3.243 \(\int \frac{\left (a+b x^3\right )^3}{x^7} \, dx\)

Optimal. Leaf size=38 \[ -\frac{a^3}{6 x^6}-\frac{a^2 b}{x^3}+3 a b^2 \log (x)+\frac{b^3 x^3}{3} \]

[Out]

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

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Rubi [A]  time = 0.052123, antiderivative size = 38, 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 \[ -\frac{a^3}{6 x^6}-\frac{a^2 b}{x^3}+3 a b^2 \log (x)+\frac{b^3 x^3}{3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^3/x^7,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{3}}{6 x^{6}} - \frac{a^{2} b}{x^{3}} + a b^{2} \log{\left (x^{3} \right )} + \frac{\int ^{x^{3}} b^{3}\, dx}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**3/x**7,x)

[Out]

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

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Mathematica [A]  time = 0.00733977, size = 38, normalized size = 1. \[ -\frac{a^3}{6 x^6}-\frac{a^2 b}{x^3}+3 a b^2 \log (x)+\frac{b^3 x^3}{3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^3/x^7,x]

[Out]

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

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Maple [A]  time = 0.008, size = 35, normalized size = 0.9 \[ -{\frac{{a}^{3}}{6\,{x}^{6}}}-{\frac{{a}^{2}b}{{x}^{3}}}+{\frac{{b}^{3}{x}^{3}}{3}}+3\,a{b}^{2}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^3/x^7,x)

[Out]

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

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Maxima [A]  time = 1.4371, size = 49, normalized size = 1.29 \[ \frac{1}{3} \, b^{3} x^{3} + a b^{2} \log \left (x^{3}\right ) - \frac{6 \, a^{2} b x^{3} + a^{3}}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^3/x^7,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.220965, size = 53, normalized size = 1.39 \[ \frac{2 \, b^{3} x^{9} + 18 \, a b^{2} x^{6} \log \left (x\right ) - 6 \, a^{2} b x^{3} - a^{3}}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^3/x^7,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.42477, size = 36, normalized size = 0.95 \[ 3 a b^{2} \log{\left (x \right )} + \frac{b^{3} x^{3}}{3} - \frac{a^{3} + 6 a^{2} b x^{3}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**3/x**7,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.220231, size = 61, normalized size = 1.61 \[ \frac{1}{3} \, b^{3} x^{3} + 3 \, a b^{2}{\rm ln}\left ({\left | x \right |}\right ) - \frac{9 \, a b^{2} x^{6} + 6 \, a^{2} b x^{3} + a^{3}}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^3/x^7,x, algorithm="giac")

[Out]

1/3*b^3*x^3 + 3*a*b^2*ln(abs(x)) - 1/6*(9*a*b^2*x^6 + 6*a^2*b*x^3 + a^3)/x^6