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

Optimal. Leaf size=37 \[ -\frac{a^3}{3 x^3}+3 a^2 b \log (x)+a b^2 x^3+\frac{b^3 x^6}{6} \]

[Out]

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

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Rubi [A]  time = 0.0556364, antiderivative size = 37, 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 \[ -\frac{a^3}{3 x^3}+3 a^2 b \log (x)+a b^2 x^3+\frac{b^3 x^6}{6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^3/x^4,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00743577, size = 37, normalized size = 1. \[ -\frac{a^3}{3 x^3}+3 a^2 b \log (x)+a b^2 x^3+\frac{b^3 x^6}{6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^3/x^4,x]

[Out]

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

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Maple [A]  time = 0.01, size = 34, normalized size = 0.9 \[ -{\frac{{a}^{3}}{3\,{x}^{3}}}+a{b}^{2}{x}^{3}+{\frac{{b}^{3}{x}^{6}}{6}}+3\,{a}^{2}b\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^3/x^4,x)

[Out]

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

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Maxima [A]  time = 1.44406, size = 46, normalized size = 1.24 \[ \frac{1}{6} \, b^{3} x^{6} + a b^{2} x^{3} + a^{2} b \log \left (x^{3}\right ) - \frac{a^{3}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.218447, size = 51, normalized size = 1.38 \[ \frac{b^{3} x^{9} + 6 \, a b^{2} x^{6} + 18 \, a^{2} b x^{3} \log \left (x\right ) - 2 \, a^{3}}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(b^3*x^9 + 6*a*b^2*x^6 + 18*a^2*b*x^3*log(x) - 2*a^3)/x^3

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Sympy [A]  time = 1.20823, size = 34, normalized size = 0.92 \[ - \frac{a^{3}}{3 x^{3}} + 3 a^{2} b \log{\left (x \right )} + a b^{2} x^{3} + \frac{b^{3} x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**3/x**4,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.218194, size = 59, normalized size = 1.59 \[ \frac{1}{6} \, b^{3} x^{6} + a b^{2} x^{3} + 3 \, a^{2} b{\rm ln}\left ({\left | x \right |}\right ) - \frac{3 \, a^{2} b x^{3} + a^{3}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*b^3*x^6 + a*b^2*x^3 + 3*a^2*b*ln(abs(x)) - 1/3*(3*a^2*b*x^3 + a^3)/x^3