3.1040 \(\int e^{2 \tanh ^{-1}(a x)} x^3 (c-a^2 c x^2)^3 \, dx\)

Optimal. Leaf size=87 \[ \frac{1}{10} a^6 c^3 x^{10}+\frac{2}{9} a^5 c^3 x^9-\frac{1}{8} a^4 c^3 x^8-\frac{4}{7} a^3 c^3 x^7-\frac{1}{6} a^2 c^3 x^6+\frac{2}{5} a c^3 x^5+\frac{c^3 x^4}{4} \]

[Out]

(c^3*x^4)/4 + (2*a*c^3*x^5)/5 - (a^2*c^3*x^6)/6 - (4*a^3*c^3*x^7)/7 - (a^4*c^3*x^8)/8 + (2*a^5*c^3*x^9)/9 + (a
^6*c^3*x^10)/10

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Rubi [A]  time = 0.0989541, antiderivative size = 87, 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, 88} \[ \frac{1}{10} a^6 c^3 x^{10}+\frac{2}{9} a^5 c^3 x^9-\frac{1}{8} a^4 c^3 x^8-\frac{4}{7} a^3 c^3 x^7-\frac{1}{6} a^2 c^3 x^6+\frac{2}{5} a c^3 x^5+\frac{c^3 x^4}{4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^3*x^4)/4 + (2*a*c^3*x^5)/5 - (a^2*c^3*x^6)/6 - (4*a^3*c^3*x^7)/7 - (a^4*c^3*x^8)/8 + (2*a^5*c^3*x^9)/9 + (a
^6*c^3*x^10)/10

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 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int e^{2 \tanh ^{-1}(a x)} x^3 \left (c-a^2 c x^2\right )^3 \, dx &=c^3 \int x^3 (1-a x)^2 (1+a x)^4 \, dx\\ &=c^3 \int \left (x^3+2 a x^4-a^2 x^5-4 a^3 x^6-a^4 x^7+2 a^5 x^8+a^6 x^9\right ) \, dx\\ &=\frac{c^3 x^4}{4}+\frac{2}{5} a c^3 x^5-\frac{1}{6} a^2 c^3 x^6-\frac{4}{7} a^3 c^3 x^7-\frac{1}{8} a^4 c^3 x^8+\frac{2}{9} a^5 c^3 x^9+\frac{1}{10} a^6 c^3 x^{10}\\ \end{align*}

Mathematica [A]  time = 0.0334374, size = 70, normalized size = 0.8 \[ c^3 \left (\frac{a^6 x^{10}}{10}+\frac{2 a^5 x^9}{9}-\frac{a^4 x^8}{8}-\frac{4 a^3 x^7}{7}-\frac{a^2 x^6}{6}+\frac{2 a x^5}{5}+\frac{x^4}{4}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

c^3*(x^4/4 + (2*a*x^5)/5 - (a^2*x^6)/6 - (4*a^3*x^7)/7 - (a^4*x^8)/8 + (2*a^5*x^9)/9 + (a^6*x^10)/10)

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Maple [A]  time = 0.026, size = 57, normalized size = 0.7 \begin{align*}{c}^{3} \left ({\frac{{a}^{6}{x}^{10}}{10}}+{\frac{2\,{a}^{5}{x}^{9}}{9}}-{\frac{{a}^{4}{x}^{8}}{8}}-{\frac{4\,{x}^{7}{a}^{3}}{7}}-{\frac{{x}^{6}{a}^{2}}{6}}+{\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)^3,x)

[Out]

c^3*(1/10*a^6*x^10+2/9*a^5*x^9-1/8*a^4*x^8-4/7*x^7*a^3-1/6*x^6*a^2+2/5*a*x^5+1/4*x^4)

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Maxima [A]  time = 0.942435, size = 99, normalized size = 1.14 \begin{align*} \frac{1}{10} \, a^{6} c^{3} x^{10} + \frac{2}{9} \, a^{5} c^{3} x^{9} - \frac{1}{8} \, a^{4} c^{3} x^{8} - \frac{4}{7} \, a^{3} c^{3} x^{7} - \frac{1}{6} \, a^{2} c^{3} x^{6} + \frac{2}{5} \, a c^{3} x^{5} + \frac{1}{4} \, c^{3} 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)^3,x, algorithm="maxima")

[Out]

1/10*a^6*c^3*x^10 + 2/9*a^5*c^3*x^9 - 1/8*a^4*c^3*x^8 - 4/7*a^3*c^3*x^7 - 1/6*a^2*c^3*x^6 + 2/5*a*c^3*x^5 + 1/
4*c^3*x^4

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Fricas [A]  time = 1.63667, size = 163, normalized size = 1.87 \begin{align*} \frac{1}{10} \, a^{6} c^{3} x^{10} + \frac{2}{9} \, a^{5} c^{3} x^{9} - \frac{1}{8} \, a^{4} c^{3} x^{8} - \frac{4}{7} \, a^{3} c^{3} x^{7} - \frac{1}{6} \, a^{2} c^{3} x^{6} + \frac{2}{5} \, a c^{3} x^{5} + \frac{1}{4} \, c^{3} 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)^3,x, algorithm="fricas")

[Out]

1/10*a^6*c^3*x^10 + 2/9*a^5*c^3*x^9 - 1/8*a^4*c^3*x^8 - 4/7*a^3*c^3*x^7 - 1/6*a^2*c^3*x^6 + 2/5*a*c^3*x^5 + 1/
4*c^3*x^4

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Sympy [A]  time = 0.096982, size = 82, normalized size = 0.94 \begin{align*} \frac{a^{6} c^{3} x^{10}}{10} + \frac{2 a^{5} c^{3} x^{9}}{9} - \frac{a^{4} c^{3} x^{8}}{8} - \frac{4 a^{3} c^{3} x^{7}}{7} - \frac{a^{2} c^{3} x^{6}}{6} + \frac{2 a c^{3} x^{5}}{5} + \frac{c^{3} 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)**3,x)

[Out]

a**6*c**3*x**10/10 + 2*a**5*c**3*x**9/9 - a**4*c**3*x**8/8 - 4*a**3*c**3*x**7/7 - a**2*c**3*x**6/6 + 2*a*c**3*
x**5/5 + c**3*x**4/4

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Giac [A]  time = 1.12946, size = 99, normalized size = 1.14 \begin{align*} \frac{1}{10} \, a^{6} c^{3} x^{10} + \frac{2}{9} \, a^{5} c^{3} x^{9} - \frac{1}{8} \, a^{4} c^{3} x^{8} - \frac{4}{7} \, a^{3} c^{3} x^{7} - \frac{1}{6} \, a^{2} c^{3} x^{6} + \frac{2}{5} \, a c^{3} x^{5} + \frac{1}{4} \, c^{3} 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)^3,x, algorithm="giac")

[Out]

1/10*a^6*c^3*x^10 + 2/9*a^5*c^3*x^9 - 1/8*a^4*c^3*x^8 - 4/7*a^3*c^3*x^7 - 1/6*a^2*c^3*x^6 + 2/5*a*c^3*x^5 + 1/
4*c^3*x^4