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

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

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Rubi [A]  time = 0.03, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {98, 151, 12, 92, 206} \begin {gather*} -\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 ) \end {gather*}

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 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 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 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*}

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Mathematica [A]  time = 0.02, size = 71, normalized size = 0.62 \begin {gather*} -\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/24*(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]])/(x^4*Sqrt[1
- x^2])

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IntegrateAlgebraic [A]  time = 0.24, size = 100, normalized size = 0.87 \begin {gather*} \frac {\sqrt {1-x} \left (\frac {21 (1-x)^3}{(x+1)^3}-\frac {77 (1-x)^2}{(x+1)^2}+\frac {83 (1-x)}{x+1}-75\right )}{12 \sqrt {x+1} \left (\frac {1-x}{x+1}-1\right )^4}-\frac {7}{4} \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^5),x]

[Out]

(Sqrt[1 - x]*(-75 + (21*(1 - x)^3)/(1 + x)^3 - (77*(1 - x)^2)/(1 + x)^2 + (83*(1 - x))/(1 + x)))/(12*Sqrt[1 +
x]*(-1 + (1 - x)/(1 + x))^4) - (7*ArcTanh[Sqrt[1 - x]/Sqrt[1 + x]])/4

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fricas [A]  time = 1.25, size = 60, normalized size = 0.52 \begin {gather*} \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 {gather*}

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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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^5/(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]1/6*(21*(2*sqrt(x+1)/(-2*sqrt(-x+1)+2*sqrt(2))-1/2*(-2*sqrt(-x+1)+2*sqrt(2))/sqrt(x+1
))^7-308*(2*sqrt(x+1)/(-2*sqrt(-x+1)+2*sqrt(2))-1/2*(-2*sqrt(-x+1)+2*sqrt(2))/sqrt(x+1))^5+1328*(2*sqrt(x+1)/(
-2*sqrt(-x+1)+2*sqrt(2))-1/2*(-2*sqrt(-x+1)+2*sqrt(2))/sqrt(x+1))^3-4800*(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)^4-7/8*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)))+7/8*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 = 0.02, size = 94, normalized size = 0.82 \begin {gather*} -\frac {\sqrt {x +1}\, \sqrt {-x +1}\, \left (21 x^{4} \arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right )+32 \sqrt {-x^{2}+1}\, x^{3}+21 \sqrt {-x^{2}+1}\, x^{2}+16 \sqrt {-x^{2}+1}\, x +6 \sqrt {-x^{2}+1}\right )}{24 \sqrt {-x^{2}+1}\, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.95, size = 82, normalized size = 0.71 \begin {gather*} -\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 {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x+1\right )}^{3/2}}{x^5\,\sqrt {1-x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((x + 1)^(3/2)/(x^5*(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^{5} \sqrt {1 - x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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