3.537 \(\int e^{-\tanh ^{-1}(a x)} (c-\frac{c}{a x})^{5/2} \, dx\)

Optimal. Leaf size=137 \[ \frac{7 a^{3/2} x^{5/2} \left (c-\frac{c}{a x}\right )^{5/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{(1-a x)^{5/2}}+\frac{a x^2 (18-a x) \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{5/2}}{3 (1-a x)^{5/2}}-\frac{2 x \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{5/2}}{3 \sqrt{1-a x}} \]

[Out]

(-2*(c - c/(a*x))^(5/2)*x*Sqrt[1 + a*x])/(3*Sqrt[1 - a*x]) + (a*(c - c/(a*x))^(5/2)*x^2*(18 - a*x)*Sqrt[1 + a*
x])/(3*(1 - a*x)^(5/2)) + (7*a^(3/2)*(c - c/(a*x))^(5/2)*x^(5/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(1 - a*x)^(5/2)

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Rubi [A]  time = 0.154269, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6134, 6129, 98, 143, 54, 215} \[ \frac{7 a^{3/2} x^{5/2} \left (c-\frac{c}{a x}\right )^{5/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{(1-a x)^{5/2}}+\frac{a x^2 (18-a x) \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{5/2}}{3 (1-a x)^{5/2}}-\frac{2 x \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{5/2}}{3 \sqrt{1-a x}} \]

Antiderivative was successfully verified.

[In]

Int[(c - c/(a*x))^(5/2)/E^ArcTanh[a*x],x]

[Out]

(-2*(c - c/(a*x))^(5/2)*x*Sqrt[1 + a*x])/(3*Sqrt[1 - a*x]) + (a*(c - c/(a*x))^(5/2)*x^2*(18 - a*x)*Sqrt[1 + a*
x])/(3*(1 - a*x)^(5/2)) + (7*a^(3/2)*(c - c/(a*x))^(5/2)*x^(5/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(1 - a*x)^(5/2)

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 98

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

Rule 143

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

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^{-\tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^{5/2} \, dx &=\frac{\left (\left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}\right ) \int \frac{e^{-\tanh ^{-1}(a x)} (1-a x)^{5/2}}{x^{5/2}} \, dx}{(1-a x)^{5/2}}\\ &=\frac{\left (\left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}\right ) \int \frac{(1-a x)^3}{x^{5/2} \sqrt{1+a x}} \, dx}{(1-a x)^{5/2}}\\ &=-\frac{2 \left (c-\frac{c}{a x}\right )^{5/2} x \sqrt{1+a x}}{3 \sqrt{1-a x}}-\frac{\left (2 \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}\right ) \int \frac{(1-a x) \left (\frac{9 a}{2}-\frac{a^2 x}{2}\right )}{x^{3/2} \sqrt{1+a x}} \, dx}{3 (1-a x)^{5/2}}\\ &=-\frac{2 \left (c-\frac{c}{a x}\right )^{5/2} x \sqrt{1+a x}}{3 \sqrt{1-a x}}+\frac{a \left (c-\frac{c}{a x}\right )^{5/2} x^2 (18-a x) \sqrt{1+a x}}{3 (1-a x)^{5/2}}+\frac{\left (7 a^2 \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}\right ) \int \frac{1}{\sqrt{x} \sqrt{1+a x}} \, dx}{2 (1-a x)^{5/2}}\\ &=-\frac{2 \left (c-\frac{c}{a x}\right )^{5/2} x \sqrt{1+a x}}{3 \sqrt{1-a x}}+\frac{a \left (c-\frac{c}{a x}\right )^{5/2} x^2 (18-a x) \sqrt{1+a x}}{3 (1-a x)^{5/2}}+\frac{\left (7 a^2 \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+a x^2}} \, dx,x,\sqrt{x}\right )}{(1-a x)^{5/2}}\\ &=-\frac{2 \left (c-\frac{c}{a x}\right )^{5/2} x \sqrt{1+a x}}{3 \sqrt{1-a x}}+\frac{a \left (c-\frac{c}{a x}\right )^{5/2} x^2 (18-a x) \sqrt{1+a x}}{3 (1-a x)^{5/2}}+\frac{7 a^{3/2} \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{(1-a x)^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0613968, size = 87, normalized size = 0.64 \[ -\frac{c^2 \sqrt{c-\frac{c}{a x}} \left (\sqrt{a x+1} \left (3 a^2 x^2-22 a x+2\right )-21 a^{3/2} x^{3/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )\right )}{3 a^2 x \sqrt{1-a x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c - c/(a*x))^(5/2)/E^ArcTanh[a*x],x]

[Out]

-(c^2*Sqrt[c - c/(a*x)]*(Sqrt[1 + a*x]*(2 - 22*a*x + 3*a^2*x^2) - 21*a^(3/2)*x^(3/2)*ArcSinh[Sqrt[a]*Sqrt[x]])
)/(3*a^2*x*Sqrt[1 - a*x])

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Maple [A]  time = 0.159, size = 136, normalized size = 1. \begin{align*}{\frac{{c}^{2}}{6\, \left ( ax-1 \right ) x}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}\sqrt{-{a}^{2}{x}^{2}+1} \left ( 6\,{a}^{5/2}{x}^{2}\sqrt{- \left ( ax+1 \right ) x}+21\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ){x}^{2}{a}^{2}-44\,{a}^{3/2}x\sqrt{- \left ( ax+1 \right ) x}+4\,\sqrt{a}\sqrt{- \left ( ax+1 \right ) x} \right ){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((c-c/a/x)^(5/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x)

[Out]

1/6*(c*(a*x-1)/a/x)^(1/2)/x*c^2/a^(5/2)*(-a^2*x^2+1)^(1/2)*(6*a^(5/2)*x^2*(-(a*x+1)*x)^(1/2)+21*arctan(1/2/a^(
1/2)*(2*a*x+1)/(-(a*x+1)*x)^(1/2))*x^2*a^2-44*a^(3/2)*x*(-(a*x+1)*x)^(1/2)+4*a^(1/2)*(-(a*x+1)*x)^(1/2))/(a*x-
1)/(-(a*x+1)*x)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a^{2} x^{2} + 1}{\left (c - \frac{c}{a x}\right )}^{\frac{5}{2}}}{a x + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a/x)^(5/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-a^2*x^2 + 1)*(c - c/(a*x))^(5/2)/(a*x + 1), x)

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Fricas [A]  time = 2.46867, size = 680, normalized size = 4.96 \begin{align*} \left [\frac{21 \,{\left (a^{2} c^{2} x^{2} - a c^{2} x\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 (3 \, a^{2} c^{2} x^{2} - 22 \, a c^{2} x + 2 \, c^{2}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{12 \,{\left (a^{3} x^{2} - a^{2} x\right )}}, -\frac{21 \,{\left (a^{2} c^{2} x^{2} - a c^{2} x\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 (3 \, a^{2} c^{2} x^{2} - 22 \, a c^{2} x + 2 \, c^{2}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{6 \,{\left (a^{3} x^{2} - a^{2} x\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a/x)^(5/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

[1/12*(21*(a^2*c^2*x^2 - a*c^2*x)*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*(3*a^2*c^2*x^2 - 22*a*c^2*x + 2*c^2)*sqrt(-a^2*x^2 + 1)
*sqrt((a*c*x - c)/(a*x)))/(a^3*x^2 - a^2*x), -1/6*(21*(a^2*c^2*x^2 - a*c^2*x)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 +
1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*(3*a^2*c^2*x^2 - 22*a*c^2*x + 2*c^2)*sq
rt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*x^2 - a^2*x)]

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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((c-c/a/x)**(5/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a^{2} x^{2} + 1}{\left (c - \frac{c}{a x}\right )}^{\frac{5}{2}}}{a x + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a/x)^(5/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(-a^2*x^2 + 1)*(c - c/(a*x))^(5/2)/(a*x + 1), x)