3.1042 \(\int e^{2 \tanh ^{-1}(a x)} x (c-a^2 c x^2)^3 \, dx\)

Optimal. Leaf size=69 \[ \frac{c^3 (a x+1)^8}{8 a^2}-\frac{5 c^3 (a x+1)^7}{7 a^2}+\frac{4 c^3 (a x+1)^6}{3 a^2}-\frac{4 c^3 (a x+1)^5}{5 a^2} \]

[Out]

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

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Rubi [A]  time = 0.0799643, antiderivative size = 69, 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, 77} \[ \frac{c^3 (a x+1)^8}{8 a^2}-\frac{5 c^3 (a x+1)^7}{7 a^2}+\frac{4 c^3 (a x+1)^6}{3 a^2}-\frac{4 c^3 (a x+1)^5}{5 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.0297239, size = 70, normalized size = 1.01 \[ c^3 \left (\frac{a^6 x^8}{8}+\frac{2 a^5 x^7}{7}-\frac{a^4 x^6}{6}-\frac{4 a^3 x^5}{5}-\frac{a^2 x^4}{4}+\frac{2 a x^3}{3}+\frac{x^2}{2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.024, size = 57, normalized size = 0.8 \begin{align*}{c}^{3} \left ({\frac{{x}^{8}{a}^{6}}{8}}+{\frac{2\,{a}^{5}{x}^{7}}{7}}-{\frac{{x}^{6}{a}^{4}}{6}}-{\frac{4\,{x}^{5}{a}^{3}}{5}}-{\frac{{a}^{2}{x}^{4}}{4}}+{\frac{2\,{x}^{3}a}{3}}+{\frac{{x}^{2}}{2}} \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*(-a^2*c*x^2+c)^3,x)

[Out]

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

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Maxima [A]  time = 0.961356, size = 99, normalized size = 1.43 \begin{align*} \frac{1}{8} \, a^{6} c^{3} x^{8} + \frac{2}{7} \, a^{5} c^{3} x^{7} - \frac{1}{6} \, a^{4} c^{3} x^{6} - \frac{4}{5} \, a^{3} c^{3} x^{5} - \frac{1}{4} \, a^{2} c^{3} x^{4} + \frac{2}{3} \, a c^{3} x^{3} + \frac{1}{2} \, c^{3} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*x*(-a^2*c*x^2+c)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.71556, size = 161, normalized size = 2.33 \begin{align*} \frac{1}{8} \, a^{6} c^{3} x^{8} + \frac{2}{7} \, a^{5} c^{3} x^{7} - \frac{1}{6} \, a^{4} c^{3} x^{6} - \frac{4}{5} \, a^{3} c^{3} x^{5} - \frac{1}{4} \, a^{2} c^{3} x^{4} + \frac{2}{3} \, a c^{3} x^{3} + \frac{1}{2} \, c^{3} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*x*(-a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.096283, size = 82, normalized size = 1.19 \begin{align*} \frac{a^{6} c^{3} x^{8}}{8} + \frac{2 a^{5} c^{3} x^{7}}{7} - \frac{a^{4} c^{3} x^{6}}{6} - \frac{4 a^{3} c^{3} x^{5}}{5} - \frac{a^{2} c^{3} x^{4}}{4} + \frac{2 a c^{3} x^{3}}{3} + \frac{c^{3} x^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**2/(-a**2*x**2+1)*x*(-a**2*c*x**2+c)**3,x)

[Out]

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

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Giac [A]  time = 1.13904, size = 99, normalized size = 1.43 \begin{align*} \frac{1}{8} \, a^{6} c^{3} x^{8} + \frac{2}{7} \, a^{5} c^{3} x^{7} - \frac{1}{6} \, a^{4} c^{3} x^{6} - \frac{4}{5} \, a^{3} c^{3} x^{5} - \frac{1}{4} \, a^{2} c^{3} x^{4} + \frac{2}{3} \, a c^{3} x^{3} + \frac{1}{2} \, c^{3} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*x*(-a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

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