3.229 \(\int \frac{\left (a+b x^3\right )^2}{x^7} \, dx\)

Optimal. Leaf size=26 \[ -\frac{a^2}{6 x^6}-\frac{2 a b}{3 x^3}+b^2 \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0375561, antiderivative size = 26, 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 \[ -\frac{a^2}{6 x^6}-\frac{2 a b}{3 x^3}+b^2 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^2/x^7,x]

[Out]

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

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Rubi in Sympy [A]  time = 6.44806, size = 27, normalized size = 1.04 \[ - \frac{a^{2}}{6 x^{6}} - \frac{2 a b}{3 x^{3}} + \frac{b^{2} \log{\left (x^{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**2/x**7,x)

[Out]

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

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Mathematica [A]  time = 0.00134169, size = 26, normalized size = 1. \[ -\frac{a^2}{6 x^6}-\frac{2 a b}{3 x^3}+b^2 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^2/x^7,x]

[Out]

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

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Maple [A]  time = 0.008, size = 23, normalized size = 0.9 \[ -{\frac{{a}^{2}}{6\,{x}^{6}}}-{\frac{2\,ab}{3\,{x}^{3}}}+{b}^{2}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^2/x^7,x)

[Out]

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

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Maxima [A]  time = 1.44112, size = 35, normalized size = 1.35 \[ \frac{1}{3} \, b^{2} \log \left (x^{3}\right ) - \frac{4 \, a b x^{3} + a^{2}}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^2/x^7,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.28273, size = 38, normalized size = 1.46 \[ \frac{6 \, b^{2} x^{6} \log \left (x\right ) - 4 \, a b x^{3} - a^{2}}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^2/x^7,x, algorithm="fricas")

[Out]

1/6*(6*b^2*x^6*log(x) - 4*a*b*x^3 - a^2)/x^6

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Sympy [A]  time = 1.3735, size = 22, normalized size = 0.85 \[ b^{2} \log{\left (x \right )} - \frac{a^{2} + 4 a b x^{3}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**2/x**7,x)

[Out]

b**2*log(x) - (a**2 + 4*a*b*x**3)/(6*x**6)

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GIAC/XCAS [A]  time = 0.258013, size = 43, normalized size = 1.65 \[ b^{2}{\rm ln}\left ({\left | x \right |}\right ) - \frac{3 \, b^{2} x^{6} + 4 \, a b x^{3} + a^{2}}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^2/x^7,x, algorithm="giac")

[Out]

b^2*ln(abs(x)) - 1/6*(3*b^2*x^6 + 4*a*b*x^3 + a^2)/x^6