3.1277 \(\int \frac{x^{14}}{\left (a+b x^5\right )^2} \, dx\)

Optimal. Leaf size=46 \[ -\frac{a^2}{5 b^3 \left (a+b x^5\right )}-\frac{2 a \log \left (a+b x^5\right )}{5 b^3}+\frac{x^5}{5 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0771601, antiderivative size = 46, 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 \[ -\frac{a^2}{5 b^3 \left (a+b x^5\right )}-\frac{2 a \log \left (a+b x^5\right )}{5 b^3}+\frac{x^5}{5 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^14/(a + b*x^5)^2,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2}}{5 b^{3} \left (a + b x^{5}\right )} - \frac{2 a \log{\left (a + b x^{5} \right )}}{5 b^{3}} + \frac{\int ^{x^{5}} \frac{1}{b^{2}}\, dx}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**14/(b*x**5+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0279227, size = 38, normalized size = 0.83 \[ \frac{-\frac{a^2}{a+b x^5}-2 a \log \left (a+b x^5\right )+b x^5}{5 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^14/(a + b*x^5)^2,x]

[Out]

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

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Maple [A]  time = 0.008, size = 41, normalized size = 0.9 \[{\frac{{x}^{5}}{5\,{b}^{2}}}-{\frac{{a}^{2}}{5\,{b}^{3} \left ( b{x}^{5}+a \right ) }}-{\frac{2\,a\ln \left ( b{x}^{5}+a \right ) }{5\,{b}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^14/(b*x^5+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.43658, size = 58, normalized size = 1.26 \[ \frac{x^{5}}{5 \, b^{2}} - \frac{a^{2}}{5 \,{\left (b^{4} x^{5} + a b^{3}\right )}} - \frac{2 \, a \log \left (b x^{5} + a\right )}{5 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/(b*x^5 + a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.212215, size = 76, normalized size = 1.65 \[ \frac{b^{2} x^{10} + a b x^{5} - a^{2} - 2 \,{\left (a b x^{5} + a^{2}\right )} \log \left (b x^{5} + a\right )}{5 \,{\left (b^{4} x^{5} + a b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/(b*x^5 + a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.78886, size = 42, normalized size = 0.91 \[ - \frac{a^{2}}{5 a b^{3} + 5 b^{4} x^{5}} - \frac{2 a \log{\left (a + b x^{5} \right )}}{5 b^{3}} + \frac{x^{5}}{5 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**14/(b*x**5+a)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.229926, size = 66, normalized size = 1.43 \[ \frac{x^{5}}{5 \, b^{2}} - \frac{2 \, a{\rm ln}\left ({\left | b x^{5} + a \right |}\right )}{5 \, b^{3}} + \frac{2 \, a b x^{5} + a^{2}}{5 \,{\left (b x^{5} + a\right )} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/(b*x^5 + a)^2,x, algorithm="giac")

[Out]

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