3.8.29 \(\int \frac {(1+x)^{3/2}}{\sqrt {1-x} x^2} \, dx\) [729]

Optimal. Leaf size=44 \[ -\frac {\sqrt {1-x} \sqrt {1+x}}{x}+\sin ^{-1}(x)-2 \tanh ^{-1}\left (\sqrt {1-x} \sqrt {1+x}\right ) \]

[Out]

arcsin(x)-2*arctanh((1-x)^(1/2)*(1+x)^(1/2))-(1-x)^(1/2)*(1+x)^(1/2)/x

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Rubi [A]
time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {100, 163, 41, 222, 94, 212} \begin {gather*} \text {ArcSin}(x)-\frac {\sqrt {1-x} \sqrt {x+1}}{x}-2 \tanh ^{-1}\left (\sqrt {1-x} \sqrt {x+1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x)^(3/2)/(Sqrt[1 - x]*x^2),x]

[Out]

-((Sqrt[1 - x]*Sqrt[1 + x])/x) + ArcSin[x] - 2*ArcTanh[Sqrt[1 - x]*Sqrt[1 + x]]

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 94

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 163

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin {align*} \int \frac {(1+x)^{3/2}}{\sqrt {1-x} x^2} \, dx &=-\frac {\sqrt {1-x} \sqrt {1+x}}{x}-\int \frac {-2-x}{\sqrt {1-x} x \sqrt {1+x}} \, dx\\ &=-\frac {\sqrt {1-x} \sqrt {1+x}}{x}+2 \int \frac {1}{\sqrt {1-x} x \sqrt {1+x}} \, dx+\int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=-\frac {\sqrt {1-x} \sqrt {1+x}}{x}-2 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x} \sqrt {1+x}\right )+\int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\frac {\sqrt {1-x} \sqrt {1+x}}{x}+\sin ^{-1}(x)-2 \tanh ^{-1}\left (\sqrt {1-x} \sqrt {1+x}\right )\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 59, normalized size = 1.34 \begin {gather*} -\frac {\sqrt {1-x^2}+2 x \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right )+4 x \tanh ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)^(3/2)/(Sqrt[1 - x]*x^2),x]

[Out]

-((Sqrt[1 - x^2] + 2*x*ArcTan[Sqrt[1 - x]/Sqrt[1 + x]] + 4*x*ArcTanh[Sqrt[1 - x]/Sqrt[1 + x]])/x)

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Maple [A]
time = 0.08, size = 55, normalized size = 1.25

method	result	size
default	\(\frac {\sqrt {1+x}\, \sqrt {1-x}\, \left (\arcsin \left (x \right ) x -2 \arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right ) x -\sqrt {-x^{2}+1}\right )}{x \sqrt {-x^{2}+1}}\)	\(55\)
risch	\(\frac {\sqrt {1+x}\, \left (-1+x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{x \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}+\frac {\left (\arcsin \left (x \right )-2 \arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right )\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{\sqrt {1-x}\, \sqrt {1+x}}\)	\(81\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)^(3/2)/x^2/(1-x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

(1+x)^(1/2)*(1-x)^(1/2)*(arcsin(x)*x-2*arctanh(1/(-x^2+1)^(1/2))*x-(-x^2+1)^(1/2))/x/(-x^2+1)^(1/2)

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Maxima [A]
time = 0.49, size = 42, normalized size = 0.95 \begin {gather*} -\frac {\sqrt {-x^{2} + 1}}{x} + \arcsin \left (x\right ) - 2 \, \log \left (\frac {2 \, \sqrt {-x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 1)/x + arcsin(x) - 2*log(2*sqrt(-x^2 + 1)/abs(x) + 2/abs(x))

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Fricas [A]
time = 1.33, size = 65, normalized size = 1.48 \begin {gather*} -\frac {2 \, x \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) - 2 \, x \log \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + \sqrt {x + 1} \sqrt {-x + 1}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*x*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) - 2*x*log((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) + sqrt(x + 1)*sqrt(
-x + 1))/x

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x + 1\right )^{\frac {3}{2}}}{x^{2} \sqrt {1 - x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((x + 1)**(3/2)/(x**2*sqrt(1 - x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, choosing root of [1,0,-4,0,%%%{
4,[2]%%%}] at parameters values [5.38357630698]Warning, choosing root of [1,0,-4,0,%%%{4,[2]%%%}] at parameter
s values [81.11954429

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (x+1\right )}^{3/2}}{x^2\,\sqrt {1-x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)^(3/2)/(x^2*(1 - x)^(1/2)),x)

[Out]

int((x + 1)^(3/2)/(x^2*(1 - x)^(1/2)), x)

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