3.588 \(\int \frac{e^{3 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^3} \, dx\)

Optimal. Leaf size=191 \[ -\frac{4 a^2 \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-a x}}+\frac{4 \sqrt{2} a^{5/2} \sqrt{x} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{\sqrt{1-a x}}-\frac{2 (a x+1)^{5/2} \sqrt{c-\frac{c}{a x}}}{5 x^2 \sqrt{1-a x}}-\frac{2 a (a x+1)^{3/2} \sqrt{c-\frac{c}{a x}}}{3 x \sqrt{1-a x}} \]

[Out]

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

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Rubi [A]  time = 0.279413, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6134, 6129, 96, 94, 93, 206} \[ -\frac{4 a^2 \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-a x}}+\frac{4 \sqrt{2} a^{5/2} \sqrt{x} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{\sqrt{1-a x}}-\frac{2 (a x+1)^{5/2} \sqrt{c-\frac{c}{a x}}}{5 x^2 \sqrt{1-a x}}-\frac{2 a (a x+1)^{3/2} \sqrt{c-\frac{c}{a x}}}{3 x \sqrt{1-a x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6134

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

Rule 6129

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Dist[c^p, Int[(u*(1 + (d*x)/c)
^p*(1 + a*x)^(n/2))/(1 - a*x)^(n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ
[p] || GtQ[c, 0])

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

Mathematica [A]  time = 0.0599593, size = 101, normalized size = 0.53 \[ \frac{2 \sqrt{c-\frac{c}{a x}} \left (30 \sqrt{2} a^{5/2} x^{5/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )-\sqrt{a x+1} \left (38 a^2 x^2+11 a x+3\right )\right )}{15 x^2 \sqrt{1-a x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.145, size = 181, normalized size = 1. \begin{align*}{\frac{\sqrt{2}}{15\,{x}^{2} \left ( ax-1 \right ) }\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}\sqrt{-{a}^{2}{x}^{2}+1} \left ( 38\,{a}^{2}\sqrt{2}\sqrt{-{a}^{-1}}{x}^{2}\sqrt{- \left ( ax+1 \right ) x}+30\,{a}^{2}\ln \left ({\frac{1}{ax-1} \left ( 2\,\sqrt{2}\sqrt{-{a}^{-1}}\sqrt{- \left ( ax+1 \right ) x}a-3\,ax-1 \right ) } \right ){x}^{3}+11\,x\sqrt{- \left ( ax+1 \right ) x}a\sqrt{2}\sqrt{-{a}^{-1}}+3\,\sqrt{- \left ( ax+1 \right ) x}\sqrt{2}\sqrt{-{a}^{-1}} \right ){\frac{1}{\sqrt{- \left ( ax+1 \right ) x}}}{\frac{1}{\sqrt{-{a}^{-1}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.57177, size = 728, normalized size = 3.81 \begin{align*} \left [\frac{15 \, \sqrt{2}{\left (a^{3} x^{3} - a^{2} x^{2}\right )} \sqrt{-c} \log \left (-\frac{17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x + 4 \, \sqrt{2}{\left (3 \, a^{2} x^{2} + a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 2 \,{\left (38 \, a^{2} x^{2} + 11 \, a x + 3\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{15 \,{\left (a x^{3} - x^{2}\right )}}, -\frac{2 \,{\left (15 \, \sqrt{2}{\left (a^{3} x^{3} - a^{2} x^{2}\right )} \sqrt{c} \arctan \left (\frac{2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) -{\left (38 \, a^{2} x^{2} + 11 \, a x + 3\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}\right )}}{15 \,{\left (a x^{3} - x^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/15*(15*sqrt(2)*(a^3*x^3 - a^2*x^2)*sqrt(-c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 13*a*c*x + 4*sqrt(2)*(3*a^2*
x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) + 2*(38
*a^2*x^2 + 11*a*x + 3)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a*x^3 - x^2), -2/15*(15*sqrt(2)*(a^3*x^3 -
a^2*x^2)*sqrt(c)*arctan(2*sqrt(2)*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c
*x - c)) - (38*a^2*x^2 + 11*a*x + 3)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a*x^3 - x^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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