3.629 \(\int \frac{\left (a+b x^4\right )^2}{x^5} \, dx\)

Optimal. Leaf size=27 \[ -\frac{a^2}{4 x^4}+2 a b \log (x)+\frac{b^2 x^4}{4} \]

[Out]

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

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Rubi [A]  time = 0.038412, 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 \[ -\frac{a^2}{4 x^4}+2 a b \log (x)+\frac{b^2 x^4}{4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^2/x^5,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**2/x**5,x)

[Out]

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

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Mathematica [A]  time = 0.00136633, size = 27, normalized size = 1. \[ -\frac{a^2}{4 x^4}+2 a b \log (x)+\frac{b^2 x^4}{4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^2/x^5,x]

[Out]

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

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Maple [A]  time = 0.008, size = 24, normalized size = 0.9 \[ -{\frac{{a}^{2}}{4\,{x}^{4}}}+{\frac{{b}^{2}{x}^{4}}{4}}+2\,ab\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^2/x^5,x)

[Out]

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

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Maxima [A]  time = 1.41748, size = 34, normalized size = 1.26 \[ \frac{1}{4} \, b^{2} x^{4} + \frac{1}{2} \, a b \log \left (x^{4}\right ) - \frac{a^{2}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^2/x^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.224365, size = 36, normalized size = 1.33 \[ \frac{b^{2} x^{8} + 8 \, a b x^{4} \log \left (x\right ) - a^{2}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^2/x^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.17824, size = 24, normalized size = 0.89 \[ - \frac{a^{2}}{4 x^{4}} + 2 a b \log{\left (x \right )} + \frac{b^{2} x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**2/x**5,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.232262, size = 45, normalized size = 1.67 \[ \frac{1}{4} \, b^{2} x^{4} + \frac{1}{2} \, a b{\rm ln}\left (x^{4}\right ) - \frac{2 \, a b x^{4} + a^{2}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^2/x^5,x, algorithm="giac")

[Out]

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