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

Optimal. Leaf size=87 \[ \frac {14 \sqrt {x+1}}{3 \sqrt {1-x}}-\frac {5 \sqrt {x+1}}{3 \sqrt {1-x} x}+\frac {2 \sqrt {x+1}}{3 (1-x)^{3/2} x}-3 \tanh ^{-1}\left (\sqrt {1-x} \sqrt {x+1}\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 87, 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 = {99, 151, 152, 12, 92, 206} \begin {gather*} \frac {14 \sqrt {x+1}}{3 \sqrt {1-x}}-\frac {5 \sqrt {x+1}}{3 \sqrt {1-x} x}+\frac {2 \sqrt {x+1}}{3 (1-x)^{3/2} x}-3 \tanh ^{-1}\left (\sqrt {1-x} \sqrt {x+1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 99

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[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (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 152

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] && IntegersQ[2*m, 2*n, 2*p]

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 {\sqrt {1+x}}{(1-x)^{5/2} x^2} \, dx &=\frac {2 \sqrt {1+x}}{3 (1-x)^{3/2} x}-\frac {2}{3} \int \frac {-\frac {5}{2}-2 x}{(1-x)^{3/2} x^2 \sqrt {1+x}} \, dx\\ &=\frac {2 \sqrt {1+x}}{3 (1-x)^{3/2} x}-\frac {5 \sqrt {1+x}}{3 \sqrt {1-x} x}+\frac {2}{3} \int \frac {\frac {9}{2}+\frac {5 x}{2}}{(1-x)^{3/2} x \sqrt {1+x}} \, dx\\ &=\frac {14 \sqrt {1+x}}{3 \sqrt {1-x}}+\frac {2 \sqrt {1+x}}{3 (1-x)^{3/2} x}-\frac {5 \sqrt {1+x}}{3 \sqrt {1-x} x}-\frac {2}{3} \int -\frac {9}{2 \sqrt {1-x} x \sqrt {1+x}} \, dx\\ &=\frac {14 \sqrt {1+x}}{3 \sqrt {1-x}}+\frac {2 \sqrt {1+x}}{3 (1-x)^{3/2} x}-\frac {5 \sqrt {1+x}}{3 \sqrt {1-x} x}+3 \int \frac {1}{\sqrt {1-x} x \sqrt {1+x}} \, dx\\ &=\frac {14 \sqrt {1+x}}{3 \sqrt {1-x}}+\frac {2 \sqrt {1+x}}{3 (1-x)^{3/2} x}-\frac {5 \sqrt {1+x}}{3 \sqrt {1-x} x}-3 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x} \sqrt {1+x}\right )\\ &=\frac {14 \sqrt {1+x}}{3 \sqrt {1-x}}+\frac {2 \sqrt {1+x}}{3 (1-x)^{3/2} x}-\frac {5 \sqrt {1+x}}{3 \sqrt {1-x} x}-3 \tanh ^{-1}\left (\sqrt {1-x} \sqrt {1+x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 67, normalized size = 0.77 \begin {gather*} \frac {14 x^3-5 x^2-9 (x-1) \sqrt {1-x^2} x \tanh ^{-1}\left (\sqrt {1-x^2}\right )-16 x+3}{3 (x-1) x \sqrt {1-x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(3 - 16*x - 5*x^2 + 14*x^3 - 9*(-1 + x)*x*Sqrt[1 - x^2]*ArcTanh[Sqrt[1 - x^2]])/(3*(-1 + x)*x*Sqrt[1 - x^2])

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IntegrateAlgebraic [A]  time = 0.11, size = 83, normalized size = 0.95 \begin {gather*} \frac {\sqrt {x+1} \left (\frac {(x+1)^2}{(1-x)^2}+\frac {11 (x+1)}{1-x}-18\right )}{3 \sqrt {1-x} \left (\frac {x+1}{1-x}-1\right )}-6 \tanh ^{-1}\left (\frac {\sqrt {x+1}}{\sqrt {1-x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 + x]*(-18 + (11*(1 + x))/(1 - x) + (1 + x)^2/(1 - x)^2))/(3*Sqrt[1 - x]*(-1 + (1 + x)/(1 - x))) - 6*Ar
cTanh[Sqrt[1 + x]/Sqrt[1 - x]]

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fricas [A]  time = 1.48, size = 84, normalized size = 0.97 \begin {gather*} \frac {13 \, x^{3} - 26 \, x^{2} - {\left (14 \, x^{2} - 19 \, x + 3\right )} \sqrt {x + 1} \sqrt {-x + 1} + 9 \, {\left (x^{3} - 2 \, x^{2} + x\right )} \log \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + 13 \, x}{3 \, {\left (x^{3} - 2 \, x^{2} + x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(13*x^3 - 26*x^2 - (14*x^2 - 19*x + 3)*sqrt(x + 1)*sqrt(-x + 1) + 9*(x^3 - 2*x^2 + x)*log((sqrt(x + 1)*sqr
t(-x + 1) - 1)/x) + 13*x)/(x^3 - 2*x^2 + 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)^(1/2)/(1-x)^(5/2)/x^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]-4*(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)-3*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*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)))+2*(-11/6*sqrt(x+1)*sqrt(x+1)+4)*sqrt(x+1)*sqrt(-x+1)/(-x+1)^2

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maple [A]  time = 0.01, size = 113, normalized size = 1.30 \begin {gather*} -\frac {\left (9 x^{3} \arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right )-18 x^{2} \arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right )+14 \sqrt {-x^{2}+1}\, x^{2}+9 x \arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right )-19 \sqrt {-x^{2}+1}\, x +3 \sqrt {-x^{2}+1}\right ) \sqrt {-x +1}\, \sqrt {x +1}}{3 \left (x -1\right )^{2} \sqrt {-x^{2}+1}\, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(9*arctanh(1/(-x^2+1)^(1/2))*x^3-18*arctanh(1/(-x^2+1)^(1/2))*x^2+14*(-x^2+1)^(1/2)*x^2+9*arctanh(1/(-x^2
+1)^(1/2))*x-19*(-x^2+1)^(1/2)*x+3*(-x^2+1)^(1/2))*(-x+1)^(1/2)*(x+1)^(1/2)/x/(x-1)^2/(-x^2+1)^(1/2)

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maxima [A]  time = 0.60, size = 86, normalized size = 0.99 \begin {gather*} \frac {14 \, x}{3 \, \sqrt {-x^{2} + 1}} + \frac {3}{\sqrt {-x^{2} + 1}} + \frac {7 \, x}{3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {4}{3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} - \frac {1}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}} x} - 3 \, \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)^(1/2)/(1-x)^(5/2)/x^2,x, algorithm="maxima")

[Out]

14/3*x/sqrt(-x^2 + 1) + 3/sqrt(-x^2 + 1) + 7/3*x/(-x^2 + 1)^(3/2) + 4/3/(-x^2 + 1)^(3/2) - 1/((-x^2 + 1)^(3/2)
*x) - 3*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 {\sqrt {x+1}}{x^2\,{\left (1-x\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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