3.609 \(\int e^{-3 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^2 \, dx\)

Optimal. Leaf size=218 \[ \frac{119 x \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{8 a^2 \sqrt{1-a x}}-\frac{119 \sqrt{x} \sqrt{c-\frac{c}{a x}} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{8 a^{5/2} \sqrt{1-a x}}+\frac{x^3 \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-a x}}+\frac{8 x^3 \sqrt{c-\frac{c}{a x}}}{\sqrt{1-a x} \sqrt{a x+1}}-\frac{119 x^2 \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{12 a \sqrt{1-a x}} \]

[Out]

(8*Sqrt[c - c/(a*x)]*x^3)/(Sqrt[1 - a*x]*Sqrt[1 + a*x]) + (119*Sqrt[c - c/(a*x)]*x*Sqrt[1 + a*x])/(8*a^2*Sqrt[
1 - a*x]) - (119*Sqrt[c - c/(a*x)]*x^2*Sqrt[1 + a*x])/(12*a*Sqrt[1 - a*x]) + (Sqrt[c - c/(a*x)]*x^3*Sqrt[1 + a
*x])/(3*Sqrt[1 - a*x]) - (119*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcSinh[Sqrt[a]*Sqrt[x]])/(8*a^(5/2)*Sqrt[1 - a*x])

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Rubi [A]  time = 0.250208, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {6134, 6129, 89, 80, 50, 54, 215} \[ \frac{119 x \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{8 a^2 \sqrt{1-a x}}-\frac{119 \sqrt{x} \sqrt{c-\frac{c}{a x}} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{8 a^{5/2} \sqrt{1-a x}}+\frac{x^3 \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-a x}}+\frac{8 x^3 \sqrt{c-\frac{c}{a x}}}{\sqrt{1-a x} \sqrt{a x+1}}-\frac{119 x^2 \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{12 a \sqrt{1-a x}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[c - c/(a*x)]*x^2)/E^(3*ArcTanh[a*x]),x]

[Out]

(8*Sqrt[c - c/(a*x)]*x^3)/(Sqrt[1 - a*x]*Sqrt[1 + a*x]) + (119*Sqrt[c - c/(a*x)]*x*Sqrt[1 + a*x])/(8*a^2*Sqrt[
1 - a*x]) - (119*Sqrt[c - c/(a*x)]*x^2*Sqrt[1 + a*x])/(12*a*Sqrt[1 - a*x]) + (Sqrt[c - c/(a*x)]*x^3*Sqrt[1 + a
*x])/(3*Sqrt[1 - a*x]) - (119*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcSinh[Sqrt[a]*Sqrt[x]])/(8*a^(5/2)*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 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 80

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

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int e^{-3 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^2 \, dx &=\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int e^{-3 \tanh ^{-1}(a x)} x^{3/2} \sqrt{1-a x} \, dx}{\sqrt{1-a x}}\\ &=\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{x^{3/2} (1-a x)^2}{(1+a x)^{3/2}} \, dx}{\sqrt{1-a x}}\\ &=\frac{8 \sqrt{c-\frac{c}{a x}} x^3}{\sqrt{1-a x} \sqrt{1+a x}}-\frac{\left (2 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{x^{3/2} \left (\frac{19 a^2}{2}-\frac{a^3 x}{2}\right )}{\sqrt{1+a x}} \, dx}{a^2 \sqrt{1-a x}}\\ &=\frac{8 \sqrt{c-\frac{c}{a x}} x^3}{\sqrt{1-a x} \sqrt{1+a x}}+\frac{\sqrt{c-\frac{c}{a x}} x^3 \sqrt{1+a x}}{3 \sqrt{1-a x}}-\frac{\left (119 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{x^{3/2}}{\sqrt{1+a x}} \, dx}{6 \sqrt{1-a x}}\\ &=\frac{8 \sqrt{c-\frac{c}{a x}} x^3}{\sqrt{1-a x} \sqrt{1+a x}}-\frac{119 \sqrt{c-\frac{c}{a x}} x^2 \sqrt{1+a x}}{12 a \sqrt{1-a x}}+\frac{\sqrt{c-\frac{c}{a x}} x^3 \sqrt{1+a x}}{3 \sqrt{1-a x}}+\frac{\left (119 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{\sqrt{x}}{\sqrt{1+a x}} \, dx}{8 a \sqrt{1-a x}}\\ &=\frac{8 \sqrt{c-\frac{c}{a x}} x^3}{\sqrt{1-a x} \sqrt{1+a x}}+\frac{119 \sqrt{c-\frac{c}{a x}} x \sqrt{1+a x}}{8 a^2 \sqrt{1-a x}}-\frac{119 \sqrt{c-\frac{c}{a x}} x^2 \sqrt{1+a x}}{12 a \sqrt{1-a x}}+\frac{\sqrt{c-\frac{c}{a x}} x^3 \sqrt{1+a x}}{3 \sqrt{1-a x}}-\frac{\left (119 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{1}{\sqrt{x} \sqrt{1+a x}} \, dx}{16 a^2 \sqrt{1-a x}}\\ &=\frac{8 \sqrt{c-\frac{c}{a x}} x^3}{\sqrt{1-a x} \sqrt{1+a x}}+\frac{119 \sqrt{c-\frac{c}{a x}} x \sqrt{1+a x}}{8 a^2 \sqrt{1-a x}}-\frac{119 \sqrt{c-\frac{c}{a x}} x^2 \sqrt{1+a x}}{12 a \sqrt{1-a x}}+\frac{\sqrt{c-\frac{c}{a x}} x^3 \sqrt{1+a x}}{3 \sqrt{1-a x}}-\frac{\left (119 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+a x^2}} \, dx,x,\sqrt{x}\right )}{8 a^2 \sqrt{1-a x}}\\ &=\frac{8 \sqrt{c-\frac{c}{a x}} x^3}{\sqrt{1-a x} \sqrt{1+a x}}+\frac{119 \sqrt{c-\frac{c}{a x}} x \sqrt{1+a x}}{8 a^2 \sqrt{1-a x}}-\frac{119 \sqrt{c-\frac{c}{a x}} x^2 \sqrt{1+a x}}{12 a \sqrt{1-a x}}+\frac{\sqrt{c-\frac{c}{a x}} x^3 \sqrt{1+a x}}{3 \sqrt{1-a x}}-\frac{119 \sqrt{c-\frac{c}{a x}} \sqrt{x} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{8 a^{5/2} \sqrt{1-a x}}\\ \end{align*}

Mathematica [A]  time = 0.0587841, size = 100, normalized size = 0.46 \[ \frac{\sqrt{x} \sqrt{c-\frac{c}{a x}} \left (\sqrt{a} \sqrt{x} \left (8 a^3 x^3-38 a^2 x^2+119 a x+357\right )-357 \sqrt{a x+1} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )\right )}{24 a^{5/2} \sqrt{1-a^2 x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(Sqrt[c - c/(a*x)]*x^2)/E^(3*ArcTanh[a*x]),x]

[Out]

(Sqrt[c - c/(a*x)]*Sqrt[x]*(Sqrt[a]*Sqrt[x]*(357 + 119*a*x - 38*a^2*x^2 + 8*a^3*x^3) - 357*Sqrt[1 + a*x]*ArcSi
nh[Sqrt[a]*Sqrt[x]]))/(24*a^(5/2)*Sqrt[1 - a^2*x^2])

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Maple [A]  time = 0.138, size = 176, normalized size = 0.8 \begin{align*} -{\frac{x}{ \left ( 48\,ax+48 \right ) \left ( ax-1 \right ) }\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 16\,{a}^{7/2}{x}^{3}\sqrt{- \left ( ax+1 \right ) x}-76\,{a}^{5/2}{x}^{2}\sqrt{- \left ( ax+1 \right ) x}+238\,{a}^{3/2}x\sqrt{- \left ( ax+1 \right ) x}+357\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ) xa+714\,\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}+357\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ) \right ) \sqrt{-{a}^{2}{x}^{2}+1}{a}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{- \left ( ax+1 \right ) x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*(c*(a*x-1)/a/x)^(1/2)*x*(16*a^(7/2)*x^3*(-(a*x+1)*x)^(1/2)-76*a^(5/2)*x^2*(-(a*x+1)*x)^(1/2)+238*a^(3/2)
*x*(-(a*x+1)*x)^(1/2)+357*arctan(1/2/a^(1/2)*(2*a*x+1)/(-(a*x+1)*x)^(1/2))*x*a+714*a^(1/2)*(-(a*x+1)*x)^(1/2)+
357*arctan(1/2/a^(1/2)*(2*a*x+1)/(-(a*x+1)*x)^(1/2)))/a^(5/2)*(-a^2*x^2+1)^(1/2)/(a*x+1)/(-(a*x+1)*x)^(1/2)/(a
*x-1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.52247, size = 683, normalized size = 3.13 \begin{align*} \left [\frac{357 \,{\left (a^{2} x^{2} - 1\right )} \sqrt{-c} \log \left (-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \,{\left (2 \, 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 x - 1}\right ) - 4 \,{\left (8 \, a^{4} x^{4} - 38 \, a^{3} x^{3} + 119 \, a^{2} x^{2} + 357 \, a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{96 \,{\left (a^{5} x^{2} - a^{3}\right )}}, \frac{357 \,{\left (a^{2} x^{2} - 1\right )} \sqrt{c} \arctan \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \,{\left (8 \, a^{4} x^{4} - 38 \, a^{3} x^{3} + 119 \, a^{2} x^{2} + 357 \, a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{48 \,{\left (a^{5} x^{2} - a^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/96*(357*(a^2*x^2 - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c
)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) - 4*(8*a^4*x^4 - 38*a^3*x^3 + 119*a^2*x^2 + 357*a*x)*sqrt(-a^2*x^2 +
1)*sqrt((a*c*x - c)/(a*x)))/(a^5*x^2 - a^3), 1/48*(357*(a^2*x^2 - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sq
rt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*(8*a^4*x^4 - 38*a^3*x^3 + 119*a^2*x^2 + 357*a*x
)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^5*x^2 - a^3)]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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