3.732 \(\int \frac{(1+x)^{3/2}}{\sqrt{1-x} x^5} \, dx\)

Optimal. Leaf size=115 \[ -\frac{7 \sqrt{1-x} \sqrt{x+1}}{8 x^2}-\frac{2 \sqrt{1-x} \sqrt{x+1}}{3 x^3}-\frac{\sqrt{1-x} \sqrt{x+1}}{4 x^4}-\frac{4 \sqrt{1-x} \sqrt{x+1}}{3 x}-\frac{7}{8} \tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]

[Out]

-(Sqrt[1 - x]*Sqrt[1 + x])/(4*x^4) - (2*Sqrt[1 - x]*Sqrt[1 + x])/(3*x^3) - (7*Sqrt[1 - x]*Sqrt[1 + x])/(8*x^2)
- (4*Sqrt[1 - x]*Sqrt[1 + x])/(3*x) - (7*ArcTanh[Sqrt[1 - x]*Sqrt[1 + x]])/8

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Rubi [A]  time = 0.0332794, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {98, 151, 12, 92, 206} \[ -\frac{7 \sqrt{1-x} \sqrt{x+1}}{8 x^2}-\frac{2 \sqrt{1-x} \sqrt{x+1}}{3 x^3}-\frac{\sqrt{1-x} \sqrt{x+1}}{4 x^4}-\frac{4 \sqrt{1-x} \sqrt{x+1}}{3 x}-\frac{7}{8} \tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - x]*Sqrt[1 + x])/(4*x^4) - (2*Sqrt[1 - x]*Sqrt[1 + x])/(3*x^3) - (7*Sqrt[1 - x]*Sqrt[1 + x])/(8*x^2)
- (4*Sqrt[1 - x]*Sqrt[1 + x])/(3*x) - (7*ArcTanh[Sqrt[1 - x]*Sqrt[1 + x]])/8

Rule 98

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 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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^5} \, dx &=-\frac{\sqrt{1-x} \sqrt{1+x}}{4 x^4}-\frac{1}{4} \int \frac{-8-7 x}{\sqrt{1-x} x^4 \sqrt{1+x}} \, dx\\ &=-\frac{\sqrt{1-x} \sqrt{1+x}}{4 x^4}-\frac{2 \sqrt{1-x} \sqrt{1+x}}{3 x^3}+\frac{1}{12} \int \frac{21+16 x}{\sqrt{1-x} x^3 \sqrt{1+x}} \, dx\\ &=-\frac{\sqrt{1-x} \sqrt{1+x}}{4 x^4}-\frac{2 \sqrt{1-x} \sqrt{1+x}}{3 x^3}-\frac{7 \sqrt{1-x} \sqrt{1+x}}{8 x^2}-\frac{1}{24} \int \frac{-32-21 x}{\sqrt{1-x} x^2 \sqrt{1+x}} \, dx\\ &=-\frac{\sqrt{1-x} \sqrt{1+x}}{4 x^4}-\frac{2 \sqrt{1-x} \sqrt{1+x}}{3 x^3}-\frac{7 \sqrt{1-x} \sqrt{1+x}}{8 x^2}-\frac{4 \sqrt{1-x} \sqrt{1+x}}{3 x}+\frac{1}{24} \int \frac{21}{\sqrt{1-x} x \sqrt{1+x}} \, dx\\ &=-\frac{\sqrt{1-x} \sqrt{1+x}}{4 x^4}-\frac{2 \sqrt{1-x} \sqrt{1+x}}{3 x^3}-\frac{7 \sqrt{1-x} \sqrt{1+x}}{8 x^2}-\frac{4 \sqrt{1-x} \sqrt{1+x}}{3 x}+\frac{7}{8} \int \frac{1}{\sqrt{1-x} x \sqrt{1+x}} \, dx\\ &=-\frac{\sqrt{1-x} \sqrt{1+x}}{4 x^4}-\frac{2 \sqrt{1-x} \sqrt{1+x}}{3 x^3}-\frac{7 \sqrt{1-x} \sqrt{1+x}}{8 x^2}-\frac{4 \sqrt{1-x} \sqrt{1+x}}{3 x}-\frac{7}{8} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1-x} \sqrt{1+x}\right )\\ &=-\frac{\sqrt{1-x} \sqrt{1+x}}{4 x^4}-\frac{2 \sqrt{1-x} \sqrt{1+x}}{3 x^3}-\frac{7 \sqrt{1-x} \sqrt{1+x}}{8 x^2}-\frac{4 \sqrt{1-x} \sqrt{1+x}}{3 x}-\frac{7}{8} \tanh ^{-1}\left (\sqrt{1-x} \sqrt{1+x}\right )\\ \end{align*}

Mathematica [A]  time = 0.022774, size = 71, normalized size = 0.62 \[ -\frac{-32 x^5-21 x^4+16 x^3+15 x^2+21 \sqrt{1-x^2} x^4 \tanh ^{-1}\left (\sqrt{1-x^2}\right )+16 x+6}{24 x^4 \sqrt{1-x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(6 + 16*x + 15*x^2 + 16*x^3 - 21*x^4 - 32*x^5 + 21*x^4*Sqrt[1 - x^2]*ArcTanh[Sqrt[1 - x^2]])/(24*x^4*Sqrt[1 -
x^2])

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Maple [A]  time = 0.012, size = 94, normalized size = 0.8 \begin{align*} -{\frac{1}{24\,{x}^{4}}\sqrt{1-x}\sqrt{1+x} \left ( 21\,{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ){x}^{4}+32\,{x}^{3}\sqrt{-{x}^{2}+1}+21\,{x}^{2}\sqrt{-{x}^{2}+1}+16\,x\sqrt{-{x}^{2}+1}+6\,\sqrt{-{x}^{2}+1} \right ){\frac{1}{\sqrt{-{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)^(3/2)/x^5/(1-x)^(1/2),x)

[Out]

-1/24*(1+x)^(1/2)*(1-x)^(1/2)*(21*arctanh(1/(-x^2+1)^(1/2))*x^4+32*x^3*(-x^2+1)^(1/2)+21*x^2*(-x^2+1)^(1/2)+16
*x*(-x^2+1)^(1/2)+6*(-x^2+1)^(1/2))/x^4/(-x^2+1)^(1/2)

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Maxima [A]  time = 1.59662, size = 111, normalized size = 0.97 \begin{align*} -\frac{4 \, \sqrt{-x^{2} + 1}}{3 \, x} - \frac{7 \, \sqrt{-x^{2} + 1}}{8 \, x^{2}} - \frac{2 \, \sqrt{-x^{2} + 1}}{3 \, x^{3}} - \frac{\sqrt{-x^{2} + 1}}{4 \, x^{4}} - \frac{7}{8} \, \log \left (\frac{2 \, \sqrt{-x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(3/2)/x^5/(1-x)^(1/2),x, algorithm="maxima")

[Out]

-4/3*sqrt(-x^2 + 1)/x - 7/8*sqrt(-x^2 + 1)/x^2 - 2/3*sqrt(-x^2 + 1)/x^3 - 1/4*sqrt(-x^2 + 1)/x^4 - 7/8*log(2*s
qrt(-x^2 + 1)/abs(x) + 2/abs(x))

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Fricas [A]  time = 1.8085, size = 153, normalized size = 1.33 \begin{align*} \frac{21 \, x^{4} \log \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) -{\left (32 \, x^{3} + 21 \, x^{2} + 16 \, x + 6\right )} \sqrt{x + 1} \sqrt{-x + 1}}{24 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(3/2)/x^5/(1-x)^(1/2),x, algorithm="fricas")

[Out]

1/24*(21*x^4*log((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) - (32*x^3 + 21*x^2 + 16*x + 6)*sqrt(x + 1)*sqrt(-x + 1))/x^
4

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)**(3/2)/x**5/(1-x)**(1/2),x)

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(3/2)/x^5/(1-x)^(1/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError