3.25 \(\int \frac{1}{x^2 \left (a x+b x^3\right )^2} \, dx\)

Optimal. Leaf size=68 \[ \frac{5 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2}}+\frac{5 b}{2 a^3 x}-\frac{5}{6 a^2 x^3}+\frac{1}{2 a x^3 \left (a+b x^2\right )} \]

[Out]

-5/(6*a^2*x^3) + (5*b)/(2*a^3*x) + 1/(2*a*x^3*(a + b*x^2)) + (5*b^(3/2)*ArcTan[(
Sqrt[b]*x)/Sqrt[a]])/(2*a^(7/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.0703671, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{5 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2}}+\frac{5 b}{2 a^3 x}-\frac{5}{6 a^2 x^3}+\frac{1}{2 a x^3 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a*x + b*x^3)^2),x]

[Out]

-5/(6*a^2*x^3) + (5*b)/(2*a^3*x) + 1/(2*a*x^3*(a + b*x^2)) + (5*b^(3/2)*ArcTan[(
Sqrt[b]*x)/Sqrt[a]])/(2*a^(7/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.5361, size = 61, normalized size = 0.9 \[ \frac{1}{2 a x^{3} \left (a + b x^{2}\right )} - \frac{5}{6 a^{2} x^{3}} + \frac{5 b}{2 a^{3} x} + \frac{5 b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(2*a*x**3*(a + b*x**2)) - 5/(6*a**2*x**3) + 5*b/(2*a**3*x) + 5*b**(3/2)*atan(s
qrt(b)*x/sqrt(a))/(2*a**(7/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0701812, size = 67, normalized size = 0.99 \[ \frac{5 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2}}+\frac{b^2 x}{2 a^3 \left (a+b x^2\right )}+\frac{2 b}{a^3 x}-\frac{1}{3 a^2 x^3} \]

Antiderivative was successfully verified.

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

[Out]

-1/(3*a^2*x^3) + (2*b)/(a^3*x) + (b^2*x)/(2*a^3*(a + b*x^2)) + (5*b^(3/2)*ArcTan
[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(7/2))

_______________________________________________________________________________________

Maple [A]  time = 0.016, size = 59, normalized size = 0.9 \[ -{\frac{1}{3\,{a}^{2}{x}^{3}}}+2\,{\frac{b}{{a}^{3}x}}+{\frac{{b}^{2}x}{2\,{a}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{5\,{b}^{2}}{2\,{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x^3+a*x)^2,x)

[Out]

-1/3/a^2/x^3+2*b/a^3/x+1/2/a^3*b^2*x/(b*x^2+a)+5/2/a^3*b^2/(a*b)^(1/2)*arctan(x*
b/(a*b)^(1/2))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.215621, size = 1, normalized size = 0.01 \[ \left [\frac{30 \, b^{2} x^{4} + 20 \, a b x^{2} + 15 \,{\left (b^{2} x^{5} + a b x^{3}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) - 4 \, a^{2}}{12 \,{\left (a^{3} b x^{5} + a^{4} x^{3}\right )}}, \frac{15 \, b^{2} x^{4} + 10 \, a b x^{2} + 15 \,{\left (b^{2} x^{5} + a b x^{3}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right ) - 2 \, a^{2}}{6 \,{\left (a^{3} b x^{5} + a^{4} x^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/12*(30*b^2*x^4 + 20*a*b*x^2 + 15*(b^2*x^5 + a*b*x^3)*sqrt(-b/a)*log((b*x^2 +
2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) - 4*a^2)/(a^3*b*x^5 + a^4*x^3), 1/6*(15*b^2*x
^4 + 10*a*b*x^2 + 15*(b^2*x^5 + a*b*x^3)*sqrt(b/a)*arctan(b*x/(a*sqrt(b/a))) - 2
*a^2)/(a^3*b*x^5 + a^4*x^3)]

_______________________________________________________________________________________

Sympy [A]  time = 2.12167, size = 114, normalized size = 1.68 \[ - \frac{5 \sqrt{- \frac{b^{3}}{a^{7}}} \log{\left (- \frac{a^{4} \sqrt{- \frac{b^{3}}{a^{7}}}}{b^{2}} + x \right )}}{4} + \frac{5 \sqrt{- \frac{b^{3}}{a^{7}}} \log{\left (\frac{a^{4} \sqrt{- \frac{b^{3}}{a^{7}}}}{b^{2}} + x \right )}}{4} + \frac{- 2 a^{2} + 10 a b x^{2} + 15 b^{2} x^{4}}{6 a^{4} x^{3} + 6 a^{3} b x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*sqrt(-b**3/a**7)*log(-a**4*sqrt(-b**3/a**7)/b**2 + x)/4 + 5*sqrt(-b**3/a**7)*
log(a**4*sqrt(-b**3/a**7)/b**2 + x)/4 + (-2*a**2 + 10*a*b*x**2 + 15*b**2*x**4)/(
6*a**4*x**3 + 6*a**3*b*x**5)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.219085, size = 80, normalized size = 1.18 \[ \frac{5 \, b^{2} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{3}} + \frac{b^{2} x}{2 \,{\left (b x^{2} + a\right )} a^{3}} + \frac{6 \, b x^{2} - a}{3 \, a^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/2*b^2*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^3) + 1/2*b^2*x/((b*x^2 + a)*a^3) + 1/
3*(6*b*x^2 - a)/(a^3*x^3)