3.1438 \(\int \frac{\left (a+b x^7\right )^2}{x^8} \, dx\)

Optimal. Leaf size=27 \[ -\frac{a^2}{7 x^7}+2 a b \log (x)+\frac{b^2 x^7}{7} \]

[Out]

-a^2/(7*x^7) + (b^2*x^7)/7 + 2*a*b*Log[x]

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Rubi [A]  time = 0.0393784, antiderivative size = 27, 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 \[ -\frac{a^2}{7 x^7}+2 a b \log (x)+\frac{b^2 x^7}{7} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^7)^2/x^8,x]

[Out]

-a^2/(7*x^7) + (b^2*x^7)/7 + 2*a*b*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2}}{7 x^{7}} + \frac{2 a b \log{\left (x^{7} \right )}}{7} + \frac{\int ^{x^{7}} b^{2}\, dx}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**7+a)**2/x**8,x)

[Out]

-a**2/(7*x**7) + 2*a*b*log(x**7)/7 + Integral(b**2, (x, x**7))/7

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Mathematica [A]  time = 0.00138905, size = 27, normalized size = 1. \[ -\frac{a^2}{7 x^7}+2 a b \log (x)+\frac{b^2 x^7}{7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^7)^2/x^8,x]

[Out]

-a^2/(7*x^7) + (b^2*x^7)/7 + 2*a*b*Log[x]

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Maple [A]  time = 0.009, size = 24, normalized size = 0.9 \[ -{\frac{{a}^{2}}{7\,{x}^{7}}}+{\frac{{b}^{2}{x}^{7}}{7}}+2\,ab\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^7+a)^2/x^8,x)

[Out]

-1/7*a^2/x^7+1/7*b^2*x^7+2*a*b*ln(x)

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Maxima [A]  time = 1.43824, size = 34, normalized size = 1.26 \[ \frac{1}{7} \, b^{2} x^{7} + \frac{2}{7} \, a b \log \left (x^{7}\right ) - \frac{a^{2}}{7 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^7 + a)^2/x^8,x, algorithm="maxima")

[Out]

1/7*b^2*x^7 + 2/7*a*b*log(x^7) - 1/7*a^2/x^7

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Fricas [A]  time = 0.214779, size = 36, normalized size = 1.33 \[ \frac{b^{2} x^{14} + 14 \, a b x^{7} \log \left (x\right ) - a^{2}}{7 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^7 + a)^2/x^8,x, algorithm="fricas")

[Out]

1/7*(b^2*x^14 + 14*a*b*x^7*log(x) - a^2)/x^7

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Sympy [A]  time = 1.29469, size = 24, normalized size = 0.89 \[ - \frac{a^{2}}{7 x^{7}} + 2 a b \log{\left (x \right )} + \frac{b^{2} x^{7}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**7+a)**2/x**8,x)

[Out]

-a**2/(7*x**7) + 2*a*b*log(x) + b**2*x**7/7

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GIAC/XCAS [A]  time = 0.223603, size = 43, normalized size = 1.59 \[ \frac{1}{7} \, b^{2} x^{7} + 2 \, a b{\rm ln}\left ({\left | x \right |}\right ) - \frac{2 \, a b x^{7} + a^{2}}{7 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^7 + a)^2/x^8,x, algorithm="giac")

[Out]

1/7*b^2*x^7 + 2*a*b*ln(abs(x)) - 1/7*(2*a*b*x^7 + a^2)/x^7