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

Optimal. Leaf size=39 \[ a^3 \log (x)+\frac{3}{4} a^2 b x^4+\frac{3}{8} a b^2 x^8+\frac{b^3 x^{12}}{12} \]

[Out]

(3*a^2*b*x^4)/4 + (3*a*b^2*x^8)/8 + (b^3*x^12)/12 + a^3*Log[x]

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Rubi [A]  time = 0.0468644, antiderivative size = 39, 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 \[ a^3 \log (x)+\frac{3}{4} a^2 b x^4+\frac{3}{8} a b^2 x^8+\frac{b^3 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^3/x,x]

[Out]

(3*a^2*b*x^4)/4 + (3*a*b^2*x^8)/8 + (b^3*x^12)/12 + a^3*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{3} \log{\left (x^{4} \right )}}{4} + \frac{3 a^{2} b x^{4}}{4} + \frac{3 a b^{2} \int ^{x^{4}} x\, dx}{4} + \frac{b^{3} x^{12}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*log(x**4)/4 + 3*a**2*b*x**4/4 + 3*a*b**2*Integral(x, (x, x**4))/4 + b**3*x*
*12/12

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Mathematica [A]  time = 0.00707066, size = 39, normalized size = 1. \[ a^3 \log (x)+\frac{3}{4} a^2 b x^4+\frac{3}{8} a b^2 x^8+\frac{b^3 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^3/x,x]

[Out]

(3*a^2*b*x^4)/4 + (3*a*b^2*x^8)/8 + (b^3*x^12)/12 + a^3*Log[x]

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Maple [A]  time = 0.003, size = 34, normalized size = 0.9 \[{\frac{3\,{a}^{2}b{x}^{4}}{4}}+{\frac{3\,a{b}^{2}{x}^{8}}{8}}+{\frac{{b}^{3}{x}^{12}}{12}}+{a}^{3}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^3/x,x)

[Out]

3/4*a^2*b*x^4+3/8*a*b^2*x^8+1/12*b^3*x^12+a^3*ln(x)

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Maxima [A]  time = 1.43534, size = 49, normalized size = 1.26 \[ \frac{1}{12} \, b^{3} x^{12} + \frac{3}{8} \, a b^{2} x^{8} + \frac{3}{4} \, a^{2} b x^{4} + \frac{1}{4} \, a^{3} \log \left (x^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.226609, size = 45, normalized size = 1.15 \[ \frac{1}{12} \, b^{3} x^{12} + \frac{3}{8} \, a b^{2} x^{8} + \frac{3}{4} \, a^{2} b x^{4} + a^{3} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.07238, size = 37, normalized size = 0.95 \[ a^{3} \log{\left (x \right )} + \frac{3 a^{2} b x^{4}}{4} + \frac{3 a b^{2} x^{8}}{8} + \frac{b^{3} x^{12}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**3/x,x)

[Out]

a**3*log(x) + 3*a**2*b*x**4/4 + 3*a*b**2*x**8/8 + b**3*x**12/12

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GIAC/XCAS [A]  time = 0.228728, size = 49, normalized size = 1.26 \[ \frac{1}{12} \, b^{3} x^{12} + \frac{3}{8} \, a b^{2} x^{8} + \frac{3}{4} \, a^{2} b x^{4} + \frac{1}{4} \, a^{3}{\rm ln}\left (x^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*b^3*x^12 + 3/8*a*b^2*x^8 + 3/4*a^2*b*x^4 + 1/4*a^3*ln(x^4)