3.1182 \(\int e^{4 \tanh ^{-1}(a x)} (c-a^2 c x^2)^5 \, dx\)

Optimal. Leaf size=66 \[ -\frac{c^5 (a x+1)^{11}}{11 a}+\frac{3 c^5 (a x+1)^{10}}{5 a}-\frac{4 c^5 (a x+1)^9}{3 a}+\frac{c^5 (a x+1)^8}{a} \]

[Out]

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

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Rubi [A]  time = 0.0643129, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {6140, 43} \[ -\frac{c^5 (a x+1)^{11}}{11 a}+\frac{3 c^5 (a x+1)^{10}}{5 a}-\frac{4 c^5 (a x+1)^9}{3 a}+\frac{c^5 (a x+1)^8}{a} \]

Antiderivative was successfully verified.

[In]

Int[E^(4*ArcTanh[a*x])*(c - a^2*c*x^2)^5,x]

[Out]

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

Rule 6140

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[(1 - a*x)^(p - n/2)*
(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, 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^{4 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx &=c^5 \int (1-a x)^3 (1+a x)^7 \, dx\\ &=c^5 \int \left (8 (1+a x)^7-12 (1+a x)^8+6 (1+a x)^9-(1+a x)^{10}\right ) \, dx\\ &=\frac{c^5 (1+a x)^8}{a}-\frac{4 c^5 (1+a x)^9}{3 a}+\frac{3 c^5 (1+a x)^{10}}{5 a}-\frac{c^5 (1+a x)^{11}}{11 a}\\ \end{align*}

Mathematica [A]  time = 0.0322227, size = 39, normalized size = 0.59 \[ -\frac{c^5 (a x+1)^8 \left (15 a^3 x^3-54 a^2 x^2+67 a x-29\right )}{165 a} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(4*ArcTanh[a*x])*(c - a^2*c*x^2)^5,x]

[Out]

-(c^5*(1 + a*x)^8*(-29 + 67*a*x - 54*a^2*x^2 + 15*a^3*x^3))/(165*a)

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Maple [A]  time = 0.026, size = 75, normalized size = 1.1 \begin{align*}{c}^{5} \left ( -{\frac{{a}^{10}{x}^{11}}{11}}-{\frac{2\,{a}^{9}{x}^{10}}{5}}-{\frac{{x}^{9}{a}^{8}}{3}}+{a}^{7}{x}^{8}+2\,{x}^{7}{a}^{6}-{\frac{14\,{x}^{5}{a}^{4}}{5}}-2\,{x}^{4}{a}^{3}+{x}^{3}{a}^{2}+2\,a{x}^{2}+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^4/(-a^2*x^2+1)^2*(-a^2*c*x^2+c)^5,x)

[Out]

c^5*(-1/11*a^10*x^11-2/5*a^9*x^10-1/3*x^9*a^8+a^7*x^8+2*x^7*a^6-14/5*x^5*a^4-2*x^4*a^3+x^3*a^2+2*a*x^2+x)

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Maxima [A]  time = 0.970398, size = 136, normalized size = 2.06 \begin{align*} -\frac{1}{11} \, a^{10} c^{5} x^{11} - \frac{2}{5} \, a^{9} c^{5} x^{10} - \frac{1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} + 2 \, a^{6} c^{5} x^{7} - \frac{14}{5} \, a^{4} c^{5} x^{5} - 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} + 2 \, a c^{5} x^{2} + c^{5} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^4/(-a^2*x^2+1)^2*(-a^2*c*x^2+c)^5,x, algorithm="maxima")

[Out]

-1/11*a^10*c^5*x^11 - 2/5*a^9*c^5*x^10 - 1/3*a^8*c^5*x^9 + a^7*c^5*x^8 + 2*a^6*c^5*x^7 - 14/5*a^4*c^5*x^5 - 2*
a^3*c^5*x^4 + a^2*c^5*x^3 + 2*a*c^5*x^2 + c^5*x

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Fricas [A]  time = 2.26109, size = 215, normalized size = 3.26 \begin{align*} -\frac{1}{11} \, a^{10} c^{5} x^{11} - \frac{2}{5} \, a^{9} c^{5} x^{10} - \frac{1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} + 2 \, a^{6} c^{5} x^{7} - \frac{14}{5} \, a^{4} c^{5} x^{5} - 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} + 2 \, a c^{5} x^{2} + c^{5} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^4/(-a^2*x^2+1)^2*(-a^2*c*x^2+c)^5,x, algorithm="fricas")

[Out]

-1/11*a^10*c^5*x^11 - 2/5*a^9*c^5*x^10 - 1/3*a^8*c^5*x^9 + a^7*c^5*x^8 + 2*a^6*c^5*x^7 - 14/5*a^4*c^5*x^5 - 2*
a^3*c^5*x^4 + a^2*c^5*x^3 + 2*a*c^5*x^2 + c^5*x

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Sympy [B]  time = 0.122364, size = 109, normalized size = 1.65 \begin{align*} - \frac{a^{10} c^{5} x^{11}}{11} - \frac{2 a^{9} c^{5} x^{10}}{5} - \frac{a^{8} c^{5} x^{9}}{3} + a^{7} c^{5} x^{8} + 2 a^{6} c^{5} x^{7} - \frac{14 a^{4} c^{5} x^{5}}{5} - 2 a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} + 2 a c^{5} x^{2} + c^{5} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**4/(-a**2*x**2+1)**2*(-a**2*c*x**2+c)**5,x)

[Out]

-a**10*c**5*x**11/11 - 2*a**9*c**5*x**10/5 - a**8*c**5*x**9/3 + a**7*c**5*x**8 + 2*a**6*c**5*x**7 - 14*a**4*c*
*5*x**5/5 - 2*a**3*c**5*x**4 + a**2*c**5*x**3 + 2*a*c**5*x**2 + c**5*x

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Giac [A]  time = 1.14567, size = 136, normalized size = 2.06 \begin{align*} -\frac{1}{11} \, a^{10} c^{5} x^{11} - \frac{2}{5} \, a^{9} c^{5} x^{10} - \frac{1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} + 2 \, a^{6} c^{5} x^{7} - \frac{14}{5} \, a^{4} c^{5} x^{5} - 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} + 2 \, a c^{5} x^{2} + c^{5} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^4/(-a^2*x^2+1)^2*(-a^2*c*x^2+c)^5,x, algorithm="giac")

[Out]

-1/11*a^10*c^5*x^11 - 2/5*a^9*c^5*x^10 - 1/3*a^8*c^5*x^9 + a^7*c^5*x^8 + 2*a^6*c^5*x^7 - 14/5*a^4*c^5*x^5 - 2*
a^3*c^5*x^4 + a^2*c^5*x^3 + 2*a*c^5*x^2 + c^5*x