3.1265 \(\int \frac{x^{24}}{a+b x^5} \, dx\)

Optimal. Leaf size=66 \[ \frac{a^4 \log \left (a+b x^5\right )}{5 b^5}-\frac{a^3 x^5}{5 b^4}+\frac{a^2 x^{10}}{10 b^3}-\frac{a x^{15}}{15 b^2}+\frac{x^{20}}{20 b} \]

[Out]

-(a^3*x^5)/(5*b^4) + (a^2*x^10)/(10*b^3) - (a*x^15)/(15*b^2) + x^20/(20*b) + (a^
4*Log[a + b*x^5])/(5*b^5)

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Rubi [A]  time = 0.0995864, antiderivative size = 66, 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 \[ \frac{a^4 \log \left (a+b x^5\right )}{5 b^5}-\frac{a^3 x^5}{5 b^4}+\frac{a^2 x^{10}}{10 b^3}-\frac{a x^{15}}{15 b^2}+\frac{x^{20}}{20 b} \]

Antiderivative was successfully verified.

[In]  Int[x^24/(a + b*x^5),x]

[Out]

-(a^3*x^5)/(5*b^4) + (a^2*x^10)/(10*b^3) - (a*x^15)/(15*b^2) + x^20/(20*b) + (a^
4*Log[a + b*x^5])/(5*b^5)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{4} \log{\left (a + b x^{5} \right )}}{5 b^{5}} + \frac{a^{2} \int ^{x^{5}} x\, dx}{5 b^{3}} - \frac{a x^{15}}{15 b^{2}} + \frac{x^{20}}{20 b} - \frac{\int ^{x^{5}} a^{3}\, dx}{5 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**24/(b*x**5+a),x)

[Out]

a**4*log(a + b*x**5)/(5*b**5) + a**2*Integral(x, (x, x**5))/(5*b**3) - a*x**15/(
15*b**2) + x**20/(20*b) - Integral(a**3, (x, x**5))/(5*b**4)

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Mathematica [A]  time = 0.0131004, size = 66, normalized size = 1. \[ \frac{a^4 \log \left (a+b x^5\right )}{5 b^5}-\frac{a^3 x^5}{5 b^4}+\frac{a^2 x^{10}}{10 b^3}-\frac{a x^{15}}{15 b^2}+\frac{x^{20}}{20 b} \]

Antiderivative was successfully verified.

[In]  Integrate[x^24/(a + b*x^5),x]

[Out]

-(a^3*x^5)/(5*b^4) + (a^2*x^10)/(10*b^3) - (a*x^15)/(15*b^2) + x^20/(20*b) + (a^
4*Log[a + b*x^5])/(5*b^5)

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Maple [A]  time = 0.005, size = 57, normalized size = 0.9 \[ -{\frac{{a}^{3}{x}^{5}}{5\,{b}^{4}}}+{\frac{{a}^{2}{x}^{10}}{10\,{b}^{3}}}-{\frac{a{x}^{15}}{15\,{b}^{2}}}+{\frac{{x}^{20}}{20\,b}}+{\frac{{a}^{4}\ln \left ( b{x}^{5}+a \right ) }{5\,{b}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^24/(b*x^5+a),x)

[Out]

-1/5*a^3*x^5/b^4+1/10*a^2*x^10/b^3-1/15*a*x^15/b^2+1/20*x^20/b+1/5*a^4*ln(b*x^5+
a)/b^5

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Maxima [A]  time = 1.44604, size = 77, normalized size = 1.17 \[ \frac{a^{4} \log \left (b x^{5} + a\right )}{5 \, b^{5}} + \frac{3 \, b^{3} x^{20} - 4 \, a b^{2} x^{15} + 6 \, a^{2} b x^{10} - 12 \, a^{3} x^{5}}{60 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^24/(b*x^5 + a),x, algorithm="maxima")

[Out]

1/5*a^4*log(b*x^5 + a)/b^5 + 1/60*(3*b^3*x^20 - 4*a*b^2*x^15 + 6*a^2*b*x^10 - 12
*a^3*x^5)/b^4

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Fricas [A]  time = 0.213693, size = 76, normalized size = 1.15 \[ \frac{3 \, b^{4} x^{20} - 4 \, a b^{3} x^{15} + 6 \, a^{2} b^{2} x^{10} - 12 \, a^{3} b x^{5} + 12 \, a^{4} \log \left (b x^{5} + a\right )}{60 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^24/(b*x^5 + a),x, algorithm="fricas")

[Out]

1/60*(3*b^4*x^20 - 4*a*b^3*x^15 + 6*a^2*b^2*x^10 - 12*a^3*b*x^5 + 12*a^4*log(b*x
^5 + a))/b^5

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Sympy [A]  time = 1.61809, size = 56, normalized size = 0.85 \[ \frac{a^{4} \log{\left (a + b x^{5} \right )}}{5 b^{5}} - \frac{a^{3} x^{5}}{5 b^{4}} + \frac{a^{2} x^{10}}{10 b^{3}} - \frac{a x^{15}}{15 b^{2}} + \frac{x^{20}}{20 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**24/(b*x**5+a),x)

[Out]

a**4*log(a + b*x**5)/(5*b**5) - a**3*x**5/(5*b**4) + a**2*x**10/(10*b**3) - a*x*
*15/(15*b**2) + x**20/(20*b)

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GIAC/XCAS [A]  time = 0.227939, size = 78, normalized size = 1.18 \[ \frac{a^{4}{\rm ln}\left ({\left | b x^{5} + a \right |}\right )}{5 \, b^{5}} + \frac{3 \, b^{3} x^{20} - 4 \, a b^{2} x^{15} + 6 \, a^{2} b x^{10} - 12 \, a^{3} x^{5}}{60 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^24/(b*x^5 + a),x, algorithm="giac")

[Out]

1/5*a^4*ln(abs(b*x^5 + a))/b^5 + 1/60*(3*b^3*x^20 - 4*a*b^2*x^15 + 6*a^2*b*x^10
- 12*a^3*x^5)/b^4