3.8.5 \(\int \frac {(1+x)^{3/2}}{\sqrt {1-x} x^3} \, dx\)

Optimal. Leaf size=69 \[ -\frac {\sqrt {1-x} (x+1)^{3/2}}{2 x^2}-\frac {3 \sqrt {1-x} \sqrt {x+1}}{2 x}-\frac {3}{2} \tanh ^{-1}\left (\sqrt {1-x} \sqrt {x+1}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {94, 92, 206} \begin {gather*} -\frac {\sqrt {1-x} (x+1)^{3/2}}{2 x^2}-\frac {3 \sqrt {1-x} \sqrt {x+1}}{2 x}-\frac {3}{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^3),x]

[Out]

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

Rule 92

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 94

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

Rule 206

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

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 59, normalized size = 0.86 \begin {gather*} \frac {4 x^3+x^2-3 \sqrt {1-x^2} x^2 \tanh ^{-1}\left (\sqrt {1-x^2}\right )-4 x-1}{2 x^2 \sqrt {1-x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1 - 4*x + x^2 + 4*x^3 - 3*x^2*Sqrt[1 - x^2]*ArcTanh[Sqrt[1 - x^2]])/(2*x^2*Sqrt[1 - x^2])

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IntegrateAlgebraic [A]  time = 0.15, size = 67, normalized size = 0.97 \begin {gather*} \frac {\sqrt {1-x} \left (\frac {3 (1-x)}{x+1}-5\right )}{\sqrt {x+1} \left (\frac {1-x}{x+1}-1\right )^2}-3 \tanh ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {x+1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.33, size = 50, normalized size = 0.72 \begin {gather*} \frac {3 \, x^{2} \log \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) - {\left (4 \, x + 1\right )} \sqrt {x + 1} \sqrt {-x + 1}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(3*x^2*log((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) - (4*x + 1)*sqrt(x + 1)*sqrt(-x + 1))/x^2

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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^3/(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 [-93.616423693]Warning, choosing root of [1,0,-4,0,%%%{4,[2]%%%}] at parameter
s values [-17.8804557086]-(-6*(2*sqrt(x+1)/(-2*sqrt(-x+1)+2*sqrt(2))-1/2*(-2*sqrt(-x+1)+2*sqrt(2))/sqrt(x+1))^
3+40*(2*sqrt(x+1)/(-2*sqrt(-x+1)+2*sqrt(2))-1/2*(-2*sqrt(-x+1)+2*sqrt(2))/sqrt(x+1)))/((2*sqrt(x+1)/(-2*sqrt(-
x+1)+2*sqrt(2))-1/2*(-2*sqrt(-x+1)+2*sqrt(2))/sqrt(x+1))^2-4)^2-3/2*ln(abs(2*sqrt(x+1)/(-2*sqrt(-x+1)+2*sqrt(2
))+2-1/2*(-2*sqrt(-x+1)+2*sqrt(2))/sqrt(x+1)))+3/2*ln(abs(2*sqrt(x+1)/(-2*sqrt(-x+1)+2*sqrt(2))-2-1/2*(-2*sqrt
(-x+1)+2*sqrt(2))/sqrt(x+1)))

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maple [A]  time = 0.01, size = 64, normalized size = 0.93 \begin {gather*} -\frac {\sqrt {x +1}\, \sqrt {-x +1}\, \left (3 x^{2} \arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right )+4 \sqrt {-x^{2}+1}\, x +\sqrt {-x^{2}+1}\right )}{2 \sqrt {-x^{2}+1}\, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(x+1)^(1/2)*(-x+1)^(1/2)*(3*arctanh(1/(-x^2+1)^(1/2))*x^2+4*(-x^2+1)^(1/2)*x+(-x^2+1)^(1/2))/x^2/(-x^2+1)
^(1/2)

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maxima [A]  time = 1.95, size = 54, normalized size = 0.78 \begin {gather*} -\frac {2 \, \sqrt {-x^{2} + 1}}{x} - \frac {\sqrt {-x^{2} + 1}}{2 \, x^{2}} - \frac {3}{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^3/(1-x)^(1/2),x, algorithm="maxima")

[Out]

-2*sqrt(-x^2 + 1)/x - 1/2*sqrt(-x^2 + 1)/x^2 - 3/2*log(2*sqrt(-x^2 + 1)/abs(x) + 2/abs(x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x+1\right )}^{3/2}}{x^3\,\sqrt {1-x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((x + 1)^(3/2)/(x^3*(1 - x)^(1/2)), 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^{3} \sqrt {1 - x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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