3.371 \(\int e^{\tanh ^{-1}(x)} x (1+x)^{3/2} \, dx\)

Optimal. Leaf size=49 \[ \frac {2}{7} (1-x)^{7/2}-2 (1-x)^{5/2}+\frac {16}{3} (1-x)^{3/2}-8 \sqrt {1-x} \]

[Out]

16/3*(1-x)^(3/2)-2*(1-x)^(5/2)+2/7*(1-x)^(7/2)-8*(1-x)^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6129, 77} \[ \frac {2}{7} (1-x)^{7/2}-2 (1-x)^{5/2}+\frac {16}{3} (1-x)^{3/2}-8 \sqrt {1-x} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[x]*x*(1 + x)^(3/2),x]

[Out]

-8*Sqrt[1 - x] + (16*(1 - x)^(3/2))/3 - 2*(1 - x)^(5/2) + (2*(1 - x)^(7/2))/7

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]))))

Rule 6129

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Dist[c^p, Int[(u*(1 + (d*x)/c)
^p*(1 + a*x)^(n/2))/(1 - a*x)^(n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ
[p] || GtQ[c, 0])

Rubi steps

\begin {align*} \int e^{\tanh ^{-1}(x)} x (1+x)^{3/2} \, dx &=\int \frac {x (1+x)^2}{\sqrt {1-x}} \, dx\\ &=\int \left (\frac {4}{\sqrt {1-x}}-8 \sqrt {1-x}+5 (1-x)^{3/2}-(1-x)^{5/2}\right ) \, dx\\ &=-8 \sqrt {1-x}+\frac {16}{3} (1-x)^{3/2}-2 (1-x)^{5/2}+\frac {2}{7} (1-x)^{7/2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 28, normalized size = 0.57 \[ -\frac {2}{21} \sqrt {1-x} \left (3 x^3+12 x^2+23 x+46\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcTanh[x]*x*(1 + x)^(3/2),x]

[Out]

(-2*Sqrt[1 - x]*(46 + 23*x + 12*x^2 + 3*x^3))/21

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fricas [A]  time = 0.62, size = 31, normalized size = 0.63 \[ -\frac {2 \, {\left (3 \, x^{3} + 12 \, x^{2} + 23 \, x + 46\right )} \sqrt {-x^{2} + 1}}{21 \, \sqrt {x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/21*(3*x^3 + 12*x^2 + 23*x + 46)*sqrt(-x^2 + 1)/sqrt(x + 1)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(t_nostep)]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.03, size = 35, normalized size = 0.71 \[ \frac {2 \left (-1+x \right ) \left (3 x^{3}+12 x^{2}+23 x +46\right ) \sqrt {1+x}}{21 \sqrt {-x^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*(-1+x)*(3*x^3+12*x^2+23*x+46)*(1+x)^(1/2)/(-x^2+1)^(1/2)

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maxima [A]  time = 0.34, size = 29, normalized size = 0.59 \[ \frac {2 \, {\left (3 \, x^{4} + 9 \, x^{3} + 11 \, x^{2} + 23 \, x - 46\right )}}{21 \, \sqrt {-x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*(3*x^4 + 9*x^3 + 11*x^2 + 23*x - 46)/sqrt(-x + 1)

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mupad [B]  time = 0.92, size = 52, normalized size = 1.06 \[ -\frac {\sqrt {1-x^2}\,\left (\frac {46\,x\,\sqrt {x+1}}{21}+\frac {92\,\sqrt {x+1}}{21}+\frac {8\,x^2\,\sqrt {x+1}}{7}+\frac {2\,x^3\,\sqrt {x+1}}{7}\right )}{x+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((1 - x^2)^(1/2)*((46*x*(x + 1)^(1/2))/21 + (92*(x + 1)^(1/2))/21 + (8*x^2*(x + 1)^(1/2))/7 + (2*x^3*(x + 1)^
(1/2))/7))/(x + 1)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \left (x + 1\right )^{\frac {5}{2}}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*(x + 1)**(5/2)/sqrt(-(x - 1)*(x + 1)), x)

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