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

Optimal. Leaf size=26 \[ a^2 \log (x)+\frac{1}{2} a b x^4+\frac{b^2 x^8}{8} \]

[Out]

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

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Rubi [A]  time = 0.0343828, antiderivative size = 26, 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 \[ a^2 \log (x)+\frac{1}{2} a b x^4+\frac{b^2 x^8}{8} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^2/x,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00134809, size = 26, normalized size = 1. \[ a^2 \log (x)+\frac{1}{2} a b x^4+\frac{b^2 x^8}{8} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^2/x,x]

[Out]

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

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Maple [A]  time = 0.003, size = 23, normalized size = 0.9 \[{\frac{ab{x}^{4}}{2}}+{\frac{{b}^{2}{x}^{8}}{8}}+{a}^{2}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^2/x,x)

[Out]

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

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Maxima [A]  time = 1.41446, size = 34, normalized size = 1.31 \[ \frac{1}{8} \, b^{2} x^{8} + \frac{1}{2} \, a b x^{4} + \frac{1}{4} \, a^{2} \log \left (x^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.224095, size = 30, normalized size = 1.15 \[ \frac{1}{8} \, b^{2} x^{8} + \frac{1}{2} \, a b x^{4} + a^{2} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.01544, size = 22, normalized size = 0.85 \[ a^{2} \log{\left (x \right )} + \frac{a b x^{4}}{2} + \frac{b^{2} x^{8}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**2/x,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.221118, size = 34, normalized size = 1.31 \[ \frac{1}{8} \, b^{2} x^{8} + \frac{1}{2} \, a b x^{4} + \frac{1}{4} \, a^{2}{\rm ln}\left (x^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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