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

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

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 29, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.02, 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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fricas [A]  time = 1.17, size = 50, normalized size = 1.72 \[ \frac {2 \, {\left (a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{2} x - a\right )}} \]

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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giac [A]  time = 0.16, size = 48, normalized size = 1.66 \[ \frac {2 \, {\left (\frac {2 \, \sqrt {2} \sqrt {-c}}{\mathrm {sgn}\relax (c)} - \frac {{\left (-a c x - c\right )}^{\frac {3}{2}}}{c \mathrm {sgn}\left (-a c x - c\right )}\right )}}{3 \, a} \]

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

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maple [A]  time = 0.03, size = 35, normalized size = 1.21 \[ \frac {2 \left (a x +1\right ) \sqrt {-a c x +c}}{3 \sqrt {\frac {a x -1}{a x +1}}\, a} \]

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 = 0.34, size = 26, normalized size = 0.90 \[ \frac {2 \, {\left (a \sqrt {-c} x + \sqrt {-c}\right )} \sqrt {a x + 1}}{3 \, a} \]

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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mupad [B]  time = 0.00, size = 43, normalized size = 1.48 \[ \frac {2\,\sqrt {c-a\,c\,x}\,{\left (a\,x+1\right )}^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{3\,a\,\left (a\,x-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- c \left (a x - 1\right )}}{\sqrt {\frac {a x - 1}{a x + 1}}}\, dx \]

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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