### 3.337 $$\int e^{-\coth ^{-1}(a x)} x \sqrt{c-a c x} \, dx$$

Optimal. Leaf size=104 $-\frac{2 x \sqrt{1-\frac{1}{a^2 x^2}} (c-a c x)^{3/2}}{5 a c}-\frac{2 x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-a c x}}{5 a}-\frac{8 c x \sqrt{1-\frac{1}{a^2 x^2}}}{5 a \sqrt{c-a c x}}$

[Out]

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

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Rubi [A]  time = 0.193505, antiderivative size = 137, normalized size of antiderivative = 1.32, number of steps used = 5, number of rules used = 5, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.238, Rules used = {6176, 6181, 78, 45, 37} $\frac{12 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{5 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{2 x^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}}}-\frac{6 x \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{5 a \sqrt{1-\frac{1}{a x}}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(12*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(5*a^2*Sqrt[1 - 1/(a*x)]) - (6*Sqrt[1 + 1/(a*x)]*x*Sqrt[c - a*c*x])/(5*
a*Sqrt[1 - 1/(a*x)]) + (2*Sqrt[1 + 1/(a*x)]*x^2*Sqrt[c - a*c*x])/(5*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 \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^{3/2} \, 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^{7/2} \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^2 \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}}}+\frac{\left (9 \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 )}{5 a \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{6 \sqrt{1+\frac{1}{a x}} x \sqrt{c-a c x}}{5 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{1+\frac{1}{a x}} x^2 \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}}}-\frac{\left (6 \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 )}{5 a^2 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{12 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{5 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{6 \sqrt{1+\frac{1}{a x}} x \sqrt{c-a c x}}{5 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{1+\frac{1}{a x}} x^2 \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}}}\\ \end{align*}

Mathematica [A]  time = 0.0307322, size = 58, normalized size = 0.56 $\frac{2 \sqrt{\frac{1}{a x}+1} \left (a^2 x^2-3 a x+6\right ) \sqrt{c-a c x}}{5 a^2 \sqrt{1-\frac{1}{a x}}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x]*(6 - 3*a*x + a^2*x^2))/(5*a^2*Sqrt[1 - 1/(a*x)])

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Maple [A]  time = 0.04, size = 55, normalized size = 0.5 \begin{align*}{\frac{ \left ( 2\,ax+2 \right ) \left ({a}^{2}{x}^{2}-3\,ax+6 \right ) }{5\,{a}^{2} \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*(-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(1/2),x)

[Out]

2/5*(a*x+1)*(a^2*x^2-3*a*x+6)*(-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(1/2)/a^2/(a*x-1)

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Maxima [A]  time = 1.12581, size = 93, normalized size = 0.89 \begin{align*} \frac{2 \,{\left (a^{3} \sqrt{-c} x^{3} - 2 \, a^{2} \sqrt{-c} x^{2} + 3 \, a \sqrt{-c} x + 6 \, \sqrt{-c}\right )}{\left (a x - 1\right )}}{5 \,{\left (a^{3} x - a^{2}\right )} \sqrt{a x + 1}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(a^3*sqrt(-c)*x^3 - 2*a^2*sqrt(-c)*x^2 + 3*a*sqrt(-c)*x + 6*sqrt(-c))*(a*x - 1)/((a^3*x - a^2)*sqrt(a*x +
1))

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Fricas [A]  time = 1.60257, size = 130, normalized size = 1.25 \begin{align*} \frac{2 \,{\left (a^{3} x^{3} - 2 \, a^{2} x^{2} + 3 \, a x + 6\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{5 \,{\left (a^{3} x - a^{2}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(a^3*x^3 - 2*a^2*x^2 + 3*a*x + 6)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1))/(a^3*x - a^2)

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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*(-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 = 1.16256, size = 120, normalized size = 1.15 \begin{align*} \frac{2 \,{\left (\frac{4 \, \sqrt{2} \sqrt{-c} c^{2}}{a} - \frac{{\left (a c x + c\right )}^{2} \sqrt{-a c x - c} + 5 \,{\left (-a c x - c\right )}^{\frac{3}{2}} c + 10 \, \sqrt{-a c x - c} c^{2}}{a}\right )}{\left | c \right |} \mathrm{sgn}\left (a x + 1\right )}{5 \, a c^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(4*sqrt(2)*sqrt(-c)*c^2/a - ((a*c*x + c)^2*sqrt(-a*c*x - c) + 5*(-a*c*x - c)^(3/2)*c + 10*sqrt(-a*c*x - c)
*c^2)/a)*abs(c)*sgn(a*x + 1)/(a*c^3)