3.1028 \(\int e^{2 \tanh ^{-1}(a x)} x^4 (c-a^2 c x^2)^2 \, dx\)

Optimal. Leaf size=48 \[ -\frac{1}{9} a^4 c^2 x^9-\frac{1}{4} a^3 c^2 x^8+\frac{1}{3} a c^2 x^6+\frac{c^2 x^5}{5} \]

[Out]

(c^2*x^5)/5 + (a*c^2*x^6)/3 - (a^3*c^2*x^8)/4 - (a^4*c^2*x^9)/9

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Rubi [A]  time = 0.0881659, antiderivative size = 48, 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} \[ -\frac{1}{9} a^4 c^2 x^9-\frac{1}{4} a^3 c^2 x^8+\frac{1}{3} a c^2 x^6+\frac{c^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcTanh[a*x])*x^4*(c - a^2*c*x^2)^2,x]

[Out]

(c^2*x^5)/5 + (a*c^2*x^6)/3 - (a^3*c^2*x^8)/4 - (a^4*c^2*x^9)/9

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^4 \left (c-a^2 c x^2\right )^2 \, dx &=c^2 \int x^4 (1-a x) (1+a x)^3 \, dx\\ &=c^2 \int \left (x^4+2 a x^5-2 a^3 x^7-a^4 x^8\right ) \, dx\\ &=\frac{c^2 x^5}{5}+\frac{1}{3} a c^2 x^6-\frac{1}{4} a^3 c^2 x^8-\frac{1}{9} a^4 c^2 x^9\\ \end{align*}

Mathematica [A]  time = 0.024491, size = 32, normalized size = 0.67 \[ -\frac{1}{180} c^2 x^5 \left (20 a^4 x^4+45 a^3 x^3-60 a x-36\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^(2*ArcTanh[a*x])*x^4*(c - a^2*c*x^2)^2,x]

[Out]

-(c^2*x^5*(-36 - 60*a*x + 45*a^3*x^3 + 20*a^4*x^4))/180

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Maple [A]  time = 0.024, size = 33, normalized size = 0.7 \begin{align*}{c}^{2} \left ( -{\frac{{a}^{4}{x}^{9}}{9}}-{\frac{{a}^{3}{x}^{8}}{4}}+{\frac{{x}^{6}a}{3}}+{\frac{{x}^{5}}{5}} \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^4*(-a^2*c*x^2+c)^2,x)

[Out]

c^2*(-1/9*a^4*x^9-1/4*a^3*x^8+1/3*x^6*a+1/5*x^5)

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Maxima [A]  time = 0.940892, size = 54, normalized size = 1.12 \begin{align*} -\frac{1}{9} \, a^{4} c^{2} x^{9} - \frac{1}{4} \, a^{3} c^{2} x^{8} + \frac{1}{3} \, a c^{2} x^{6} + \frac{1}{5} \, c^{2} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*x^4*(-a^2*c*x^2+c)^2,x, algorithm="maxima")

[Out]

-1/9*a^4*c^2*x^9 - 1/4*a^3*c^2*x^8 + 1/3*a*c^2*x^6 + 1/5*c^2*x^5

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Fricas [A]  time = 1.66844, size = 89, normalized size = 1.85 \begin{align*} -\frac{1}{9} \, a^{4} c^{2} x^{9} - \frac{1}{4} \, a^{3} c^{2} x^{8} + \frac{1}{3} \, a c^{2} x^{6} + \frac{1}{5} \, c^{2} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*x^4*(-a^2*c*x^2+c)^2,x, algorithm="fricas")

[Out]

-1/9*a^4*c^2*x^9 - 1/4*a^3*c^2*x^8 + 1/3*a*c^2*x^6 + 1/5*c^2*x^5

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Sympy [A]  time = 0.08837, size = 41, normalized size = 0.85 \begin{align*} - \frac{a^{4} c^{2} x^{9}}{9} - \frac{a^{3} c^{2} x^{8}}{4} + \frac{a c^{2} x^{6}}{3} + \frac{c^{2} x^{5}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**2/(-a**2*x**2+1)*x**4*(-a**2*c*x**2+c)**2,x)

[Out]

-a**4*c**2*x**9/9 - a**3*c**2*x**8/4 + a*c**2*x**6/3 + c**2*x**5/5

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Giac [A]  time = 1.14486, size = 54, normalized size = 1.12 \begin{align*} -\frac{1}{9} \, a^{4} c^{2} x^{9} - \frac{1}{4} \, a^{3} c^{2} x^{8} + \frac{1}{3} \, a c^{2} x^{6} + \frac{1}{5} \, c^{2} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*x^4*(-a^2*c*x^2+c)^2,x, algorithm="giac")

[Out]

-1/9*a^4*c^2*x^9 - 1/4*a^3*c^2*x^8 + 1/3*a*c^2*x^6 + 1/5*c^2*x^5