3.6.72 \(\int \frac {1}{x^8 (a+b x^3)^{2/3}} \, dx\) [572]

Optimal. Leaf size=68 \[ -\frac {\sqrt [3]{a+b x^3}}{7 a x^7}+\frac {3 b \sqrt [3]{a+b x^3}}{14 a^2 x^4}-\frac {9 b^2 \sqrt [3]{a+b x^3}}{14 a^3 x} \]

[Out]

-1/7*(b*x^3+a)^(1/3)/a/x^7+3/14*b*(b*x^3+a)^(1/3)/a^2/x^4-9/14*b^2*(b*x^3+a)^(1/3)/a^3/x

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Rubi [A]
time = 0.01, 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 = {277, 270} \begin {gather*} -\frac {9 b^2 \sqrt [3]{a+b x^3}}{14 a^3 x}+\frac {3 b \sqrt [3]{a+b x^3}}{14 a^2 x^4}-\frac {\sqrt [3]{a+b x^3}}{7 a x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^8*(a + b*x^3)^(2/3)),x]

[Out]

-1/7*(a + b*x^3)^(1/3)/(a*x^7) + (3*b*(a + b*x^3)^(1/3))/(14*a^2*x^4) - (9*b^2*(a + b*x^3)^(1/3))/(14*a^3*x)

Rule 270

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 277

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^8 \left (a+b x^3\right )^{2/3}} \, dx &=-\frac {\sqrt [3]{a+b x^3}}{7 a x^7}-\frac {(6 b) \int \frac {1}{x^5 \left (a+b x^3\right )^{2/3}} \, dx}{7 a}\\ &=-\frac {\sqrt [3]{a+b x^3}}{7 a x^7}+\frac {3 b \sqrt [3]{a+b x^3}}{14 a^2 x^4}+\frac {\left (9 b^2\right ) \int \frac {1}{x^2 \left (a+b x^3\right )^{2/3}} \, dx}{14 a^2}\\ &=-\frac {\sqrt [3]{a+b x^3}}{7 a x^7}+\frac {3 b \sqrt [3]{a+b x^3}}{14 a^2 x^4}-\frac {9 b^2 \sqrt [3]{a+b x^3}}{14 a^3 x}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 42, normalized size = 0.62 \begin {gather*} \frac {\sqrt [3]{a+b x^3} \left (-2 a^2+3 a b x^3-9 b^2 x^6\right )}{14 a^3 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^8*(a + b*x^3)^(2/3)),x]

[Out]

((a + b*x^3)^(1/3)*(-2*a^2 + 3*a*b*x^3 - 9*b^2*x^6))/(14*a^3*x^7)

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Maple [A]
time = 0.14, size = 39, normalized size = 0.57

method	result	size
gosper	\(-\frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}} \left (9 b^{2} x^{6}-3 a b \,x^{3}+2 a^{2}\right )}{14 a^{3} x^{7}}\)	\(39\)
trager	\(-\frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}} \left (9 b^{2} x^{6}-3 a b \,x^{3}+2 a^{2}\right )}{14 a^{3} x^{7}}\)	\(39\)
risch	\(-\frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}} \left (9 b^{2} x^{6}-3 a b \,x^{3}+2 a^{2}\right )}{14 a^{3} x^{7}}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^8/(b*x^3+a)^(2/3),x,method=_RETURNVERBOSE)

[Out]

-1/14*(b*x^3+a)^(1/3)*(9*b^2*x^6-3*a*b*x^3+2*a^2)/a^3/x^7

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Maxima [A]
time = 0.29, size = 52, normalized size = 0.76 \begin {gather*} -\frac {\frac {14 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{2}}{x} - \frac {7 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}} b}{x^{4}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {7}{3}}}{x^{7}}}{14 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(b*x^3+a)^(2/3),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.36, size = 38, normalized size = 0.56 \begin {gather*} -\frac {{\left (9 \, b^{2} x^{6} - 3 \, a b x^{3} + 2 \, a^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{14 \, a^{3} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(b*x^3+a)^(2/3),x, algorithm="fricas")

[Out]

-1/14*(9*b^2*x^6 - 3*a*b*x^3 + 2*a^2)*(b*x^3 + a)^(1/3)/(a^3*x^7)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 406 vs. \(2 (60) = 120\).
time = 0.70, size = 406, normalized size = 5.97 \begin {gather*} \frac {4 a^{4} b^{\frac {13}{3}} \sqrt [3]{\frac {a}{b x^{3}} + 1} \Gamma \left (- \frac {7}{3}\right )}{27 a^{5} b^{4} x^{6} \Gamma \left (\frac {2}{3}\right ) + 54 a^{4} b^{5} x^{9} \Gamma \left (\frac {2}{3}\right ) + 27 a^{3} b^{6} x^{12} \Gamma \left (\frac {2}{3}\right )} + \frac {2 a^{3} b^{\frac {16}{3}} x^{3} \sqrt [3]{\frac {a}{b x^{3}} + 1} \Gamma \left (- \frac {7}{3}\right )}{27 a^{5} b^{4} x^{6} \Gamma \left (\frac {2}{3}\right ) + 54 a^{4} b^{5} x^{9} \Gamma \left (\frac {2}{3}\right ) + 27 a^{3} b^{6} x^{12} \Gamma \left (\frac {2}{3}\right )} + \frac {10 a^{2} b^{\frac {19}{3}} x^{6} \sqrt [3]{\frac {a}{b x^{3}} + 1} \Gamma \left (- \frac {7}{3}\right )}{27 a^{5} b^{4} x^{6} \Gamma \left (\frac {2}{3}\right ) + 54 a^{4} b^{5} x^{9} \Gamma \left (\frac {2}{3}\right ) + 27 a^{3} b^{6} x^{12} \Gamma \left (\frac {2}{3}\right )} + \frac {30 a b^{\frac {22}{3}} x^{9} \sqrt [3]{\frac {a}{b x^{3}} + 1} \Gamma \left (- \frac {7}{3}\right )}{27 a^{5} b^{4} x^{6} \Gamma \left (\frac {2}{3}\right ) + 54 a^{4} b^{5} x^{9} \Gamma \left (\frac {2}{3}\right ) + 27 a^{3} b^{6} x^{12} \Gamma \left (\frac {2}{3}\right )} + \frac {18 b^{\frac {25}{3}} x^{12} \sqrt [3]{\frac {a}{b x^{3}} + 1} \Gamma \left (- \frac {7}{3}\right )}{27 a^{5} b^{4} x^{6} \Gamma \left (\frac {2}{3}\right ) + 54 a^{4} b^{5} x^{9} \Gamma \left (\frac {2}{3}\right ) + 27 a^{3} b^{6} x^{12} \Gamma \left (\frac {2}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**8/(b*x**3+a)**(2/3),x)

[Out]

4*a**4*b**(13/3)*(a/(b*x**3) + 1)**(1/3)*gamma(-7/3)/(27*a**5*b**4*x**6*gamma(2/3) + 54*a**4*b**5*x**9*gamma(2
/3) + 27*a**3*b**6*x**12*gamma(2/3)) + 2*a**3*b**(16/3)*x**3*(a/(b*x**3) + 1)**(1/3)*gamma(-7/3)/(27*a**5*b**4
*x**6*gamma(2/3) + 54*a**4*b**5*x**9*gamma(2/3) + 27*a**3*b**6*x**12*gamma(2/3)) + 10*a**2*b**(19/3)*x**6*(a/(
b*x**3) + 1)**(1/3)*gamma(-7/3)/(27*a**5*b**4*x**6*gamma(2/3) + 54*a**4*b**5*x**9*gamma(2/3) + 27*a**3*b**6*x*
*12*gamma(2/3)) + 30*a*b**(22/3)*x**9*(a/(b*x**3) + 1)**(1/3)*gamma(-7/3)/(27*a**5*b**4*x**6*gamma(2/3) + 54*a
**4*b**5*x**9*gamma(2/3) + 27*a**3*b**6*x**12*gamma(2/3)) + 18*b**(25/3)*x**12*(a/(b*x**3) + 1)**(1/3)*gamma(-
7/3)/(27*a**5*b**4*x**6*gamma(2/3) + 54*a**4*b**5*x**9*gamma(2/3) + 27*a**3*b**6*x**12*gamma(2/3))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(b*x^3+a)^(2/3),x, algorithm="giac")

[Out]

integrate(1/((b*x^3 + a)^(2/3)*x^8), x)

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Mupad [B]
time = 1.15, size = 38, normalized size = 0.56 \begin {gather*} -\frac {{\left (b\,x^3+a\right )}^{1/3}\,\left (2\,a^2-3\,a\,b\,x^3+9\,b^2\,x^6\right )}{14\,a^3\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^8*(a + b*x^3)^(2/3)),x)

[Out]

-((a + b*x^3)^(1/3)*(2*a^2 + 9*b^2*x^6 - 3*a*b*x^3))/(14*a^3*x^7)

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