3.1267 \(\int \frac{x^{14}}{a+b x^5} \, dx\)

Optimal. Leaf size=40 \[ \frac{a^2 \log \left (a+b x^5\right )}{5 b^3}-\frac{a x^5}{5 b^2}+\frac{x^{10}}{10 b} \]

[Out]

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

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Rubi [A]  time = 0.0598736, antiderivative size = 40, 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 \[ \frac{a^2 \log \left (a+b x^5\right )}{5 b^3}-\frac{a x^5}{5 b^2}+\frac{x^{10}}{10 b} \]

Antiderivative was successfully verified.

[In]  Int[x^14/(a + b*x^5),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{2} \log{\left (a + b x^{5} \right )}}{5 b^{3}} + \frac{\int ^{x^{5}} x\, dx}{5 b} - \frac{\int ^{x^{5}} a\, dx}{5 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**14/(b*x**5+a),x)

[Out]

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

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Mathematica [A]  time = 0.010259, size = 40, normalized size = 1. \[ \frac{a^2 \log \left (a+b x^5\right )}{5 b^3}-\frac{a x^5}{5 b^2}+\frac{x^{10}}{10 b} \]

Antiderivative was successfully verified.

[In]  Integrate[x^14/(a + b*x^5),x]

[Out]

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

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Maple [A]  time = 0.003, size = 35, normalized size = 0.9 \[ -{\frac{a{x}^{5}}{5\,{b}^{2}}}+{\frac{{x}^{10}}{10\,b}}+{\frac{{a}^{2}\ln \left ( b{x}^{5}+a \right ) }{5\,{b}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^14/(b*x^5+a),x)

[Out]

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

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Maxima [A]  time = 1.44261, size = 46, normalized size = 1.15 \[ \frac{a^{2} \log \left (b x^{5} + a\right )}{5 \, b^{3}} + \frac{b x^{10} - 2 \, a x^{5}}{10 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/(b*x^5 + a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.212382, size = 45, normalized size = 1.12 \[ \frac{b^{2} x^{10} - 2 \, a b x^{5} + 2 \, a^{2} \log \left (b x^{5} + a\right )}{10 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/(b*x^5 + a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.51939, size = 32, normalized size = 0.8 \[ \frac{a^{2} \log{\left (a + b x^{5} \right )}}{5 b^{3}} - \frac{a x^{5}}{5 b^{2}} + \frac{x^{10}}{10 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**14/(b*x**5+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.222877, size = 47, normalized size = 1.18 \[ \frac{a^{2}{\rm ln}\left ({\left | b x^{5} + a \right |}\right )}{5 \, b^{3}} + \frac{b x^{10} - 2 \, a x^{5}}{10 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/(b*x^5 + a),x, algorithm="giac")

[Out]

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