3.226 \(\int \frac{\left (a+b x^3\right )^2}{x^4} \, dx\)

Optimal. Leaf size=27 \[ -\frac{a^2}{3 x^3}+2 a b \log (x)+\frac{b^2 x^3}{3} \]

[Out]

-a^2/(3*x^3) + (b^2*x^3)/3 + 2*a*b*Log[x]

_______________________________________________________________________________________

Rubi [A]  time = 0.0396798, antiderivative size = 27, 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 \[ -\frac{a^2}{3 x^3}+2 a b \log (x)+\frac{b^2 x^3}{3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^2/x^4,x]

[Out]

-a^2/(3*x^3) + (b^2*x^3)/3 + 2*a*b*Log[x]

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2}}{3 x^{3}} + \frac{2 a b \log{\left (x^{3} \right )}}{3} + \frac{\int ^{x^{3}} b^{2}\, dx}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**2/x**4,x)

[Out]

-a**2/(3*x**3) + 2*a*b*log(x**3)/3 + Integral(b**2, (x, x**3))/3

_______________________________________________________________________________________

Mathematica [A]  time = 0.00138169, size = 27, normalized size = 1. \[ -\frac{a^2}{3 x^3}+2 a b \log (x)+\frac{b^2 x^3}{3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^2/x^4,x]

[Out]

-a^2/(3*x^3) + (b^2*x^3)/3 + 2*a*b*Log[x]

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 24, normalized size = 0.9 \[ -{\frac{{a}^{2}}{3\,{x}^{3}}}+{\frac{{b}^{2}{x}^{3}}{3}}+2\,ab\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^2/x^4,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.43302, size = 34, normalized size = 1.26 \[ \frac{1}{3} \, b^{2} x^{3} + \frac{2}{3} \, a b \log \left (x^{3}\right ) - \frac{a^{2}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^2/x^4,x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.255162, size = 36, normalized size = 1.33 \[ \frac{b^{2} x^{6} + 6 \, a b x^{3} \log \left (x\right ) - a^{2}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^2/x^4,x, algorithm="fricas")

[Out]

1/3*(b^2*x^6 + 6*a*b*x^3*log(x) - a^2)/x^3

_______________________________________________________________________________________

Sympy [A]  time = 1.21082, size = 24, normalized size = 0.89 \[ - \frac{a^{2}}{3 x^{3}} + 2 a b \log{\left (x \right )} + \frac{b^{2} x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**2/x**4,x)

[Out]

-a**2/(3*x**3) + 2*a*b*log(x) + b**2*x**3/3

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.219086, size = 43, normalized size = 1.59 \[ \frac{1}{3} \, b^{2} x^{3} + 2 \, a b{\rm ln}\left ({\left | x \right |}\right ) - \frac{2 \, a b x^{3} + a^{2}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^2/x^4,x, algorithm="giac")

[Out]

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