3.4.72 \(\int e^{\tanh ^{-1}(x)} (1+x)^{3/2} \, dx\) [372]

Optimal. Leaf size=38 \[ -8 \sqrt {1-x}+\frac {8}{3} (1-x)^{3/2}-\frac {2}{5} (1-x)^{5/2} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6264, 45} \begin {gather*} -\frac {2}{5} (1-x)^{5/2}+\frac {8}{3} (1-x)^{3/2}-8 \sqrt {1-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[x]*(1 + x)^(3/2),x]

[Out]

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

Rule 45

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])

Rule 6264

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

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Mathematica [A]
time = 0.01, size = 23, normalized size = 0.61 \begin {gather*} -\frac {2}{15} \sqrt {1-x} \left (43+14 x+3 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - x]*(43 + 14*x + 3*x^2))/15

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Maple [A]
time = 0.77, size = 27, normalized size = 0.71

method	result	size
default	\(-\frac {2 \sqrt {-x^{2}+1}\, \left (3 x^{2}+14 x +43\right )}{15 \sqrt {1+x}}\)	\(27\)
gosper	\(\frac {2 \left (x -1\right ) \left (3 x^{2}+14 x +43\right ) \sqrt {1+x}}{15 \sqrt {-x^{2}+1}}\)	\(30\)
risch	\(\frac {2 \sqrt {\frac {-x^{2}+1}{1+x}}\, \sqrt {1+x}\, \left (3 x^{2}+14 x +43\right ) \left (x -1\right )}{15 \sqrt {-x^{2}+1}\, \sqrt {1-x}}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)^(5/2)/(-x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/15/(1+x)^(1/2)*(-x^2+1)^(1/2)*(3*x^2+14*x+43)

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Maxima [A]
time = 0.27, size = 24, normalized size = 0.63 \begin {gather*} \frac {2 \, {\left (3 \, x^{3} + 11 \, x^{2} + 29 \, x - 43\right )}}{15 \, \sqrt {-x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*x^3 + 11*x^2 + 29*x - 43)/sqrt(-x + 1)

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Fricas [A]
time = 0.34, size = 26, normalized size = 0.68 \begin {gather*} -\frac {2 \, {\left (3 \, x^{2} + 14 \, x + 43\right )} \sqrt {-x^{2} + 1}}{15 \, \sqrt {x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(3*x^2 + 14*x + 43)*sqrt(-x^2 + 1)/sqrt(x + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(t_

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Mupad [B]
time = 0.88, size = 45, normalized size = 1.18 \begin {gather*} -\frac {6\,x^2\,\sqrt {1-x^2}+28\,x\,\sqrt {1-x^2}+86\,\sqrt {1-x^2}}{15\,\sqrt {x+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)^(5/2)/(1 - x^2)^(1/2),x)

[Out]

-(6*x^2*(1 - x^2)^(1/2) + 28*x*(1 - x^2)^(1/2) + 86*(1 - x^2)^(1/2))/(15*(x + 1)^(1/2))

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