3.1273 \(\int \frac{1}{x^{16} \left (a+b x^5\right )} \, dx\)

Optimal. Leaf size=63 \[ \frac{b^3 \log \left (a+b x^5\right )}{5 a^4}-\frac{b^3 \log (x)}{a^4}-\frac{b^2}{5 a^3 x^5}+\frac{b}{10 a^2 x^{10}}-\frac{1}{15 a x^{15}} \]

[Out]

-1/(15*a*x^15) + b/(10*a^2*x^10) - b^2/(5*a^3*x^5) - (b^3*Log[x])/a^4 + (b^3*Log
[a + b*x^5])/(5*a^4)

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Rubi [A]  time = 0.0801135, antiderivative size = 63, 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 \[ \frac{b^3 \log \left (a+b x^5\right )}{5 a^4}-\frac{b^3 \log (x)}{a^4}-\frac{b^2}{5 a^3 x^5}+\frac{b}{10 a^2 x^{10}}-\frac{1}{15 a x^{15}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^16*(a + b*x^5)),x]

[Out]

-1/(15*a*x^15) + b/(10*a^2*x^10) - b^2/(5*a^3*x^5) - (b^3*Log[x])/a^4 + (b^3*Log
[a + b*x^5])/(5*a^4)

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Rubi in Sympy [A]  time = 12.5634, size = 60, normalized size = 0.95 \[ - \frac{1}{15 a x^{15}} + \frac{b}{10 a^{2} x^{10}} - \frac{b^{2}}{5 a^{3} x^{5}} - \frac{b^{3} \log{\left (x^{5} \right )}}{5 a^{4}} + \frac{b^{3} \log{\left (a + b x^{5} \right )}}{5 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**16/(b*x**5+a),x)

[Out]

-1/(15*a*x**15) + b/(10*a**2*x**10) - b**2/(5*a**3*x**5) - b**3*log(x**5)/(5*a**
4) + b**3*log(a + b*x**5)/(5*a**4)

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Mathematica [A]  time = 0.0121859, size = 63, normalized size = 1. \[ \frac{b^3 \log \left (a+b x^5\right )}{5 a^4}-\frac{b^3 \log (x)}{a^4}-\frac{b^2}{5 a^3 x^5}+\frac{b}{10 a^2 x^{10}}-\frac{1}{15 a x^{15}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^16*(a + b*x^5)),x]

[Out]

-1/(15*a*x^15) + b/(10*a^2*x^10) - b^2/(5*a^3*x^5) - (b^3*Log[x])/a^4 + (b^3*Log
[a + b*x^5])/(5*a^4)

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Maple [A]  time = 0.011, size = 56, normalized size = 0.9 \[ -{\frac{1}{15\,a{x}^{15}}}+{\frac{b}{10\,{a}^{2}{x}^{10}}}-{\frac{{b}^{2}}{5\,{a}^{3}{x}^{5}}}-{\frac{{b}^{3}\ln \left ( x \right ) }{{a}^{4}}}+{\frac{{b}^{3}\ln \left ( b{x}^{5}+a \right ) }{5\,{a}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^16/(b*x^5+a),x)

[Out]

-1/15/a/x^15+1/10*b/a^2/x^10-1/5*b^2/a^3/x^5-b^3*ln(x)/a^4+1/5*b^3*ln(b*x^5+a)/a
^4

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Maxima [A]  time = 1.41469, size = 78, normalized size = 1.24 \[ \frac{b^{3} \log \left (b x^{5} + a\right )}{5 \, a^{4}} - \frac{b^{3} \log \left (x^{5}\right )}{5 \, a^{4}} - \frac{6 \, b^{2} x^{10} - 3 \, a b x^{5} + 2 \, a^{2}}{30 \, a^{3} x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^5 + a)*x^16),x, algorithm="maxima")

[Out]

1/5*b^3*log(b*x^5 + a)/a^4 - 1/5*b^3*log(x^5)/a^4 - 1/30*(6*b^2*x^10 - 3*a*b*x^5
+ 2*a^2)/(a^3*x^15)

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Fricas [A]  time = 0.222387, size = 78, normalized size = 1.24 \[ \frac{6 \, b^{3} x^{15} \log \left (b x^{5} + a\right ) - 30 \, b^{3} x^{15} \log \left (x\right ) - 6 \, a b^{2} x^{10} + 3 \, a^{2} b x^{5} - 2 \, a^{3}}{30 \, a^{4} x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^5 + a)*x^16),x, algorithm="fricas")

[Out]

1/30*(6*b^3*x^15*log(b*x^5 + a) - 30*b^3*x^15*log(x) - 6*a*b^2*x^10 + 3*a^2*b*x^
5 - 2*a^3)/(a^4*x^15)

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Sympy [A]  time = 157.098, size = 56, normalized size = 0.89 \[ - \frac{2 a^{2} - 3 a b x^{5} + 6 b^{2} x^{10}}{30 a^{3} x^{15}} - \frac{b^{3} \log{\left (x \right )}}{a^{4}} + \frac{b^{3} \log{\left (\frac{a}{b} + x^{5} \right )}}{5 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**16/(b*x**5+a),x)

[Out]

-(2*a**2 - 3*a*b*x**5 + 6*b**2*x**10)/(30*a**3*x**15) - b**3*log(x)/a**4 + b**3*
log(a/b + x**5)/(5*a**4)

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GIAC/XCAS [A]  time = 0.238044, size = 93, normalized size = 1.48 \[ \frac{b^{3}{\rm ln}\left ({\left | b x^{5} + a \right |}\right )}{5 \, a^{4}} - \frac{b^{3}{\rm ln}\left ({\left | x \right |}\right )}{a^{4}} + \frac{11 \, b^{3} x^{15} - 6 \, a b^{2} x^{10} + 3 \, a^{2} b x^{5} - 2 \, a^{3}}{30 \, a^{4} x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^5 + a)*x^16),x, algorithm="giac")

[Out]

1/5*b^3*ln(abs(b*x^5 + a))/a^4 - b^3*ln(abs(x))/a^4 + 1/30*(11*b^3*x^15 - 6*a*b^
2*x^10 + 3*a^2*b*x^5 - 2*a^3)/(a^4*x^15)