3.1278 \(\int \frac{x^9}{\left (a+b x^5\right )^2} \, dx\)

Optimal. Leaf size=33 \[ \frac{a}{5 b^2 \left (a+b x^5\right )}+\frac{\log \left (a+b x^5\right )}{5 b^2} \]

[Out]

a/(5*b^2*(a + b*x^5)) + Log[a + b*x^5]/(5*b^2)

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Rubi [A]  time = 0.0593079, antiderivative size = 33, 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 \[ \frac{a}{5 b^2 \left (a+b x^5\right )}+\frac{\log \left (a+b x^5\right )}{5 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^9/(a + b*x^5)^2,x]

[Out]

a/(5*b^2*(a + b*x^5)) + Log[a + b*x^5]/(5*b^2)

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Rubi in Sympy [A]  time = 7.77699, size = 26, normalized size = 0.79 \[ \frac{a}{5 b^{2} \left (a + b x^{5}\right )} + \frac{\log{\left (a + b x^{5} \right )}}{5 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9/(b*x**5+a)**2,x)

[Out]

a/(5*b**2*(a + b*x**5)) + log(a + b*x**5)/(5*b**2)

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Mathematica [A]  time = 0.0155022, size = 27, normalized size = 0.82 \[ \frac{\frac{a}{a+b x^5}+\log \left (a+b x^5\right )}{5 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^9/(a + b*x^5)^2,x]

[Out]

(a/(a + b*x^5) + Log[a + b*x^5])/(5*b^2)

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Maple [A]  time = 0.007, size = 30, normalized size = 0.9 \[{\frac{a}{5\,{b}^{2} \left ( b{x}^{5}+a \right ) }}+{\frac{\ln \left ( b{x}^{5}+a \right ) }{5\,{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9/(b*x^5+a)^2,x)

[Out]

1/5*a/b^2/(b*x^5+a)+1/5*ln(b*x^5+a)/b^2

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Maxima [A]  time = 1.42643, size = 43, normalized size = 1.3 \[ \frac{a}{5 \,{\left (b^{3} x^{5} + a b^{2}\right )}} + \frac{\log \left (b x^{5} + a\right )}{5 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(b*x^5 + a)^2,x, algorithm="maxima")

[Out]

1/5*a/(b^3*x^5 + a*b^2) + 1/5*log(b*x^5 + a)/b^2

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Fricas [A]  time = 0.216524, size = 47, normalized size = 1.42 \[ \frac{{\left (b x^{5} + a\right )} \log \left (b x^{5} + a\right ) + a}{5 \,{\left (b^{3} x^{5} + a b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(b*x^5 + a)^2,x, algorithm="fricas")

[Out]

1/5*((b*x^5 + a)*log(b*x^5 + a) + a)/(b^3*x^5 + a*b^2)

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Sympy [A]  time = 2.54323, size = 29, normalized size = 0.88 \[ \frac{a}{5 a b^{2} + 5 b^{3} x^{5}} + \frac{\log{\left (a + b x^{5} \right )}}{5 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9/(b*x**5+a)**2,x)

[Out]

a/(5*a*b**2 + 5*b**3*x**5) + log(a + b*x**5)/(5*b**2)

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GIAC/XCAS [A]  time = 0.231069, size = 65, normalized size = 1.97 \[ -\frac{\frac{{\rm ln}\left (\frac{{\left | b x^{5} + a \right |}}{{\left (b x^{5} + a\right )}^{2}{\left | b \right |}}\right )}{b} - \frac{a}{{\left (b x^{5} + a\right )} b}}{5 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(b*x^5 + a)^2,x, algorithm="giac")

[Out]

-1/5*(ln(abs(b*x^5 + a)/((b*x^5 + a)^2*abs(b)))/b - a/((b*x^5 + a)*b))/b