3.1275 \(\int \frac{x^{24}}{\left (a+b x^5\right )^2} \, dx\)

Optimal. Leaf size=72 \[ -\frac{a^4}{5 b^5 \left (a+b x^5\right )}-\frac{4 a^3 \log \left (a+b x^5\right )}{5 b^5}+\frac{3 a^2 x^5}{5 b^4}-\frac{a x^{10}}{5 b^3}+\frac{x^{15}}{15 b^2} \]

[Out]

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

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Rubi [A]  time = 0.119689, antiderivative size = 72, 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 \[ -\frac{a^4}{5 b^5 \left (a+b x^5\right )}-\frac{4 a^3 \log \left (a+b x^5\right )}{5 b^5}+\frac{3 a^2 x^5}{5 b^4}-\frac{a x^{10}}{5 b^3}+\frac{x^{15}}{15 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^24/(a + b*x^5)^2,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**24/(b*x**5+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.044246, size = 60, normalized size = 0.83 \[ \frac{-\frac{3 a^4}{a+b x^5}-12 a^3 \log \left (a+b x^5\right )+9 a^2 b x^5-3 a b^2 x^{10}+b^3 x^{15}}{15 b^5} \]

Antiderivative was successfully verified.

[In]  Integrate[x^24/(a + b*x^5)^2,x]

[Out]

(9*a^2*b*x^5 - 3*a*b^2*x^10 + b^3*x^15 - (3*a^4)/(a + b*x^5) - 12*a^3*Log[a + b*
x^5])/(15*b^5)

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Maple [A]  time = 0.01, size = 63, normalized size = 0.9 \[{\frac{3\,{x}^{5}{a}^{2}}{5\,{b}^{4}}}-{\frac{a{x}^{10}}{5\,{b}^{3}}}+{\frac{{x}^{15}}{15\,{b}^{2}}}-{\frac{{a}^{4}}{5\,{b}^{5} \left ( b{x}^{5}+a \right ) }}-{\frac{4\,{a}^{3}\ln \left ( b{x}^{5}+a \right ) }{5\,{b}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^24/(b*x^5+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.43222, size = 88, normalized size = 1.22 \[ -\frac{a^{4}}{5 \,{\left (b^{6} x^{5} + a b^{5}\right )}} - \frac{4 \, a^{3} \log \left (b x^{5} + a\right )}{5 \, b^{5}} + \frac{b^{2} x^{15} - 3 \, a b x^{10} + 9 \, a^{2} x^{5}}{15 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^24/(b*x^5 + a)^2,x, algorithm="maxima")

[Out]

-1/5*a^4/(b^6*x^5 + a*b^5) - 4/5*a^3*log(b*x^5 + a)/b^5 + 1/15*(b^2*x^15 - 3*a*b
*x^10 + 9*a^2*x^5)/b^4

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Fricas [A]  time = 0.214202, size = 109, normalized size = 1.51 \[ \frac{b^{4} x^{20} - 2 \, a b^{3} x^{15} + 6 \, a^{2} b^{2} x^{10} + 9 \, a^{3} b x^{5} - 3 \, a^{4} - 12 \,{\left (a^{3} b x^{5} + a^{4}\right )} \log \left (b x^{5} + a\right )}{15 \,{\left (b^{6} x^{5} + a b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^24/(b*x^5 + a)^2,x, algorithm="fricas")

[Out]

1/15*(b^4*x^20 - 2*a*b^3*x^15 + 6*a^2*b^2*x^10 + 9*a^3*b*x^5 - 3*a^4 - 12*(a^3*b
*x^5 + a^4)*log(b*x^5 + a))/(b^6*x^5 + a*b^5)

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Sympy [A]  time = 2.96634, size = 68, normalized size = 0.94 \[ - \frac{a^{4}}{5 a b^{5} + 5 b^{6} x^{5}} - \frac{4 a^{3} \log{\left (a + b x^{5} \right )}}{5 b^{5}} + \frac{3 a^{2} x^{5}}{5 b^{4}} - \frac{a x^{10}}{5 b^{3}} + \frac{x^{15}}{15 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**24/(b*x**5+a)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.233235, size = 108, normalized size = 1.5 \[ -\frac{4 \, a^{3}{\rm ln}\left ({\left | b x^{5} + a \right |}\right )}{5 \, b^{5}} + \frac{b^{4} x^{15} - 3 \, a b^{3} x^{10} + 9 \, a^{2} b^{2} x^{5}}{15 \, b^{6}} + \frac{4 \, a^{3} b x^{5} + 3 \, a^{4}}{5 \,{\left (b x^{5} + a\right )} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^24/(b*x^5 + a)^2,x, algorithm="giac")

[Out]

-4/5*a^3*ln(abs(b*x^5 + a))/b^5 + 1/15*(b^4*x^15 - 3*a*b^3*x^10 + 9*a^2*b^2*x^5)
/b^6 + 1/5*(4*a^3*b*x^5 + 3*a^4)/((b*x^5 + a)*b^5)