3.641 \(\int \frac{\left (a+b x^4\right )^3}{x^5} \, dx\)

Optimal. Leaf size=40 \[ -\frac{a^3}{4 x^4}+3 a^2 b \log (x)+\frac{3}{4} a b^2 x^4+\frac{b^3 x^8}{8} \]

[Out]

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

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Rubi [A]  time = 0.0533335, antiderivative size = 40, 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 \[ -\frac{a^3}{4 x^4}+3 a^2 b \log (x)+\frac{3}{4} a b^2 x^4+\frac{b^3 x^8}{8} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^3/x^5,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**3/x**5,x)

[Out]

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

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Mathematica [A]  time = 0.00794614, size = 40, normalized size = 1. \[ -\frac{a^3}{4 x^4}+3 a^2 b \log (x)+\frac{3}{4} a b^2 x^4+\frac{b^3 x^8}{8} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^3/x^5,x]

[Out]

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

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Maple [A]  time = 0.007, size = 35, normalized size = 0.9 \[ -{\frac{{a}^{3}}{4\,{x}^{4}}}+{\frac{3\,a{b}^{2}{x}^{4}}{4}}+{\frac{{b}^{3}{x}^{8}}{8}}+3\,{a}^{2}b\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^3/x^5,x)

[Out]

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

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Maxima [A]  time = 1.43044, size = 49, normalized size = 1.22 \[ \frac{1}{8} \, b^{3} x^{8} + \frac{3}{4} \, a b^{2} x^{4} + \frac{3}{4} \, a^{2} b \log \left (x^{4}\right ) - \frac{a^{3}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^3/x^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.223746, size = 51, normalized size = 1.27 \[ \frac{b^{3} x^{12} + 6 \, a b^{2} x^{8} + 24 \, a^{2} b x^{4} \log \left (x\right ) - 2 \, a^{3}}{8 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^3/x^5,x, algorithm="fricas")

[Out]

1/8*(b^3*x^12 + 6*a*b^2*x^8 + 24*a^2*b*x^4*log(x) - 2*a^3)/x^4

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Sympy [A]  time = 1.25327, size = 37, normalized size = 0.92 \[ - \frac{a^{3}}{4 x^{4}} + 3 a^{2} b \log{\left (x \right )} + \frac{3 a b^{2} x^{4}}{4} + \frac{b^{3} x^{8}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**3/x**5,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.218691, size = 62, normalized size = 1.55 \[ \frac{1}{8} \, b^{3} x^{8} + \frac{3}{4} \, a b^{2} x^{4} + \frac{3}{4} \, a^{2} b{\rm ln}\left (x^{4}\right ) - \frac{3 \, a^{2} b x^{4} + a^{3}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^3/x^5,x, algorithm="giac")

[Out]

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