3.511 \(\int e^{\tanh ^{-1}(a x)} (c-\frac{c}{a x})^{7/2} \, dx\)

Optimal. Leaf size=181 \[ -\frac{a^2 x^3 \sqrt{a x+1} (31 a x+18) \left (c-\frac{c}{a x}\right )^{7/2}}{15 (1-a x)^{7/2}}+\frac{5 a^{5/2} x^{7/2} \left (c-\frac{c}{a x}\right )^{7/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{(1-a x)^{7/2}}+\frac{2 a x^2 \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{7/2}}{3 (1-a x)^{3/2}}-\frac{2 x \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{7/2}}{5 \sqrt{1-a x}} \]

[Out]

(2*a*(c - c/(a*x))^(7/2)*x^2*Sqrt[1 + a*x])/(3*(1 - a*x)^(3/2)) - (2*(c - c/(a*x))^(7/2)*x*Sqrt[1 + a*x])/(5*S
qrt[1 - a*x]) - (a^2*(c - c/(a*x))^(7/2)*x^3*Sqrt[1 + a*x]*(18 + 31*a*x))/(15*(1 - a*x)^(7/2)) + (5*a^(5/2)*(c
- c/(a*x))^(7/2)*x^(7/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(1 - a*x)^(7/2)

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Rubi [A]  time = 0.165372, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {6134, 6129, 97, 150, 143, 54, 215} \[ -\frac{a^2 x^3 \sqrt{a x+1} (31 a x+18) \left (c-\frac{c}{a x}\right )^{7/2}}{15 (1-a x)^{7/2}}+\frac{5 a^{5/2} x^{7/2} \left (c-\frac{c}{a x}\right )^{7/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{(1-a x)^{7/2}}+\frac{2 a x^2 \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{7/2}}{3 (1-a x)^{3/2}}-\frac{2 x \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{7/2}}{5 \sqrt{1-a x}} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]*(c - c/(a*x))^(7/2),x]

[Out]

(2*a*(c - c/(a*x))^(7/2)*x^2*Sqrt[1 + a*x])/(3*(1 - a*x)^(3/2)) - (2*(c - c/(a*x))^(7/2)*x*Sqrt[1 + a*x])/(5*S
qrt[1 - a*x]) - (a^2*(c - c/(a*x))^(7/2)*x^3*Sqrt[1 + a*x]*(18 + 31*a*x))/(15*(1 - a*x)^(7/2)) + (5*a^(5/2)*(c
- c/(a*x))^(7/2)*x^(7/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(1 - a*x)^(7/2)

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 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 )^{7/2} \, dx &=\frac{\left (\left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}\right ) \int \frac{e^{\tanh ^{-1}(a x)} (1-a x)^{7/2}}{x^{7/2}} \, dx}{(1-a x)^{7/2}}\\ &=\frac{\left (\left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}\right ) \int \frac{(1-a x)^3 \sqrt{1+a x}}{x^{7/2}} \, dx}{(1-a x)^{7/2}}\\ &=-\frac{2 \left (c-\frac{c}{a x}\right )^{7/2} x \sqrt{1+a x}}{5 \sqrt{1-a x}}+\frac{\left (2 \left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}\right ) \int \frac{(1-a x)^2 \left (-\frac{5 a}{2}-\frac{7 a^2 x}{2}\right )}{x^{5/2} \sqrt{1+a x}} \, dx}{5 (1-a x)^{7/2}}\\ &=\frac{2 a \left (c-\frac{c}{a x}\right )^{7/2} x^2 \sqrt{1+a x}}{3 (1-a x)^{3/2}}-\frac{2 \left (c-\frac{c}{a x}\right )^{7/2} x \sqrt{1+a x}}{5 \sqrt{1-a x}}+\frac{\left (4 \left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}\right ) \int \frac{(1-a x) \left (\frac{9 a^2}{4}+\frac{31 a^3 x}{4}\right )}{x^{3/2} \sqrt{1+a x}} \, dx}{15 (1-a x)^{7/2}}\\ &=\frac{2 a \left (c-\frac{c}{a x}\right )^{7/2} x^2 \sqrt{1+a x}}{3 (1-a x)^{3/2}}-\frac{2 \left (c-\frac{c}{a x}\right )^{7/2} x \sqrt{1+a x}}{5 \sqrt{1-a x}}-\frac{a^2 \left (c-\frac{c}{a x}\right )^{7/2} x^3 \sqrt{1+a x} (18+31 a x)}{15 (1-a x)^{7/2}}+\frac{\left (5 a^3 \left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}\right ) \int \frac{1}{\sqrt{x} \sqrt{1+a x}} \, dx}{2 (1-a x)^{7/2}}\\ &=\frac{2 a \left (c-\frac{c}{a x}\right )^{7/2} x^2 \sqrt{1+a x}}{3 (1-a x)^{3/2}}-\frac{2 \left (c-\frac{c}{a x}\right )^{7/2} x \sqrt{1+a x}}{5 \sqrt{1-a x}}-\frac{a^2 \left (c-\frac{c}{a x}\right )^{7/2} x^3 \sqrt{1+a x} (18+31 a x)}{15 (1-a x)^{7/2}}+\frac{\left (5 a^3 \left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+a x^2}} \, dx,x,\sqrt{x}\right )}{(1-a x)^{7/2}}\\ &=\frac{2 a \left (c-\frac{c}{a x}\right )^{7/2} x^2 \sqrt{1+a x}}{3 (1-a x)^{3/2}}-\frac{2 \left (c-\frac{c}{a x}\right )^{7/2} x \sqrt{1+a x}}{5 \sqrt{1-a x}}-\frac{a^2 \left (c-\frac{c}{a x}\right )^{7/2} x^3 \sqrt{1+a x} (18+31 a x)}{15 (1-a x)^{7/2}}+\frac{5 a^{5/2} \left (c-\frac{c}{a x}\right )^{7/2} x^{7/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{(1-a x)^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.074095, size = 95, normalized size = 0.52 \[ \frac{c^3 \sqrt{c-\frac{c}{a x}} \left (\sqrt{a x+1} \left (15 a^3 x^3+56 a^2 x^2-28 a x+6\right )-75 a^{5/2} x^{5/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )\right )}{15 a^3 x^2 \sqrt{1-a x}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcTanh[a*x]*(c - c/(a*x))^(7/2),x]

[Out]

(c^3*Sqrt[c - c/(a*x)]*(Sqrt[1 + a*x]*(6 - 28*a*x + 56*a^2*x^2 + 15*a^3*x^3) - 75*a^(5/2)*x^(5/2)*ArcSinh[Sqrt
[a]*Sqrt[x]]))/(15*a^3*x^2*Sqrt[1 - a*x])

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Maple [A]  time = 0.145, size = 154, normalized size = 0.9 \begin{align*} -{\frac{{c}^{3}}{30\,{x}^{2} \left ( ax-1 \right ) }\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}\sqrt{-{a}^{2}{x}^{2}+1} \left ( 30\,{a}^{7/2}{x}^{3}\sqrt{- \left ( ax+1 \right ) x}+75\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ){x}^{3}{a}^{3}+112\,{a}^{5/2}{x}^{2}\sqrt{- \left ( ax+1 \right ) x}-56\,{a}^{3/2}x\sqrt{- \left ( ax+1 \right ) x}+12\,\sqrt{a}\sqrt{- \left ( ax+1 \right ) x} \right ){a}^{-{\frac{7}{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)/(-a^2*x^2+1)^(1/2)*(c-c/a/x)^(7/2),x)

[Out]

-1/30*(c*(a*x-1)/a/x)^(1/2)/x^2*c^3/a^(7/2)*(-a^2*x^2+1)^(1/2)*(30*a^(7/2)*x^3*(-(a*x+1)*x)^(1/2)+75*arctan(1/
2/a^(1/2)*(2*a*x+1)/(-(a*x+1)*x)^(1/2))*x^3*a^3+112*a^(5/2)*x^2*(-(a*x+1)*x)^(1/2)-56*a^(3/2)*x*(-(a*x+1)*x)^(
1/2)+12*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{{\left (a x + 1\right )}{\left (c - \frac{c}{a x}\right )}^{\frac{7}{2}}}{\sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a/x)^(7/2),x, algorithm="maxima")

[Out]

integrate((a*x + 1)*(c - c/(a*x))^(7/2)/sqrt(-a^2*x^2 + 1), x)

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Fricas [A]  time = 2.92512, size = 745, normalized size = 4.12 \begin{align*} \left [\frac{75 \,{\left (a^{3} c^{3} x^{3} - a^{2} c^{3} x^{2}\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 (15 \, a^{3} c^{3} x^{3} + 56 \, a^{2} c^{3} x^{2} - 28 \, a c^{3} x + 6 \, c^{3}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{60 \,{\left (a^{4} x^{3} - a^{3} x^{2}\right )}}, \frac{75 \,{\left (a^{3} c^{3} x^{3} - a^{2} c^{3} x^{2}\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 (15 \, a^{3} c^{3} x^{3} + 56 \, a^{2} c^{3} x^{2} - 28 \, a c^{3} x + 6 \, c^{3}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{30 \,{\left (a^{4} x^{3} - a^{3} x^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a/x)^(7/2),x, algorithm="fricas")

[Out]

[1/60*(75*(a^3*c^3*x^3 - a^2*c^3*x^2)*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*(15*a^3*c^3*x^3 + 56*a^2*c^3*x^2 - 28*a*c^3*x + 6*c
^3)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*x^3 - a^3*x^2), 1/30*(75*(a^3*c^3*x^3 - a^2*c^3*x^2)*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*(15*a^3*c^3
*x^3 + 56*a^2*c^3*x^2 - 28*a*c^3*x + 6*c^3)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*x^3 - a^3*x^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((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a/x)**(7/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )}{\left (c - \frac{c}{a x}\right )}^{\frac{7}{2}}}{\sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a/x)^(7/2),x, algorithm="giac")

[Out]

integrate((a*x + 1)*(c - c/(a*x))^(7/2)/sqrt(-a^2*x^2 + 1), x)