3.6.21 \(\int \frac {\sqrt [3]{a+b x^3}}{x^{11}} \, dx\) [521]

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

[Out]

-1/10*(b*x^3+a)^(4/3)/a/x^10+3/35*b*(b*x^3+a)^(4/3)/a^2/x^7-9/140*b^2*(b*x^3+a)^(4/3)/a^3/x^4

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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 \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}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^(1/3)/x^11,x]

[Out]

-1/10*(a + b*x^3)^(4/3)/(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 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 {\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*}

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Mathematica [A]
time = 0.10, size = 53, normalized size = 0.78 \begin {gather*} \frac {\sqrt [3]{a+b x^3} \left (-14 a^3-2 a^2 b x^3+3 a b^2 x^6-9 b^3 x^9\right )}{140 a^3 x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^(1/3)/x^11,x]

[Out]

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

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Maple [A]
time = 0.15, size = 39, normalized size = 0.57

method	result	size
gosper	\(-\frac {\left (b \,x^{3}+a \right )^{\frac {4}{3}} \left (9 b^{2} x^{6}-12 a b \,x^{3}+14 a^{2}\right )}{140 a^{3} x^{10}}\)	\(39\)
trager	\(-\frac {\left (9 b^{3} x^{9}-3 a \,b^{2} x^{6}+2 a^{2} b \,x^{3}+14 a^{3}\right ) \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{140 a^{3} x^{10}}\)	\(50\)
risch	\(-\frac {\left (9 b^{3} x^{9}-3 a \,b^{2} x^{6}+2 a^{2} b \,x^{3}+14 a^{3}\right ) \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{140 a^{3} x^{10}}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 52, normalized size = 0.76 \begin {gather*} -\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 {gather*}

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 = 0.38, size = 49, normalized size = 0.72 \begin {gather*} -\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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 520 vs. \(2 (61) = 122\).
time = 0.77, size = 520, normalized size = 7.65 \begin {gather*} \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 {gather*}

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 = 0.00, 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((b*x^3+a)^(1/3)/x^11,x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(1/3)/x^11, x)

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Mupad [B]
time = 1.35, size = 73, normalized size = 1.07 \begin {gather*} \frac {3\,b^2\,{\left (b\,x^3+a\right )}^{1/3}}{140\,a^2\,x^4}-\frac {b\,{\left (b\,x^3+a\right )}^{1/3}}{70\,a\,x^7}-\frac {9\,b^3\,{\left (b\,x^3+a\right )}^{1/3}}{140\,a^3\,x}-\frac {{\left (b\,x^3+a\right )}^{1/3}}{10\,x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^(1/3)/x^11,x)

[Out]

(3*b^2*(a + b*x^3)^(1/3))/(140*a^2*x^4) - (b*(a + b*x^3)^(1/3))/(70*a*x^7) - (9*b^3*(a + b*x^3)^(1/3))/(140*a^
3*x) - (a + b*x^3)^(1/3)/(10*x^10)

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