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

Optimal. Leaf size=50 \[ a^4 \log (x)+2 a^3 b x^2+\frac{3}{2} a^2 b^2 x^4+\frac{2}{3} a b^3 x^6+\frac{b^4 x^8}{8} \]

[Out]

2*a^3*b*x^2 + (3*a^2*b^2*x^4)/2 + (2*a*b^3*x^6)/3 + (b^4*x^8)/8 + a^4*Log[x]

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Rubi [A]  time = 0.0819044, antiderivative size = 50, 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 \[ a^4 \log (x)+2 a^3 b x^2+\frac{3}{2} a^2 b^2 x^4+\frac{2}{3} a b^3 x^6+\frac{b^4 x^8}{8} \]

Antiderivative was successfully verified.

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

[Out]

2*a^3*b*x^2 + (3*a^2*b^2*x^4)/2 + (2*a*b^3*x^6)/3 + (b^4*x^8)/8 + a^4*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{4} \log{\left (x^{2} \right )}}{2} + 2 a^{3} b x^{2} + 3 a^{2} b^{2} \int ^{x^{2}} x\, dx + \frac{2 a b^{3} x^{6}}{3} + \frac{b^{4} x^{8}}{8} \]

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,x)

[Out]

a**4*log(x**2)/2 + 2*a**3*b*x**2 + 3*a**2*b**2*Integral(x, (x, x**2)) + 2*a*b**3
*x**6/3 + b**4*x**8/8

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Mathematica [A]  time = 0.00748856, size = 50, normalized size = 1. \[ a^4 \log (x)+2 a^3 b x^2+\frac{3}{2} a^2 b^2 x^4+\frac{2}{3} a b^3 x^6+\frac{b^4 x^8}{8} \]

Antiderivative was successfully verified.

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

[Out]

2*a^3*b*x^2 + (3*a^2*b^2*x^4)/2 + (2*a*b^3*x^6)/3 + (b^4*x^8)/8 + a^4*Log[x]

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Maple [A]  time = 0.003, size = 45, normalized size = 0.9 \[ 2\,{a}^{3}b{x}^{2}+{\frac{3\,{a}^{2}{b}^{2}{x}^{4}}{2}}+{\frac{2\,a{b}^{3}{x}^{6}}{3}}+{\frac{{b}^{4}{x}^{8}}{8}}+{a}^{4}\ln \left ( x \right ) \]

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,x)

[Out]

2*a^3*b*x^2+3/2*a^2*b^2*x^4+2/3*a*b^3*x^6+1/8*b^4*x^8+a^4*ln(x)

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Maxima [A]  time = 0.707176, size = 63, normalized size = 1.26 \[ \frac{1}{8} \, b^{4} x^{8} + \frac{2}{3} \, a b^{3} x^{6} + \frac{3}{2} \, a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + \frac{1}{2} \, a^{4} \log \left (x^{2}\right ) \]

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,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.26777, size = 59, normalized size = 1.18 \[ \frac{1}{8} \, b^{4} x^{8} + \frac{2}{3} \, a b^{3} x^{6} + \frac{3}{2} \, a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4} \log \left (x\right ) \]

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,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.11365, size = 49, normalized size = 0.98 \[ a^{4} \log{\left (x \right )} + 2 a^{3} b x^{2} + \frac{3 a^{2} b^{2} x^{4}}{2} + \frac{2 a b^{3} x^{6}}{3} + \frac{b^{4} x^{8}}{8} \]

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,x)

[Out]

a**4*log(x) + 2*a**3*b*x**2 + 3*a**2*b**2*x**4/2 + 2*a*b**3*x**6/3 + b**4*x**8/8

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GIAC/XCAS [A]  time = 0.269664, size = 63, normalized size = 1.26 \[ \frac{1}{8} \, b^{4} x^{8} + \frac{2}{3} \, a b^{3} x^{6} + \frac{3}{2} \, a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + \frac{1}{2} \, a^{4}{\rm ln}\left (x^{2}\right ) \]

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,x, algorithm="giac")

[Out]

1/8*b^4*x^8 + 2/3*a*b^3*x^6 + 3/2*a^2*b^2*x^4 + 2*a^3*b*x^2 + 1/2*a^4*ln(x^2)