### 3.230 $$\int e^{\coth ^{-1}(a x)} \sqrt{c-a c x} \, dx$$

Optimal. Leaf size=29 $\frac{2 (a x+1) \sqrt{c-a c x} e^{\coth ^{-1}(a x)}}{3 a}$

[Out]

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

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Rubi [A]  time = 0.0341808, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.056, Rules used = {6174} $\frac{2 (a x+1) \sqrt{c-a c x} e^{\coth ^{-1}(a x)}}{3 a}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^ArcCoth[a*x]*Sqrt[c - a*c*x],x]

[Out]

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

Rule 6174

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Simp[((1 + a*x)*(c + d*x)^p*E^(n*Arc
Coth[a*x]))/(a*(p + 1)), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a*c + d, 0] && EqQ[p, n/2] &&  !IntegerQ[n/2]

Rubi steps

\begin{align*} \int e^{\coth ^{-1}(a x)} \sqrt{c-a c x} \, dx &=\frac{2 e^{\coth ^{-1}(a x)} (1+a x) \sqrt{c-a c x}}{3 a}\\ \end{align*}

Mathematica [A]  time = 0.0198929, size = 43, normalized size = 1.48 $\frac{2 x \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.04, size = 35, normalized size = 1.2 \begin{align*}{\frac{2\,ax+2}{3\,a}\sqrt{-acx+c}{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.06179, size = 35, normalized size = 1.21 \begin{align*} \frac{2 \,{\left (a \sqrt{-c} x + \sqrt{-c}\right )} \sqrt{a x + 1}}{3 \, a} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.51021, size = 111, normalized size = 3.83 \begin{align*} \frac{2 \,{\left (a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{3 \,{\left (a^{2} x - a\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- c \left (a x - 1\right )}}{\sqrt{\frac{a x - 1}{a x + 1}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-c*(a*x - 1))/sqrt((a*x - 1)/(a*x + 1)), x)

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Giac [A]  time = 1.16605, size = 66, normalized size = 2.28 \begin{align*} \frac{2 \,{\left (\frac{2 \, \sqrt{2} \sqrt{-c} c}{\mathrm{sgn}\left (c\right )} - \frac{{\left (-a c x - c\right )}^{\frac{3}{2}}}{\mathrm{sgn}\left (-a c x - c\right )}\right )}}{3 \, a c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(2*sqrt(2)*sqrt(-c)*c/sgn(c) - (-a*c*x - c)^(3/2)/sgn(-a*c*x - c))/(a*c)