3.542 \(\int \frac{(a+b x^3)^{2/3}}{x^{12}} \, dx\)

Optimal. Leaf size=68 \[ -\frac{9 b^2 \left (a+b x^3\right )^{5/3}}{220 a^3 x^5}+\frac{3 b \left (a+b x^3\right )^{5/3}}{44 a^2 x^8}-\frac{\left (a+b x^3\right )^{5/3}}{11 a x^{11}} \]

[Out]

-(a + b*x^3)^(5/3)/(11*a*x^11) + (3*b*(a + b*x^3)^(5/3))/(44*a^2*x^8) - (9*b^2*(a + b*x^3)^(5/3))/(220*a^3*x^5
)

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Rubi [A]  time = 0.0191239, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 264} \[ -\frac{9 b^2 \left (a+b x^3\right )^{5/3}}{220 a^3 x^5}+\frac{3 b \left (a+b x^3\right )^{5/3}}{44 a^2 x^8}-\frac{\left (a+b x^3\right )^{5/3}}{11 a x^{11}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^(2/3)/x^12,x]

[Out]

-(a + b*x^3)^(5/3)/(11*a*x^11) + (3*b*(a + b*x^3)^(5/3))/(44*a^2*x^8) - (9*b^2*(a + b*x^3)^(5/3))/(220*a^3*x^5
)

Rule 271

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

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 \frac{\left (a+b x^3\right )^{2/3}}{x^{12}} \, dx &=-\frac{\left (a+b x^3\right )^{5/3}}{11 a x^{11}}-\frac{(6 b) \int \frac{\left (a+b x^3\right )^{2/3}}{x^9} \, dx}{11 a}\\ &=-\frac{\left (a+b x^3\right )^{5/3}}{11 a x^{11}}+\frac{3 b \left (a+b x^3\right )^{5/3}}{44 a^2 x^8}+\frac{\left (9 b^2\right ) \int \frac{\left (a+b x^3\right )^{2/3}}{x^6} \, dx}{44 a^2}\\ &=-\frac{\left (a+b x^3\right )^{5/3}}{11 a x^{11}}+\frac{3 b \left (a+b x^3\right )^{5/3}}{44 a^2 x^8}-\frac{9 b^2 \left (a+b x^3\right )^{5/3}}{220 a^3 x^5}\\ \end{align*}

Mathematica [A]  time = 0.0101776, size = 42, normalized size = 0.62 \[ -\frac{\left (a+b x^3\right )^{5/3} \left (20 a^2-15 a b x^3+9 b^2 x^6\right )}{220 a^3 x^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^(2/3)/x^12,x]

[Out]

-((a + b*x^3)^(5/3)*(20*a^2 - 15*a*b*x^3 + 9*b^2*x^6))/(220*a^3*x^11)

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Maple [A]  time = 0.004, size = 39, normalized size = 0.6 \begin{align*} -{\frac{9\,{b}^{2}{x}^{6}-15\,{x}^{3}ab+20\,{a}^{2}}{220\,{x}^{11}{a}^{3}} \left ( b{x}^{3}+a \right ) ^{{\frac{5}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^(2/3)/x^12,x)

[Out]

-1/220*(b*x^3+a)^(5/3)*(9*b^2*x^6-15*a*b*x^3+20*a^2)/x^11/a^3

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Maxima [A]  time = 1.40869, size = 70, normalized size = 1.03 \begin{align*} -\frac{\frac{44 \,{\left (b x^{3} + a\right )}^{\frac{5}{3}} b^{2}}{x^{5}} - \frac{55 \,{\left (b x^{3} + a\right )}^{\frac{8}{3}} b}{x^{8}} + \frac{20 \,{\left (b x^{3} + a\right )}^{\frac{11}{3}}}{x^{11}}}{220 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(2/3)/x^12,x, algorithm="maxima")

[Out]

-1/220*(44*(b*x^3 + a)^(5/3)*b^2/x^5 - 55*(b*x^3 + a)^(8/3)*b/x^8 + 20*(b*x^3 + a)^(11/3)/x^11)/a^3

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Fricas [A]  time = 1.77188, size = 116, normalized size = 1.71 \begin{align*} -\frac{{\left (9 \, b^{3} x^{9} - 6 \, a b^{2} x^{6} + 5 \, a^{2} b x^{3} + 20 \, a^{3}\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{220 \, a^{3} x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(2/3)/x^12,x, algorithm="fricas")

[Out]

-1/220*(9*b^3*x^9 - 6*a*b^2*x^6 + 5*a^2*b*x^3 + 20*a^3)*(b*x^3 + a)^(2/3)/(a^3*x^11)

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Sympy [B]  time = 3.41313, size = 520, normalized size = 7.65 \begin{align*} \frac{40 a^{5} b^{\frac{14}{3}} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{11}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac{2}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac{2}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac{2}{3}\right )} + \frac{90 a^{4} b^{\frac{17}{3}} x^{3} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{11}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac{2}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac{2}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac{2}{3}\right )} + \frac{48 a^{3} b^{\frac{20}{3}} x^{6} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{11}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac{2}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac{2}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac{2}{3}\right )} + \frac{4 a^{2} b^{\frac{23}{3}} x^{9} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{11}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac{2}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac{2}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac{2}{3}\right )} + \frac{24 a b^{\frac{26}{3}} x^{12} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{11}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac{2}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac{2}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac{2}{3}\right )} + \frac{18 b^{\frac{29}{3}} x^{15} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{11}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac{2}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac{2}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac{2}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**(2/3)/x**12,x)

[Out]

40*a**5*b**(14/3)*(a/(b*x**3) + 1)**(2/3)*gamma(-11/3)/(27*a**5*b**4*x**9*gamma(-2/3) + 54*a**4*b**5*x**12*gam
ma(-2/3) + 27*a**3*b**6*x**15*gamma(-2/3)) + 90*a**4*b**(17/3)*x**3*(a/(b*x**3) + 1)**(2/3)*gamma(-11/3)/(27*a
**5*b**4*x**9*gamma(-2/3) + 54*a**4*b**5*x**12*gamma(-2/3) + 27*a**3*b**6*x**15*gamma(-2/3)) + 48*a**3*b**(20/
3)*x**6*(a/(b*x**3) + 1)**(2/3)*gamma(-11/3)/(27*a**5*b**4*x**9*gamma(-2/3) + 54*a**4*b**5*x**12*gamma(-2/3) +
27*a**3*b**6*x**15*gamma(-2/3)) + 4*a**2*b**(23/3)*x**9*(a/(b*x**3) + 1)**(2/3)*gamma(-11/3)/(27*a**5*b**4*x*
*9*gamma(-2/3) + 54*a**4*b**5*x**12*gamma(-2/3) + 27*a**3*b**6*x**15*gamma(-2/3)) + 24*a*b**(26/3)*x**12*(a/(b
*x**3) + 1)**(2/3)*gamma(-11/3)/(27*a**5*b**4*x**9*gamma(-2/3) + 54*a**4*b**5*x**12*gamma(-2/3) + 27*a**3*b**6
*x**15*gamma(-2/3)) + 18*b**(29/3)*x**15*(a/(b*x**3) + 1)**(2/3)*gamma(-11/3)/(27*a**5*b**4*x**9*gamma(-2/3) +
54*a**4*b**5*x**12*gamma(-2/3) + 27*a**3*b**6*x**15*gamma(-2/3))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{x^{12}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(2/3)/x^12,x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(2/3)/x^12, x)