3.312 $$\int e^{3 \coth ^{-1}(a x)} x \sqrt{c-a c x} \, dx$$

Optimal. Leaf size=211 $\frac{4 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{a^2 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{a^{5/2} \sqrt{1-\frac{1}{a x}}}+\frac{2 x^2 \left (\frac{1}{a x}+1\right )^{5/2} \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}}}+\frac{2 x \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{3 a \sqrt{1-\frac{1}{a x}}}$

[Out]

(4*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(a^2*Sqrt[1 - 1/(a*x)]) + (2*(1 + 1/(a*x))^(3/2)*x*Sqrt[c - a*c*x])/(3*a
*Sqrt[1 - 1/(a*x)]) + (2*(1 + 1/(a*x))^(5/2)*x^2*Sqrt[c - a*c*x])/(5*Sqrt[1 - 1/(a*x)]) - (4*Sqrt[2]*Sqrt[x^(-
1)]*Sqrt[c - a*c*x]*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/(a^(5/2)*Sqrt[1 - 1/(a*x)])

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Rubi [A]  time = 0.236304, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.286, Rules used = {6176, 6181, 96, 94, 93, 206} $\frac{4 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{a^2 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{a^{5/2} \sqrt{1-\frac{1}{a x}}}+\frac{2 x^2 \left (\frac{1}{a x}+1\right )^{5/2} \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}}}+\frac{2 x \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{3 a \sqrt{1-\frac{1}{a x}}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(4*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(a^2*Sqrt[1 - 1/(a*x)]) + (2*(1 + 1/(a*x))^(3/2)*x*Sqrt[c - a*c*x])/(3*a
*Sqrt[1 - 1/(a*x)]) + (2*(1 + 1/(a*x))^(5/2)*x^2*Sqrt[c - a*c*x])/(5*Sqrt[1 - 1/(a*x)]) - (4*Sqrt[2]*Sqrt[x^(-
1)]*Sqrt[c - a*c*x]*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/(a^(5/2)*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 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 94

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int e^{3 \coth ^{-1}(a x)} x \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^{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{\left (1+\frac{x}{a}\right )^{3/2}}{x^{7/2} \left (1-\frac{x}{a}\right )} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{2 \left (1+\frac{1}{a x}\right )^{5/2} x^2 \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}}}-\frac{\left (\sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{3/2}}{x^{5/2} \left (1-\frac{x}{a}\right )} \, dx,x,\frac{1}{x}\right )}{a \sqrt{1-\frac{1}{a x}}}\\ &=\frac{2 \left (1+\frac{1}{a x}\right )^{3/2} x \sqrt{c-a c x}}{3 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \left (1+\frac{1}{a x}\right )^{5/2} x^2 \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}}}-\frac{\left (2 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}}}{x^{3/2} \left (1-\frac{x}{a}\right )} \, dx,x,\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{4 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{a^2 \sqrt{1-\frac{1}{a x}}}+\frac{2 \left (1+\frac{1}{a x}\right )^{3/2} x \sqrt{c-a c x}}{3 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \left (1+\frac{1}{a x}\right )^{5/2} x^2 \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}}}-\frac{\left (4 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{a^3 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{4 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{a^2 \sqrt{1-\frac{1}{a x}}}+\frac{2 \left (1+\frac{1}{a x}\right )^{3/2} x \sqrt{c-a c x}}{3 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \left (1+\frac{1}{a x}\right )^{5/2} x^2 \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}}}-\frac{\left (8 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{2 x^2}{a}} \, dx,x,\frac{\sqrt{\frac{1}{x}}}{\sqrt{1+\frac{1}{a x}}}\right )}{a^3 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{4 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{a^2 \sqrt{1-\frac{1}{a x}}}+\frac{2 \left (1+\frac{1}{a x}\right )^{3/2} x \sqrt{c-a c x}}{3 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \left (1+\frac{1}{a x}\right )^{5/2} x^2 \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{1+\frac{1}{a x}}}\right )}{a^{5/2} \sqrt{1-\frac{1}{a x}}}\\ \end{align*}

Mathematica [A]  time = 0.0740351, size = 114, normalized size = 0.54 $\frac{2 \sqrt{c-a c x} \left (\sqrt{a} \sqrt{\frac{1}{a x}+1} \left (3 a^2 x^2+11 a x+38\right )-30 \sqrt{2} \sqrt{\frac{1}{x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )\right )}{15 a^{5/2} \sqrt{1-\frac{1}{a x}}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*Sqrt[c - a*c*x]*(Sqrt[a]*Sqrt[1 + 1/(a*x)]*(38 + 11*a*x + 3*a^2*x^2) - 30*Sqrt[2]*Sqrt[x^(-1)]*ArcTanh[(Sqr
t[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/(15*a^(5/2)*Sqrt[1 - 1/(a*x)])

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Maple [A]  time = 0.178, size = 125, normalized size = 0.6 \begin{align*} -{\frac{2\,ax-2}{ \left ( 15\,ax+15 \right ){a}^{2}}\sqrt{-c \left ( ax-1 \right ) } \left ( -3\,{x}^{2}{a}^{2}\sqrt{-c \left ( ax+1 \right ) }+30\,\sqrt{c}\sqrt{2}\arctan \left ( 1/2\,{\frac{\sqrt{-c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) -11\,xa\sqrt{-c \left ( ax+1 \right ) }-38\,\sqrt{-c \left ( ax+1 \right ) } \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{-c \left ( ax+1 \right ) }}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15/((a*x-1)/(a*x+1))^(3/2)*(a*x-1)/(a*x+1)*(-c*(a*x-1))^(1/2)*(-3*x^2*a^2*(-c*(a*x+1))^(1/2)+30*c^(1/2)*2^(
1/2)*arctan(1/2*(-c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))-11*x*a*(-c*(a*x+1))^(1/2)-38*(-c*(a*x+1))^(1/2))/(-c*(a*x+
1))^(1/2)/a^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a c x + c} x}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.63742, size = 655, normalized size = 3.1 \begin{align*} \left [\frac{2 \,{\left (15 \, \sqrt{2}{\left (a x - 1\right )} \sqrt{-c} \log \left (-\frac{a^{2} c x^{2} + 2 \, a c x + 2 \, \sqrt{2} \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) +{\left (3 \, a^{3} x^{3} + 14 \, a^{2} x^{2} + 49 \, a x + 38\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{15 \,{\left (a^{3} x - a^{2}\right )}}, -\frac{2 \,{\left (30 \, \sqrt{2}{\left (a x - 1\right )} \sqrt{c} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}}}{a c x - c}\right ) -{\left (3 \, a^{3} x^{3} + 14 \, a^{2} x^{2} + 49 \, a x + 38\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{15 \,{\left (a^{3} x - a^{2}\right )}}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[2/15*(15*sqrt(2)*(a*x - 1)*sqrt(-c)*log(-(a^2*c*x^2 + 2*a*c*x + 2*sqrt(2)*sqrt(-a*c*x + c)*(a*x + 1)*sqrt(-c)
*sqrt((a*x - 1)/(a*x + 1)) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + (3*a^3*x^3 + 14*a^2*x^2 + 49*a*x + 38)*sqrt(-a*c*x
+ c)*sqrt((a*x - 1)/(a*x + 1)))/(a^3*x - a^2), -2/15*(30*sqrt(2)*(a*x - 1)*sqrt(c)*arctan(sqrt(2)*sqrt(-a*c*x
+ c)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))/(a*c*x - c)) - (3*a^3*x^3 + 14*a^2*x^2 + 49*a*x + 38)*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(1/((a*x-1)/(a*x+1))**(3/2)*x*(-a*c*x+c)**(1/2),x)

[Out]

Timed out

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Giac [C]  time = 1.2015, size = 180, normalized size = 0.85 \begin{align*} -\frac{60 i \, \sqrt{2} \sqrt{-c} \arctan \left (-i\right ) - 104 \, \sqrt{2} \sqrt{-c}}{15 \, a^{2} \mathrm{sgn}\left (c\right )} - \frac{2 \,{\left (30 \, \sqrt{2} c^{\frac{5}{2}} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x - c}}{2 \, \sqrt{c}}\right ) - 3 \,{\left (a c x + c\right )}^{2} \sqrt{-a c x - c} + 5 \,{\left (-a c x - c\right )}^{\frac{3}{2}} c - 30 \, \sqrt{-a c x - c} c^{2}\right )}}{15 \, a^{2} c^{2} \mathrm{sgn}\left (-a c x - c\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(60*I*sqrt(2)*sqrt(-c)*arctan(-I) - 104*sqrt(2)*sqrt(-c))/(a^2*sgn(c)) - 2/15*(30*sqrt(2)*c^(5/2)*arctan
(1/2*sqrt(2)*sqrt(-a*c*x - c)/sqrt(c)) - 3*(a*c*x + c)^2*sqrt(-a*c*x - c) + 5*(-a*c*x - c)^(3/2)*c - 30*sqrt(-
a*c*x - c)*c^2)/(a^2*c^2*sgn(-a*c*x - c))