3.1268 \(\int \frac{x^9}{a+b x^5} \, dx\)

Optimal. Leaf size=27 \[ \frac{x^5}{5 b}-\frac{a \log \left (a+b x^5\right )}{5 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0444636, 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 \[ \frac{x^5}{5 b}-\frac{a \log \left (a+b x^5\right )}{5 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^9/(a + b*x^5),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a \log{\left (a + b x^{5} \right )}}{5 b^{2}} + \frac{\int ^{x^{5}} \frac{1}{b}\, dx}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*log(a + b*x**5)/(5*b**2) + Integral(1/b, (x, x**5))/5

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Mathematica [A]  time = 0.00691355, size = 27, normalized size = 1. \[ \frac{x^5}{5 b}-\frac{a \log \left (a+b x^5\right )}{5 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^9/(a + b*x^5),x]

[Out]

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

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Maple [A]  time = 0.003, size = 24, normalized size = 0.9 \[{\frac{{x}^{5}}{5\,b}}-{\frac{a\ln \left ( b{x}^{5}+a \right ) }{5\,{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9/(b*x^5+a),x)

[Out]

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

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Maxima [A]  time = 1.46607, size = 31, normalized size = 1.15 \[ \frac{x^{5}}{5 \, b} - \frac{a \log \left (b x^{5} + a\right )}{5 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*x^5/b - 1/5*a*log(b*x^5 + a)/b^2

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Fricas [A]  time = 0.21727, size = 30, normalized size = 1.11 \[ \frac{b x^{5} - a \log \left (b x^{5} + a\right )}{5 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*(b*x^5 - a*log(b*x^5 + a))/b^2

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Sympy [A]  time = 1.42749, size = 20, normalized size = 0.74 \[ - \frac{a \log{\left (a + b x^{5} \right )}}{5 b^{2}} + \frac{x^{5}}{5 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9/(b*x**5+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.22287, size = 32, normalized size = 1.19 \[ \frac{x^{5}}{5 \, b} - \frac{a{\rm ln}\left ({\left | b x^{5} + a \right |}\right )}{5 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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