3.2719 \(\int x^{-4-3 p} \left (a+b x^3\right )^p \, dx\)

Optimal. Leaf size=30 \[ -\frac{x^{-3 (p+1)} \left (a+b x^3\right )^{p+1}}{3 a (p+1)} \]

[Out]

-(a + b*x^3)^(1 + p)/(3*a*(1 + p)*x^(3*(1 + p)))

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Rubi [A]  time = 0.0273358, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{x^{-3 (p+1)} \left (a+b x^3\right )^{p+1}}{3 a (p+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^(-4 - 3*p)*(a + b*x^3)^p,x]

[Out]

-(a + b*x^3)^(1 + p)/(3*a*(1 + p)*x^(3*(1 + p)))

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Rubi in Sympy [A]  time = 3.42212, size = 26, normalized size = 0.87 \[ - \frac{x^{- 3 p - 3} \left (a + b x^{3}\right )^{p + 1}}{3 a \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-4-3*p)*(b*x**3+a)**p,x)

[Out]

-x**(-3*p - 3)*(a + b*x**3)**(p + 1)/(3*a*(p + 1))

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Mathematica [A]  time = 0.0449106, size = 30, normalized size = 1. \[ -\frac{x^{-3 p-3} \left (a+b x^3\right )^{p+1}}{3 a p+3 a} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-4 - 3*p)*(a + b*x^3)^p,x]

[Out]

-((x^(-3 - 3*p)*(a + b*x^3)^(1 + p))/(3*a + 3*a*p))

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Maple [A]  time = 0.004, size = 29, normalized size = 1. \[ -{\frac{{x}^{-3-3\,p} \left ( b{x}^{3}+a \right ) ^{1+p}}{3\,a \left ( 1+p \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-4-3*p)*(b*x^3+a)^p,x)

[Out]

-1/3*x^(-3-3*p)*(b*x^3+a)^(1+p)/a/(1+p)

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Maxima [A]  time = 1.38232, size = 50, normalized size = 1.67 \[ -\frac{{\left (b x^{3} + a\right )} e^{\left (p \log \left (b x^{3} + a\right ) - 3 \, p \log \left (x\right )\right )}}{3 \, a{\left (p + 1\right )} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^p*x^(-3*p - 4),x, algorithm="maxima")

[Out]

-1/3*(b*x^3 + a)*e^(p*log(b*x^3 + a) - 3*p*log(x))/(a*(p + 1)*x^3)

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Fricas [A]  time = 0.241068, size = 46, normalized size = 1.53 \[ -\frac{{\left (b x^{4} + a x\right )}{\left (b x^{3} + a\right )}^{p} x^{-3 \, p - 4}}{3 \,{\left (a p + a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^p*x^(-3*p - 4),x, algorithm="fricas")

[Out]

-1/3*(b*x^4 + a*x)*(b*x^3 + a)^p*x^(-3*p - 4)/(a*p + a)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-4-3*p)*(b*x**3+a)**p,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{3} + a\right )}^{p} x^{-3 \, p - 4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^p*x^(-3*p - 4),x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^p*x^(-3*p - 4), x)