3.1029 \(\int e^{2 \tanh ^{-1}(a x)} x^3 (c-a^2 c x^2)^2 \, dx\)

Optimal. Leaf size=48 \[ -\frac{1}{8} a^4 c^2 x^8-\frac{2}{7} a^3 c^2 x^7+\frac{2}{5} a c^2 x^5+\frac{c^2 x^4}{4} \]

[Out]

(c^2*x^4)/4 + (2*a*c^2*x^5)/5 - (2*a^3*c^2*x^7)/7 - (a^4*c^2*x^8)/8

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Rubi [A]  time = 0.0892372, 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} \[ -\frac{1}{8} a^4 c^2 x^8-\frac{2}{7} a^3 c^2 x^7+\frac{2}{5} a c^2 x^5+\frac{c^2 x^4}{4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*x^4)/4 + (2*a*c^2*x^5)/5 - (2*a^3*c^2*x^7)/7 - (a^4*c^2*x^8)/8

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

Mathematica [A]  time = 0.0222726, size = 40, normalized size = 0.83 \[ c^2 \left (-\frac{1}{8} a^4 x^8-\frac{2 a^3 x^7}{7}+\frac{2 a x^5}{5}+\frac{x^4}{4}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

c^2*(x^4/4 + (2*a*x^5)/5 - (2*a^3*x^7)/7 - (a^4*x^8)/8)

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Maple [A]  time = 0.023, size = 33, normalized size = 0.7 \begin{align*}{c}^{2} \left ( -{\frac{{a}^{4}{x}^{8}}{8}}-{\frac{2\,{x}^{7}{a}^{3}}{7}}+{\frac{2\,a{x}^{5}}{5}}+{\frac{{x}^{4}}{4}} \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^3*(-a^2*c*x^2+c)^2,x)

[Out]

c^2*(-1/8*a^4*x^8-2/7*x^7*a^3+2/5*a*x^5+1/4*x^4)

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Maxima [A]  time = 0.944045, size = 54, normalized size = 1.12 \begin{align*} -\frac{1}{8} \, a^{4} c^{2} x^{8} - \frac{2}{7} \, a^{3} c^{2} x^{7} + \frac{2}{5} \, a c^{2} x^{5} + \frac{1}{4} \, c^{2} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*a^4*c^2*x^8 - 2/7*a^3*c^2*x^7 + 2/5*a*c^2*x^5 + 1/4*c^2*x^4

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Fricas [A]  time = 1.63187, size = 89, normalized size = 1.85 \begin{align*} -\frac{1}{8} \, a^{4} c^{2} x^{8} - \frac{2}{7} \, a^{3} c^{2} x^{7} + \frac{2}{5} \, a c^{2} x^{5} + \frac{1}{4} \, c^{2} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*a^4*c^2*x^8 - 2/7*a^3*c^2*x^7 + 2/5*a*c^2*x^5 + 1/4*c^2*x^4

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Sympy [A]  time = 0.088677, size = 44, normalized size = 0.92 \begin{align*} - \frac{a^{4} c^{2} x^{8}}{8} - \frac{2 a^{3} c^{2} x^{7}}{7} + \frac{2 a c^{2} x^{5}}{5} + \frac{c^{2} x^{4}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**4*c**2*x**8/8 - 2*a**3*c**2*x**7/7 + 2*a*c**2*x**5/5 + c**2*x**4/4

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Giac [A]  time = 1.1107, size = 54, normalized size = 1.12 \begin{align*} -\frac{1}{8} \, a^{4} c^{2} x^{8} - \frac{2}{7} \, a^{3} c^{2} x^{7} + \frac{2}{5} \, a c^{2} x^{5} + \frac{1}{4} \, c^{2} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*a^4*c^2*x^8 - 2/7*a^3*c^2*x^7 + 2/5*a*c^2*x^5 + 1/4*c^2*x^4