3.1030 \(\int e^{2 \tanh ^{-1}(a x)} x^2 (c-a^2 c x^2)^2 \, dx\)

Optimal. Leaf size=48 \[ -\frac{1}{7} a^4 c^2 x^7-\frac{1}{3} a^3 c^2 x^6+\frac{1}{2} a c^2 x^4+\frac{c^2 x^3}{3} \]

[Out]

(c^2*x^3)/3 + (a*c^2*x^4)/2 - (a^3*c^2*x^6)/3 - (a^4*c^2*x^7)/7

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Rubi [A]  time = 0.0896065, 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} \[ -\frac{1}{7} a^4 c^2 x^7-\frac{1}{3} a^3 c^2 x^6+\frac{1}{2} a c^2 x^4+\frac{c^2 x^3}{3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*x^3)/3 + (a*c^2*x^4)/2 - (a^3*c^2*x^6)/3 - (a^4*c^2*x^7)/7

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

Mathematica [A]  time = 0.0210124, size = 32, normalized size = 0.67 \[ -\frac{1}{42} c^2 x^3 \left (6 a^4 x^4+14 a^3 x^3-21 a x-14\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c^2*x^3*(-14 - 21*a*x + 14*a^3*x^3 + 6*a^4*x^4))/42

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Maple [A]  time = 0.025, size = 33, normalized size = 0.7 \begin{align*}{c}^{2} \left ( -{\frac{{x}^{7}{a}^{4}}{7}}-{\frac{{x}^{6}{a}^{3}}{3}}+{\frac{{x}^{4}a}{2}}+{\frac{{x}^{3}}{3}} \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^2*(-a^2*c*x^2+c)^2,x)

[Out]

c^2*(-1/7*x^7*a^4-1/3*x^6*a^3+1/2*x^4*a+1/3*x^3)

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Maxima [A]  time = 0.944307, size = 54, normalized size = 1.12 \begin{align*} -\frac{1}{7} \, a^{4} c^{2} x^{7} - \frac{1}{3} \, a^{3} c^{2} x^{6} + \frac{1}{2} \, a c^{2} x^{4} + \frac{1}{3} \, c^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*a^4*c^2*x^7 - 1/3*a^3*c^2*x^6 + 1/2*a*c^2*x^4 + 1/3*c^2*x^3

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Fricas [A]  time = 1.68789, size = 89, normalized size = 1.85 \begin{align*} -\frac{1}{7} \, a^{4} c^{2} x^{7} - \frac{1}{3} \, a^{3} c^{2} x^{6} + \frac{1}{2} \, a c^{2} x^{4} + \frac{1}{3} \, c^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*a^4*c^2*x^7 - 1/3*a^3*c^2*x^6 + 1/2*a*c^2*x^4 + 1/3*c^2*x^3

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Sympy [A]  time = 0.08737, size = 41, normalized size = 0.85 \begin{align*} - \frac{a^{4} c^{2} x^{7}}{7} - \frac{a^{3} c^{2} x^{6}}{3} + \frac{a c^{2} x^{4}}{2} + \frac{c^{2} x^{3}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**4*c**2*x**7/7 - a**3*c**2*x**6/3 + a*c**2*x**4/2 + c**2*x**3/3

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Giac [A]  time = 1.12429, size = 54, normalized size = 1.12 \begin{align*} -\frac{1}{7} \, a^{4} c^{2} x^{7} - \frac{1}{3} \, a^{3} c^{2} x^{6} + \frac{1}{2} \, a c^{2} x^{4} + \frac{1}{3} \, c^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*a^4*c^2*x^7 - 1/3*a^3*c^2*x^6 + 1/2*a*c^2*x^4 + 1/3*c^2*x^3