3.527 \(\int e^{3 \tanh ^{-1}(a x)} (c-\frac{c}{a x})^{9/2} \, dx\)

Optimal. Leaf size=223 \[ \frac{3 a^2 x^3 (6-17 a x) (a x+1)^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{35 (1-a x)^{9/2}}-\frac{3 a^3 x^4 \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{9/2}}{(1-a x)^{9/2}}+\frac{3 a^{7/2} x^{9/2} \left (c-\frac{c}{a x}\right )^{9/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{(1-a x)^{9/2}}+\frac{6 a x^2 (a x+1)^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{35 (1-a x)^{5/2}}-\frac{2 x (a x+1)^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{7 (1-a x)^{3/2}} \]

[Out]

(-3*a^3*(c - c/(a*x))^(9/2)*x^4*Sqrt[1 + a*x])/(1 - a*x)^(9/2) + (3*a^2*(c - c/(a*x))^(9/2)*x^3*(6 - 17*a*x)*(
1 + a*x)^(3/2))/(35*(1 - a*x)^(9/2)) + (6*a*(c - c/(a*x))^(9/2)*x^2*(1 + a*x)^(3/2))/(35*(1 - a*x)^(5/2)) - (2
*(c - c/(a*x))^(9/2)*x*(1 + a*x)^(3/2))/(7*(1 - a*x)^(3/2)) + (3*a^(7/2)*(c - c/(a*x))^(9/2)*x^(9/2)*ArcSinh[S
qrt[a]*Sqrt[x]])/(1 - a*x)^(9/2)

________________________________________________________________________________________

Rubi [A]  time = 0.183927, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6134, 6129, 97, 150, 143, 47, 54, 215} \[ \frac{3 a^2 x^3 (6-17 a x) (a x+1)^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{35 (1-a x)^{9/2}}-\frac{3 a^3 x^4 \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{9/2}}{(1-a x)^{9/2}}+\frac{3 a^{7/2} x^{9/2} \left (c-\frac{c}{a x}\right )^{9/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{(1-a x)^{9/2}}+\frac{6 a x^2 (a x+1)^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{35 (1-a x)^{5/2}}-\frac{2 x (a x+1)^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{7 (1-a x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a*x])*(c - c/(a*x))^(9/2),x]

[Out]

(-3*a^3*(c - c/(a*x))^(9/2)*x^4*Sqrt[1 + a*x])/(1 - a*x)^(9/2) + (3*a^2*(c - c/(a*x))^(9/2)*x^3*(6 - 17*a*x)*(
1 + a*x)^(3/2))/(35*(1 - a*x)^(9/2)) + (6*a*(c - c/(a*x))^(9/2)*x^2*(1 + a*x)^(3/2))/(35*(1 - a*x)^(5/2)) - (2
*(c - c/(a*x))^(9/2)*x*(1 + a*x)^(3/2))/(7*(1 - a*x)^(3/2)) + (3*a^(7/2)*(c - c/(a*x))^(9/2)*x^(9/2)*ArcSinh[S
qrt[a]*Sqrt[x]])/(1 - a*x)^(9/2)

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 97

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)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

Rule 150

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*(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 - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]

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 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 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)} \left (c-\frac{c}{a x}\right )^{9/2} \, dx &=\frac{\left (\left (c-\frac{c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac{e^{3 \tanh ^{-1}(a x)} (1-a x)^{9/2}}{x^{9/2}} \, dx}{(1-a x)^{9/2}}\\ &=\frac{\left (\left (c-\frac{c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac{(1-a x)^3 (1+a x)^{3/2}}{x^{9/2}} \, dx}{(1-a x)^{9/2}}\\ &=-\frac{2 \left (c-\frac{c}{a x}\right )^{9/2} x (1+a x)^{3/2}}{7 (1-a x)^{3/2}}+\frac{\left (2 \left (c-\frac{c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac{(1-a x)^2 \sqrt{1+a x} \left (-\frac{3 a}{2}-\frac{9 a^2 x}{2}\right )}{x^{7/2}} \, dx}{7 (1-a x)^{9/2}}\\ &=\frac{6 a \left (c-\frac{c}{a x}\right )^{9/2} x^2 (1+a x)^{3/2}}{35 (1-a x)^{5/2}}-\frac{2 \left (c-\frac{c}{a x}\right )^{9/2} x (1+a x)^{3/2}}{7 (1-a x)^{3/2}}+\frac{\left (4 \left (c-\frac{c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac{(1-a x) \sqrt{1+a x} \left (-\frac{27 a^2}{4}+\frac{51 a^3 x}{4}\right )}{x^{5/2}} \, dx}{35 (1-a x)^{9/2}}\\ &=\frac{3 a^2 \left (c-\frac{c}{a x}\right )^{9/2} x^3 (6-17 a x) (1+a x)^{3/2}}{35 (1-a x)^{9/2}}+\frac{6 a \left (c-\frac{c}{a x}\right )^{9/2} x^2 (1+a x)^{3/2}}{35 (1-a x)^{5/2}}-\frac{2 \left (c-\frac{c}{a x}\right )^{9/2} x (1+a x)^{3/2}}{7 (1-a x)^{3/2}}+\frac{\left (3 a^3 \left (c-\frac{c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac{\sqrt{1+a x}}{x^{3/2}} \, dx}{2 (1-a x)^{9/2}}\\ &=-\frac{3 a^3 \left (c-\frac{c}{a x}\right )^{9/2} x^4 \sqrt{1+a x}}{(1-a x)^{9/2}}+\frac{3 a^2 \left (c-\frac{c}{a x}\right )^{9/2} x^3 (6-17 a x) (1+a x)^{3/2}}{35 (1-a x)^{9/2}}+\frac{6 a \left (c-\frac{c}{a x}\right )^{9/2} x^2 (1+a x)^{3/2}}{35 (1-a x)^{5/2}}-\frac{2 \left (c-\frac{c}{a x}\right )^{9/2} x (1+a x)^{3/2}}{7 (1-a x)^{3/2}}+\frac{\left (3 a^4 \left (c-\frac{c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac{1}{\sqrt{x} \sqrt{1+a x}} \, dx}{2 (1-a x)^{9/2}}\\ &=-\frac{3 a^3 \left (c-\frac{c}{a x}\right )^{9/2} x^4 \sqrt{1+a x}}{(1-a x)^{9/2}}+\frac{3 a^2 \left (c-\frac{c}{a x}\right )^{9/2} x^3 (6-17 a x) (1+a x)^{3/2}}{35 (1-a x)^{9/2}}+\frac{6 a \left (c-\frac{c}{a x}\right )^{9/2} x^2 (1+a x)^{3/2}}{35 (1-a x)^{5/2}}-\frac{2 \left (c-\frac{c}{a x}\right )^{9/2} x (1+a x)^{3/2}}{7 (1-a x)^{3/2}}+\frac{\left (3 a^4 \left (c-\frac{c}{a x}\right )^{9/2} x^{9/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+a x^2}} \, dx,x,\sqrt{x}\right )}{(1-a x)^{9/2}}\\ &=-\frac{3 a^3 \left (c-\frac{c}{a x}\right )^{9/2} x^4 \sqrt{1+a x}}{(1-a x)^{9/2}}+\frac{3 a^2 \left (c-\frac{c}{a x}\right )^{9/2} x^3 (6-17 a x) (1+a x)^{3/2}}{35 (1-a x)^{9/2}}+\frac{6 a \left (c-\frac{c}{a x}\right )^{9/2} x^2 (1+a x)^{3/2}}{35 (1-a x)^{5/2}}-\frac{2 \left (c-\frac{c}{a x}\right )^{9/2} x (1+a x)^{3/2}}{7 (1-a x)^{3/2}}+\frac{3 a^{7/2} \left (c-\frac{c}{a x}\right )^{9/2} x^{9/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{(1-a x)^{9/2}}\\ \end{align*}

Mathematica [C]  time = 0.0637342, size = 85, normalized size = 0.38 \[ -\frac{c^4 \sqrt{c-\frac{c}{a x}} \left (35 a^2 x^2 \text{Hypergeometric2F1}\left (-\frac{3}{2},-\frac{3}{2},-\frac{1}{2},-a x\right )+\left (35 a^2 x^2-46 a x+10\right ) (a x+1)^{5/2}\right )}{35 a^4 x^3 \sqrt{1-a x}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(3*ArcTanh[a*x])*(c - c/(a*x))^(9/2),x]

[Out]

-(c^4*Sqrt[c - c/(a*x)]*((1 + a*x)^(5/2)*(10 - 46*a*x + 35*a^2*x^2) + 35*a^2*x^2*Hypergeometric2F1[-3/2, -3/2,
-1/2, -(a*x)]))/(35*a^4*x^3*Sqrt[1 - a*x])

________________________________________________________________________________________

Maple [A]  time = 0.137, size = 172, normalized size = 0.8 \begin{align*}{\frac{{c}^{4}}{70\,{x}^{3} \left ( ax-1 \right ) }\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}\sqrt{-{a}^{2}{x}^{2}+1} \left ( 70\,{a}^{9/2}\sqrt{- \left ( ax+1 \right ) x}{x}^{4}+105\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ){x}^{4}{a}^{4}+328\,{a}^{7/2}{x}^{3}\sqrt{- \left ( ax+1 \right ) x}-24\,{a}^{5/2}{x}^{2}\sqrt{- \left ( ax+1 \right ) x}-52\,{a}^{3/2}x\sqrt{- \left ( ax+1 \right ) x}+20\,\sqrt{a}\sqrt{- \left ( ax+1 \right ) x} \right ){a}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{- \left ( ax+1 \right ) x}}}} \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)^(9/2),x)

[Out]

1/70*(c*(a*x-1)/a/x)^(1/2)/x^3*c^4/a^(9/2)*(-a^2*x^2+1)^(1/2)*(70*a^(9/2)*(-(a*x+1)*x)^(1/2)*x^4+105*arctan(1/
2/a^(1/2)*(2*a*x+1)/(-(a*x+1)*x)^(1/2))*x^4*a^4+328*a^(7/2)*x^3*(-(a*x+1)*x)^(1/2)-24*a^(5/2)*x^2*(-(a*x+1)*x)
^(1/2)-52*a^(3/2)*x*(-(a*x+1)*x)^(1/2)+20*a^(1/2)*(-(a*x+1)*x)^(1/2))/(a*x-1)/(-(a*x+1)*x)^(1/2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )}^{3}{\left (c - \frac{c}{a x}\right )}^{\frac{9}{2}}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{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)^(9/2),x, algorithm="maxima")

[Out]

integrate((a*x + 1)^3*(c - c/(a*x))^(9/2)/(-a^2*x^2 + 1)^(3/2), x)

________________________________________________________________________________________

Fricas [A]  time = 2.15417, size = 802, normalized size = 3.6 \begin{align*} \left [\frac{105 \,{\left (a^{4} c^{4} x^{4} - a^{3} c^{4} x^{3}\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 (35 \, a^{4} c^{4} x^{4} + 164 \, a^{3} c^{4} x^{3} - 12 \, a^{2} c^{4} x^{2} - 26 \, a c^{4} x + 10 \, c^{4}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{140 \,{\left (a^{5} x^{4} - a^{4} x^{3}\right )}}, -\frac{105 \,{\left (a^{4} c^{4} x^{4} - a^{3} c^{4} x^{3}\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 (35 \, a^{4} c^{4} x^{4} + 164 \, a^{3} c^{4} x^{3} - 12 \, a^{2} c^{4} x^{2} - 26 \, a c^{4} x + 10 \, c^{4}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{70 \,{\left (a^{5} x^{4} - a^{4} x^{3}\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)^(9/2),x, algorithm="fricas")

[Out]

[1/140*(105*(a^4*c^4*x^4 - a^3*c^4*x^3)*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*(35*a^4*c^4*x^4 + 164*a^3*c^4*x^3 - 12*a^2*c^4*x^
2 - 26*a*c^4*x + 10*c^4)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^5*x^4 - a^4*x^3), -1/70*(105*(a^4*c^4*
x^4 - a^3*c^4*x^3)*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*(35*a^4*c^4*x^4 + 164*a^3*c^4*x^3 - 12*a^2*c^4*x^2 - 26*a*c^4*x + 10*c^4)*sqrt(-a^2*x^2 + 1)*sqrt(
(a*c*x - c)/(a*x)))/(a^5*x^4 - a^4*x^3)]

________________________________________________________________________________________

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((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a/x)**(9/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )}^{3}{\left (c - \frac{c}{a x}\right )}^{\frac{9}{2}}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{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)^(9/2),x, algorithm="giac")

[Out]

integrate((a*x + 1)^3*(c - c/(a*x))^(9/2)/(-a^2*x^2 + 1)^(3/2), x)