3.1 \(\int (a x^3+b x^6)^{5/3} \, dx\)

Optimal. Leaf size=52 \[ \frac{\left (a x^3+b x^6\right )^{8/3}}{11 b x^5}-\frac{3 a \left (a x^3+b x^6\right )^{8/3}}{88 b^2 x^8} \]

[Out]

(-3*a*(a*x^3 + b*x^6)^(8/3))/(88*b^2*x^8) + (a*x^3 + b*x^6)^(8/3)/(11*b*x^5)

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Rubi [A]  time = 0.0482703, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2002, 2014} \[ \frac{\left (a x^3+b x^6\right )^{8/3}}{11 b x^5}-\frac{3 a \left (a x^3+b x^6\right )^{8/3}}{88 b^2 x^8} \]

Antiderivative was successfully verified.

[In]

Int[(a*x^3 + b*x^6)^(5/3),x]

[Out]

(-3*a*(a*x^3 + b*x^6)^(8/3))/(88*b^2*x^8) + (a*x^3 + b*x^6)^(8/3)/(11*b*x^5)

Rule 2002

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + b*x^n)^(p + 1)/(a*(j*p + 1)*x^(j -
1)), x] - Dist[(b*(n*p + n - j + 1))/(a*(j*p + 1)), Int[x^(n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, j,
n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && ILtQ[Simplify[(n*p + n - j + 1)/(n - j)], 0] && NeQ[j*p + 1, 0]

Rule 2014

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

Rubi steps

\begin{align*} \int \left (a x^3+b x^6\right )^{5/3} \, dx &=\frac{\left (a x^3+b x^6\right )^{8/3}}{11 b x^5}-\frac{(3 a) \int \frac{\left (a x^3+b x^6\right )^{5/3}}{x^3} \, dx}{11 b}\\ &=-\frac{3 a \left (a x^3+b x^6\right )^{8/3}}{88 b^2 x^8}+\frac{\left (a x^3+b x^6\right )^{8/3}}{11 b x^5}\\ \end{align*}

Mathematica [A]  time = 0.0248177, size = 42, normalized size = 0.81 \[ \frac{x \left (a+b x^3\right )^3 \left (8 b x^3-3 a\right )}{88 b^2 \sqrt [3]{x^3 \left (a+b x^3\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*x^3 + b*x^6)^(5/3),x]

[Out]

(x*(a + b*x^3)^3*(-3*a + 8*b*x^3))/(88*b^2*(x^3*(a + b*x^3))^(1/3))

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Maple [A]  time = 0.029, size = 39, normalized size = 0.8 \begin{align*} -{\frac{ \left ( b{x}^{3}+a \right ) \left ( -8\,b{x}^{3}+3\,a \right ) }{88\,{b}^{2}{x}^{5}} \left ( b{x}^{6}+a{x}^{3} \right ) ^{{\frac{5}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^6+a*x^3)^(5/3),x)

[Out]

-1/88*(b*x^3+a)*(-8*b*x^3+3*a)*(b*x^6+a*x^3)^(5/3)/b^2/x^5

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Maxima [A]  time = 1.02154, size = 62, normalized size = 1.19 \begin{align*} \frac{{\left (8 \, b^{3} x^{9} + 13 \, a b^{2} x^{6} + 2 \, a^{2} b x^{3} - 3 \, a^{3}\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{88 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^6+a*x^3)^(5/3),x, algorithm="maxima")

[Out]

1/88*(8*b^3*x^9 + 13*a*b^2*x^6 + 2*a^2*b*x^3 - 3*a^3)*(b*x^3 + a)^(2/3)/b^2

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Fricas [A]  time = 2.20542, size = 117, normalized size = 2.25 \begin{align*} \frac{{\left (8 \, b^{3} x^{9} + 13 \, a b^{2} x^{6} + 2 \, a^{2} b x^{3} - 3 \, a^{3}\right )}{\left (b x^{6} + a x^{3}\right )}^{\frac{2}{3}}}{88 \, b^{2} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^6+a*x^3)^(5/3),x, algorithm="fricas")

[Out]

1/88*(8*b^3*x^9 + 13*a*b^2*x^6 + 2*a^2*b*x^3 - 3*a^3)*(b*x^6 + a*x^3)^(2/3)/(b^2*x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a x^{3} + b x^{6}\right )^{\frac{5}{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**6+a*x**3)**(5/3),x)

[Out]

Integral((a*x**3 + b*x**6)**(5/3), x)

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Giac [A]  time = 1.14666, size = 107, normalized size = 2.06 \begin{align*} \frac{\frac{11 \,{\left (5 \,{\left (b x^{3} + a\right )}^{\frac{8}{3}} - 8 \,{\left (b x^{3} + a\right )}^{\frac{5}{3}} a\right )} a}{b} + \frac{2 \,{\left (20 \,{\left (b x^{3} + a\right )}^{\frac{11}{3}} - 55 \,{\left (b x^{3} + a\right )}^{\frac{8}{3}} a + 44 \,{\left (b x^{3} + a\right )}^{\frac{5}{3}} a^{2}\right )}}{b}}{440 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^6+a*x^3)^(5/3),x, algorithm="giac")

[Out]

1/440*(11*(5*(b*x^3 + a)^(8/3) - 8*(b*x^3 + a)^(5/3)*a)*a/b + 2*(20*(b*x^3 + a)^(11/3) - 55*(b*x^3 + a)^(8/3)*
a + 44*(b*x^3 + a)^(5/3)*a^2)/b)/b