3.560 \(\int \frac {1}{x^{12} \sqrt [3]{a+b x^3}} \, dx\)

Optimal. Leaf size=92 \[ \frac {81 b^3 \left (a+b x^3\right )^{2/3}}{440 a^4 x^2}-\frac {27 b^2 \left (a+b x^3\right )^{2/3}}{220 a^3 x^5}+\frac {9 b \left (a+b x^3\right )^{2/3}}{88 a^2 x^8}-\frac {\left (a+b x^3\right )^{2/3}}{11 a x^{11}} \]

[Out]

-1/11*(b*x^3+a)^(2/3)/a/x^11+9/88*b*(b*x^3+a)^(2/3)/a^2/x^8-27/220*b^2*(b*x^3+a)^(2/3)/a^3/x^5+81/440*b^3*(b*x
^3+a)^(2/3)/a^4/x^2

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Rubi [A]  time = 0.03, 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 = {271, 264} \[ \frac {81 b^3 \left (a+b x^3\right )^{2/3}}{440 a^4 x^2}-\frac {27 b^2 \left (a+b x^3\right )^{2/3}}{220 a^3 x^5}+\frac {9 b \left (a+b x^3\right )^{2/3}}{88 a^2 x^8}-\frac {\left (a+b x^3\right )^{2/3}}{11 a x^{11}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^12*(a + b*x^3)^(1/3)),x]

[Out]

-(a + b*x^3)^(2/3)/(11*a*x^11) + (9*b*(a + b*x^3)^(2/3))/(88*a^2*x^8) - (27*b^2*(a + b*x^3)^(2/3))/(220*a^3*x^
5) + (81*b^3*(a + b*x^3)^(2/3))/(440*a^4*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^{12} \sqrt [3]{a+b x^3}} \, dx &=-\frac {\left (a+b x^3\right )^{2/3}}{11 a x^{11}}-\frac {(9 b) \int \frac {1}{x^9 \sqrt [3]{a+b x^3}} \, dx}{11 a}\\ &=-\frac {\left (a+b x^3\right )^{2/3}}{11 a x^{11}}+\frac {9 b \left (a+b x^3\right )^{2/3}}{88 a^2 x^8}+\frac {\left (27 b^2\right ) \int \frac {1}{x^6 \sqrt [3]{a+b x^3}} \, dx}{44 a^2}\\ &=-\frac {\left (a+b x^3\right )^{2/3}}{11 a x^{11}}+\frac {9 b \left (a+b x^3\right )^{2/3}}{88 a^2 x^8}-\frac {27 b^2 \left (a+b x^3\right )^{2/3}}{220 a^3 x^5}-\frac {\left (81 b^3\right ) \int \frac {1}{x^3 \sqrt [3]{a+b x^3}} \, dx}{220 a^3}\\ &=-\frac {\left (a+b x^3\right )^{2/3}}{11 a x^{11}}+\frac {9 b \left (a+b x^3\right )^{2/3}}{88 a^2 x^8}-\frac {27 b^2 \left (a+b x^3\right )^{2/3}}{220 a^3 x^5}+\frac {81 b^3 \left (a+b x^3\right )^{2/3}}{440 a^4 x^2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 53, normalized size = 0.58 \[ \frac {\left (a+b x^3\right )^{2/3} \left (-40 a^3+45 a^2 b x^3-54 a b^2 x^6+81 b^3 x^9\right )}{440 a^4 x^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^12*(a + b*x^3)^(1/3)),x]

[Out]

((a + b*x^3)^(2/3)*(-40*a^3 + 45*a^2*b*x^3 - 54*a*b^2*x^6 + 81*b^3*x^9))/(440*a^4*x^11)

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fricas [A]  time = 0.61, size = 49, normalized size = 0.53 \[ \frac {{\left (81 \, b^{3} x^{9} - 54 \, a b^{2} x^{6} + 45 \, a^{2} b x^{3} - 40 \, a^{3}\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{440 \, a^{4} x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^12/(b*x^3+a)^(1/3),x, algorithm="fricas")

[Out]

1/440*(81*b^3*x^9 - 54*a*b^2*x^6 + 45*a^2*b*x^3 - 40*a^3)*(b*x^3 + a)^(2/3)/(a^4*x^11)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} x^{12}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^12/(b*x^3+a)^(1/3),x, algorithm="giac")

[Out]

integrate(1/((b*x^3 + a)^(1/3)*x^12), x)

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maple [A]  time = 0.01, size = 50, normalized size = 0.54 \[ -\frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} \left (-81 b^{3} x^{9}+54 a \,b^{2} x^{6}-45 a^{2} b \,x^{3}+40 a^{3}\right )}{440 a^{4} x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^12/(b*x^3+a)^(1/3),x)

[Out]

-1/440*(b*x^3+a)^(2/3)*(-81*b^3*x^9+54*a*b^2*x^6-45*a^2*b*x^3+40*a^3)/x^11/a^4

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maxima [A]  time = 1.29, size = 69, normalized size = 0.75 \[ \frac {\frac {220 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} b^{3}}{x^{2}} - \frac {264 \, {\left (b x^{3} + a\right )}^{\frac {5}{3}} b^{2}}{x^{5}} + \frac {165 \, {\left (b x^{3} + a\right )}^{\frac {8}{3}} b}{x^{8}} - \frac {40 \, {\left (b x^{3} + a\right )}^{\frac {11}{3}}}{x^{11}}}{440 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^12/(b*x^3+a)^(1/3),x, algorithm="maxima")

[Out]

1/440*(220*(b*x^3 + a)^(2/3)*b^3/x^2 - 264*(b*x^3 + a)^(5/3)*b^2/x^5 + 165*(b*x^3 + a)^(8/3)*b/x^8 - 40*(b*x^3
+ a)^(11/3)/x^11)/a^4

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mupad [B]  time = 1.20, size = 76, normalized size = 0.83 \[ \frac {9\,b\,{\left (b\,x^3+a\right )}^{2/3}}{88\,a^2\,x^8}-\frac {{\left (b\,x^3+a\right )}^{2/3}}{11\,a\,x^{11}}+\frac {81\,b^3\,{\left (b\,x^3+a\right )}^{2/3}}{440\,a^4\,x^2}-\frac {27\,b^2\,{\left (b\,x^3+a\right )}^{2/3}}{220\,a^3\,x^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^12*(a + b*x^3)^(1/3)),x)

[Out]

(9*b*(a + b*x^3)^(2/3))/(88*a^2*x^8) - (a + b*x^3)^(2/3)/(11*a*x^11) + (81*b^3*(a + b*x^3)^(2/3))/(440*a^4*x^2
) - (27*b^2*(a + b*x^3)^(2/3))/(220*a^3*x^5)

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sympy [B]  time = 3.44, size = 692, normalized size = 7.52 \[ - \frac {80 a^{6} b^{\frac {29}{3}} \left (\frac {a}{b x^{3}} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {11}{3}\right )}{81 a^{7} b^{9} x^{9} \Gamma \left (\frac {1}{3}\right ) + 243 a^{6} b^{10} x^{12} \Gamma \left (\frac {1}{3}\right ) + 243 a^{5} b^{11} x^{15} \Gamma \left (\frac {1}{3}\right ) + 81 a^{4} b^{12} x^{18} \Gamma \left (\frac {1}{3}\right )} - \frac {150 a^{5} b^{\frac {32}{3}} x^{3} \left (\frac {a}{b x^{3}} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {11}{3}\right )}{81 a^{7} b^{9} x^{9} \Gamma \left (\frac {1}{3}\right ) + 243 a^{6} b^{10} x^{12} \Gamma \left (\frac {1}{3}\right ) + 243 a^{5} b^{11} x^{15} \Gamma \left (\frac {1}{3}\right ) + 81 a^{4} b^{12} x^{18} \Gamma \left (\frac {1}{3}\right )} - \frac {78 a^{4} b^{\frac {35}{3}} x^{6} \left (\frac {a}{b x^{3}} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {11}{3}\right )}{81 a^{7} b^{9} x^{9} \Gamma \left (\frac {1}{3}\right ) + 243 a^{6} b^{10} x^{12} \Gamma \left (\frac {1}{3}\right ) + 243 a^{5} b^{11} x^{15} \Gamma \left (\frac {1}{3}\right ) + 81 a^{4} b^{12} x^{18} \Gamma \left (\frac {1}{3}\right )} + \frac {28 a^{3} b^{\frac {38}{3}} x^{9} \left (\frac {a}{b x^{3}} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {11}{3}\right )}{81 a^{7} b^{9} x^{9} \Gamma \left (\frac {1}{3}\right ) + 243 a^{6} b^{10} x^{12} \Gamma \left (\frac {1}{3}\right ) + 243 a^{5} b^{11} x^{15} \Gamma \left (\frac {1}{3}\right ) + 81 a^{4} b^{12} x^{18} \Gamma \left (\frac {1}{3}\right )} + \frac {252 a^{2} b^{\frac {41}{3}} x^{12} \left (\frac {a}{b x^{3}} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {11}{3}\right )}{81 a^{7} b^{9} x^{9} \Gamma \left (\frac {1}{3}\right ) + 243 a^{6} b^{10} x^{12} \Gamma \left (\frac {1}{3}\right ) + 243 a^{5} b^{11} x^{15} \Gamma \left (\frac {1}{3}\right ) + 81 a^{4} b^{12} x^{18} \Gamma \left (\frac {1}{3}\right )} + \frac {378 a b^{\frac {44}{3}} x^{15} \left (\frac {a}{b x^{3}} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {11}{3}\right )}{81 a^{7} b^{9} x^{9} \Gamma \left (\frac {1}{3}\right ) + 243 a^{6} b^{10} x^{12} \Gamma \left (\frac {1}{3}\right ) + 243 a^{5} b^{11} x^{15} \Gamma \left (\frac {1}{3}\right ) + 81 a^{4} b^{12} x^{18} \Gamma \left (\frac {1}{3}\right )} + \frac {162 b^{\frac {47}{3}} x^{18} \left (\frac {a}{b x^{3}} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {11}{3}\right )}{81 a^{7} b^{9} x^{9} \Gamma \left (\frac {1}{3}\right ) + 243 a^{6} b^{10} x^{12} \Gamma \left (\frac {1}{3}\right ) + 243 a^{5} b^{11} x^{15} \Gamma \left (\frac {1}{3}\right ) + 81 a^{4} b^{12} x^{18} \Gamma \left (\frac {1}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**12/(b*x**3+a)**(1/3),x)

[Out]

-80*a**6*b**(29/3)*(a/(b*x**3) + 1)**(2/3)*gamma(-11/3)/(81*a**7*b**9*x**9*gamma(1/3) + 243*a**6*b**10*x**12*g
amma(1/3) + 243*a**5*b**11*x**15*gamma(1/3) + 81*a**4*b**12*x**18*gamma(1/3)) - 150*a**5*b**(32/3)*x**3*(a/(b*
x**3) + 1)**(2/3)*gamma(-11/3)/(81*a**7*b**9*x**9*gamma(1/3) + 243*a**6*b**10*x**12*gamma(1/3) + 243*a**5*b**1
1*x**15*gamma(1/3) + 81*a**4*b**12*x**18*gamma(1/3)) - 78*a**4*b**(35/3)*x**6*(a/(b*x**3) + 1)**(2/3)*gamma(-1
1/3)/(81*a**7*b**9*x**9*gamma(1/3) + 243*a**6*b**10*x**12*gamma(1/3) + 243*a**5*b**11*x**15*gamma(1/3) + 81*a*
*4*b**12*x**18*gamma(1/3)) + 28*a**3*b**(38/3)*x**9*(a/(b*x**3) + 1)**(2/3)*gamma(-11/3)/(81*a**7*b**9*x**9*ga
mma(1/3) + 243*a**6*b**10*x**12*gamma(1/3) + 243*a**5*b**11*x**15*gamma(1/3) + 81*a**4*b**12*x**18*gamma(1/3))
+ 252*a**2*b**(41/3)*x**12*(a/(b*x**3) + 1)**(2/3)*gamma(-11/3)/(81*a**7*b**9*x**9*gamma(1/3) + 243*a**6*b**1
0*x**12*gamma(1/3) + 243*a**5*b**11*x**15*gamma(1/3) + 81*a**4*b**12*x**18*gamma(1/3)) + 378*a*b**(44/3)*x**15
*(a/(b*x**3) + 1)**(2/3)*gamma(-11/3)/(81*a**7*b**9*x**9*gamma(1/3) + 243*a**6*b**10*x**12*gamma(1/3) + 243*a*
*5*b**11*x**15*gamma(1/3) + 81*a**4*b**12*x**18*gamma(1/3)) + 162*b**(47/3)*x**18*(a/(b*x**3) + 1)**(2/3)*gamm
a(-11/3)/(81*a**7*b**9*x**9*gamma(1/3) + 243*a**6*b**10*x**12*gamma(1/3) + 243*a**5*b**11*x**15*gamma(1/3) + 8
1*a**4*b**12*x**18*gamma(1/3))

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