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

Optimal. Leaf size=112 \[ -\frac{7 \sqrt{x+1}}{6 \sqrt{1-x} x^2}+\frac{2 \sqrt{x+1}}{3 (1-x)^{3/2} x^2}+\frac{26 \sqrt{x+1}}{3 \sqrt{1-x}}-\frac{19 \sqrt{x+1}}{6 \sqrt{1-x} x}-\frac{11}{2} \tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]

[Out]

(26*Sqrt[1 + x])/(3*Sqrt[1 - x]) + (2*Sqrt[1 + x])/(3*(1 - x)^(3/2)*x^2) - (7*Sq
rt[1 + x])/(6*Sqrt[1 - x]*x^2) - (19*Sqrt[1 + x])/(6*Sqrt[1 - x]*x) - (11*ArcTan
h[Sqrt[1 - x]*Sqrt[1 + x]])/2

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Rubi [A]  time = 0.209634, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{7 \sqrt{x+1}}{6 \sqrt{1-x} x^2}+\frac{2 \sqrt{x+1}}{3 (1-x)^{3/2} x^2}+\frac{26 \sqrt{x+1}}{3 \sqrt{1-x}}-\frac{19 \sqrt{x+1}}{6 \sqrt{1-x} x}-\frac{11}{2} \tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(26*Sqrt[1 + x])/(3*Sqrt[1 - x]) + (2*Sqrt[1 + x])/(3*(1 - x)^(3/2)*x^2) - (7*Sq
rt[1 + x])/(6*Sqrt[1 - x]*x^2) - (19*Sqrt[1 + x])/(6*Sqrt[1 - x]*x) - (11*ArcTan
h[Sqrt[1 - x]*Sqrt[1 + x]])/2

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Rubi in Sympy [A]  time = 15.9353, size = 87, normalized size = 0.78 \[ - \frac{11 \operatorname{atanh}{\left (\sqrt{- x + 1} \sqrt{x + 1} \right )}}{2} + \frac{26 \sqrt{x + 1}}{3 \sqrt{- x + 1}} + \frac{19 \sqrt{x + 1}}{6 \left (- x + 1\right )^{\frac{3}{2}}} - \frac{2 \sqrt{x + 1}}{x \left (- x + 1\right )^{\frac{3}{2}}} - \frac{\sqrt{x + 1}}{2 x^{2} \left (- x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x)**(1/2)/(1-x)**(5/2)/x**3,x)

[Out]

-11*atanh(sqrt(-x + 1)*sqrt(x + 1))/2 + 26*sqrt(x + 1)/(3*sqrt(-x + 1)) + 19*sqr
t(x + 1)/(6*(-x + 1)**(3/2)) - 2*sqrt(x + 1)/(x*(-x + 1)**(3/2)) - sqrt(x + 1)/(
2*x**2*(-x + 1)**(3/2))

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Mathematica [A]  time = 0.0743509, size = 61, normalized size = 0.54 \[ \frac{1}{6} \left (-33 \log \left (\sqrt{1-x^2}+1\right )-\frac{\sqrt{1-x^2} \left (52 x^3-71 x^2+12 x+3\right )}{(x-1)^2 x^2}+33 \log (x)\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

(-((Sqrt[1 - x^2]*(3 + 12*x - 71*x^2 + 52*x^3))/((-1 + x)^2*x^2)) + 33*Log[x] -
33*Log[1 + Sqrt[1 - x^2]])/6

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Maple [A]  time = 0.019, size = 129, normalized size = 1.2 \[ -{\frac{1}{6\,{x}^{2} \left ( -1+x \right ) ^{2}} \left ( 33\,{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ){x}^{4}-66\,{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ){x}^{3}+52\,{x}^{3}\sqrt{-{x}^{2}+1}+33\,{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ){x}^{2}-71\,{x}^{2}\sqrt{-{x}^{2}+1}+12\,x\sqrt{-{x}^{2}+1}+3\,\sqrt{-{x}^{2}+1} \right ) \sqrt{1-x}\sqrt{1+x}{\frac{1}{\sqrt{-{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(33*arctanh(1/(-x^2+1)^(1/2))*x^4-66*arctanh(1/(-x^2+1)^(1/2))*x^3+52*x^3*(
-x^2+1)^(1/2)+33*arctanh(1/(-x^2+1)^(1/2))*x^2-71*x^2*(-x^2+1)^(1/2)+12*x*(-x^2+
1)^(1/2)+3*(-x^2+1)^(1/2))*(1-x)^(1/2)*(1+x)^(1/2)/x^2/(-1+x)^2/(-x^2+1)^(1/2)

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Maxima [A]  time = 1.34868, size = 135, normalized size = 1.21 \[ \frac{26 \, x}{3 \, \sqrt{-x^{2} + 1}} + \frac{11}{2 \, \sqrt{-x^{2} + 1}} + \frac{13 \, x}{3 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} + \frac{11}{6 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} - \frac{3}{{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x} - \frac{1}{2 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x^{2}} - \frac{11}{2} \, \log \left (\frac{2 \, \sqrt{-x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x + 1)/(x^3*(-x + 1)^(5/2)),x, algorithm="maxima")

[Out]

26/3*x/sqrt(-x^2 + 1) + 11/2/sqrt(-x^2 + 1) + 13/3*x/(-x^2 + 1)^(3/2) + 11/6/(-x
^2 + 1)^(3/2) - 3/((-x^2 + 1)^(3/2)*x) - 1/2/((-x^2 + 1)^(3/2)*x^2) - 11/2*log(2
*sqrt(-x^2 + 1)/abs(x) + 2/abs(x))

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Fricas [A]  time = 0.230051, size = 308, normalized size = 2.75 \[ -\frac{14 \, x^{7} - 303 \, x^{6} + 227 \, x^{5} + 591 \, x^{4} - 429 \, x^{3} - 240 \, x^{2} +{\left (90 \, x^{6} - 53 \, x^{5} - 480 \, x^{4} + 375 \, x^{3} + 228 \, x^{2} - 108 \, x - 24\right )} \sqrt{x + 1} \sqrt{-x + 1} - 33 \,{\left (x^{7} + 2 \, x^{6} - 11 \, x^{5} + 4 \, x^{4} + 12 \, x^{3} - 8 \, x^{2} -{\left (x^{6} - 5 \, x^{5} + 12 \, x^{3} - 8 \, x^{2}\right )} \sqrt{x + 1} \sqrt{-x + 1}\right )} \log \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 108 \, x + 24}{6 \,{\left (x^{7} + 2 \, x^{6} - 11 \, x^{5} + 4 \, x^{4} + 12 \, x^{3} - 8 \, x^{2} -{\left (x^{6} - 5 \, x^{5} + 12 \, x^{3} - 8 \, x^{2}\right )} \sqrt{x + 1} \sqrt{-x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x + 1)/(x^3*(-x + 1)^(5/2)),x, algorithm="fricas")

[Out]

-1/6*(14*x^7 - 303*x^6 + 227*x^5 + 591*x^4 - 429*x^3 - 240*x^2 + (90*x^6 - 53*x^
5 - 480*x^4 + 375*x^3 + 228*x^2 - 108*x - 24)*sqrt(x + 1)*sqrt(-x + 1) - 33*(x^7
+ 2*x^6 - 11*x^5 + 4*x^4 + 12*x^3 - 8*x^2 - (x^6 - 5*x^5 + 12*x^3 - 8*x^2)*sqrt
(x + 1)*sqrt(-x + 1))*log((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) + 108*x + 24)/(x^7 +
2*x^6 - 11*x^5 + 4*x^4 + 12*x^3 - 8*x^2 - (x^6 - 5*x^5 + 12*x^3 - 8*x^2)*sqrt(x
+ 1)*sqrt(-x + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x + 1)/(x^3*(-x + 1)^(5/2)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError