3.2726 \(\int x^{-4-3 p} (a+b x^3)^p \, dx\)

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)))

________________________________________________________________________________________

Rubi [A]  time = 0.0076007, 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, Rules used = {264} \[ -\frac{x^{-3 (p+1)} \left (a+b x^3\right )^{p+1}}{3 a (p+1)} \]

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)))

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int x^{-4-3 p} \left (a+b x^3\right )^p \, dx &=-\frac{x^{-3 (1+p)} \left (a+b x^3\right )^{1+p}}{3 a (1+p)}\\ \end{align*}

Mathematica [A]  time = 0.0135981, size = 29, normalized size = 0.97 \[ \frac{x^{-3 p-3} \left (a+b x^3\right )^{p+1}}{a (-3 p-3)} \]

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))/(a*(-3 - 3*p))

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 29, normalized size = 1. \begin{align*} -{\frac{{x}^{-3-3\,p} \left ( b{x}^{3}+a \right ) ^{1+p}}{3\,a \left ( 1+p \right ) }} \end{align*}

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)

________________________________________________________________________________________

Maxima [A]  time = 0.964313, size = 50, normalized size = 1.67 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

Fricas [A]  time = 1.37221, size = 77, normalized size = 2.57 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{3} + a\right )}^{p} x^{-3 \, p - 4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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