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

Optimal. Leaf size=122 \[ -\frac{7}{6 x^{3/2}}+\frac{1}{2 x^{3/2} \left (x^2+1\right )}+\frac{7 \log \left (x-\sqrt{2} \sqrt{x}+1\right )}{8 \sqrt{2}}-\frac{7 \log \left (x+\sqrt{2} \sqrt{x}+1\right )}{8 \sqrt{2}}+\frac{7 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{4 \sqrt{2}}-\frac{7 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{4 \sqrt{2}} \]

[Out]

-7/(6*x^(3/2)) + 1/(2*x^(3/2)*(1 + x^2)) + (7*ArcTan[1 - Sqrt[2]*Sqrt[x]])/(4*Sq
rt[2]) - (7*ArcTan[1 + Sqrt[2]*Sqrt[x]])/(4*Sqrt[2]) + (7*Log[1 - Sqrt[2]*Sqrt[x
] + x])/(8*Sqrt[2]) - (7*Log[1 + Sqrt[2]*Sqrt[x] + x])/(8*Sqrt[2])

_______________________________________________________________________________________

Rubi [A]  time = 0.149198, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.692 \[ -\frac{7}{6 x^{3/2}}+\frac{1}{2 x^{3/2} \left (x^2+1\right )}+\frac{7 \log \left (x-\sqrt{2} \sqrt{x}+1\right )}{8 \sqrt{2}}-\frac{7 \log \left (x+\sqrt{2} \sqrt{x}+1\right )}{8 \sqrt{2}}+\frac{7 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{4 \sqrt{2}}-\frac{7 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{4 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(5/2)*(1 + x^2)^2),x]

[Out]

-7/(6*x^(3/2)) + 1/(2*x^(3/2)*(1 + x^2)) + (7*ArcTan[1 - Sqrt[2]*Sqrt[x]])/(4*Sq
rt[2]) - (7*ArcTan[1 + Sqrt[2]*Sqrt[x]])/(4*Sqrt[2]) + (7*Log[1 - Sqrt[2]*Sqrt[x
] + x])/(8*Sqrt[2]) - (7*Log[1 + Sqrt[2]*Sqrt[x] + x])/(8*Sqrt[2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 20.0951, size = 112, normalized size = 0.92 \[ \frac{7 \sqrt{2} \log{\left (- \sqrt{2} \sqrt{x} + x + 1 \right )}}{16} - \frac{7 \sqrt{2} \log{\left (\sqrt{2} \sqrt{x} + x + 1 \right )}}{16} - \frac{7 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{8} - \frac{7 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{8} - \frac{7}{6 x^{\frac{3}{2}}} + \frac{1}{2 x^{\frac{3}{2}} \left (x^{2} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1)/16 - 7*sqrt(2)*log(sqrt(2)*sqrt(x) + x +
1)/16 - 7*sqrt(2)*atan(sqrt(2)*sqrt(x) - 1)/8 - 7*sqrt(2)*atan(sqrt(2)*sqrt(x)
+ 1)/8 - 7/(6*x**(3/2)) + 1/(2*x**(3/2)*(x**2 + 1))

_______________________________________________________________________________________

Mathematica [A]  time = 0.161299, size = 114, normalized size = 0.93 \[ \frac{1}{48} \left (-\frac{32}{x^{3/2}}-\frac{24 \sqrt{x}}{x^2+1}+21 \sqrt{2} \log \left (x-\sqrt{2} \sqrt{x}+1\right )-21 \sqrt{2} \log \left (x+\sqrt{2} \sqrt{x}+1\right )+42 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )-42 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-32/x^(3/2) - (24*Sqrt[x])/(1 + x^2) + 42*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[x]] -
42*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[x]] + 21*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[x] + x
] - 21*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[x] + x])/48

_______________________________________________________________________________________

Maple [A]  time = 0.015, size = 79, normalized size = 0.7 \[ -{\frac{2}{3}{x}^{-{\frac{3}{2}}}}-{\frac{1}{2\,{x}^{2}+2}\sqrt{x}}-{\frac{7\,\sqrt{2}}{8}\arctan \left ( \sqrt{2}\sqrt{x}-1 \right ) }-{\frac{7\,\sqrt{2}}{16}\ln \left ({1 \left ( 1+x+\sqrt{2}\sqrt{x} \right ) \left ( 1+x-\sqrt{2}\sqrt{x} \right ) ^{-1}} \right ) }-{\frac{7\,\sqrt{2}}{8}\arctan \left ( 1+\sqrt{2}\sqrt{x} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(5/2)/(x^2+1)^2,x)

[Out]

-2/3/x^(3/2)-1/2*x^(1/2)/(x^2+1)-7/8*arctan(2^(1/2)*x^(1/2)-1)*2^(1/2)-7/16*2^(1
/2)*ln((1+x+2^(1/2)*x^(1/2))/(1+x-2^(1/2)*x^(1/2)))-7/8*arctan(1+2^(1/2)*x^(1/2)
)*2^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.51589, size = 124, normalized size = 1.02 \[ -\frac{7}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) - \frac{7}{8} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) - \frac{7}{16} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{7}{16} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{7 \, x^{2} + 4}{6 \,{\left (x^{\frac{7}{2}} + x^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7/8*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) - 7/8*sqrt(2)*arctan(-1/2
*sqrt(2)*(sqrt(2) - 2*sqrt(x))) - 7/16*sqrt(2)*log(sqrt(2)*sqrt(x) + x + 1) + 7/
16*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1) - 1/6*(7*x^2 + 4)/(x^(7/2) + x^(3/2))

_______________________________________________________________________________________

Fricas [A]  time = 0.249191, size = 208, normalized size = 1.7 \[ \frac{84 \, \sqrt{2}{\left (x^{4} + x^{2}\right )} \arctan \left (\frac{1}{\sqrt{2} \sqrt{x} + \sqrt{2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2} + 1}\right ) + 84 \, \sqrt{2}{\left (x^{4} + x^{2}\right )} \arctan \left (\frac{1}{\sqrt{2} \sqrt{x} + \sqrt{-2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2} - 1}\right ) - 21 \, \sqrt{2}{\left (x^{4} + x^{2}\right )} \log \left (2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2\right ) + 21 \, \sqrt{2}{\left (x^{4} + x^{2}\right )} \log \left (-2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2\right ) - 8 \,{\left (7 \, x^{2} + 4\right )} \sqrt{x}}{48 \,{\left (x^{4} + x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(84*sqrt(2)*(x^4 + x^2)*arctan(1/(sqrt(2)*sqrt(x) + sqrt(2*sqrt(2)*sqrt(x)
+ 2*x + 2) + 1)) + 84*sqrt(2)*(x^4 + x^2)*arctan(1/(sqrt(2)*sqrt(x) + sqrt(-2*sq
rt(2)*sqrt(x) + 2*x + 2) - 1)) - 21*sqrt(2)*(x^4 + x^2)*log(2*sqrt(2)*sqrt(x) +
2*x + 2) + 21*sqrt(2)*(x^4 + x^2)*log(-2*sqrt(2)*sqrt(x) + 2*x + 2) - 8*(7*x^2 +
4)*sqrt(x))/(x^4 + x^2)

_______________________________________________________________________________________

Sympy [A]  time = 62.7663, size = 366, normalized size = 3. \[ \frac{21 \sqrt{2} x^{\frac{7}{2}} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{48 x^{\frac{7}{2}} + 48 x^{\frac{3}{2}}} - \frac{21 \sqrt{2} x^{\frac{7}{2}} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{48 x^{\frac{7}{2}} + 48 x^{\frac{3}{2}}} - \frac{42 \sqrt{2} x^{\frac{7}{2}} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{48 x^{\frac{7}{2}} + 48 x^{\frac{3}{2}}} - \frac{42 \sqrt{2} x^{\frac{7}{2}} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{48 x^{\frac{7}{2}} + 48 x^{\frac{3}{2}}} + \frac{21 \sqrt{2} x^{\frac{3}{2}} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{48 x^{\frac{7}{2}} + 48 x^{\frac{3}{2}}} - \frac{21 \sqrt{2} x^{\frac{3}{2}} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{48 x^{\frac{7}{2}} + 48 x^{\frac{3}{2}}} - \frac{42 \sqrt{2} x^{\frac{3}{2}} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{48 x^{\frac{7}{2}} + 48 x^{\frac{3}{2}}} - \frac{42 \sqrt{2} x^{\frac{3}{2}} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{48 x^{\frac{7}{2}} + 48 x^{\frac{3}{2}}} - \frac{56 x^{2}}{48 x^{\frac{7}{2}} + 48 x^{\frac{3}{2}}} - \frac{32}{48 x^{\frac{7}{2}} + 48 x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

21*sqrt(2)*x**(7/2)*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(48*x**(7/2) + 48*x**(3/2)
) - 21*sqrt(2)*x**(7/2)*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(48*x**(7/2) + 48*x**(3
/2)) - 42*sqrt(2)*x**(7/2)*atan(sqrt(2)*sqrt(x) - 1)/(48*x**(7/2) + 48*x**(3/2))
- 42*sqrt(2)*x**(7/2)*atan(sqrt(2)*sqrt(x) + 1)/(48*x**(7/2) + 48*x**(3/2)) + 2
1*sqrt(2)*x**(3/2)*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(48*x**(7/2) + 48*x**(3/2))
- 21*sqrt(2)*x**(3/2)*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(48*x**(7/2) + 48*x**(3/
2)) - 42*sqrt(2)*x**(3/2)*atan(sqrt(2)*sqrt(x) - 1)/(48*x**(7/2) + 48*x**(3/2))
- 42*sqrt(2)*x**(3/2)*atan(sqrt(2)*sqrt(x) + 1)/(48*x**(7/2) + 48*x**(3/2)) - 56
*x**2/(48*x**(7/2) + 48*x**(3/2)) - 32/(48*x**(7/2) + 48*x**(3/2))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.223687, size = 123, normalized size = 1.01 \[ -\frac{7}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) - \frac{7}{8} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) - \frac{7}{16} \, \sqrt{2}{\rm ln}\left (\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{7}{16} \, \sqrt{2}{\rm ln}\left (-\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{\sqrt{x}}{2 \,{\left (x^{2} + 1\right )}} - \frac{2}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7/8*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) - 7/8*sqrt(2)*arctan(-1/2
*sqrt(2)*(sqrt(2) - 2*sqrt(x))) - 7/16*sqrt(2)*ln(sqrt(2)*sqrt(x) + x + 1) + 7/1
6*sqrt(2)*ln(-sqrt(2)*sqrt(x) + x + 1) - 1/2*sqrt(x)/(x^2 + 1) - 2/3/x^(3/2)