3.23 \(\int \frac{\left (a+b x^2\right )^2}{x^5} \, dx\)

Optimal. Leaf size=24 \[ -\frac{a^2}{4 x^4}-\frac{a b}{x^2}+b^2 \log (x) \]

[Out]

-a^2/(4*x^4) - (a*b)/x^2 + b^2*Log[x]

_______________________________________________________________________________________

Rubi [A]  time = 0.0409143, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{a^2}{4 x^4}-\frac{a b}{x^2}+b^2 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^2/x^5,x]

[Out]

-a^2/(4*x^4) - (a*b)/x^2 + b^2*Log[x]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.16817, size = 24, normalized size = 1. \[ - \frac{a^{2}}{4 x^{4}} - \frac{a b}{x^{2}} + \frac{b^{2} \log{\left (x^{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**2/(4*x**4) - a*b/x**2 + b**2*log(x**2)/2

_______________________________________________________________________________________

Mathematica [A]  time = 0.00150968, size = 24, normalized size = 1. \[ -\frac{a^2}{4 x^4}-\frac{a b}{x^2}+b^2 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^2/x^5,x]

[Out]

-a^2/(4*x^4) - (a*b)/x^2 + b^2*Log[x]

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 23, normalized size = 1. \[ -{\frac{{a}^{2}}{4\,{x}^{4}}}-{\frac{ab}{{x}^{2}}}+{b}^{2}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2/x^5,x)

[Out]

-1/4*a^2/x^4-a*b/x^2+b^2*ln(x)

_______________________________________________________________________________________

Maxima [A]  time = 1.34151, size = 35, normalized size = 1.46 \[ \frac{1}{2} \, b^{2} \log \left (x^{2}\right ) - \frac{4 \, a b x^{2} + a^{2}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*b^2*log(x^2) - 1/4*(4*a*b*x^2 + a^2)/x^4

_______________________________________________________________________________________

Fricas [A]  time = 0.202645, size = 38, normalized size = 1.58 \[ \frac{4 \, b^{2} x^{4} \log \left (x\right ) - 4 \, a b x^{2} - a^{2}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(4*b^2*x^4*log(x) - 4*a*b*x^2 - a^2)/x^4

_______________________________________________________________________________________

Sympy [A]  time = 1.27951, size = 22, normalized size = 0.92 \[ b^{2} \log{\left (x \right )} - \frac{a^{2} + 4 a b x^{2}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2/x**5,x)

[Out]

b**2*log(x) - (a**2 + 4*a*b*x**2)/(4*x**4)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.210985, size = 46, normalized size = 1.92 \[ \frac{1}{2} \, b^{2}{\rm ln}\left (x^{2}\right ) - \frac{3 \, b^{2} x^{4} + 4 \, a b x^{2} + a^{2}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*b^2*ln(x^2) - 1/4*(3*b^2*x^4 + 4*a*b*x^2 + a^2)/x^4