x-1)/(a*x+1))**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.20292, size = 104, normalized size = 0.53 \begin{align*} \frac{2 \,{\left ({\left (a c x + c\right )}^{2} \sqrt{-a c x - c} + 10 \,{\left (-a c x - c\right )}^{\frac{3}{2}} c + 60 \, \sqrt{-a c x - c} c^{2} - \frac{40 \, c^{3}}{\sqrt{-a c x - c}}\right )}{\left | c \right |}}{5 \, a c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*((a*c*x + c)^2*sqrt(-a*c*x - c) + 10*(-a*c*x - c)^(3/2)*c + 60*sqrt(-a*c*x - c)*c^2 - 40*c^3/sqrt(-a*c*x -
 c))*abs(c)/(a*c^2)