3.318 \(\int \frac{x^8}{a+b x^3} \, dx\)

Optimal. Leaf size=40 \[ \frac{a^2 \log \left (a+b x^3\right )}{3 b^3}-\frac{a x^3}{3 b^2}+\frac{x^6}{6 b} \]

[Out]

-(a*x^3)/(3*b^2) + x^6/(6*b) + (a^2*Log[a + b*x^3])/(3*b^3)

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Rubi [A]  time = 0.0643188, 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 \[ \frac{a^2 \log \left (a+b x^3\right )}{3 b^3}-\frac{a x^3}{3 b^2}+\frac{x^6}{6 b} \]

Antiderivative was successfully verified.

[In]  Int[x^8/(a + b*x^3),x]

[Out]

-(a*x^3)/(3*b^2) + x^6/(6*b) + (a^2*Log[a + b*x^3])/(3*b^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(b*x**3+a),x)

[Out]

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

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Mathematica [A]  time = 0.00997771, size = 40, normalized size = 1. \[ \frac{a^2 \log \left (a+b x^3\right )}{3 b^3}-\frac{a x^3}{3 b^2}+\frac{x^6}{6 b} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/(a + b*x^3),x]

[Out]

-(a*x^3)/(3*b^2) + x^6/(6*b) + (a^2*Log[a + b*x^3])/(3*b^3)

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Maple [A]  time = 0.004, size = 35, normalized size = 0.9 \[ -{\frac{a{x}^{3}}{3\,{b}^{2}}}+{\frac{{x}^{6}}{6\,b}}+{\frac{{a}^{2}\ln \left ( b{x}^{3}+a \right ) }{3\,{b}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(b*x^3+a),x)

[Out]

-1/3*a*x^3/b^2+1/6*x^6/b+1/3*a^2*ln(b*x^3+a)/b^3

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Maxima [A]  time = 1.44188, size = 46, normalized size = 1.15 \[ \frac{a^{2} \log \left (b x^{3} + a\right )}{3 \, b^{3}} + \frac{b x^{6} - 2 \, a x^{3}}{6 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(b*x^3 + a),x, algorithm="maxima")

[Out]

1/3*a^2*log(b*x^3 + a)/b^3 + 1/6*(b*x^6 - 2*a*x^3)/b^2

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Fricas [A]  time = 0.209634, size = 45, normalized size = 1.12 \[ \frac{b^{2} x^{6} - 2 \, a b x^{3} + 2 \, a^{2} \log \left (b x^{3} + a\right )}{6 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(b*x^3 + a),x, algorithm="fricas")

[Out]

1/6*(b^2*x^6 - 2*a*b*x^3 + 2*a^2*log(b*x^3 + a))/b^3

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Sympy [A]  time = 1.29637, size = 32, normalized size = 0.8 \[ \frac{a^{2} \log{\left (a + b x^{3} \right )}}{3 b^{3}} - \frac{a x^{3}}{3 b^{2}} + \frac{x^{6}}{6 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(b*x**3+a),x)

[Out]

a**2*log(a + b*x**3)/(3*b**3) - a*x**3/(3*b**2) + x**6/(6*b)

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GIAC/XCAS [A]  time = 0.225013, size = 47, normalized size = 1.18 \[ \frac{a^{2}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} + \frac{b x^{6} - 2 \, a x^{3}}{6 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(b*x^3 + a),x, algorithm="giac")

[Out]

1/3*a^2*ln(abs(b*x^3 + a))/b^3 + 1/6*(b*x^6 - 2*a*x^3)/b^2