3.350 \(\int e^{-3 \coth ^{-1}(a x)} x^3 \sqrt{c-a c x} \, dx\)

Optimal. Leaf size=281 \[ \frac{164 x^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{15 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{82 x^2 \sqrt{c-a c x}}{9 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{656 x \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{45 a^3 \sqrt{1-\frac{1}{a x}}}+\frac{1312 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{45 a^4 \sqrt{1-\frac{1}{a x}}}+\frac{2 x^4 \sqrt{c-a c x}}{9 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{8 x^3 \sqrt{c-a c x}}{9 a \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}} \]

[Out]

(1312*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(45*a^4*Sqrt[1 - 1/(a*x)]) - (656*Sqrt[1 + 1/(a*x)]*x*Sqrt[c - a*c*x]
)/(45*a^3*Sqrt[1 - 1/(a*x)]) - (82*x^2*Sqrt[c - a*c*x])/(9*a^2*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) + (164*Sqr
t[1 + 1/(a*x)]*x^2*Sqrt[c - a*c*x])/(15*a^2*Sqrt[1 - 1/(a*x)]) - (8*x^3*Sqrt[c - a*c*x])/(9*a*Sqrt[1 - 1/(a*x)
]*Sqrt[1 + 1/(a*x)]) + (2*x^4*Sqrt[c - a*c*x])/(9*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])

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Rubi [A]  time = 0.286094, antiderivative size = 281, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {6176, 6181, 89, 78, 45, 37} \[ \frac{164 x^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{15 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{82 x^2 \sqrt{c-a c x}}{9 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{656 x \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{45 a^3 \sqrt{1-\frac{1}{a x}}}+\frac{1312 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{45 a^4 \sqrt{1-\frac{1}{a x}}}+\frac{2 x^4 \sqrt{c-a c x}}{9 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{8 x^3 \sqrt{c-a c x}}{9 a \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1312*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(45*a^4*Sqrt[1 - 1/(a*x)]) - (656*Sqrt[1 + 1/(a*x)]*x*Sqrt[c - a*c*x]
)/(45*a^3*Sqrt[1 - 1/(a*x)]) - (82*x^2*Sqrt[c - a*c*x])/(9*a^2*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) + (164*Sqr
t[1 + 1/(a*x)]*x^2*Sqrt[c - a*c*x])/(15*a^2*Sqrt[1 - 1/(a*x)]) - (8*x^3*Sqrt[c - a*c*x])/(9*a*Sqrt[1 - 1/(a*x)
]*Sqrt[1 + 1/(a*x)]) + (2*x^4*Sqrt[c - a*c*x])/(9*Sqrt[1 - 1/(a*x)]*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 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 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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)} x^3 \sqrt{c-a c x} \, dx &=\frac{\sqrt{c-a c x} \int e^{-3 \coth ^{-1}(a x)} \sqrt{1-\frac{1}{a x}} x^{7/2} \, dx}{\sqrt{1-\frac{1}{a x}} \sqrt{x}}\\ &=-\frac{\left (\sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^2}{x^{11/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{2 x^4 \sqrt{c-a c x}}{9 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{\left (2 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{-\frac{14}{a}+\frac{9 x}{2 a^2}}{x^{9/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{9 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{8 x^3 \sqrt{c-a c x}}{9 a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{2 x^4 \sqrt{c-a c x}}{9 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{\left (41 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{x^{7/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{9 a^2 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{82 x^2 \sqrt{c-a c x}}{9 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{8 x^3 \sqrt{c-a c x}}{9 a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{2 x^4 \sqrt{c-a c x}}{9 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{\left (82 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{x^{7/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{3 a^2 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{82 x^2 \sqrt{c-a c x}}{9 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{164 \sqrt{1+\frac{1}{a x}} x^2 \sqrt{c-a c x}}{15 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{8 x^3 \sqrt{c-a c x}}{9 a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{2 x^4 \sqrt{c-a c x}}{9 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{\left (328 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{x^{5/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{15 a^3 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{656 \sqrt{1+\frac{1}{a x}} x \sqrt{c-a c x}}{45 a^3 \sqrt{1-\frac{1}{a x}}}-\frac{82 x^2 \sqrt{c-a c x}}{9 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{164 \sqrt{1+\frac{1}{a x}} x^2 \sqrt{c-a c x}}{15 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{8 x^3 \sqrt{c-a c x}}{9 a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{2 x^4 \sqrt{c-a c x}}{9 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{\left (656 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{x^{3/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{45 a^4 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{1312 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{45 a^4 \sqrt{1-\frac{1}{a x}}}-\frac{656 \sqrt{1+\frac{1}{a x}} x \sqrt{c-a c x}}{45 a^3 \sqrt{1-\frac{1}{a x}}}-\frac{82 x^2 \sqrt{c-a c x}}{9 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{164 \sqrt{1+\frac{1}{a x}} x^2 \sqrt{c-a c x}}{15 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{8 x^3 \sqrt{c-a c x}}{9 a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{2 x^4 \sqrt{c-a c x}}{9 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}\\ \end{align*}

Mathematica [A]  time = 0.0414572, size = 73, normalized size = 0.26 \[ \frac{2 \left (5 a^5 x^5-20 a^4 x^4+41 a^3 x^3-82 a^2 x^2+328 a x+656\right ) \sqrt{c-a c x}}{45 a^5 x \sqrt{1-\frac{1}{a^2 x^2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c - a*c*x]*(656 + 328*a*x - 82*a^2*x^2 + 41*a^3*x^3 - 20*a^4*x^4 + 5*a^5*x^5))/(45*a^5*Sqrt[1 - 1/(a^2
*x^2)]*x)

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Maple [A]  time = 0.051, size = 80, normalized size = 0.3 \begin{align*}{\frac{ \left ( 2\,ax+2 \right ) \left ( 5\,{x}^{5}{a}^{5}-20\,{x}^{4}{a}^{4}+41\,{x}^{3}{a}^{3}-82\,{a}^{2}{x}^{2}+328\,ax+656 \right ) }{45\,{a}^{4} \left ( ax-1 \right ) ^{2}}\sqrt{-acx+c} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45*(a*x+1)*(5*a^5*x^5-20*a^4*x^4+41*a^3*x^3-82*a^2*x^2+328*a*x+656)*(-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2)
/a^4/(a*x-1)^2

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Maxima [A]  time = 1.12839, size = 158, normalized size = 0.56 \begin{align*} \frac{2 \,{\left (5 \, a^{6} \sqrt{-c} x^{6} - 15 \, a^{5} \sqrt{-c} x^{5} + 21 \, a^{4} \sqrt{-c} x^{4} - 41 \, a^{3} \sqrt{-c} x^{3} + 246 \, a^{2} \sqrt{-c} x^{2} + 984 \, a \sqrt{-c} x + 656 \, \sqrt{-c}\right )}{\left (a x - 1\right )}^{2}}{45 \,{\left (a^{6} x^{2} - 2 \, a^{5} x + a^{4}\right )}{\left (a x + 1\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45*(5*a^6*sqrt(-c)*x^6 - 15*a^5*sqrt(-c)*x^5 + 21*a^4*sqrt(-c)*x^4 - 41*a^3*sqrt(-c)*x^3 + 246*a^2*sqrt(-c)*
x^2 + 984*a*sqrt(-c)*x + 656*sqrt(-c))*(a*x - 1)^2/((a^6*x^2 - 2*a^5*x + a^4)*(a*x + 1)^(3/2))

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Fricas [A]  time = 1.39528, size = 176, normalized size = 0.63 \begin{align*} \frac{2 \,{\left (5 \, a^{5} x^{5} - 20 \, a^{4} x^{4} + 41 \, a^{3} x^{3} - 82 \, a^{2} x^{2} + 328 \, a x + 656\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{45 \,{\left (a^{5} x - a^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45*(5*a^5*x^5 - 20*a^4*x^4 + 41*a^3*x^3 - 82*a^2*x^2 + 328*a*x + 656)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x +
1))/(a^5*x - a^4)

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

[Out]

Timed out

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Giac [A]  time = 1.30864, size = 184, normalized size = 0.65 \begin{align*} -\frac{2 \,{\left (5 \,{\left (a c x + c\right )}^{4} \sqrt{-a c x - c}{\left | c \right |} - 45 \,{\left (a c x + c\right )}^{3} \sqrt{-a c x - c} c{\left | c \right |} + 171 \,{\left (a c x + c\right )}^{2} \sqrt{-a c x - c} c^{2}{\left | c \right |} + 375 \,{\left (-a c x - c\right )}^{\frac{3}{2}} c^{3}{\left | c \right |} + 720 \, \sqrt{-a c x - c} c^{4}{\left | c \right |} - \frac{180 \, c^{5}{\left | c \right |}}{\sqrt{-a c x - c}}\right )}}{45 \, a^{4} c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/45*(5*(a*c*x + c)^4*sqrt(-a*c*x - c)*abs(c) - 45*(a*c*x + c)^3*sqrt(-a*c*x - c)*c*abs(c) + 171*(a*c*x + c)^
2*sqrt(-a*c*x - c)*c^2*abs(c) + 375*(-a*c*x - c)^(3/2)*c^3*abs(c) + 720*sqrt(-a*c*x - c)*c^4*abs(c) - 180*c^5*
abs(c)/sqrt(-a*c*x - c))/(a^4*c^5)