3.555 \(\int e^{-3 \tanh ^{-1}(a x)} (c-\frac{c}{a x})^{7/2} \, dx\)

Optimal. Leaf size=225 \[ -\frac{a^3 x^4 (2525-427 a x) \left (c-\frac{c}{a x}\right )^{7/2}}{15 (1-a x)^{7/2} \sqrt{a x+1}}-\frac{398 a^2 x^3 \left (c-\frac{c}{a x}\right )^{7/2}}{15 (1-a x)^{3/2} \sqrt{a x+1}}+\frac{13 a^{5/2} x^{7/2} \left (c-\frac{c}{a x}\right )^{7/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{(1-a x)^{7/2}}+\frac{38 a x^2 \left (c-\frac{c}{a x}\right )^{7/2}}{15 \sqrt{1-a x} \sqrt{a x+1}}-\frac{2 x \sqrt{1-a x} \left (c-\frac{c}{a x}\right )^{7/2}}{5 \sqrt{a x+1}} \]

[Out]

-(a^3*(c - c/(a*x))^(7/2)*x^4*(2525 - 427*a*x))/(15*(1 - a*x)^(7/2)*Sqrt[1 + a*x]) - (398*a^2*(c - c/(a*x))^(7
/2)*x^3)/(15*(1 - a*x)^(3/2)*Sqrt[1 + a*x]) + (38*a*(c - c/(a*x))^(7/2)*x^2)/(15*Sqrt[1 - a*x]*Sqrt[1 + a*x])
- (2*(c - c/(a*x))^(7/2)*x*Sqrt[1 - a*x])/(5*Sqrt[1 + a*x]) + (13*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 = 0.206663, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {6134, 6129, 98, 150, 143, 54, 215} \[ -\frac{a^3 x^4 (2525-427 a x) \left (c-\frac{c}{a x}\right )^{7/2}}{15 (1-a x)^{7/2} \sqrt{a x+1}}-\frac{398 a^2 x^3 \left (c-\frac{c}{a x}\right )^{7/2}}{15 (1-a x)^{3/2} \sqrt{a x+1}}+\frac{13 a^{5/2} x^{7/2} \left (c-\frac{c}{a x}\right )^{7/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{(1-a x)^{7/2}}+\frac{38 a x^2 \left (c-\frac{c}{a x}\right )^{7/2}}{15 \sqrt{1-a x} \sqrt{a x+1}}-\frac{2 x \sqrt{1-a x} \left (c-\frac{c}{a x}\right )^{7/2}}{5 \sqrt{a x+1}} \]

Antiderivative was successfully verified.

[In]

Int[(c - c/(a*x))^(7/2)/E^(3*ArcTanh[a*x]),x]

[Out]

-(a^3*(c - c/(a*x))^(7/2)*x^4*(2525 - 427*a*x))/(15*(1 - a*x)^(7/2)*Sqrt[1 + a*x]) - (398*a^2*(c - c/(a*x))^(7
/2)*x^3)/(15*(1 - a*x)^(3/2)*Sqrt[1 + a*x]) + (38*a*(c - c/(a*x))^(7/2)*x^2)/(15*Sqrt[1 - a*x]*Sqrt[1 + a*x])
- (2*(c - c/(a*x))^(7/2)*x*Sqrt[1 - a*x])/(5*Sqrt[1 + a*x]) + (13*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 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 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^{-3 \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^{-3 \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)^5}{x^{7/2} (1+a x)^{3/2}} \, 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)^3 \left (\frac{19 a}{2}-\frac{3 a^2 x}{2}\right )}{x^{5/2} (1+a x)^{3/2}} \, dx}{5 (1-a x)^{7/2}}\\ &=\frac{38 a \left (c-\frac{c}{a x}\right )^{7/2} x^2}{15 \sqrt{1-a x} \sqrt{1+a x}}-\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)^2 \left (-\frac{199 a^2}{4}-\frac{29 a^3 x}{4}\right )}{x^{3/2} (1+a x)^{3/2}} \, dx}{15 (1-a x)^{7/2}}\\ &=-\frac{398 a^2 \left (c-\frac{c}{a x}\right )^{7/2} x^3}{15 (1-a x)^{3/2} \sqrt{1+a x}}+\frac{38 a \left (c-\frac{c}{a x}\right )^{7/2} x^2}{15 \sqrt{1-a x} \sqrt{1+a x}}-\frac{2 \left (c-\frac{c}{a x}\right )^{7/2} x \sqrt{1-a x}}{5 \sqrt{1+a x}}-\frac{\left (8 \left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}\right ) \int \frac{(1-a x) \left (\frac{1165 a^3}{8}+\frac{427 a^4 x}{8}\right )}{\sqrt{x} (1+a x)^{3/2}} \, dx}{15 (1-a x)^{7/2}}\\ &=-\frac{a^3 \left (c-\frac{c}{a x}\right )^{7/2} x^4 (2525-427 a x)}{15 (1-a x)^{7/2} \sqrt{1+a x}}-\frac{398 a^2 \left (c-\frac{c}{a x}\right )^{7/2} x^3}{15 (1-a x)^{3/2} \sqrt{1+a x}}+\frac{38 a \left (c-\frac{c}{a x}\right )^{7/2} x^2}{15 \sqrt{1-a x} \sqrt{1+a x}}-\frac{2 \left (c-\frac{c}{a x}\right )^{7/2} x \sqrt{1-a x}}{5 \sqrt{1+a x}}+\frac{\left (13 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{a^3 \left (c-\frac{c}{a x}\right )^{7/2} x^4 (2525-427 a x)}{15 (1-a x)^{7/2} \sqrt{1+a x}}-\frac{398 a^2 \left (c-\frac{c}{a x}\right )^{7/2} x^3}{15 (1-a x)^{3/2} \sqrt{1+a x}}+\frac{38 a \left (c-\frac{c}{a x}\right )^{7/2} x^2}{15 \sqrt{1-a x} \sqrt{1+a x}}-\frac{2 \left (c-\frac{c}{a x}\right )^{7/2} x \sqrt{1-a x}}{5 \sqrt{1+a x}}+\frac{\left (13 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{a^3 \left (c-\frac{c}{a x}\right )^{7/2} x^4 (2525-427 a x)}{15 (1-a x)^{7/2} \sqrt{1+a x}}-\frac{398 a^2 \left (c-\frac{c}{a x}\right )^{7/2} x^3}{15 (1-a x)^{3/2} \sqrt{1+a x}}+\frac{38 a \left (c-\frac{c}{a x}\right )^{7/2} x^2}{15 \sqrt{1-a x} \sqrt{1+a x}}-\frac{2 \left (c-\frac{c}{a x}\right )^{7/2} x \sqrt{1-a x}}{5 \sqrt{1+a x}}+\frac{13 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 [C]  time = 3.54371, size = 200, normalized size = 0.89 \[ -\frac{c^3 x \sqrt{c-\frac{c}{a x}} \left (1040 a^4 x^4 (a x-1)^3 \sqrt{a x+1} \sqrt{-a x (a x+1)} \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{9}{2},\frac{11}{2},-a x\right )-3 \sqrt{-a x (a x+1)} \left (520 a^6 x^6-2470 a^5 x^5+6325 a^4 x^4-28706 a^3 x^3-21508 a^2 x^2+19192 a x-6921\right )+585 \left (19 a^4 x^4-86 a^3 x^3+70 a x-35\right ) \sin ^{-1}\left (\sqrt{-a x}\right )\right )}{720 (-a x)^{7/2} \sqrt{a x+1} \sqrt{1-a^2 x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c - c/(a*x))^(7/2)/E^(3*ArcTanh[a*x]),x]

[Out]

-(c^3*Sqrt[c - c/(a*x)]*x*(-3*Sqrt[-(a*x*(1 + a*x))]*(-6921 + 19192*a*x - 21508*a^2*x^2 - 28706*a^3*x^3 + 6325
*a^4*x^4 - 2470*a^5*x^5 + 520*a^6*x^6) + 585*(-35 + 70*a*x - 86*a^3*x^3 + 19*a^4*x^4)*ArcSin[Sqrt[-(a*x)]] + 1
040*a^4*x^4*(-1 + a*x)^3*Sqrt[1 + a*x]*Sqrt[-(a*x*(1 + a*x))]*Hypergeometric2F1[3/2, 9/2, 11/2, -(a*x)]))/(720
*(-(a*x))^(7/2)*Sqrt[1 + a*x]*Sqrt[1 - a^2*x^2])

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Maple [A]  time = 0.153, size = 209, normalized size = 0.9 \begin{align*} -{\frac{{c}^{3}}{30\,{x}^{2} \left ( ax+1 \right ) \left ( ax-1 \right ) }\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 30\,{a}^{9/2}\sqrt{- \left ( ax+1 \right ) x}{x}^{4}+195\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ){x}^{4}{a}^{4}+3182\,{a}^{7/2}{x}^{3}\sqrt{- \left ( ax+1 \right ) x}+195\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ){x}^{3}{a}^{3}+1096\,{a}^{5/2}{x}^{2}\sqrt{- \left ( ax+1 \right ) x}-124\,{a}^{3/2}x\sqrt{- \left ( ax+1 \right ) x}+12\,\sqrt{a}\sqrt{- \left ( ax+1 \right ) x} \right ) \sqrt{-{a}^{2}{x}^{2}+1}{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((c-c/a/x)^(7/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x)

[Out]

-1/30*(c*(a*x-1)/a/x)^(1/2)/x^2*c^3/a^(7/2)*(30*a^(9/2)*(-(a*x+1)*x)^(1/2)*x^4+195*arctan(1/2/a^(1/2)*(2*a*x+1
)/(-(a*x+1)*x)^(1/2))*x^4*a^4+3182*a^(7/2)*x^3*(-(a*x+1)*x)^(1/2)+195*arctan(1/2/a^(1/2)*(2*a*x+1)/(-(a*x+1)*x
)^(1/2))*x^3*a^3+1096*a^(5/2)*x^2*(-(a*x+1)*x)^(1/2)-124*a^(3/2)*x*(-(a*x+1)*x)^(1/2)+12*a^(1/2)*(-(a*x+1)*x)^
(1/2))*(-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}}{\left (c - \frac{c}{a x}\right )}^{\frac{7}{2}}}{{\left (a x + 1\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a/x)^(7/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="maxima")

[Out]

integrate((-a^2*x^2 + 1)^(3/2)*(c - c/(a*x))^(7/2)/(a*x + 1)^3, x)

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Fricas [A]  time = 2.21649, size = 802, normalized size = 3.56 \begin{align*} \left [\frac{195 \,{\left (a^{4} c^{3} x^{4} - 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^{4} c^{3} x^{4} + 1591 \, a^{3} c^{3} x^{3} + 548 \, a^{2} c^{3} x^{2} - 62 \, 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^{5} x^{4} - a^{3} x^{2}\right )}}, \frac{195 \,{\left (a^{4} c^{3} x^{4} - 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^{4} c^{3} x^{4} + 1591 \, a^{3} c^{3} x^{3} + 548 \, a^{2} c^{3} x^{2} - 62 \, 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^{5} x^{4} - a^{3} x^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a/x)^(7/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="fricas")

[Out]

[1/60*(195*(a^4*c^3*x^4 - 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^4*c^3*x^4 + 1591*a^3*c^3*x^3 + 548*a^2*c^3*x
^2 - 62*a*c^3*x + 6*c^3)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^5*x^4 - a^3*x^2), 1/30*(195*(a^4*c^3*x
^4 - 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^4*c^3*x^4 + 1591*a^3*c^3*x^3 + 548*a^2*c^3*x^2 - 62*a*c^3*x + 6*c^3)*sqrt(-a^2*x^2 + 1)*sqrt(
(a*c*x - c)/(a*x)))/(a^5*x^4 - 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((c-c/a/x)**(7/2)/(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}}{\left (c - \frac{c}{a x}\right )}^{\frac{7}{2}}}{{\left (a x + 1\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a/x)^(7/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="giac")

[Out]

integrate((-a^2*x^2 + 1)^(3/2)*(c - c/(a*x))^(7/2)/(a*x + 1)^3, x)