3.521 \(\int \frac{\sqrt [3]{a+b x^3}}{x^{11}} \, dx\)

Optimal. Leaf size=68 \[ -\frac{9 b^2 \left (a+b x^3\right )^{4/3}}{140 a^3 x^4}+\frac{3 b \left (a+b x^3\right )^{4/3}}{35 a^2 x^7}-\frac{\left (a+b x^3\right )^{4/3}}{10 a x^{10}} \]

[Out]

-(a + b*x^3)^(4/3)/(10*a*x^10) + (3*b*(a + b*x^3)^(4/3))/(35*a^2*x^7) - (9*b^2*(a + b*x^3)^(4/3))/(140*a^3*x^4
)

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Rubi [A]  time = 0.0194384, 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} \[ -\frac{9 b^2 \left (a+b x^3\right )^{4/3}}{140 a^3 x^4}+\frac{3 b \left (a+b x^3\right )^{4/3}}{35 a^2 x^7}-\frac{\left (a+b x^3\right )^{4/3}}{10 a x^{10}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^(1/3)/x^11,x]

[Out]

-(a + b*x^3)^(4/3)/(10*a*x^10) + (3*b*(a + b*x^3)^(4/3))/(35*a^2*x^7) - (9*b^2*(a + b*x^3)^(4/3))/(140*a^3*x^4
)

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{\sqrt [3]{a+b x^3}}{x^{11}} \, dx &=-\frac{\left (a+b x^3\right )^{4/3}}{10 a x^{10}}-\frac{(3 b) \int \frac{\sqrt [3]{a+b x^3}}{x^8} \, dx}{5 a}\\ &=-\frac{\left (a+b x^3\right )^{4/3}}{10 a x^{10}}+\frac{3 b \left (a+b x^3\right )^{4/3}}{35 a^2 x^7}+\frac{\left (9 b^2\right ) \int \frac{\sqrt [3]{a+b x^3}}{x^5} \, dx}{35 a^2}\\ &=-\frac{\left (a+b x^3\right )^{4/3}}{10 a x^{10}}+\frac{3 b \left (a+b x^3\right )^{4/3}}{35 a^2 x^7}-\frac{9 b^2 \left (a+b x^3\right )^{4/3}}{140 a^3 x^4}\\ \end{align*}

Mathematica [A]  time = 0.0093234, size = 42, normalized size = 0.62 \[ -\frac{\left (a+b x^3\right )^{4/3} \left (14 a^2-12 a b x^3+9 b^2 x^6\right )}{140 a^3 x^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^(1/3)/x^11,x]

[Out]

-((a + b*x^3)^(4/3)*(14*a^2 - 12*a*b*x^3 + 9*b^2*x^6))/(140*a^3*x^10)

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Maple [A]  time = 0.003, size = 39, normalized size = 0.6 \begin{align*} -{\frac{9\,{b}^{2}{x}^{6}-12\,{x}^{3}ab+14\,{a}^{2}}{140\,{x}^{10}{a}^{3}} \left ( b{x}^{3}+a \right ) ^{{\frac{4}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^(1/3)/x^11,x)

[Out]

-1/140*(b*x^3+a)^(4/3)*(9*b^2*x^6-12*a*b*x^3+14*a^2)/x^10/a^3

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Maxima [A]  time = 0.988183, size = 70, normalized size = 1.03 \begin{align*} -\frac{\frac{35 \,{\left (b x^{3} + a\right )}^{\frac{4}{3}} b^{2}}{x^{4}} - \frac{40 \,{\left (b x^{3} + a\right )}^{\frac{7}{3}} b}{x^{7}} + \frac{14 \,{\left (b x^{3} + a\right )}^{\frac{10}{3}}}{x^{10}}}{140 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(1/3)/x^11,x, algorithm="maxima")

[Out]

-1/140*(35*(b*x^3 + a)^(4/3)*b^2/x^4 - 40*(b*x^3 + a)^(7/3)*b/x^7 + 14*(b*x^3 + a)^(10/3)/x^10)/a^3

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Fricas [A]  time = 2.01224, size = 116, normalized size = 1.71 \begin{align*} -\frac{{\left (9 \, b^{3} x^{9} - 3 \, a b^{2} x^{6} + 2 \, a^{2} b x^{3} + 14 \, a^{3}\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{140 \, a^{3} x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(1/3)/x^11,x, algorithm="fricas")

[Out]

-1/140*(9*b^3*x^9 - 3*a*b^2*x^6 + 2*a^2*b*x^3 + 14*a^3)*(b*x^3 + a)^(1/3)/(a^3*x^10)

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Sympy [B]  time = 2.53442, size = 520, normalized size = 7.65 \begin{align*} \frac{28 a^{5} b^{\frac{13}{3}} \sqrt [3]{\frac{a}{b x^{3}} + 1} \Gamma \left (- \frac{10}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac{1}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac{1}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac{1}{3}\right )} + \frac{60 a^{4} b^{\frac{16}{3}} x^{3} \sqrt [3]{\frac{a}{b x^{3}} + 1} \Gamma \left (- \frac{10}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac{1}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac{1}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac{1}{3}\right )} + \frac{30 a^{3} b^{\frac{19}{3}} x^{6} \sqrt [3]{\frac{a}{b x^{3}} + 1} \Gamma \left (- \frac{10}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac{1}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac{1}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac{1}{3}\right )} + \frac{10 a^{2} b^{\frac{22}{3}} x^{9} \sqrt [3]{\frac{a}{b x^{3}} + 1} \Gamma \left (- \frac{10}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac{1}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac{1}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac{1}{3}\right )} + \frac{30 a b^{\frac{25}{3}} x^{12} \sqrt [3]{\frac{a}{b x^{3}} + 1} \Gamma \left (- \frac{10}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac{1}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac{1}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac{1}{3}\right )} + \frac{18 b^{\frac{28}{3}} x^{15} \sqrt [3]{\frac{a}{b x^{3}} + 1} \Gamma \left (- \frac{10}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac{1}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac{1}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac{1}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**(1/3)/x**11,x)

[Out]

28*a**5*b**(13/3)*(a/(b*x**3) + 1)**(1/3)*gamma(-10/3)/(27*a**5*b**4*x**9*gamma(-1/3) + 54*a**4*b**5*x**12*gam
ma(-1/3) + 27*a**3*b**6*x**15*gamma(-1/3)) + 60*a**4*b**(16/3)*x**3*(a/(b*x**3) + 1)**(1/3)*gamma(-10/3)/(27*a
**5*b**4*x**9*gamma(-1/3) + 54*a**4*b**5*x**12*gamma(-1/3) + 27*a**3*b**6*x**15*gamma(-1/3)) + 30*a**3*b**(19/
3)*x**6*(a/(b*x**3) + 1)**(1/3)*gamma(-10/3)/(27*a**5*b**4*x**9*gamma(-1/3) + 54*a**4*b**5*x**12*gamma(-1/3) +
27*a**3*b**6*x**15*gamma(-1/3)) + 10*a**2*b**(22/3)*x**9*(a/(b*x**3) + 1)**(1/3)*gamma(-10/3)/(27*a**5*b**4*x
**9*gamma(-1/3) + 54*a**4*b**5*x**12*gamma(-1/3) + 27*a**3*b**6*x**15*gamma(-1/3)) + 30*a*b**(25/3)*x**12*(a/(
b*x**3) + 1)**(1/3)*gamma(-10/3)/(27*a**5*b**4*x**9*gamma(-1/3) + 54*a**4*b**5*x**12*gamma(-1/3) + 27*a**3*b**
6*x**15*gamma(-1/3)) + 18*b**(28/3)*x**15*(a/(b*x**3) + 1)**(1/3)*gamma(-10/3)/(27*a**5*b**4*x**9*gamma(-1/3)
+ 54*a**4*b**5*x**12*gamma(-1/3) + 27*a**3*b**6*x**15*gamma(-1/3))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x^{11}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(1/3)/x^11,x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(1/3)/x^11, x)