3.1041 \(\int e^{2 \tanh ^{-1}(a x)} x^2 (c-a^2 c x^2)^3 \, dx\)

Optimal. Leaf size=84 \[ \frac{c^3 (a x+1)^9}{9 a^3}-\frac{3 c^3 (a x+1)^8}{4 a^3}+\frac{13 c^3 (a x+1)^7}{7 a^3}-\frac{2 c^3 (a x+1)^6}{a^3}+\frac{4 c^3 (a x+1)^5}{5 a^3} \]

[Out]

(4*c^3*(1 + a*x)^5)/(5*a^3) - (2*c^3*(1 + a*x)^6)/a^3 + (13*c^3*(1 + a*x)^7)/(7*a^3) - (3*c^3*(1 + a*x)^8)/(4*
a^3) + (c^3*(1 + a*x)^9)/(9*a^3)

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Rubi [A]  time = 0.10281, antiderivative size = 84, 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} \[ \frac{c^3 (a x+1)^9}{9 a^3}-\frac{3 c^3 (a x+1)^8}{4 a^3}+\frac{13 c^3 (a x+1)^7}{7 a^3}-\frac{2 c^3 (a x+1)^6}{a^3}+\frac{4 c^3 (a x+1)^5}{5 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*c^3*(1 + a*x)^5)/(5*a^3) - (2*c^3*(1 + a*x)^6)/a^3 + (13*c^3*(1 + a*x)^7)/(7*a^3) - (3*c^3*(1 + a*x)^8)/(4*
a^3) + (c^3*(1 + a*x)^9)/(9*a^3)

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^2 \left (c-a^2 c x^2\right )^3 \, dx &=c^3 \int x^2 (1-a x)^2 (1+a x)^4 \, dx\\ &=c^3 \int \left (\frac{4 (1+a x)^4}{a^2}-\frac{12 (1+a x)^5}{a^2}+\frac{13 (1+a x)^6}{a^2}-\frac{6 (1+a x)^7}{a^2}+\frac{(1+a x)^8}{a^2}\right ) \, dx\\ &=\frac{4 c^3 (1+a x)^5}{5 a^3}-\frac{2 c^3 (1+a x)^6}{a^3}+\frac{13 c^3 (1+a x)^7}{7 a^3}-\frac{3 c^3 (1+a x)^8}{4 a^3}+\frac{c^3 (1+a x)^9}{9 a^3}\\ \end{align*}

Mathematica [A]  time = 0.0317254, size = 70, normalized size = 0.83 \[ c^3 \left (\frac{a^6 x^9}{9}+\frac{a^5 x^8}{4}-\frac{a^4 x^7}{7}-\frac{2 a^3 x^6}{3}-\frac{a^2 x^5}{5}+\frac{a x^4}{2}+\frac{x^3}{3}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

[Out]

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

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Maxima [A]  time = 0.945745, size = 99, normalized size = 1.18 \begin{align*} \frac{1}{9} \, a^{6} c^{3} x^{9} + \frac{1}{4} \, a^{5} c^{3} x^{8} - \frac{1}{7} \, a^{4} c^{3} x^{7} - \frac{2}{3} \, a^{3} c^{3} x^{6} - \frac{1}{5} \, a^{2} c^{3} x^{5} + \frac{1}{2} \, a c^{3} x^{4} + \frac{1}{3} \, c^{3} 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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.61706, size = 161, normalized size = 1.92 \begin{align*} \frac{1}{9} \, a^{6} c^{3} x^{9} + \frac{1}{4} \, a^{5} c^{3} x^{8} - \frac{1}{7} \, a^{4} c^{3} x^{7} - \frac{2}{3} \, a^{3} c^{3} x^{6} - \frac{1}{5} \, a^{2} c^{3} x^{5} + \frac{1}{2} \, a c^{3} x^{4} + \frac{1}{3} \, c^{3} 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)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.097526, size = 78, normalized size = 0.93 \begin{align*} \frac{a^{6} c^{3} x^{9}}{9} + \frac{a^{5} c^{3} x^{8}}{4} - \frac{a^{4} c^{3} x^{7}}{7} - \frac{2 a^{3} c^{3} x^{6}}{3} - \frac{a^{2} c^{3} x^{5}}{5} + \frac{a c^{3} x^{4}}{2} + \frac{c^{3} 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)**3,x)

[Out]

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

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Giac [A]  time = 1.1377, size = 99, normalized size = 1.18 \begin{align*} \frac{1}{9} \, a^{6} c^{3} x^{9} + \frac{1}{4} \, a^{5} c^{3} x^{8} - \frac{1}{7} \, a^{4} c^{3} x^{7} - \frac{2}{3} \, a^{3} c^{3} x^{6} - \frac{1}{5} \, a^{2} c^{3} x^{5} + \frac{1}{2} \, a c^{3} x^{4} + \frac{1}{3} \, c^{3} 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)^3,x, algorithm="giac")

[Out]

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