3.336 \(\int e^{-\coth ^{-1}(a x)} x^2 \sqrt{c-a c x} \, dx\)

Optimal. Leaf size=142 \[ -\frac{2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} (c-a c x)^{3/2}}{7 a c}+\frac{6 x \sqrt{1-\frac{1}{a^2 x^2}} (c-a c x)^{3/2}}{35 a^2 c}+\frac{38 x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-a c x}}{105 a^2}+\frac{152 c x \sqrt{1-\frac{1}{a^2 x^2}}}{105 a^2 \sqrt{c-a c x}} \]

[Out]

(152*c*Sqrt[1 - 1/(a^2*x^2)]*x)/(105*a^2*Sqrt[c - a*c*x]) + (38*Sqrt[1 - 1/(a^2*x^2)]*x*Sqrt[c - a*c*x])/(105*
a^2) + (6*Sqrt[1 - 1/(a^2*x^2)]*x*(c - a*c*x)^(3/2))/(35*a^2*c) - (2*Sqrt[1 - 1/(a^2*x^2)]*x^2*(c - a*c*x)^(3/
2))/(7*a*c)

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Rubi [A]  time = 0.236783, antiderivative size = 185, normalized size of antiderivative = 1.3, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {6176, 6181, 78, 45, 37} \[ \frac{104 x \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{105 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{208 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{105 a^3 \sqrt{1-\frac{1}{a x}}}+\frac{2 x^3 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{7 \sqrt{1-\frac{1}{a x}}}-\frac{26 x^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{35 a \sqrt{1-\frac{1}{a x}}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-208*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(105*a^3*Sqrt[1 - 1/(a*x)]) + (104*Sqrt[1 + 1/(a*x)]*x*Sqrt[c - a*c*x
])/(105*a^2*Sqrt[1 - 1/(a*x)]) - (26*Sqrt[1 + 1/(a*x)]*x^2*Sqrt[c - a*c*x])/(35*a*Sqrt[1 - 1/(a*x)]) + (2*Sqrt
[1 + 1/(a*x)]*x^3*Sqrt[c - a*c*x])/(7*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 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^{-\coth ^{-1}(a x)} x^2 \sqrt{c-a c x} \, dx &=\frac{\sqrt{c-a c x} \int e^{-\coth ^{-1}(a x)} \sqrt{1-\frac{1}{a x}} x^{5/2} \, 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{1-\frac{x}{a}}{x^{9/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{2 \sqrt{1+\frac{1}{a x}} x^3 \sqrt{c-a c x}}{7 \sqrt{1-\frac{1}{a x}}}+\frac{\left (13 \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 )}{7 a \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{26 \sqrt{1+\frac{1}{a x}} x^2 \sqrt{c-a c x}}{35 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{1+\frac{1}{a x}} x^3 \sqrt{c-a c x}}{7 \sqrt{1-\frac{1}{a x}}}-\frac{\left (52 \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 )}{35 a^2 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{104 \sqrt{1+\frac{1}{a x}} x \sqrt{c-a c x}}{105 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{26 \sqrt{1+\frac{1}{a x}} x^2 \sqrt{c-a c x}}{35 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{1+\frac{1}{a x}} x^3 \sqrt{c-a c x}}{7 \sqrt{1-\frac{1}{a x}}}+\frac{\left (104 \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 )}{105 a^3 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{208 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{105 a^3 \sqrt{1-\frac{1}{a x}}}+\frac{104 \sqrt{1+\frac{1}{a x}} x \sqrt{c-a c x}}{105 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{26 \sqrt{1+\frac{1}{a x}} x^2 \sqrt{c-a c x}}{35 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{1+\frac{1}{a x}} x^3 \sqrt{c-a c x}}{7 \sqrt{1-\frac{1}{a x}}}\\ \end{align*}

Mathematica [A]  time = 0.0344716, size = 67, normalized size = 0.47 \[ \frac{2 \sqrt{\frac{1}{a x}+1} \left (15 a^3 x^3-39 a^2 x^2+52 a x-104\right ) \sqrt{c-a c x}}{105 a^3 \sqrt{1-\frac{1}{a x}}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x]*(-104 + 52*a*x - 39*a^2*x^2 + 15*a^3*x^3))/(105*a^3*Sqrt[1 - 1/(a*x)])

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Maple [A]  time = 0.043, size = 64, normalized size = 0.5 \begin{align*}{\frac{ \left ( 2\,ax+2 \right ) \left ( 15\,{x}^{3}{a}^{3}-39\,{a}^{2}{x}^{2}+52\,ax-104 \right ) }{105\,{a}^{3} \left ( ax-1 \right ) }\sqrt{-acx+c}\sqrt{{\frac{ax-1}{ax+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(a*x+1)*(15*a^3*x^3-39*a^2*x^2+52*a*x-104)*(-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(1/2)/a^3/(a*x-1)

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Maxima [A]  time = 1.10097, size = 112, normalized size = 0.79 \begin{align*} \frac{2 \,{\left (15 \, a^{4} \sqrt{-c} x^{4} - 24 \, a^{3} \sqrt{-c} x^{3} + 13 \, a^{2} \sqrt{-c} x^{2} - 52 \, a \sqrt{-c} x - 104 \, \sqrt{-c}\right )}{\left (a x - 1\right )}}{105 \,{\left (a^{4} x - a^{3}\right )} \sqrt{a x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(15*a^4*sqrt(-c)*x^4 - 24*a^3*sqrt(-c)*x^3 + 13*a^2*sqrt(-c)*x^2 - 52*a*sqrt(-c)*x - 104*sqrt(-c))*(a*x
- 1)/((a^4*x - a^3)*sqrt(a*x + 1))

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Fricas [A]  time = 1.63865, size = 159, normalized size = 1.12 \begin{align*} \frac{2 \,{\left (15 \, a^{4} x^{4} - 24 \, a^{3} x^{3} + 13 \, a^{2} x^{2} - 52 \, a x - 104\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{105 \,{\left (a^{4} x - a^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(15*a^4*x^4 - 24*a^3*x^3 + 13*a^2*x^2 - 52*a*x - 104)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1))/(a^4*x
- a^3)

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

[Out]

Timed out

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Giac [A]  time = 1.17414, size = 150, normalized size = 1.06 \begin{align*} -\frac{2 \,{\left (\frac{76 \, \sqrt{2} \sqrt{-c} c}{a^{3}} + \frac{15 \,{\left (a c x + c\right )}^{3} \sqrt{-a c x - c} - 84 \,{\left (a c x + c\right )}^{2} \sqrt{-a c x - c} c - 175 \,{\left (-a c x - c\right )}^{\frac{3}{2}} c^{2} - 210 \, \sqrt{-a c x - c} c^{3}}{a^{3} c^{2}}\right )}{\left | c \right |} \mathrm{sgn}\left (a x + 1\right )}{105 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/105*(76*sqrt(2)*sqrt(-c)*c/a^3 + (15*(a*c*x + c)^3*sqrt(-a*c*x - c) - 84*(a*c*x + c)^2*sqrt(-a*c*x - c)*c -
175*(-a*c*x - c)^(3/2)*c^2 - 210*sqrt(-a*c*x - c)*c^3)/(a^3*c^2))*abs(c)*sgn(a*x + 1)/c^2