3.2436 \(\int \frac{1}{(a+\frac{b}{\sqrt [3]{x}})^2 x^5} \, dx\)

Optimal. Leaf size=183 \[ -\frac{27 a^8}{2 b^{10} x^{2/3}}-\frac{21 a^6}{4 b^8 x^{4/3}}+\frac{18 a^5}{5 b^7 x^{5/3}}-\frac{5 a^4}{2 b^6 x^2}+\frac{12 a^3}{7 b^5 x^{7/3}}-\frac{9 a^2}{8 b^4 x^{8/3}}+\frac{3 a^{10}}{b^{11} \left (a \sqrt [3]{x}+b\right )}+\frac{30 a^9}{b^{11} \sqrt [3]{x}}+\frac{8 a^7}{b^9 x}-\frac{33 a^{10} \log \left (a \sqrt [3]{x}+b\right )}{b^{12}}+\frac{11 a^{10} \log (x)}{b^{12}}+\frac{2 a}{3 b^3 x^3}-\frac{3}{10 b^2 x^{10/3}} \]

[Out]

(3*a^10)/(b^11*(b + a*x^(1/3))) - 3/(10*b^2*x^(10/3)) + (2*a)/(3*b^3*x^3) - (9*a^2)/(8*b^4*x^(8/3)) + (12*a^3)
/(7*b^5*x^(7/3)) - (5*a^4)/(2*b^6*x^2) + (18*a^5)/(5*b^7*x^(5/3)) - (21*a^6)/(4*b^8*x^(4/3)) + (8*a^7)/(b^9*x)
- (27*a^8)/(2*b^10*x^(2/3)) + (30*a^9)/(b^11*x^(1/3)) - (33*a^10*Log[b + a*x^(1/3)])/b^12 + (11*a^10*Log[x])/
b^12

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Rubi [A]  time = 0.131665, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {263, 266, 44} \[ -\frac{27 a^8}{2 b^{10} x^{2/3}}-\frac{21 a^6}{4 b^8 x^{4/3}}+\frac{18 a^5}{5 b^7 x^{5/3}}-\frac{5 a^4}{2 b^6 x^2}+\frac{12 a^3}{7 b^5 x^{7/3}}-\frac{9 a^2}{8 b^4 x^{8/3}}+\frac{3 a^{10}}{b^{11} \left (a \sqrt [3]{x}+b\right )}+\frac{30 a^9}{b^{11} \sqrt [3]{x}}+\frac{8 a^7}{b^9 x}-\frac{33 a^{10} \log \left (a \sqrt [3]{x}+b\right )}{b^{12}}+\frac{11 a^{10} \log (x)}{b^{12}}+\frac{2 a}{3 b^3 x^3}-\frac{3}{10 b^2 x^{10/3}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^(1/3))^2*x^5),x]

[Out]

(3*a^10)/(b^11*(b + a*x^(1/3))) - 3/(10*b^2*x^(10/3)) + (2*a)/(3*b^3*x^3) - (9*a^2)/(8*b^4*x^(8/3)) + (12*a^3)
/(7*b^5*x^(7/3)) - (5*a^4)/(2*b^6*x^2) + (18*a^5)/(5*b^7*x^(5/3)) - (21*a^6)/(4*b^8*x^(4/3)) + (8*a^7)/(b^9*x)
- (27*a^8)/(2*b^10*x^(2/3)) + (30*a^9)/(b^11*x^(1/3)) - (33*a^10*Log[b + a*x^(1/3)])/b^12 + (11*a^10*Log[x])/
b^12

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{\left (a+\frac{b}{\sqrt [3]{x}}\right )^2 x^5} \, dx &=\int \frac{1}{\left (b+a \sqrt [3]{x}\right )^2 x^{13/3}} \, dx\\ &=3 \operatorname{Subst}\left (\int \frac{1}{x^{11} (b+a x)^2} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{1}{b^2 x^{11}}-\frac{2 a}{b^3 x^{10}}+\frac{3 a^2}{b^4 x^9}-\frac{4 a^3}{b^5 x^8}+\frac{5 a^4}{b^6 x^7}-\frac{6 a^5}{b^7 x^6}+\frac{7 a^6}{b^8 x^5}-\frac{8 a^7}{b^9 x^4}+\frac{9 a^8}{b^{10} x^3}-\frac{10 a^9}{b^{11} x^2}+\frac{11 a^{10}}{b^{12} x}-\frac{a^{11}}{b^{11} (b+a x)^2}-\frac{11 a^{11}}{b^{12} (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3 a^{10}}{b^{11} \left (b+a \sqrt [3]{x}\right )}-\frac{3}{10 b^2 x^{10/3}}+\frac{2 a}{3 b^3 x^3}-\frac{9 a^2}{8 b^4 x^{8/3}}+\frac{12 a^3}{7 b^5 x^{7/3}}-\frac{5 a^4}{2 b^6 x^2}+\frac{18 a^5}{5 b^7 x^{5/3}}-\frac{21 a^6}{4 b^8 x^{4/3}}+\frac{8 a^7}{b^9 x}-\frac{27 a^8}{2 b^{10} x^{2/3}}+\frac{30 a^9}{b^{11} \sqrt [3]{x}}-\frac{33 a^{10} \log \left (b+a \sqrt [3]{x}\right )}{b^{12}}+\frac{11 a^{10} \log (x)}{b^{12}}\\ \end{align*}

Mathematica [A]  time = 0.157093, size = 173, normalized size = 0.95 \[ -\frac{27 a^8}{2 b^{10} x^{2/3}}-\frac{21 a^6}{4 b^8 x^{4/3}}+\frac{18 a^5}{5 b^7 x^{5/3}}-\frac{5 a^4}{2 b^6 x^2}+\frac{12 a^3}{7 b^5 x^{7/3}}-\frac{9 a^2}{8 b^4 x^{8/3}}-\frac{3 a^{11}}{b^{12} \left (a+\frac{b}{\sqrt [3]{x}}\right )}+\frac{30 a^9}{b^{11} \sqrt [3]{x}}+\frac{8 a^7}{b^9 x}-\frac{33 a^{10} \log \left (a+\frac{b}{\sqrt [3]{x}}\right )}{b^{12}}+\frac{2 a}{3 b^3 x^3}-\frac{3}{10 b^2 x^{10/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^(1/3))^2*x^5),x]

[Out]

(-3*a^11)/(b^12*(a + b/x^(1/3))) - 3/(10*b^2*x^(10/3)) + (2*a)/(3*b^3*x^3) - (9*a^2)/(8*b^4*x^(8/3)) + (12*a^3
)/(7*b^5*x^(7/3)) - (5*a^4)/(2*b^6*x^2) + (18*a^5)/(5*b^7*x^(5/3)) - (21*a^6)/(4*b^8*x^(4/3)) + (8*a^7)/(b^9*x
) - (27*a^8)/(2*b^10*x^(2/3)) + (30*a^9)/(b^11*x^(1/3)) - (33*a^10*Log[a + b/x^(1/3)])/b^12

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Maple [A]  time = 0.014, size = 150, normalized size = 0.8 \begin{align*} 3\,{\frac{{a}^{10}}{{b}^{11} \left ( b+a\sqrt [3]{x} \right ) }}-{\frac{3}{10\,{b}^{2}}{x}^{-{\frac{10}{3}}}}+{\frac{2\,a}{3\,{b}^{3}{x}^{3}}}-{\frac{9\,{a}^{2}}{8\,{b}^{4}}{x}^{-{\frac{8}{3}}}}+{\frac{12\,{a}^{3}}{7\,{b}^{5}}{x}^{-{\frac{7}{3}}}}-{\frac{5\,{a}^{4}}{2\,{b}^{6}{x}^{2}}}+{\frac{18\,{a}^{5}}{5\,{b}^{7}}{x}^{-{\frac{5}{3}}}}-{\frac{21\,{a}^{6}}{4\,{b}^{8}}{x}^{-{\frac{4}{3}}}}+8\,{\frac{{a}^{7}}{{b}^{9}x}}-{\frac{27\,{a}^{8}}{2\,{b}^{10}}{x}^{-{\frac{2}{3}}}}+30\,{\frac{{a}^{9}}{{b}^{11}\sqrt [3]{x}}}-33\,{\frac{{a}^{10}\ln \left ( b+a\sqrt [3]{x} \right ) }{{b}^{12}}}+11\,{\frac{{a}^{10}\ln \left ( x \right ) }{{b}^{12}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x^(1/3))^2/x^5,x)

[Out]

3*a^10/b^11/(b+a*x^(1/3))-3/10/b^2/x^(10/3)+2/3*a/b^3/x^3-9/8*a^2/b^4/x^(8/3)+12/7*a^3/b^5/x^(7/3)-5/2*a^4/b^6
/x^2+18/5*a^5/b^7/x^(5/3)-21/4*a^6/b^8/x^(4/3)+8*a^7/b^9/x-27/2*a^8/b^10/x^(2/3)+30*a^9/b^11/x^(1/3)-33*a^10*l
n(b+a*x^(1/3))/b^12+11*a^10*ln(x)/b^12

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Maxima [A]  time = 1.018, size = 266, normalized size = 1.45 \begin{align*} -\frac{33 \, a^{10} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{b^{12}} - \frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{10}}{10 \, b^{12}} + \frac{11 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{9} a}{3 \, b^{12}} - \frac{165 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{8} a^{2}}{8 \, b^{12}} + \frac{495 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{7} a^{3}}{7 \, b^{12}} - \frac{165 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{6} a^{4}}{b^{12}} + \frac{1386 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{5} a^{5}}{5 \, b^{12}} - \frac{693 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{4} a^{6}}{2 \, b^{12}} + \frac{330 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{3} a^{7}}{b^{12}} - \frac{495 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2} a^{8}}{2 \, b^{12}} + \frac{165 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} a^{9}}{b^{12}} - \frac{3 \, a^{11}}{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} b^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^(1/3))^2/x^5,x, algorithm="maxima")

[Out]

-33*a^10*log(a + b/x^(1/3))/b^12 - 3/10*(a + b/x^(1/3))^10/b^12 + 11/3*(a + b/x^(1/3))^9*a/b^12 - 165/8*(a + b
/x^(1/3))^8*a^2/b^12 + 495/7*(a + b/x^(1/3))^7*a^3/b^12 - 165*(a + b/x^(1/3))^6*a^4/b^12 + 1386/5*(a + b/x^(1/
3))^5*a^5/b^12 - 693/2*(a + b/x^(1/3))^4*a^6/b^12 + 330*(a + b/x^(1/3))^3*a^7/b^12 - 495/2*(a + b/x^(1/3))^2*a
^8/b^12 + 165*(a + b/x^(1/3))*a^9/b^12 - 3*a^11/((a + b/x^(1/3))*b^12)

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Fricas [A]  time = 1.59244, size = 517, normalized size = 2.83 \begin{align*} \frac{9240 \, a^{10} b^{3} x^{4} + 4620 \, a^{7} b^{6} x^{3} - 1540 \, a^{4} b^{9} x^{2} + 560 \, a b^{12} x - 27720 \,{\left (a^{13} x^{5} + a^{10} b^{3} x^{4}\right )} \log \left (a x^{\frac{1}{3}} + b\right ) + 27720 \,{\left (a^{13} x^{5} + a^{10} b^{3} x^{4}\right )} \log \left (x^{\frac{1}{3}}\right ) + 18 \,{\left (1540 \, a^{12} b x^{4} + 1155 \, a^{9} b^{4} x^{3} - 165 \, a^{6} b^{7} x^{2} + 66 \, a^{3} b^{10} x - 14 \, b^{13}\right )} x^{\frac{2}{3}} - 63 \,{\left (220 \, a^{11} b^{2} x^{4} + 132 \, a^{8} b^{5} x^{3} - 33 \, a^{5} b^{8} x^{2} + 15 \, a^{2} b^{11} x\right )} x^{\frac{1}{3}}}{840 \,{\left (a^{3} b^{12} x^{5} + b^{15} x^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^(1/3))^2/x^5,x, algorithm="fricas")

[Out]

1/840*(9240*a^10*b^3*x^4 + 4620*a^7*b^6*x^3 - 1540*a^4*b^9*x^2 + 560*a*b^12*x - 27720*(a^13*x^5 + a^10*b^3*x^4
)*log(a*x^(1/3) + b) + 27720*(a^13*x^5 + a^10*b^3*x^4)*log(x^(1/3)) + 18*(1540*a^12*b*x^4 + 1155*a^9*b^4*x^3 -
165*a^6*b^7*x^2 + 66*a^3*b^10*x - 14*b^13)*x^(2/3) - 63*(220*a^11*b^2*x^4 + 132*a^8*b^5*x^3 - 33*a^5*b^8*x^2
+ 15*a^2*b^11*x)*x^(1/3))/(a^3*b^12*x^5 + b^15*x^4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**(1/3))**2/x**5,x)

[Out]

Timed out

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Giac [A]  time = 1.21263, size = 211, normalized size = 1.15 \begin{align*} -\frac{33 \, a^{10} \log \left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{b^{12}} + \frac{11 \, a^{10} \log \left ({\left | x \right |}\right )}{b^{12}} + \frac{27720 \, a^{10} b x^{\frac{10}{3}} + 13860 \, a^{9} b^{2} x^{3} - 4620 \, a^{8} b^{3} x^{\frac{8}{3}} + 2310 \, a^{7} b^{4} x^{\frac{7}{3}} - 1386 \, a^{6} b^{5} x^{2} + 924 \, a^{5} b^{6} x^{\frac{5}{3}} - 660 \, a^{4} b^{7} x^{\frac{4}{3}} + 495 \, a^{3} b^{8} x - 385 \, a^{2} b^{9} x^{\frac{2}{3}} + 308 \, a b^{10} x^{\frac{1}{3}} - 252 \, b^{11}}{840 \,{\left (a x^{\frac{1}{3}} + b\right )} b^{12} x^{\frac{10}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^(1/3))^2/x^5,x, algorithm="giac")

[Out]

-33*a^10*log(abs(a*x^(1/3) + b))/b^12 + 11*a^10*log(abs(x))/b^12 + 1/840*(27720*a^10*b*x^(10/3) + 13860*a^9*b^
2*x^3 - 4620*a^8*b^3*x^(8/3) + 2310*a^7*b^4*x^(7/3) - 1386*a^6*b^5*x^2 + 924*a^5*b^6*x^(5/3) - 660*a^4*b^7*x^(
4/3) + 495*a^3*b^8*x - 385*a^2*b^9*x^(2/3) + 308*a*b^10*x^(1/3) - 252*b^11)/((a*x^(1/3) + b)*b^12*x^(10/3))