3.335 \(\int \frac{1}{x^{5/2} \left (1+x^2\right )^3} \, dx\)

Optimal. Leaf size=138 \[ -\frac{77}{48 x^{3/2}}+\frac{11}{16 x^{3/2} \left (x^2+1\right )}+\frac{1}{4 x^{3/2} \left (x^2+1\right )^2}+\frac{77 \log \left (x-\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}-\frac{77 \log \left (x+\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}+\frac{77 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}-\frac{77 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{32 \sqrt{2}} \]

[Out]

-77/(48*x^(3/2)) + 1/(4*x^(3/2)*(1 + x^2)^2) + 11/(16*x^(3/2)*(1 + x^2)) + (77*A
rcTan[1 - Sqrt[2]*Sqrt[x]])/(32*Sqrt[2]) - (77*ArcTan[1 + Sqrt[2]*Sqrt[x]])/(32*
Sqrt[2]) + (77*Log[1 - Sqrt[2]*Sqrt[x] + x])/(64*Sqrt[2]) - (77*Log[1 + Sqrt[2]*
Sqrt[x] + x])/(64*Sqrt[2])

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Rubi [A]  time = 0.170225, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.692 \[ -\frac{77}{48 x^{3/2}}+\frac{11}{16 x^{3/2} \left (x^2+1\right )}+\frac{1}{4 x^{3/2} \left (x^2+1\right )^2}+\frac{77 \log \left (x-\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}-\frac{77 \log \left (x+\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}+\frac{77 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}-\frac{77 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{32 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(5/2)*(1 + x^2)^3),x]

[Out]

-77/(48*x^(3/2)) + 1/(4*x^(3/2)*(1 + x^2)^2) + 11/(16*x^(3/2)*(1 + x^2)) + (77*A
rcTan[1 - Sqrt[2]*Sqrt[x]])/(32*Sqrt[2]) - (77*ArcTan[1 + Sqrt[2]*Sqrt[x]])/(32*
Sqrt[2]) + (77*Log[1 - Sqrt[2]*Sqrt[x] + x])/(64*Sqrt[2]) - (77*Log[1 + Sqrt[2]*
Sqrt[x] + x])/(64*Sqrt[2])

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Rubi in Sympy [A]  time = 22.2736, size = 128, normalized size = 0.93 \[ \frac{77 \sqrt{2} \log{\left (- \sqrt{2} \sqrt{x} + x + 1 \right )}}{128} - \frac{77 \sqrt{2} \log{\left (\sqrt{2} \sqrt{x} + x + 1 \right )}}{128} - \frac{77 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{64} - \frac{77 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{64} - \frac{77}{48 x^{\frac{3}{2}}} + \frac{11}{16 x^{\frac{3}{2}} \left (x^{2} + 1\right )} + \frac{1}{4 x^{\frac{3}{2}} \left (x^{2} + 1\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

77*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1)/128 - 77*sqrt(2)*log(sqrt(2)*sqrt(x) +
x + 1)/128 - 77*sqrt(2)*atan(sqrt(2)*sqrt(x) - 1)/64 - 77*sqrt(2)*atan(sqrt(2)*s
qrt(x) + 1)/64 - 77/(48*x**(3/2)) + 11/(16*x**(3/2)*(x**2 + 1)) + 1/(4*x**(3/2)*
(x**2 + 1)**2)

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Mathematica [A]  time = 0.105269, size = 128, normalized size = 0.93 \[ \frac{1}{384} \left (-\frac{256}{x^{3/2}}-\frac{360 \sqrt{x}}{x^2+1}-\frac{96 \sqrt{x}}{\left (x^2+1\right )^2}+231 \sqrt{2} \log \left (x-\sqrt{2} \sqrt{x}+1\right )-231 \sqrt{2} \log \left (x+\sqrt{2} \sqrt{x}+1\right )+462 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )-462 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-256/x^(3/2) - (96*Sqrt[x])/(1 + x^2)^2 - (360*Sqrt[x])/(1 + x^2) + 462*Sqrt[2]
*ArcTan[1 - Sqrt[2]*Sqrt[x]] - 462*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[x]] + 231*Sqr
t[2]*Log[1 - Sqrt[2]*Sqrt[x] + x] - 231*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[x] + x])/38
4

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Maple [A]  time = 0.016, size = 87, normalized size = 0.6 \[ -{\frac{2}{3}{x}^{-{\frac{3}{2}}}}-2\,{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ({\frac{15\,{x}^{5/2}}{32}}+{\frac{19\,\sqrt{x}}{32}} \right ) }-{\frac{77\,\sqrt{2}}{64}\arctan \left ( \sqrt{2}\sqrt{x}-1 \right ) }-{\frac{77\,\sqrt{2}}{128}\ln \left ({1 \left ( 1+x+\sqrt{2}\sqrt{x} \right ) \left ( 1+x-\sqrt{2}\sqrt{x} \right ) ^{-1}} \right ) }-{\frac{77\,\sqrt{2}}{64}\arctan \left ( 1+\sqrt{2}\sqrt{x} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(5/2)/(x^2+1)^3,x)

[Out]

-2/3/x^(3/2)-2*(15/32*x^(5/2)+19/32*x^(1/2))/(x^2+1)^2-77/64*arctan(2^(1/2)*x^(1
/2)-1)*2^(1/2)-77/128*2^(1/2)*ln((1+x+2^(1/2)*x^(1/2))/(1+x-2^(1/2)*x^(1/2)))-77
/64*arctan(1+2^(1/2)*x^(1/2))*2^(1/2)

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Maxima [A]  time = 1.50314, size = 138, normalized size = 1. \[ -\frac{77}{64} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) - \frac{77}{64} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) - \frac{77}{128} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{77}{128} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{77 \, x^{4} + 121 \, x^{2} + 32}{48 \,{\left (x^{\frac{11}{2}} + 2 \, x^{\frac{7}{2}} + x^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-77/64*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) - 77/64*sqrt(2)*arctan(
-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(x))) - 77/128*sqrt(2)*log(sqrt(2)*sqrt(x) + x + 1
) + 77/128*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1) - 1/48*(77*x^4 + 121*x^2 + 32)/
(x^(11/2) + 2*x^(7/2) + x^(3/2))

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Fricas [A]  time = 0.256738, size = 248, normalized size = 1.8 \[ \frac{924 \, \sqrt{2}{\left (x^{6} + 2 \, x^{4} + x^{2}\right )} \arctan \left (\frac{1}{\sqrt{2} \sqrt{x} + \sqrt{2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2} + 1}\right ) + 924 \, \sqrt{2}{\left (x^{6} + 2 \, x^{4} + x^{2}\right )} \arctan \left (\frac{1}{\sqrt{2} \sqrt{x} + \sqrt{-2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2} - 1}\right ) - 231 \, \sqrt{2}{\left (x^{6} + 2 \, x^{4} + x^{2}\right )} \log \left (2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2\right ) + 231 \, \sqrt{2}{\left (x^{6} + 2 \, x^{4} + x^{2}\right )} \log \left (-2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2\right ) - 8 \,{\left (77 \, x^{4} + 121 \, x^{2} + 32\right )} \sqrt{x}}{384 \,{\left (x^{6} + 2 \, x^{4} + x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/384*(924*sqrt(2)*(x^6 + 2*x^4 + x^2)*arctan(1/(sqrt(2)*sqrt(x) + sqrt(2*sqrt(2
)*sqrt(x) + 2*x + 2) + 1)) + 924*sqrt(2)*(x^6 + 2*x^4 + x^2)*arctan(1/(sqrt(2)*s
qrt(x) + sqrt(-2*sqrt(2)*sqrt(x) + 2*x + 2) - 1)) - 231*sqrt(2)*(x^6 + 2*x^4 + x
^2)*log(2*sqrt(2)*sqrt(x) + 2*x + 2) + 231*sqrt(2)*(x^6 + 2*x^4 + x^2)*log(-2*sq
rt(2)*sqrt(x) + 2*x + 2) - 8*(77*x^4 + 121*x^2 + 32)*sqrt(x))/(x^6 + 2*x^4 + x^2
)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.212246, size = 134, normalized size = 0.97 \[ -\frac{77}{64} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) - \frac{77}{64} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) - \frac{77}{128} \, \sqrt{2}{\rm ln}\left (\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{77}{128} \, \sqrt{2}{\rm ln}\left (-\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{15 \, x^{\frac{5}{2}} + 19 \, \sqrt{x}}{16 \,{\left (x^{2} + 1\right )}^{2}} - \frac{2}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-77/64*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) - 77/64*sqrt(2)*arctan(
-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(x))) - 77/128*sqrt(2)*ln(sqrt(2)*sqrt(x) + x + 1)
+ 77/128*sqrt(2)*ln(-sqrt(2)*sqrt(x) + x + 1) - 1/16*(15*x^(5/2) + 19*sqrt(x))/
(x^2 + 1)^2 - 2/3/x^(3/2)