3.826 \(\int e^{2 \tanh ^{-1}(a+b x)} x^4 \, dx\)

Optimal. Leaf size=83 \[ -\frac{2 (1-a) x^3}{3 b^2}-\frac{(1-a)^2 x^2}{b^3}-\frac{2 (1-a)^3 x}{b^4}-\frac{2 (1-a)^4 \log (-a-b x+1)}{b^5}-\frac{x^4}{2 b}-\frac{x^5}{5} \]

[Out]

(-2*(1 - a)^3*x)/b^4 - ((1 - a)^2*x^2)/b^3 - (2*(1 - a)*x^3)/(3*b^2) - x^4/(2*b) - x^5/5 - (2*(1 - a)^4*Log[1
- a - b*x])/b^5

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Rubi [A]  time = 0.0834926, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {6163, 77} \[ -\frac{2 (1-a) x^3}{3 b^2}-\frac{(1-a)^2 x^2}{b^3}-\frac{2 (1-a)^3 x}{b^4}-\frac{2 (1-a)^4 \log (-a-b x+1)}{b^5}-\frac{x^4}{2 b}-\frac{x^5}{5} \]

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcTanh[a + b*x])*x^4,x]

[Out]

(-2*(1 - a)^3*x)/b^4 - ((1 - a)^2*x^2)/b^3 - (2*(1 - a)*x^3)/(3*b^2) - x^4/(2*b) - x^5/5 - (2*(1 - a)^4*Log[1
- a - b*x])/b^5

Rule 6163

Int[E^(ArcTanh[(c_.)*((a_) + (b_.)*(x_))]*(n_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[((d + e*x)^m*(1
+ a*c + b*c*x)^(n/2))/(1 - a*c - b*c*x)^(n/2), x] /; FreeQ[{a, b, c, d, e, m, n}, x]

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+b x)} x^4 \, dx &=\int \frac{x^4 (1+a+b x)}{1-a-b x} \, dx\\ &=\int \left (\frac{2 (-1+a)^3}{b^4}-\frac{2 (-1+a)^2 x}{b^3}+\frac{2 (-1+a) x^2}{b^2}-\frac{2 x^3}{b}-x^4-\frac{2 (-1+a)^4}{b^4 (-1+a+b x)}\right ) \, dx\\ &=-\frac{2 (1-a)^3 x}{b^4}-\frac{(1-a)^2 x^2}{b^3}-\frac{2 (1-a) x^3}{3 b^2}-\frac{x^4}{2 b}-\frac{x^5}{5}-\frac{2 (1-a)^4 \log (1-a-b x)}{b^5}\\ \end{align*}

Mathematica [A]  time = 0.0636001, size = 75, normalized size = 0.9 \[ \frac{2 (a-1) x^3}{3 b^2}-\frac{(a-1)^2 x^2}{b^3}+\frac{2 (a-1)^3 x}{b^4}-\frac{2 (a-1)^4 \log (-a-b x+1)}{b^5}-\frac{x^4}{2 b}-\frac{x^5}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(2*ArcTanh[a + b*x])*x^4,x]

[Out]

(2*(-1 + a)^3*x)/b^4 - ((-1 + a)^2*x^2)/b^3 + (2*(-1 + a)*x^3)/(3*b^2) - x^4/(2*b) - x^5/5 - (2*(-1 + a)^4*Log
[1 - a - b*x])/b^5

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Maple [B]  time = 0.03, size = 161, normalized size = 1.9 \begin{align*} -{\frac{{x}^{5}}{5}}-{\frac{{x}^{4}}{2\,b}}+{\frac{2\,{x}^{3}a}{3\,{b}^{2}}}-{\frac{2\,{x}^{3}}{3\,{b}^{2}}}-{\frac{{a}^{2}{x}^{2}}{{b}^{3}}}+2\,{\frac{a{x}^{2}}{{b}^{3}}}+2\,{\frac{x{a}^{3}}{{b}^{4}}}-{\frac{{x}^{2}}{{b}^{3}}}-6\,{\frac{{a}^{2}x}{{b}^{4}}}+6\,{\frac{ax}{{b}^{4}}}-2\,{\frac{x}{{b}^{4}}}-2\,{\frac{\ln \left ( bx+a-1 \right ){a}^{4}}{{b}^{5}}}+8\,{\frac{\ln \left ( bx+a-1 \right ){a}^{3}}{{b}^{5}}}-12\,{\frac{\ln \left ( bx+a-1 \right ){a}^{2}}{{b}^{5}}}+8\,{\frac{\ln \left ( bx+a-1 \right ) a}{{b}^{5}}}-2\,{\frac{\ln \left ( bx+a-1 \right ) }{{b}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*x^5-1/2*x^4/b+2/3/b^2*x^3*a-2/3/b^2*x^3-1/b^3*x^2*a^2+2/b^3*x^2*a+2/b^4*x*a^3-1/b^3*x^2-6/b^4*x*a^2+6/b^4
*a*x-2/b^4*x-2/b^5*ln(b*x+a-1)*a^4+8/b^5*ln(b*x+a-1)*a^3-12/b^5*ln(b*x+a-1)*a^2+8/b^5*ln(b*x+a-1)*a-2/b^5*ln(b
*x+a-1)

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Maxima [A]  time = 0.961459, size = 127, normalized size = 1.53 \begin{align*} -\frac{6 \, b^{4} x^{5} + 15 \, b^{3} x^{4} - 20 \,{\left (a - 1\right )} b^{2} x^{3} + 30 \,{\left (a^{2} - 2 \, a + 1\right )} b x^{2} - 60 \,{\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )} x}{30 \, b^{4}} - \frac{2 \,{\left (a^{4} - 4 \, a^{3} + 6 \, a^{2} - 4 \, a + 1\right )} \log \left (b x + a - 1\right )}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(6*b^4*x^5 + 15*b^3*x^4 - 20*(a - 1)*b^2*x^3 + 30*(a^2 - 2*a + 1)*b*x^2 - 60*(a^3 - 3*a^2 + 3*a - 1)*x)/
b^4 - 2*(a^4 - 4*a^3 + 6*a^2 - 4*a + 1)*log(b*x + a - 1)/b^5

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Fricas [A]  time = 1.88907, size = 234, normalized size = 2.82 \begin{align*} -\frac{6 \, b^{5} x^{5} + 15 \, b^{4} x^{4} - 20 \,{\left (a - 1\right )} b^{3} x^{3} + 30 \,{\left (a^{2} - 2 \, a + 1\right )} b^{2} x^{2} - 60 \,{\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )} b x + 60 \,{\left (a^{4} - 4 \, a^{3} + 6 \, a^{2} - 4 \, a + 1\right )} \log \left (b x + a - 1\right )}{30 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(6*b^5*x^5 + 15*b^4*x^4 - 20*(a - 1)*b^3*x^3 + 30*(a^2 - 2*a + 1)*b^2*x^2 - 60*(a^3 - 3*a^2 + 3*a - 1)*b
*x + 60*(a^4 - 4*a^3 + 6*a^2 - 4*a + 1)*log(b*x + a - 1))/b^5

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Sympy [A]  time = 0.444106, size = 78, normalized size = 0.94 \begin{align*} - \frac{x^{5}}{5} - \frac{x^{4}}{2 b} + \frac{x^{3} \left (2 a - 2\right )}{3 b^{2}} - \frac{x^{2} \left (a^{2} - 2 a + 1\right )}{b^{3}} + \frac{x \left (2 a^{3} - 6 a^{2} + 6 a - 2\right )}{b^{4}} - \frac{2 \left (a - 1\right )^{4} \log{\left (a + b x - 1 \right )}}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**5/5 - x**4/(2*b) + x**3*(2*a - 2)/(3*b**2) - x**2*(a**2 - 2*a + 1)/b**3 + x*(2*a**3 - 6*a**2 + 6*a - 2)/b*
*4 - 2*(a - 1)**4*log(a + b*x - 1)/b**5

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Giac [A]  time = 1.15296, size = 165, normalized size = 1.99 \begin{align*} -\frac{2 \,{\left (a^{4} - 4 \, a^{3} + 6 \, a^{2} - 4 \, a + 1\right )} \log \left ({\left | b x + a - 1 \right |}\right )}{b^{5}} - \frac{6 \, b^{5} x^{5} + 15 \, b^{4} x^{4} - 20 \, a b^{3} x^{3} + 30 \, a^{2} b^{2} x^{2} + 20 \, b^{3} x^{3} - 60 \, a^{3} b x - 60 \, a b^{2} x^{2} + 180 \, a^{2} b x + 30 \, b^{2} x^{2} - 180 \, a b x + 60 \, b x}{30 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(a^4 - 4*a^3 + 6*a^2 - 4*a + 1)*log(abs(b*x + a - 1))/b^5 - 1/30*(6*b^5*x^5 + 15*b^4*x^4 - 20*a*b^3*x^3 + 3
0*a^2*b^2*x^2 + 20*b^3*x^3 - 60*a^3*b*x - 60*a*b^2*x^2 + 180*a^2*b*x + 30*b^2*x^2 - 180*a*b*x + 60*b*x)/b^5