3.1266 \(\int \frac{x^{19}}{a+b x^5} \, dx\)

Optimal. Leaf size=53 \[ -\frac{a^3 \log \left (a+b x^5\right )}{5 b^4}+\frac{a^2 x^5}{5 b^3}-\frac{a x^{10}}{10 b^2}+\frac{x^{15}}{15 b} \]

[Out]

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

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Rubi [A]  time = 0.0772589, antiderivative size = 53, 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 \[ -\frac{a^3 \log \left (a+b x^5\right )}{5 b^4}+\frac{a^2 x^5}{5 b^3}-\frac{a x^{10}}{10 b^2}+\frac{x^{15}}{15 b} \]

Antiderivative was successfully verified.

[In]  Int[x^19/(a + b*x^5),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**19/(b*x**5+a),x)

[Out]

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

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Mathematica [A]  time = 0.00960077, size = 53, normalized size = 1. \[ -\frac{a^3 \log \left (a+b x^5\right )}{5 b^4}+\frac{a^2 x^5}{5 b^3}-\frac{a x^{10}}{10 b^2}+\frac{x^{15}}{15 b} \]

Antiderivative was successfully verified.

[In]  Integrate[x^19/(a + b*x^5),x]

[Out]

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

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Maple [A]  time = 0.004, size = 46, normalized size = 0.9 \[{\frac{{x}^{5}{a}^{2}}{5\,{b}^{3}}}-{\frac{a{x}^{10}}{10\,{b}^{2}}}+{\frac{{x}^{15}}{15\,b}}-{\frac{{a}^{3}\ln \left ( b{x}^{5}+a \right ) }{5\,{b}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^19/(b*x^5+a),x)

[Out]

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

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Maxima [A]  time = 1.4278, size = 62, normalized size = 1.17 \[ -\frac{a^{3} \log \left (b x^{5} + a\right )}{5 \, b^{4}} + \frac{2 \, b^{2} x^{15} - 3 \, a b x^{10} + 6 \, a^{2} x^{5}}{30 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^19/(b*x^5 + a),x, algorithm="maxima")

[Out]

-1/5*a^3*log(b*x^5 + a)/b^4 + 1/30*(2*b^2*x^15 - 3*a*b*x^10 + 6*a^2*x^5)/b^3

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Fricas [A]  time = 0.215513, size = 61, normalized size = 1.15 \[ \frac{2 \, b^{3} x^{15} - 3 \, a b^{2} x^{10} + 6 \, a^{2} b x^{5} - 6 \, a^{3} \log \left (b x^{5} + a\right )}{30 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^19/(b*x^5 + a),x, algorithm="fricas")

[Out]

1/30*(2*b^3*x^15 - 3*a*b^2*x^10 + 6*a^2*b*x^5 - 6*a^3*log(b*x^5 + a))/b^4

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Sympy [A]  time = 1.51254, size = 44, normalized size = 0.83 \[ - \frac{a^{3} \log{\left (a + b x^{5} \right )}}{5 b^{4}} + \frac{a^{2} x^{5}}{5 b^{3}} - \frac{a x^{10}}{10 b^{2}} + \frac{x^{15}}{15 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**19/(b*x**5+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.227641, size = 63, normalized size = 1.19 \[ -\frac{a^{3}{\rm ln}\left ({\left | b x^{5} + a \right |}\right )}{5 \, b^{4}} + \frac{2 \, b^{2} x^{15} - 3 \, a b x^{10} + 6 \, a^{2} x^{5}}{30 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^19/(b*x^5 + a),x, algorithm="giac")

[Out]

-1/5*a^3*ln(abs(b*x^5 + a))/b^4 + 1/30*(2*b^2*x^15 - 3*a*b*x^10 + 6*a^2*x^5)/b^3