3.559 \(\int \frac {1}{x^9 \sqrt [3]{a+b x^3}} \, dx\)

Optimal. Leaf size=68 \[ -\frac {9 b^2 \left (a+b x^3\right )^{2/3}}{40 a^3 x^2}+\frac {3 b \left (a+b x^3\right )^{2/3}}{20 a^2 x^5}-\frac {\left (a+b x^3\right )^{2/3}}{8 a x^8} \]

[Out]

-1/8*(b*x^3+a)^(2/3)/a/x^8+3/20*b*(b*x^3+a)^(2/3)/a^2/x^5-9/40*b^2*(b*x^3+a)^(2/3)/a^3/x^2

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Rubi [A]  time = 0.02, antiderivative size = 68, normalized size of antiderivative = 1.00, 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} \[ -\frac {9 b^2 \left (a+b x^3\right )^{2/3}}{40 a^3 x^2}+\frac {3 b \left (a+b x^3\right )^{2/3}}{20 a^2 x^5}-\frac {\left (a+b x^3\right )^{2/3}}{8 a x^8} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^9*(a + b*x^3)^(1/3)),x]

[Out]

-(a + b*x^3)^(2/3)/(8*a*x^8) + (3*b*(a + b*x^3)^(2/3))/(20*a^2*x^5) - (9*b^2*(a + b*x^3)^(2/3))/(40*a^3*x^2)

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]

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]

Rubi steps

\begin {align*} \int \frac {1}{x^9 \sqrt [3]{a+b x^3}} \, dx &=-\frac {\left (a+b x^3\right )^{2/3}}{8 a x^8}-\frac {(3 b) \int \frac {1}{x^6 \sqrt [3]{a+b x^3}} \, dx}{4 a}\\ &=-\frac {\left (a+b x^3\right )^{2/3}}{8 a x^8}+\frac {3 b \left (a+b x^3\right )^{2/3}}{20 a^2 x^5}+\frac {\left (9 b^2\right ) \int \frac {1}{x^3 \sqrt [3]{a+b x^3}} \, dx}{20 a^2}\\ &=-\frac {\left (a+b x^3\right )^{2/3}}{8 a x^8}+\frac {3 b \left (a+b x^3\right )^{2/3}}{20 a^2 x^5}-\frac {9 b^2 \left (a+b x^3\right )^{2/3}}{40 a^3 x^2}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 42, normalized size = 0.62 \[ -\frac {\left (a+b x^3\right )^{2/3} \left (5 a^2-6 a b x^3+9 b^2 x^6\right )}{40 a^3 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^9*(a + b*x^3)^(1/3)),x]

[Out]

-1/40*((a + b*x^3)^(2/3)*(5*a^2 - 6*a*b*x^3 + 9*b^2*x^6))/(a^3*x^8)

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fricas [A]  time = 0.76, size = 38, normalized size = 0.56 \[ -\frac {{\left (9 \, b^{2} x^{6} - 6 \, a b x^{3} + 5 \, a^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{40 \, a^{3} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(b*x^3+a)^(1/3),x, algorithm="fricas")

[Out]

-1/40*(9*b^2*x^6 - 6*a*b*x^3 + 5*a^2)*(b*x^3 + a)^(2/3)/(a^3*x^8)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} x^{9}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(b*x^3+a)^(1/3),x, algorithm="giac")

[Out]

integrate(1/((b*x^3 + a)^(1/3)*x^9), x)

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maple [A]  time = 0.01, size = 39, normalized size = 0.57 \[ -\frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} \left (9 b^{2} x^{6}-6 a b \,x^{3}+5 a^{2}\right )}{40 a^{3} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^9/(b*x^3+a)^(1/3),x)

[Out]

-1/40*(b*x^3+a)^(2/3)*(9*b^2*x^6-6*a*b*x^3+5*a^2)/x^8/a^3

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maxima [A]  time = 1.31, size = 52, normalized size = 0.76 \[ -\frac {\frac {20 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} b^{2}}{x^{2}} - \frac {16 \, {\left (b x^{3} + a\right )}^{\frac {5}{3}} b}{x^{5}} + \frac {5 \, {\left (b x^{3} + a\right )}^{\frac {8}{3}}}{x^{8}}}{40 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(b*x^3+a)^(1/3),x, algorithm="maxima")

[Out]

-1/40*(20*(b*x^3 + a)^(2/3)*b^2/x^2 - 16*(b*x^3 + a)^(5/3)*b/x^5 + 5*(b*x^3 + a)^(8/3)/x^8)/a^3

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mupad [B]  time = 1.14, size = 38, normalized size = 0.56 \[ -\frac {{\left (b\,x^3+a\right )}^{2/3}\,\left (5\,a^2-6\,a\,b\,x^3+9\,b^2\,x^6\right )}{40\,a^3\,x^8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^9*(a + b*x^3)^(1/3)),x)

[Out]

-((a + b*x^3)^(2/3)*(5*a^2 + 9*b^2*x^6 - 6*a*b*x^3))/(40*a^3*x^8)

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sympy [B]  time = 2.58, size = 406, normalized size = 5.97 \[ \frac {10 a^{4} b^{\frac {14}{3}} \left (\frac {a}{b x^{3}} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{27 a^{5} b^{4} x^{6} \Gamma \left (\frac {1}{3}\right ) + 54 a^{4} b^{5} x^{9} \Gamma \left (\frac {1}{3}\right ) + 27 a^{3} b^{6} x^{12} \Gamma \left (\frac {1}{3}\right )} + \frac {8 a^{3} b^{\frac {17}{3}} x^{3} \left (\frac {a}{b x^{3}} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{27 a^{5} b^{4} x^{6} \Gamma \left (\frac {1}{3}\right ) + 54 a^{4} b^{5} x^{9} \Gamma \left (\frac {1}{3}\right ) + 27 a^{3} b^{6} x^{12} \Gamma \left (\frac {1}{3}\right )} + \frac {4 a^{2} b^{\frac {20}{3}} x^{6} \left (\frac {a}{b x^{3}} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{27 a^{5} b^{4} x^{6} \Gamma \left (\frac {1}{3}\right ) + 54 a^{4} b^{5} x^{9} \Gamma \left (\frac {1}{3}\right ) + 27 a^{3} b^{6} x^{12} \Gamma \left (\frac {1}{3}\right )} + \frac {24 a b^{\frac {23}{3}} x^{9} \left (\frac {a}{b x^{3}} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{27 a^{5} b^{4} x^{6} \Gamma \left (\frac {1}{3}\right ) + 54 a^{4} b^{5} x^{9} \Gamma \left (\frac {1}{3}\right ) + 27 a^{3} b^{6} x^{12} \Gamma \left (\frac {1}{3}\right )} + \frac {18 b^{\frac {26}{3}} x^{12} \left (\frac {a}{b x^{3}} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{27 a^{5} b^{4} x^{6} \Gamma \left (\frac {1}{3}\right ) + 54 a^{4} b^{5} x^{9} \Gamma \left (\frac {1}{3}\right ) + 27 a^{3} b^{6} x^{12} \Gamma \left (\frac {1}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**9/(b*x**3+a)**(1/3),x)

[Out]

10*a**4*b**(14/3)*(a/(b*x**3) + 1)**(2/3)*gamma(-8/3)/(27*a**5*b**4*x**6*gamma(1/3) + 54*a**4*b**5*x**9*gamma(
1/3) + 27*a**3*b**6*x**12*gamma(1/3)) + 8*a**3*b**(17/3)*x**3*(a/(b*x**3) + 1)**(2/3)*gamma(-8/3)/(27*a**5*b**
4*x**6*gamma(1/3) + 54*a**4*b**5*x**9*gamma(1/3) + 27*a**3*b**6*x**12*gamma(1/3)) + 4*a**2*b**(20/3)*x**6*(a/(
b*x**3) + 1)**(2/3)*gamma(-8/3)/(27*a**5*b**4*x**6*gamma(1/3) + 54*a**4*b**5*x**9*gamma(1/3) + 27*a**3*b**6*x*
*12*gamma(1/3)) + 24*a*b**(23/3)*x**9*(a/(b*x**3) + 1)**(2/3)*gamma(-8/3)/(27*a**5*b**4*x**6*gamma(1/3) + 54*a
**4*b**5*x**9*gamma(1/3) + 27*a**3*b**6*x**12*gamma(1/3)) + 18*b**(26/3)*x**12*(a/(b*x**3) + 1)**(2/3)*gamma(-
8/3)/(27*a**5*b**4*x**6*gamma(1/3) + 54*a**4*b**5*x**9*gamma(1/3) + 27*a**3*b**6*x**12*gamma(1/3))

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