3.1019 \(\int e^{2 \tanh ^{-1}(a x)} x^4 (c-a^2 c x^2) \, dx\)

Optimal. Leaf size=29 \[ \frac{1}{7} a^2 c x^7+\frac{1}{3} a c x^6+\frac{c x^5}{5} \]

[Out]

(c*x^5)/5 + (a*c*x^6)/3 + (a^2*c*x^7)/7

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Rubi [A]  time = 0.0546501, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {6150, 43} \[ \frac{1}{7} a^2 c x^7+\frac{1}{3} a c x^6+\frac{c x^5}{5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x^5)/5 + (a*c*x^6)/3 + (a^2*c*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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0167055, size = 22, normalized size = 0.76 \[ \frac{1}{105} c x^5 \left (15 a^2 x^2+35 a x+21\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x^5*(21 + 35*a*x + 15*a^2*x^2))/105

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Maple [A]  time = 0.026, size = 23, normalized size = 0.8 \begin{align*} c \left ({\frac{{a}^{2}{x}^{7}}{7}}+{\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),x)

[Out]

c*(1/7*a^2*x^7+1/3*x^6*a+1/5*x^5)

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Maxima [A]  time = 0.946904, size = 31, normalized size = 1.07 \begin{align*} \frac{1}{7} \, a^{2} c x^{7} + \frac{1}{3} \, a c x^{6} + \frac{1}{5} \, c 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),x, algorithm="maxima")

[Out]

1/7*a^2*c*x^7 + 1/3*a*c*x^6 + 1/5*c*x^5

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Fricas [A]  time = 1.86128, size = 55, normalized size = 1.9 \begin{align*} \frac{1}{7} \, a^{2} c x^{7} + \frac{1}{3} \, a c x^{6} + \frac{1}{5} \, c 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),x, algorithm="fricas")

[Out]

1/7*a^2*c*x^7 + 1/3*a*c*x^6 + 1/5*c*x^5

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Sympy [A]  time = 0.077603, size = 24, normalized size = 0.83 \begin{align*} \frac{a^{2} c x^{7}}{7} + \frac{a c x^{6}}{3} + \frac{c 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),x)

[Out]

a**2*c*x**7/7 + a*c*x**6/3 + c*x**5/5

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Giac [A]  time = 1.18084, size = 31, normalized size = 1.07 \begin{align*} \frac{1}{7} \, a^{2} c x^{7} + \frac{1}{3} \, a c x^{6} + \frac{1}{5} \, c 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),x, algorithm="giac")

[Out]

1/7*a^2*c*x^7 + 1/3*a*c*x^6 + 1/5*c*x^5