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

Optimal. Leaf size=48 \[ -\frac{a^4}{2 x^2}+4 a^3 b \log (x)+3 a^2 b^2 x^2+a b^3 x^4+\frac{b^4 x^6}{6} \]

[Out]

-a^4/(2*x^2) + 3*a^2*b^2*x^2 + a*b^3*x^4 + (b^4*x^6)/6 + 4*a^3*b*Log[x]

_______________________________________________________________________________________

Rubi [A]  time = 0.0916041, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{a^4}{2 x^2}+4 a^3 b \log (x)+3 a^2 b^2 x^2+a b^3 x^4+\frac{b^4 x^6}{6} \]

Antiderivative was successfully verified.

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

[Out]

-a^4/(2*x^2) + 3*a^2*b^2*x^2 + a*b^3*x^4 + (b^4*x^6)/6 + 4*a^3*b*Log[x]

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{4}}{2 x^{2}} + 2 a^{3} b \log{\left (x^{2} \right )} + 3 a^{2} b^{2} x^{2} + 2 a b^{3} \int ^{x^{2}} x\, dx + \frac{b^{4} x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**4/(2*x**2) + 2*a**3*b*log(x**2) + 3*a**2*b**2*x**2 + 2*a*b**3*Integral(x, (x
, x**2)) + b**4*x**6/6

_______________________________________________________________________________________

Mathematica [A]  time = 0.00769079, size = 48, normalized size = 1. \[ -\frac{a^4}{2 x^2}+4 a^3 b \log (x)+3 a^2 b^2 x^2+a b^3 x^4+\frac{b^4 x^6}{6} \]

Antiderivative was successfully verified.

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

[Out]

-a^4/(2*x^2) + 3*a^2*b^2*x^2 + a*b^3*x^4 + (b^4*x^6)/6 + 4*a^3*b*Log[x]

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 45, normalized size = 0.9 \[ -{\frac{{a}^{4}}{2\,{x}^{2}}}+3\,{a}^{2}{b}^{2}{x}^{2}+a{b}^{3}{x}^{4}+{\frac{{b}^{4}{x}^{6}}{6}}+4\,{a}^{3}b\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*a^4/x^2+3*a^2*b^2*x^2+a*b^3*x^4+1/6*b^4*x^6+4*a^3*b*ln(x)

_______________________________________________________________________________________

Maxima [A]  time = 0.703458, size = 62, normalized size = 1.29 \[ \frac{1}{6} \, b^{4} x^{6} + a b^{3} x^{4} + 3 \, a^{2} b^{2} x^{2} + 2 \, a^{3} b \log \left (x^{2}\right ) - \frac{a^{4}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*b^4*x^6 + a*b^3*x^4 + 3*a^2*b^2*x^2 + 2*a^3*b*log(x^2) - 1/2*a^4/x^2

_______________________________________________________________________________________

Fricas [A]  time = 0.264049, size = 66, normalized size = 1.38 \[ \frac{b^{4} x^{8} + 6 \, a b^{3} x^{6} + 18 \, a^{2} b^{2} x^{4} + 24 \, a^{3} b x^{2} \log \left (x\right ) - 3 \, a^{4}}{6 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(b^4*x^8 + 6*a*b^3*x^6 + 18*a^2*b^2*x^4 + 24*a^3*b*x^2*log(x) - 3*a^4)/x^2

_______________________________________________________________________________________

Sympy [A]  time = 1.15638, size = 46, normalized size = 0.96 \[ - \frac{a^{4}}{2 x^{2}} + 4 a^{3} b \log{\left (x \right )} + 3 a^{2} b^{2} x^{2} + a b^{3} x^{4} + \frac{b^{4} x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**4/(2*x**2) + 4*a**3*b*log(x) + 3*a**2*b**2*x**2 + a*b**3*x**4 + b**4*x**6/6

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.270609, size = 76, normalized size = 1.58 \[ \frac{1}{6} \, b^{4} x^{6} + a b^{3} x^{4} + 3 \, a^{2} b^{2} x^{2} + 2 \, a^{3} b{\rm ln}\left (x^{2}\right ) - \frac{4 \, a^{3} b x^{2} + a^{4}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*b^4*x^6 + a*b^3*x^4 + 3*a^2*b^2*x^2 + 2*a^3*b*ln(x^2) - 1/2*(4*a^3*b*x^2 + a
^4)/x^2