3.6.73 \(\int \frac {1}{x^{11} (a+b x^3)^{2/3}} \, dx\) [573]

Optimal. Leaf size=92 \[ -\frac {\sqrt [3]{a+b x^3}}{10 a x^{10}}+\frac {9 b \sqrt [3]{a+b x^3}}{70 a^2 x^7}-\frac {27 b^2 \sqrt [3]{a+b x^3}}{140 a^3 x^4}+\frac {81 b^3 \sqrt [3]{a+b x^3}}{140 a^4 x} \]

[Out]

-1/10*(b*x^3+a)^(1/3)/a/x^10+9/70*b*(b*x^3+a)^(1/3)/a^2/x^7-27/140*b^2*(b*x^3+a)^(1/3)/a^3/x^4+81/140*b^3*(b*x
^3+a)^(1/3)/a^4/x

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Rubi [A]
time = 0.02, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, 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 {81 b^3 \sqrt [3]{a+b x^3}}{140 a^4 x}-\frac {27 b^2 \sqrt [3]{a+b x^3}}{140 a^3 x^4}+\frac {9 b \sqrt [3]{a+b x^3}}{70 a^2 x^7}-\frac {\sqrt [3]{a+b x^3}}{10 a x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^11*(a + b*x^3)^(2/3)),x]

[Out]

-1/10*(a + b*x^3)^(1/3)/(a*x^10) + (9*b*(a + b*x^3)^(1/3))/(70*a^2*x^7) - (27*b^2*(a + b*x^3)^(1/3))/(140*a^3*
x^4) + (81*b^3*(a + b*x^3)^(1/3))/(140*a^4*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^{11} \left (a+b x^3\right )^{2/3}} \, dx &=-\frac {\sqrt [3]{a+b x^3}}{10 a x^{10}}-\frac {(9 b) \int \frac {1}{x^8 \left (a+b x^3\right )^{2/3}} \, dx}{10 a}\\ &=-\frac {\sqrt [3]{a+b x^3}}{10 a x^{10}}+\frac {9 b \sqrt [3]{a+b x^3}}{70 a^2 x^7}+\frac {\left (27 b^2\right ) \int \frac {1}{x^5 \left (a+b x^3\right )^{2/3}} \, dx}{35 a^2}\\ &=-\frac {\sqrt [3]{a+b x^3}}{10 a x^{10}}+\frac {9 b \sqrt [3]{a+b x^3}}{70 a^2 x^7}-\frac {27 b^2 \sqrt [3]{a+b x^3}}{140 a^3 x^4}-\frac {\left (81 b^3\right ) \int \frac {1}{x^2 \left (a+b x^3\right )^{2/3}} \, dx}{140 a^3}\\ &=-\frac {\sqrt [3]{a+b x^3}}{10 a x^{10}}+\frac {9 b \sqrt [3]{a+b x^3}}{70 a^2 x^7}-\frac {27 b^2 \sqrt [3]{a+b x^3}}{140 a^3 x^4}+\frac {81 b^3 \sqrt [3]{a+b x^3}}{140 a^4 x}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 53, normalized size = 0.58 \begin {gather*} \frac {\sqrt [3]{a+b x^3} \left (-14 a^3+18 a^2 b x^3-27 a b^2 x^6+81 b^3 x^9\right )}{140 a^4 x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^11*(a + b*x^3)^(2/3)),x]

[Out]

((a + b*x^3)^(1/3)*(-14*a^3 + 18*a^2*b*x^3 - 27*a*b^2*x^6 + 81*b^3*x^9))/(140*a^4*x^10)

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Maple [A]
time = 0.14, size = 50, normalized size = 0.54

method	result	size
gosper	\(-\frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}} \left (-81 b^{3} x^{9}+27 a \,b^{2} x^{6}-18 a^{2} b \,x^{3}+14 a^{3}\right )}{140 x^{10} a^{4}}\)	\(50\)
trager	\(-\frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}} \left (-81 b^{3} x^{9}+27 a \,b^{2} x^{6}-18 a^{2} b \,x^{3}+14 a^{3}\right )}{140 x^{10} a^{4}}\)	\(50\)
risch	\(-\frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}} \left (-81 b^{3} x^{9}+27 a \,b^{2} x^{6}-18 a^{2} b \,x^{3}+14 a^{3}\right )}{140 x^{10} a^{4}}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^11/(b*x^3+a)^(2/3),x,method=_RETURNVERBOSE)

[Out]

-1/140*(b*x^3+a)^(1/3)*(-81*b^3*x^9+27*a*b^2*x^6-18*a^2*b*x^3+14*a^3)/x^10/a^4

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Maxima [A]
time = 0.30, size = 69, normalized size = 0.75 \begin {gather*} \frac {\frac {140 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{3}}{x} - \frac {105 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}} b^{2}}{x^{4}} + \frac {60 \, {\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^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^11/(b*x^3+a)^(2/3),x, algorithm="maxima")

[Out]

1/140*(140*(b*x^3 + a)^(1/3)*b^3/x - 105*(b*x^3 + a)^(4/3)*b^2/x^4 + 60*(b*x^3 + a)^(7/3)*b/x^7 - 14*(b*x^3 +
a)^(10/3)/x^10)/a^4

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Fricas [A]
time = 0.36, size = 49, normalized size = 0.53 \begin {gather*} \frac {{\left (81 \, b^{3} x^{9} - 27 \, a b^{2} x^{6} + 18 \, a^{2} b x^{3} - 14 \, a^{3}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{140 \, a^{4} x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^11/(b*x^3+a)^(2/3),x, algorithm="fricas")

[Out]

1/140*(81*b^3*x^9 - 27*a*b^2*x^6 + 18*a^2*b*x^3 - 14*a^3)*(b*x^3 + a)^(1/3)/(a^4*x^10)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 692 vs. \(2 (83) = 166\).
time = 1.01, size = 692, normalized size = 7.52 \begin {gather*} - \frac {28 a^{6} b^{\frac {28}{3}} \sqrt [3]{\frac {a}{b x^{3}} + 1} \Gamma \left (- \frac {10}{3}\right )}{81 a^{7} b^{9} x^{9} \Gamma \left (\frac {2}{3}\right ) + 243 a^{6} b^{10} x^{12} \Gamma \left (\frac {2}{3}\right ) + 243 a^{5} b^{11} x^{15} \Gamma \left (\frac {2}{3}\right ) + 81 a^{4} b^{12} x^{18} \Gamma \left (\frac {2}{3}\right )} - \frac {48 a^{5} b^{\frac {31}{3}} x^{3} \sqrt [3]{\frac {a}{b x^{3}} + 1} \Gamma \left (- \frac {10}{3}\right )}{81 a^{7} b^{9} x^{9} \Gamma \left (\frac {2}{3}\right ) + 243 a^{6} b^{10} x^{12} \Gamma \left (\frac {2}{3}\right ) + 243 a^{5} b^{11} x^{15} \Gamma \left (\frac {2}{3}\right ) + 81 a^{4} b^{12} x^{18} \Gamma \left (\frac {2}{3}\right )} - \frac {30 a^{4} b^{\frac {34}{3}} x^{6} \sqrt [3]{\frac {a}{b x^{3}} + 1} \Gamma \left (- \frac {10}{3}\right )}{81 a^{7} b^{9} x^{9} \Gamma \left (\frac {2}{3}\right ) + 243 a^{6} b^{10} x^{12} \Gamma \left (\frac {2}{3}\right ) + 243 a^{5} b^{11} x^{15} \Gamma \left (\frac {2}{3}\right ) + 81 a^{4} b^{12} x^{18} \Gamma \left (\frac {2}{3}\right )} + \frac {80 a^{3} b^{\frac {37}{3}} x^{9} \sqrt [3]{\frac {a}{b x^{3}} + 1} \Gamma \left (- \frac {10}{3}\right )}{81 a^{7} b^{9} x^{9} \Gamma \left (\frac {2}{3}\right ) + 243 a^{6} b^{10} x^{12} \Gamma \left (\frac {2}{3}\right ) + 243 a^{5} b^{11} x^{15} \Gamma \left (\frac {2}{3}\right ) + 81 a^{4} b^{12} x^{18} \Gamma \left (\frac {2}{3}\right )} + \frac {360 a^{2} b^{\frac {40}{3}} x^{12} \sqrt [3]{\frac {a}{b x^{3}} + 1} \Gamma \left (- \frac {10}{3}\right )}{81 a^{7} b^{9} x^{9} \Gamma \left (\frac {2}{3}\right ) + 243 a^{6} b^{10} x^{12} \Gamma \left (\frac {2}{3}\right ) + 243 a^{5} b^{11} x^{15} \Gamma \left (\frac {2}{3}\right ) + 81 a^{4} b^{12} x^{18} \Gamma \left (\frac {2}{3}\right )} + \frac {432 a b^{\frac {43}{3}} x^{15} \sqrt [3]{\frac {a}{b x^{3}} + 1} \Gamma \left (- \frac {10}{3}\right )}{81 a^{7} b^{9} x^{9} \Gamma \left (\frac {2}{3}\right ) + 243 a^{6} b^{10} x^{12} \Gamma \left (\frac {2}{3}\right ) + 243 a^{5} b^{11} x^{15} \Gamma \left (\frac {2}{3}\right ) + 81 a^{4} b^{12} x^{18} \Gamma \left (\frac {2}{3}\right )} + \frac {162 b^{\frac {46}{3}} x^{18} \sqrt [3]{\frac {a}{b x^{3}} + 1} \Gamma \left (- \frac {10}{3}\right )}{81 a^{7} b^{9} x^{9} \Gamma \left (\frac {2}{3}\right ) + 243 a^{6} b^{10} x^{12} \Gamma \left (\frac {2}{3}\right ) + 243 a^{5} b^{11} x^{15} \Gamma \left (\frac {2}{3}\right ) + 81 a^{4} b^{12} x^{18} \Gamma \left (\frac {2}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**11/(b*x**3+a)**(2/3),x)

[Out]

-28*a**6*b**(28/3)*(a/(b*x**3) + 1)**(1/3)*gamma(-10/3)/(81*a**7*b**9*x**9*gamma(2/3) + 243*a**6*b**10*x**12*g
amma(2/3) + 243*a**5*b**11*x**15*gamma(2/3) + 81*a**4*b**12*x**18*gamma(2/3)) - 48*a**5*b**(31/3)*x**3*(a/(b*x
**3) + 1)**(1/3)*gamma(-10/3)/(81*a**7*b**9*x**9*gamma(2/3) + 243*a**6*b**10*x**12*gamma(2/3) + 243*a**5*b**11
*x**15*gamma(2/3) + 81*a**4*b**12*x**18*gamma(2/3)) - 30*a**4*b**(34/3)*x**6*(a/(b*x**3) + 1)**(1/3)*gamma(-10
/3)/(81*a**7*b**9*x**9*gamma(2/3) + 243*a**6*b**10*x**12*gamma(2/3) + 243*a**5*b**11*x**15*gamma(2/3) + 81*a**
4*b**12*x**18*gamma(2/3)) + 80*a**3*b**(37/3)*x**9*(a/(b*x**3) + 1)**(1/3)*gamma(-10/3)/(81*a**7*b**9*x**9*gam
ma(2/3) + 243*a**6*b**10*x**12*gamma(2/3) + 243*a**5*b**11*x**15*gamma(2/3) + 81*a**4*b**12*x**18*gamma(2/3))
+ 360*a**2*b**(40/3)*x**12*(a/(b*x**3) + 1)**(1/3)*gamma(-10/3)/(81*a**7*b**9*x**9*gamma(2/3) + 243*a**6*b**10
*x**12*gamma(2/3) + 243*a**5*b**11*x**15*gamma(2/3) + 81*a**4*b**12*x**18*gamma(2/3)) + 432*a*b**(43/3)*x**15*
(a/(b*x**3) + 1)**(1/3)*gamma(-10/3)/(81*a**7*b**9*x**9*gamma(2/3) + 243*a**6*b**10*x**12*gamma(2/3) + 243*a**
5*b**11*x**15*gamma(2/3) + 81*a**4*b**12*x**18*gamma(2/3)) + 162*b**(46/3)*x**18*(a/(b*x**3) + 1)**(1/3)*gamma
(-10/3)/(81*a**7*b**9*x**9*gamma(2/3) + 243*a**6*b**10*x**12*gamma(2/3) + 243*a**5*b**11*x**15*gamma(2/3) + 81
*a**4*b**12*x**18*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^11/(b*x^3+a)^(2/3),x, algorithm="giac")

[Out]

integrate(1/((b*x^3 + a)^(2/3)*x^11), x)

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Mupad [B]
time = 1.20, size = 76, normalized size = 0.83 \begin {gather*} \frac {9\,b\,{\left (b\,x^3+a\right )}^{1/3}}{70\,a^2\,x^7}-\frac {{\left (b\,x^3+a\right )}^{1/3}}{10\,a\,x^{10}}+\frac {81\,b^3\,{\left (b\,x^3+a\right )}^{1/3}}{140\,a^4\,x}-\frac {27\,b^2\,{\left (b\,x^3+a\right )}^{1/3}}{140\,a^3\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^11*(a + b*x^3)^(2/3)),x)

[Out]

(9*b*(a + b*x^3)^(1/3))/(70*a^2*x^7) - (a + b*x^3)^(1/3)/(10*a*x^10) + (81*b^3*(a + b*x^3)^(1/3))/(140*a^4*x)
- (27*b^2*(a + b*x^3)^(1/3))/(140*a^3*x^4)

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