3.1039 \(\int e^{2 \tanh ^{-1}(a x)} x^4 (c-a^2 c x^2)^3 \, dx\)

Optimal. Leaf size=87 \[ \frac{1}{11} a^6 c^3 x^{11}+\frac{1}{5} a^5 c^3 x^{10}-\frac{1}{9} a^4 c^3 x^9-\frac{1}{2} a^3 c^3 x^8-\frac{1}{7} a^2 c^3 x^7+\frac{1}{3} a c^3 x^6+\frac{c^3 x^5}{5} \]

[Out]

(c^3*x^5)/5 + (a*c^3*x^6)/3 - (a^2*c^3*x^7)/7 - (a^3*c^3*x^8)/2 - (a^4*c^3*x^9)/9 + (a^5*c^3*x^10)/5 + (a^6*c^
3*x^11)/11

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Rubi [A]  time = 0.10185, 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} \[ \frac{1}{11} a^6 c^3 x^{11}+\frac{1}{5} a^5 c^3 x^{10}-\frac{1}{9} a^4 c^3 x^9-\frac{1}{2} a^3 c^3 x^8-\frac{1}{7} a^2 c^3 x^7+\frac{1}{3} a c^3 x^6+\frac{c^3 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcTanh[a*x])*x^4*(c - a^2*c*x^2)^3,x]

[Out]

(c^3*x^5)/5 + (a*c^3*x^6)/3 - (a^2*c^3*x^7)/7 - (a^3*c^3*x^8)/2 - (a^4*c^3*x^9)/9 + (a^5*c^3*x^10)/5 + (a^6*c^
3*x^11)/11

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^4 \left (c-a^2 c x^2\right )^3 \, dx &=c^3 \int x^4 (1-a x)^2 (1+a x)^4 \, dx\\ &=c^3 \int \left (x^4+2 a x^5-a^2 x^6-4 a^3 x^7-a^4 x^8+2 a^5 x^9+a^6 x^{10}\right ) \, dx\\ &=\frac{c^3 x^5}{5}+\frac{1}{3} a c^3 x^6-\frac{1}{7} a^2 c^3 x^7-\frac{1}{2} a^3 c^3 x^8-\frac{1}{9} a^4 c^3 x^9+\frac{1}{5} a^5 c^3 x^{10}+\frac{1}{11} a^6 c^3 x^{11}\\ \end{align*}

Mathematica [A]  time = 0.037536, size = 70, normalized size = 0.8 \[ c^3 \left (\frac{a^6 x^{11}}{11}+\frac{a^5 x^{10}}{5}-\frac{a^4 x^9}{9}-\frac{a^3 x^8}{2}-\frac{a^2 x^7}{7}+\frac{a x^6}{3}+\frac{x^5}{5}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^(2*ArcTanh[a*x])*x^4*(c - a^2*c*x^2)^3,x]

[Out]

c^3*(x^5/5 + (a*x^6)/3 - (a^2*x^7)/7 - (a^3*x^8)/2 - (a^4*x^9)/9 + (a^5*x^10)/5 + (a^6*x^11)/11)

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Maple [A]  time = 0.026, size = 57, normalized size = 0.7 \begin{align*}{c}^{3} \left ({\frac{{a}^{6}{x}^{11}}{11}}+{\frac{{a}^{5}{x}^{10}}{5}}-{\frac{{a}^{4}{x}^{9}}{9}}-{\frac{{a}^{3}{x}^{8}}{2}}-{\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)^3,x)

[Out]

c^3*(1/11*a^6*x^11+1/5*a^5*x^10-1/9*a^4*x^9-1/2*a^3*x^8-1/7*a^2*x^7+1/3*x^6*a+1/5*x^5)

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Maxima [A]  time = 0.955658, size = 99, normalized size = 1.14 \begin{align*} \frac{1}{11} \, a^{6} c^{3} x^{11} + \frac{1}{5} \, a^{5} c^{3} x^{10} - \frac{1}{9} \, a^{4} c^{3} x^{9} - \frac{1}{2} \, a^{3} c^{3} x^{8} - \frac{1}{7} \, a^{2} c^{3} x^{7} + \frac{1}{3} \, a c^{3} x^{6} + \frac{1}{5} \, c^{3} 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)^3,x, algorithm="maxima")

[Out]

1/11*a^6*c^3*x^11 + 1/5*a^5*c^3*x^10 - 1/9*a^4*c^3*x^9 - 1/2*a^3*c^3*x^8 - 1/7*a^2*c^3*x^7 + 1/3*a*c^3*x^6 + 1
/5*c^3*x^5

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Fricas [A]  time = 1.6361, size = 165, normalized size = 1.9 \begin{align*} \frac{1}{11} \, a^{6} c^{3} x^{11} + \frac{1}{5} \, a^{5} c^{3} x^{10} - \frac{1}{9} \, a^{4} c^{3} x^{9} - \frac{1}{2} \, a^{3} c^{3} x^{8} - \frac{1}{7} \, a^{2} c^{3} x^{7} + \frac{1}{3} \, a c^{3} x^{6} + \frac{1}{5} \, c^{3} 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)^3,x, algorithm="fricas")

[Out]

1/11*a^6*c^3*x^11 + 1/5*a^5*c^3*x^10 - 1/9*a^4*c^3*x^9 - 1/2*a^3*c^3*x^8 - 1/7*a^2*c^3*x^7 + 1/3*a*c^3*x^6 + 1
/5*c^3*x^5

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Sympy [A]  time = 0.09861, size = 76, normalized size = 0.87 \begin{align*} \frac{a^{6} c^{3} x^{11}}{11} + \frac{a^{5} c^{3} x^{10}}{5} - \frac{a^{4} c^{3} x^{9}}{9} - \frac{a^{3} c^{3} x^{8}}{2} - \frac{a^{2} c^{3} x^{7}}{7} + \frac{a c^{3} x^{6}}{3} + \frac{c^{3} 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)**3,x)

[Out]

a**6*c**3*x**11/11 + a**5*c**3*x**10/5 - a**4*c**3*x**9/9 - a**3*c**3*x**8/2 - a**2*c**3*x**7/7 + a*c**3*x**6/
3 + c**3*x**5/5

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Giac [A]  time = 1.14489, size = 99, normalized size = 1.14 \begin{align*} \frac{1}{11} \, a^{6} c^{3} x^{11} + \frac{1}{5} \, a^{5} c^{3} x^{10} - \frac{1}{9} \, a^{4} c^{3} x^{9} - \frac{1}{2} \, a^{3} c^{3} x^{8} - \frac{1}{7} \, a^{2} c^{3} x^{7} + \frac{1}{3} \, a c^{3} x^{6} + \frac{1}{5} \, c^{3} 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)^3,x, algorithm="giac")

[Out]

1/11*a^6*c^3*x^11 + 1/5*a^5*c^3*x^10 - 1/9*a^4*c^3*x^9 - 1/2*a^3*c^3*x^8 - 1/7*a^2*c^3*x^7 + 1/3*a*c^3*x^6 + 1
/5*c^3*x^5