3.186 \(\int \frac{e^{3 \tanh ^{-1}(a x)}}{(c-a c x)^5} \, dx\)

Optimal. Leaf size=129 \[ \frac{2 \left (1-a^2 x^2\right )^{5/2}}{1155 a c^5 (1-a x)^5}+\frac{2 \left (1-a^2 x^2\right )^{5/2}}{231 a c^5 (1-a x)^6}+\frac{\left (1-a^2 x^2\right )^{5/2}}{33 a c^5 (1-a x)^7}+\frac{\left (1-a^2 x^2\right )^{5/2}}{11 a c^5 (1-a x)^8} \]

[Out]

(1 - a^2*x^2)^(5/2)/(11*a*c^5*(1 - a*x)^8) + (1 - a^2*x^2)^(5/2)/(33*a*c^5*(1 - a*x)^7) + (2*(1 - a^2*x^2)^(5/
2))/(231*a*c^5*(1 - a*x)^6) + (2*(1 - a^2*x^2)^(5/2))/(1155*a*c^5*(1 - a*x)^5)

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Rubi [A]  time = 0.0917504, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6127, 659, 651} \[ \frac{2 \left (1-a^2 x^2\right )^{5/2}}{1155 a c^5 (1-a x)^5}+\frac{2 \left (1-a^2 x^2\right )^{5/2}}{231 a c^5 (1-a x)^6}+\frac{\left (1-a^2 x^2\right )^{5/2}}{33 a c^5 (1-a x)^7}+\frac{\left (1-a^2 x^2\right )^{5/2}}{11 a c^5 (1-a x)^8} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a*x])/(c - a*c*x)^5,x]

[Out]

(1 - a^2*x^2)^(5/2)/(11*a*c^5*(1 - a*x)^8) + (1 - a^2*x^2)^(5/2)/(33*a*c^5*(1 - a*x)^7) + (2*(1 - a^2*x^2)^(5/
2))/(231*a*c^5*(1 - a*x)^6) + (2*(1 - a^2*x^2)^(5/2))/(1155*a*c^5*(1 - a*x)^5)

Rule 6127

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Dist[c^n, Int[(c + d*x)^(p - n)*(1 -
a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && IntegerQ[2*p]

Rule 659

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(e*(d + e*x)^m*(a + c*x^2)^(p + 1)
)/(2*c*d*(m + p + 1)), x] + Dist[Simplify[m + 2*p + 2]/(2*d*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p,
x], x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p + 2
], 0]

Rule 651

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a + c*x^2)^(p + 1))
/(2*c*d*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p
+ 2, 0]

Rubi steps

\begin{align*} \int \frac{e^{3 \tanh ^{-1}(a x)}}{(c-a c x)^5} \, dx &=c^3 \int \frac{\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^8} \, dx\\ &=\frac{\left (1-a^2 x^2\right )^{5/2}}{11 a c^5 (1-a x)^8}+\frac{1}{11} \left (3 c^2\right ) \int \frac{\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^7} \, dx\\ &=\frac{\left (1-a^2 x^2\right )^{5/2}}{11 a c^5 (1-a x)^8}+\frac{\left (1-a^2 x^2\right )^{5/2}}{33 a c^5 (1-a x)^7}+\frac{1}{33} (2 c) \int \frac{\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^6} \, dx\\ &=\frac{\left (1-a^2 x^2\right )^{5/2}}{11 a c^5 (1-a x)^8}+\frac{\left (1-a^2 x^2\right )^{5/2}}{33 a c^5 (1-a x)^7}+\frac{2 \left (1-a^2 x^2\right )^{5/2}}{231 a c^5 (1-a x)^6}+\frac{2}{231} \int \frac{\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^5} \, dx\\ &=\frac{\left (1-a^2 x^2\right )^{5/2}}{11 a c^5 (1-a x)^8}+\frac{\left (1-a^2 x^2\right )^{5/2}}{33 a c^5 (1-a x)^7}+\frac{2 \left (1-a^2 x^2\right )^{5/2}}{231 a c^5 (1-a x)^6}+\frac{2 \left (1-a^2 x^2\right )^{5/2}}{1155 a c^5 (1-a x)^5}\\ \end{align*}

Mathematica [A]  time = 0.0274414, size = 51, normalized size = 0.4 \[ -\frac{(a x+1)^{5/2} \left (2 a^3 x^3-16 a^2 x^2+61 a x-152\right )}{1155 a c^5 (1-a x)^{11/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(3*ArcTanh[a*x])/(c - a*c*x)^5,x]

[Out]

-((1 + a*x)^(5/2)*(-152 + 61*a*x - 16*a^2*x^2 + 2*a^3*x^3))/(1155*a*c^5*(1 - a*x)^(11/2))

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Maple [A]  time = 0.036, size = 57, normalized size = 0.4 \begin{align*} -{\frac{ \left ( 2\,{x}^{3}{a}^{3}-16\,{a}^{2}{x}^{2}+61\,ax-152 \right ) \left ( ax+1 \right ) ^{4}}{1155\,{c}^{5} \left ( ax-1 \right ) ^{4}a} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}} \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)/(-a*c*x+c)^5,x)

[Out]

-1/1155*(2*a^3*x^3-16*a^2*x^2+61*a*x-152)*(a*x+1)^4/(a*x-1)^4/c^5/(-a^2*x^2+1)^(3/2)/a

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Maxima [B]  time = 1.02939, size = 624, normalized size = 4.84 \begin{align*} -\frac{8}{11 \,{\left (\sqrt{-a^{2} x^{2} + 1} a^{6} c^{5} x^{5} - 5 \, \sqrt{-a^{2} x^{2} + 1} a^{5} c^{5} x^{4} + 10 \, \sqrt{-a^{2} x^{2} + 1} a^{4} c^{5} x^{3} - 10 \, \sqrt{-a^{2} x^{2} + 1} a^{3} c^{5} x^{2} + 5 \, \sqrt{-a^{2} x^{2} + 1} a^{2} c^{5} x - \sqrt{-a^{2} x^{2} + 1} a c^{5}\right )}} - \frac{28}{33 \,{\left (\sqrt{-a^{2} x^{2} + 1} a^{5} c^{5} x^{4} - 4 \, \sqrt{-a^{2} x^{2} + 1} a^{4} c^{5} x^{3} + 6 \, \sqrt{-a^{2} x^{2} + 1} a^{3} c^{5} x^{2} - 4 \, \sqrt{-a^{2} x^{2} + 1} a^{2} c^{5} x + \sqrt{-a^{2} x^{2} + 1} a c^{5}\right )}} - \frac{58}{231 \,{\left (\sqrt{-a^{2} x^{2} + 1} a^{4} c^{5} x^{3} - 3 \, \sqrt{-a^{2} x^{2} + 1} a^{3} c^{5} x^{2} + 3 \, \sqrt{-a^{2} x^{2} + 1} a^{2} c^{5} x - \sqrt{-a^{2} x^{2} + 1} a c^{5}\right )}} + \frac{1}{1155 \,{\left (\sqrt{-a^{2} x^{2} + 1} a^{3} c^{5} x^{2} - 2 \, \sqrt{-a^{2} x^{2} + 1} a^{2} c^{5} x + \sqrt{-a^{2} x^{2} + 1} a c^{5}\right )}} - \frac{1}{1155 \,{\left (\sqrt{-a^{2} x^{2} + 1} a^{2} c^{5} x - \sqrt{-a^{2} x^{2} + 1} a c^{5}\right )}} + \frac{2 \, x}{1155 \, \sqrt{-a^{2} x^{2} + 1} c^{5}} \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)/(-a*c*x+c)^5,x, algorithm="maxima")

[Out]

-8/11/(sqrt(-a^2*x^2 + 1)*a^6*c^5*x^5 - 5*sqrt(-a^2*x^2 + 1)*a^5*c^5*x^4 + 10*sqrt(-a^2*x^2 + 1)*a^4*c^5*x^3 -
10*sqrt(-a^2*x^2 + 1)*a^3*c^5*x^2 + 5*sqrt(-a^2*x^2 + 1)*a^2*c^5*x - sqrt(-a^2*x^2 + 1)*a*c^5) - 28/33/(sqrt(
-a^2*x^2 + 1)*a^5*c^5*x^4 - 4*sqrt(-a^2*x^2 + 1)*a^4*c^5*x^3 + 6*sqrt(-a^2*x^2 + 1)*a^3*c^5*x^2 - 4*sqrt(-a^2*
x^2 + 1)*a^2*c^5*x + sqrt(-a^2*x^2 + 1)*a*c^5) - 58/231/(sqrt(-a^2*x^2 + 1)*a^4*c^5*x^3 - 3*sqrt(-a^2*x^2 + 1)
*a^3*c^5*x^2 + 3*sqrt(-a^2*x^2 + 1)*a^2*c^5*x - sqrt(-a^2*x^2 + 1)*a*c^5) + 1/1155/(sqrt(-a^2*x^2 + 1)*a^3*c^5
*x^2 - 2*sqrt(-a^2*x^2 + 1)*a^2*c^5*x + sqrt(-a^2*x^2 + 1)*a*c^5) - 1/1155/(sqrt(-a^2*x^2 + 1)*a^2*c^5*x - sqr
t(-a^2*x^2 + 1)*a*c^5) + 2/1155*x/(sqrt(-a^2*x^2 + 1)*c^5)

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Fricas [A]  time = 1.7893, size = 389, normalized size = 3.02 \begin{align*} \frac{152 \, a^{6} x^{6} - 912 \, a^{5} x^{5} + 2280 \, a^{4} x^{4} - 3040 \, a^{3} x^{3} + 2280 \, a^{2} x^{2} - 912 \, a x -{\left (2 \, a^{5} x^{5} - 12 \, a^{4} x^{4} + 31 \, a^{3} x^{3} - 46 \, a^{2} x^{2} - 243 \, a x - 152\right )} \sqrt{-a^{2} x^{2} + 1} + 152}{1155 \,{\left (a^{7} c^{5} x^{6} - 6 \, a^{6} c^{5} x^{5} + 15 \, a^{5} c^{5} x^{4} - 20 \, a^{4} c^{5} x^{3} + 15 \, a^{3} c^{5} x^{2} - 6 \, a^{2} c^{5} x + a c^{5}\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)/(-a*c*x+c)^5,x, algorithm="fricas")

[Out]

1/1155*(152*a^6*x^6 - 912*a^5*x^5 + 2280*a^4*x^4 - 3040*a^3*x^3 + 2280*a^2*x^2 - 912*a*x - (2*a^5*x^5 - 12*a^4
*x^4 + 31*a^3*x^3 - 46*a^2*x^2 - 243*a*x - 152)*sqrt(-a^2*x^2 + 1) + 152)/(a^7*c^5*x^6 - 6*a^6*c^5*x^5 + 15*a^
5*c^5*x^4 - 20*a^4*c^5*x^3 + 15*a^3*c^5*x^2 - 6*a^2*c^5*x + a*c^5)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{3 a x}{- a^{7} x^{7} \sqrt{- a^{2} x^{2} + 1} + 5 a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} - 9 a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} + 5 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 5 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 9 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 5 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a^{2} x^{2}}{- a^{7} x^{7} \sqrt{- a^{2} x^{2} + 1} + 5 a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} - 9 a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} + 5 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 5 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 9 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 5 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a^{3} x^{3}}{- a^{7} x^{7} \sqrt{- a^{2} x^{2} + 1} + 5 a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} - 9 a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} + 5 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 5 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 9 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 5 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{- a^{7} x^{7} \sqrt{- a^{2} x^{2} + 1} + 5 a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} - 9 a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} + 5 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 5 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 9 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 5 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{5}} \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)/(-a*c*x+c)**5,x)

[Out]

-(Integral(3*a*x/(-a**7*x**7*sqrt(-a**2*x**2 + 1) + 5*a**6*x**6*sqrt(-a**2*x**2 + 1) - 9*a**5*x**5*sqrt(-a**2*
x**2 + 1) + 5*a**4*x**4*sqrt(-a**2*x**2 + 1) + 5*a**3*x**3*sqrt(-a**2*x**2 + 1) - 9*a**2*x**2*sqrt(-a**2*x**2
+ 1) + 5*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(3*a**2*x**2/(-a**7*x**7*sqrt(-a**2*x*
*2 + 1) + 5*a**6*x**6*sqrt(-a**2*x**2 + 1) - 9*a**5*x**5*sqrt(-a**2*x**2 + 1) + 5*a**4*x**4*sqrt(-a**2*x**2 +
1) + 5*a**3*x**3*sqrt(-a**2*x**2 + 1) - 9*a**2*x**2*sqrt(-a**2*x**2 + 1) + 5*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-
a**2*x**2 + 1)), x) + Integral(a**3*x**3/(-a**7*x**7*sqrt(-a**2*x**2 + 1) + 5*a**6*x**6*sqrt(-a**2*x**2 + 1) -
9*a**5*x**5*sqrt(-a**2*x**2 + 1) + 5*a**4*x**4*sqrt(-a**2*x**2 + 1) + 5*a**3*x**3*sqrt(-a**2*x**2 + 1) - 9*a*
*2*x**2*sqrt(-a**2*x**2 + 1) + 5*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(1/(-a**7*x**7
*sqrt(-a**2*x**2 + 1) + 5*a**6*x**6*sqrt(-a**2*x**2 + 1) - 9*a**5*x**5*sqrt(-a**2*x**2 + 1) + 5*a**4*x**4*sqrt
(-a**2*x**2 + 1) + 5*a**3*x**3*sqrt(-a**2*x**2 + 1) - 9*a**2*x**2*sqrt(-a**2*x**2 + 1) + 5*a*x*sqrt(-a**2*x**2
+ 1) - sqrt(-a**2*x**2 + 1)), x))/c**5

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (a c x - c\right )}^{5}}\,{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)/(-a*c*x+c)^5,x, algorithm="giac")

[Out]

integrate(-(a*x + 1)^3/((-a^2*x^2 + 1)^(3/2)*(a*c*x - c)^5), x)