3.1126 \(\int e^{2 \tanh ^{-1}(a x)} x^m (c-a^2 c x^2)^2 \, dx\)

Optimal. Leaf size=67 \[ -\frac{2 a^3 c^2 x^{m+4}}{m+4}-\frac{a^4 c^2 x^{m+5}}{m+5}+\frac{2 a c^2 x^{m+2}}{m+2}+\frac{c^2 x^{m+1}}{m+1} \]

[Out]

(c^2*x^(1 + m))/(1 + m) + (2*a*c^2*x^(2 + m))/(2 + m) - (2*a^3*c^2*x^(4 + m))/(4 + m) - (a^4*c^2*x^(5 + m))/(5
+ m)

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Rubi [A]  time = 0.0872407, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {6150, 75} \[ -\frac{2 a^3 c^2 x^{m+4}}{m+4}-\frac{a^4 c^2 x^{m+5}}{m+5}+\frac{2 a c^2 x^{m+2}}{m+2}+\frac{c^2 x^{m+1}}{m+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*x^(1 + m))/(1 + m) + (2*a*c^2*x^(2 + m))/(2 + m) - (2*a^3*c^2*x^(4 + m))/(4 + m) - (a^4*c^2*x^(5 + m))/(5
+ m)

Rule 6150

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

Rule 75

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rubi steps

\begin{align*} \int e^{2 \tanh ^{-1}(a x)} x^m \left (c-a^2 c x^2\right )^2 \, dx &=c^2 \int x^m (1-a x) (1+a x)^3 \, dx\\ &=c^2 \int \left (x^m+2 a x^{1+m}-2 a^3 x^{3+m}-a^4 x^{4+m}\right ) \, dx\\ &=\frac{c^2 x^{1+m}}{1+m}+\frac{2 a c^2 x^{2+m}}{2+m}-\frac{2 a^3 c^2 x^{4+m}}{4+m}-\frac{a^4 c^2 x^{5+m}}{5+m}\\ \end{align*}

Mathematica [A]  time = 0.104508, size = 69, normalized size = 1.03 \[ \frac{c^2 x^{m+1} \left (2 (m+3) \left (\frac{a^3 x^3}{m+4}+\frac{3 a^2 x^2}{m+3}+\frac{3 a x}{m+2}+\frac{1}{m+1}\right )-(a x+1)^4\right )}{m+5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*x^(1 + m)*(-(1 + a*x)^4 + 2*(3 + m)*((1 + m)^(-1) + (3*a*x)/(2 + m) + (3*a^2*x^2)/(3 + m) + (a^3*x^3)/(4
+ m))))/(5 + m)

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Maple [B]  time = 0.029, size = 146, normalized size = 2.2 \begin{align*} -{\frac{{c}^{2}{x}^{1+m} \left ({a}^{4}{m}^{3}{x}^{4}+7\,{a}^{4}{m}^{2}{x}^{4}+14\,{a}^{4}m{x}^{4}+2\,{a}^{3}{m}^{3}{x}^{3}+8\,{x}^{4}{a}^{4}+16\,{a}^{3}{m}^{2}{x}^{3}+34\,{a}^{3}m{x}^{3}+20\,{x}^{3}{a}^{3}-2\,a{m}^{3}x-20\,a{m}^{2}x-58\,amx-{m}^{3}-40\,ax-11\,{m}^{2}-38\,m-40 \right ) }{ \left ( 5+m \right ) \left ( 4+m \right ) \left ( 2+m \right ) \left ( 1+m \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^2/(-a^2*x^2+1)*x^m*(-a^2*c*x^2+c)^2,x)

[Out]

-c^2*x^(1+m)*(a^4*m^3*x^4+7*a^4*m^2*x^4+14*a^4*m*x^4+2*a^3*m^3*x^3+8*a^4*x^4+16*a^3*m^2*x^3+34*a^3*m*x^3+20*a^
3*x^3-2*a*m^3*x-20*a*m^2*x-58*a*m*x-m^3-40*a*x-11*m^2-38*m-40)/(5+m)/(4+m)/(2+m)/(1+m)

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Maxima [A]  time = 1.13279, size = 154, normalized size = 2.3 \begin{align*} -\frac{{\left ({\left (m^{3} + 7 \, m^{2} + 14 \, m + 8\right )} a^{4} c^{2} x^{5} + 2 \,{\left (m^{3} + 8 \, m^{2} + 17 \, m + 10\right )} a^{3} c^{2} x^{4} - 2 \,{\left (m^{3} + 10 \, m^{2} + 29 \, m + 20\right )} a c^{2} x^{2} -{\left (m^{3} + 11 \, m^{2} + 38 \, m + 40\right )} c^{2} x\right )} x^{m}}{m^{4} + 12 \, m^{3} + 49 \, m^{2} + 78 \, m + 40} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((m^3 + 7*m^2 + 14*m + 8)*a^4*c^2*x^5 + 2*(m^3 + 8*m^2 + 17*m + 10)*a^3*c^2*x^4 - 2*(m^3 + 10*m^2 + 29*m + 20
)*a*c^2*x^2 - (m^3 + 11*m^2 + 38*m + 40)*c^2*x)*x^m/(m^4 + 12*m^3 + 49*m^2 + 78*m + 40)

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Fricas [B]  time = 3.04994, size = 374, normalized size = 5.58 \begin{align*} -\frac{{\left ({\left (a^{4} c^{2} m^{3} + 7 \, a^{4} c^{2} m^{2} + 14 \, a^{4} c^{2} m + 8 \, a^{4} c^{2}\right )} x^{5} + 2 \,{\left (a^{3} c^{2} m^{3} + 8 \, a^{3} c^{2} m^{2} + 17 \, a^{3} c^{2} m + 10 \, a^{3} c^{2}\right )} x^{4} - 2 \,{\left (a c^{2} m^{3} + 10 \, a c^{2} m^{2} + 29 \, a c^{2} m + 20 \, a c^{2}\right )} x^{2} -{\left (c^{2} m^{3} + 11 \, c^{2} m^{2} + 38 \, c^{2} m + 40 \, c^{2}\right )} x\right )} x^{m}}{m^{4} + 12 \, m^{3} + 49 \, m^{2} + 78 \, m + 40} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((a^4*c^2*m^3 + 7*a^4*c^2*m^2 + 14*a^4*c^2*m + 8*a^4*c^2)*x^5 + 2*(a^3*c^2*m^3 + 8*a^3*c^2*m^2 + 17*a^3*c^2*m
+ 10*a^3*c^2)*x^4 - 2*(a*c^2*m^3 + 10*a*c^2*m^2 + 29*a*c^2*m + 20*a*c^2)*x^2 - (c^2*m^3 + 11*c^2*m^2 + 38*c^2
*m + 40*c^2)*x)*x^m/(m^4 + 12*m^3 + 49*m^2 + 78*m + 40)

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Sympy [A]  time = 2.19458, size = 706, normalized size = 10.54 \begin{align*} \begin{cases} - a^{4} c^{2} \log{\left (x \right )} + \frac{2 a^{3} c^{2}}{x} - \frac{2 a c^{2}}{3 x^{3}} - \frac{c^{2}}{4 x^{4}} & \text{for}\: m = -5 \\- a^{4} c^{2} x - 2 a^{3} c^{2} \log{\left (x \right )} - \frac{a c^{2}}{x^{2}} - \frac{c^{2}}{3 x^{3}} & \text{for}\: m = -4 \\- \frac{a^{4} c^{2} x^{3}}{3} - a^{3} c^{2} x^{2} + 2 a c^{2} \log{\left (x \right )} - \frac{c^{2}}{x} & \text{for}\: m = -2 \\- \frac{a^{4} c^{2} x^{4}}{4} - \frac{2 a^{3} c^{2} x^{3}}{3} + 2 a c^{2} x + c^{2} \log{\left (x \right )} & \text{for}\: m = -1 \\- \frac{a^{4} c^{2} m^{3} x^{5} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac{7 a^{4} c^{2} m^{2} x^{5} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac{14 a^{4} c^{2} m x^{5} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac{8 a^{4} c^{2} x^{5} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac{2 a^{3} c^{2} m^{3} x^{4} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac{16 a^{3} c^{2} m^{2} x^{4} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac{34 a^{3} c^{2} m x^{4} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac{20 a^{3} c^{2} x^{4} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac{2 a c^{2} m^{3} x^{2} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac{20 a c^{2} m^{2} x^{2} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac{58 a c^{2} m x^{2} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac{40 a c^{2} x^{2} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac{c^{2} m^{3} x x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac{11 c^{2} m^{2} x x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac{38 c^{2} m x x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac{40 c^{2} x x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**4*c**2*log(x) + 2*a**3*c**2/x - 2*a*c**2/(3*x**3) - c**2/(4*x**4), Eq(m, -5)), (-a**4*c**2*x -
2*a**3*c**2*log(x) - a*c**2/x**2 - c**2/(3*x**3), Eq(m, -4)), (-a**4*c**2*x**3/3 - a**3*c**2*x**2 + 2*a*c**2*l
og(x) - c**2/x, Eq(m, -2)), (-a**4*c**2*x**4/4 - 2*a**3*c**2*x**3/3 + 2*a*c**2*x + c**2*log(x), Eq(m, -1)), (-
a**4*c**2*m**3*x**5*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) - 7*a**4*c**2*m**2*x**5*x**m/(m**4 + 12*m**3 +
49*m**2 + 78*m + 40) - 14*a**4*c**2*m*x**5*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) - 8*a**4*c**2*x**5*x**
m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) - 2*a**3*c**2*m**3*x**4*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) -
16*a**3*c**2*m**2*x**4*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) - 34*a**3*c**2*m*x**4*x**m/(m**4 + 12*m**3
+ 49*m**2 + 78*m + 40) - 20*a**3*c**2*x**4*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) + 2*a*c**2*m**3*x**2*x
**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) + 20*a*c**2*m**2*x**2*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) +
58*a*c**2*m*x**2*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) + 40*a*c**2*x**2*x**m/(m**4 + 12*m**3 + 49*m**2
+ 78*m + 40) + c**2*m**3*x*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) + 11*c**2*m**2*x*x**m/(m**4 + 12*m**3 +
49*m**2 + 78*m + 40) + 38*c**2*m*x*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) + 40*c**2*x*x**m/(m**4 + 12*m*
*3 + 49*m**2 + 78*m + 40), True))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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