3.2444 \(\int \frac{1}{(a+\frac{b}{\sqrt [3]{x}})^3 x^5} \, dx\)

Optimal. Leaf size=183 \[ \frac{54 a^7}{b^{10} x^{2/3}}+\frac{63 a^5}{4 b^8 x^{4/3}}-\frac{9 a^4}{b^7 x^{5/3}}+\frac{5 a^3}{b^6 x^2}-\frac{18 a^2}{7 b^5 x^{7/3}}-\frac{30 a^9}{b^{11} \left (a \sqrt [3]{x}+b\right )}-\frac{3 a^9}{2 b^{10} \left (a \sqrt [3]{x}+b\right )^2}-\frac{135 a^8}{b^{11} \sqrt [3]{x}}-\frac{28 a^6}{b^9 x}+\frac{165 a^9 \log \left (a \sqrt [3]{x}+b\right )}{b^{12}}-\frac{55 a^9 \log (x)}{b^{12}}+\frac{9 a}{8 b^4 x^{8/3}}-\frac{1}{3 b^3 x^3} \]

[Out]

(-3*a^9)/(2*b^10*(b + a*x^(1/3))^2) - (30*a^9)/(b^11*(b + a*x^(1/3))) - 1/(3*b^3*x^3) + (9*a)/(8*b^4*x^(8/3))
- (18*a^2)/(7*b^5*x^(7/3)) + (5*a^3)/(b^6*x^2) - (9*a^4)/(b^7*x^(5/3)) + (63*a^5)/(4*b^8*x^(4/3)) - (28*a^6)/(
b^9*x) + (54*a^7)/(b^10*x^(2/3)) - (135*a^8)/(b^11*x^(1/3)) + (165*a^9*Log[b + a*x^(1/3)])/b^12 - (55*a^9*Log[
x])/b^12

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Rubi [A]  time = 0.143918, 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} \[ \frac{54 a^7}{b^{10} x^{2/3}}+\frac{63 a^5}{4 b^8 x^{4/3}}-\frac{9 a^4}{b^7 x^{5/3}}+\frac{5 a^3}{b^6 x^2}-\frac{18 a^2}{7 b^5 x^{7/3}}-\frac{30 a^9}{b^{11} \left (a \sqrt [3]{x}+b\right )}-\frac{3 a^9}{2 b^{10} \left (a \sqrt [3]{x}+b\right )^2}-\frac{135 a^8}{b^{11} \sqrt [3]{x}}-\frac{28 a^6}{b^9 x}+\frac{165 a^9 \log \left (a \sqrt [3]{x}+b\right )}{b^{12}}-\frac{55 a^9 \log (x)}{b^{12}}+\frac{9 a}{8 b^4 x^{8/3}}-\frac{1}{3 b^3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^(1/3))^3*x^5),x]

[Out]

(-3*a^9)/(2*b^10*(b + a*x^(1/3))^2) - (30*a^9)/(b^11*(b + a*x^(1/3))) - 1/(3*b^3*x^3) + (9*a)/(8*b^4*x^(8/3))
- (18*a^2)/(7*b^5*x^(7/3)) + (5*a^3)/(b^6*x^2) - (9*a^4)/(b^7*x^(5/3)) + (63*a^5)/(4*b^8*x^(4/3)) - (28*a^6)/(
b^9*x) + (54*a^7)/(b^10*x^(2/3)) - (135*a^8)/(b^11*x^(1/3)) + (165*a^9*Log[b + a*x^(1/3)])/b^12 - (55*a^9*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 )^3 x^5} \, dx &=\int \frac{1}{\left (b+a \sqrt [3]{x}\right )^3 x^4} \, dx\\ &=3 \operatorname{Subst}\left (\int \frac{1}{x^{10} (b+a x)^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{1}{b^3 x^{10}}-\frac{3 a}{b^4 x^9}+\frac{6 a^2}{b^5 x^8}-\frac{10 a^3}{b^6 x^7}+\frac{15 a^4}{b^7 x^6}-\frac{21 a^5}{b^8 x^5}+\frac{28 a^6}{b^9 x^4}-\frac{36 a^7}{b^{10} x^3}+\frac{45 a^8}{b^{11} x^2}-\frac{55 a^9}{b^{12} x}+\frac{a^{10}}{b^{10} (b+a x)^3}+\frac{10 a^{10}}{b^{11} (b+a x)^2}+\frac{55 a^{10}}{b^{12} (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{3 a^9}{2 b^{10} \left (b+a \sqrt [3]{x}\right )^2}-\frac{30 a^9}{b^{11} \left (b+a \sqrt [3]{x}\right )}-\frac{1}{3 b^3 x^3}+\frac{9 a}{8 b^4 x^{8/3}}-\frac{18 a^2}{7 b^5 x^{7/3}}+\frac{5 a^3}{b^6 x^2}-\frac{9 a^4}{b^7 x^{5/3}}+\frac{63 a^5}{4 b^8 x^{4/3}}-\frac{28 a^6}{b^9 x}+\frac{54 a^7}{b^{10} x^{2/3}}-\frac{135 a^8}{b^{11} \sqrt [3]{x}}+\frac{165 a^9 \log \left (b+a \sqrt [3]{x}\right )}{b^{12}}-\frac{55 a^9 \log (x)}{b^{12}}\\ \end{align*}

Mathematica [A]  time = 0.259117, size = 167, normalized size = 0.91 \[ -\frac{\frac{b \left (9240 a^8 b^2 x^{8/3}-2310 a^7 b^3 x^{7/3}+924 a^6 b^4 x^2-462 a^5 b^5 x^{5/3}+264 a^4 b^6 x^{4/3}+110 a^2 b^8 x^{2/3}-165 a^3 b^7 x+41580 a^9 b x^3+27720 a^{10} x^{10/3}-77 a b^9 \sqrt [3]{x}+56 b^{10}\right )}{x^3 \left (a \sqrt [3]{x}+b\right )^2}-27720 a^9 \log \left (a \sqrt [3]{x}+b\right )+9240 a^9 \log (x)}{168 b^{12}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*(56*b^10 - 77*a*b^9*x^(1/3) + 110*a^2*b^8*x^(2/3) - 165*a^3*b^7*x + 264*a^4*b^6*x^(4/3) - 462*a^5*b^5*x^(
5/3) + 924*a^6*b^4*x^2 - 2310*a^7*b^3*x^(7/3) + 9240*a^8*b^2*x^(8/3) + 41580*a^9*b*x^3 + 27720*a^10*x^(10/3)))
/((b + a*x^(1/3))^2*x^3) - 27720*a^9*Log[b + a*x^(1/3)] + 9240*a^9*Log[x])/(168*b^12)

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Maple [A]  time = 0.012, size = 156, normalized size = 0.9 \begin{align*} -{\frac{3\,{a}^{9}}{2\,{b}^{10}} \left ( b+a\sqrt [3]{x} \right ) ^{-2}}-30\,{\frac{{a}^{9}}{{b}^{11} \left ( b+a\sqrt [3]{x} \right ) }}-{\frac{1}{3\,{b}^{3}{x}^{3}}}+{\frac{9\,a}{8\,{b}^{4}}{x}^{-{\frac{8}{3}}}}-{\frac{18\,{a}^{2}}{7\,{b}^{5}}{x}^{-{\frac{7}{3}}}}+5\,{\frac{{a}^{3}}{{b}^{6}{x}^{2}}}-9\,{\frac{{a}^{4}}{{b}^{7}{x}^{5/3}}}+{\frac{63\,{a}^{5}}{4\,{b}^{8}}{x}^{-{\frac{4}{3}}}}-28\,{\frac{{a}^{6}}{{b}^{9}x}}+54\,{\frac{{a}^{7}}{{b}^{10}{x}^{2/3}}}-135\,{\frac{{a}^{8}}{{b}^{11}\sqrt [3]{x}}}+165\,{\frac{{a}^{9}\ln \left ( b+a\sqrt [3]{x} \right ) }{{b}^{12}}}-55\,{\frac{{a}^{9}\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))^3/x^5,x)

[Out]

-3/2*a^9/b^10/(b+a*x^(1/3))^2-30*a^9/b^11/(b+a*x^(1/3))-1/3/b^3/x^3+9/8*a/b^4/x^(8/3)-18/7*a^2/b^5/x^(7/3)+5*a
^3/b^6/x^2-9*a^4/b^7/x^(5/3)+63/4*a^5/b^8/x^(4/3)-28*a^6/b^9/x+54*a^7/b^10/x^(2/3)-135*a^8/b^11/x^(1/3)+165*a^
9*ln(b+a*x^(1/3))/b^12-55*a^9*ln(x)/b^12

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Maxima [A]  time = 1.01225, size = 266, normalized size = 1.45 \begin{align*} \frac{165 \, a^{9} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{b^{12}} - \frac{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{9}}{3 \, b^{12}} + \frac{33 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{8} a}{8 \, b^{12}} - \frac{165 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{7} a^{2}}{7 \, b^{12}} + \frac{165 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{6} a^{3}}{2 \, b^{12}} - \frac{198 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{5} a^{4}}{b^{12}} + \frac{693 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{4} a^{5}}{2 \, b^{12}} - \frac{462 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{3} a^{6}}{b^{12}} + \frac{495 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2} a^{7}}{b^{12}} - \frac{495 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} a^{8}}{b^{12}} + \frac{33 \, a^{10}}{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} b^{12}} - \frac{3 \, a^{11}}{2 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2} b^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

165*a^9*log(a + b/x^(1/3))/b^12 - 1/3*(a + b/x^(1/3))^9/b^12 + 33/8*(a + b/x^(1/3))^8*a/b^12 - 165/7*(a + b/x^
(1/3))^7*a^2/b^12 + 165/2*(a + b/x^(1/3))^6*a^3/b^12 - 198*(a + b/x^(1/3))^5*a^4/b^12 + 693/2*(a + b/x^(1/3))^
4*a^5/b^12 - 462*(a + b/x^(1/3))^3*a^6/b^12 + 495*(a + b/x^(1/3))^2*a^7/b^12 - 495*(a + b/x^(1/3))*a^8/b^12 +
33*a^10/((a + b/x^(1/3))*b^12) - 3/2*a^11/((a + b/x^(1/3))^2*b^12)

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Fricas [A]  time = 1.62866, size = 625, normalized size = 3.42 \begin{align*} -\frac{9240 \, a^{12} b^{3} x^{4} + 13860 \, a^{9} b^{6} x^{3} + 3080 \, a^{6} b^{9} x^{2} - 728 \, a^{3} b^{12} x + 56 \, b^{15} - 27720 \,{\left (a^{15} x^{5} + 2 \, a^{12} b^{3} x^{4} + a^{9} b^{6} x^{3}\right )} \log \left (a x^{\frac{1}{3}} + b\right ) + 27720 \,{\left (a^{15} x^{5} + 2 \, a^{12} b^{3} x^{4} + a^{9} b^{6} x^{3}\right )} \log \left (x^{\frac{1}{3}}\right ) + 18 \,{\left (1540 \, a^{14} b x^{4} + 2695 \, a^{11} b^{4} x^{3} + 990 \, a^{8} b^{7} x^{2} - 99 \, a^{5} b^{10} x + 24 \, a^{2} b^{13}\right )} x^{\frac{2}{3}} - 63 \,{\left (220 \, a^{13} b^{2} x^{4} + 352 \, a^{10} b^{5} x^{3} + 99 \, a^{7} b^{8} x^{2} - 18 \, a^{4} b^{11} x + 3 \, a b^{14}\right )} x^{\frac{1}{3}}}{168 \,{\left (a^{6} b^{12} x^{5} + 2 \, a^{3} b^{15} x^{4} + b^{18} x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/168*(9240*a^12*b^3*x^4 + 13860*a^9*b^6*x^3 + 3080*a^6*b^9*x^2 - 728*a^3*b^12*x + 56*b^15 - 27720*(a^15*x^5
+ 2*a^12*b^3*x^4 + a^9*b^6*x^3)*log(a*x^(1/3) + b) + 27720*(a^15*x^5 + 2*a^12*b^3*x^4 + a^9*b^6*x^3)*log(x^(1/
3)) + 18*(1540*a^14*b*x^4 + 2695*a^11*b^4*x^3 + 990*a^8*b^7*x^2 - 99*a^5*b^10*x + 24*a^2*b^13)*x^(2/3) - 63*(2
20*a^13*b^2*x^4 + 352*a^10*b^5*x^3 + 99*a^7*b^8*x^2 - 18*a^4*b^11*x + 3*a*b^14)*x^(1/3))/(a^6*b^12*x^5 + 2*a^3
*b^15*x^4 + b^18*x^3)

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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))**3/x**5,x)

[Out]

Timed out

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Giac [A]  time = 1.17818, size = 211, normalized size = 1.15 \begin{align*} \frac{165 \, a^{9} \log \left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{b^{12}} - \frac{55 \, a^{9} \log \left ({\left | x \right |}\right )}{b^{12}} - \frac{27720 \, a^{10} b x^{\frac{10}{3}} + 41580 \, a^{9} b^{2} x^{3} + 9240 \, a^{8} b^{3} x^{\frac{8}{3}} - 2310 \, a^{7} b^{4} x^{\frac{7}{3}} + 924 \, a^{6} b^{5} x^{2} - 462 \, a^{5} b^{6} x^{\frac{5}{3}} + 264 \, a^{4} b^{7} x^{\frac{4}{3}} - 165 \, a^{3} b^{8} x + 110 \, a^{2} b^{9} x^{\frac{2}{3}} - 77 \, a b^{10} x^{\frac{1}{3}} + 56 \, b^{11}}{168 \,{\left (a x^{\frac{1}{3}} + b\right )}^{2} b^{12} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

165*a^9*log(abs(a*x^(1/3) + b))/b^12 - 55*a^9*log(abs(x))/b^12 - 1/168*(27720*a^10*b*x^(10/3) + 41580*a^9*b^2*
x^3 + 9240*a^8*b^3*x^(8/3) - 2310*a^7*b^4*x^(7/3) + 924*a^6*b^5*x^2 - 462*a^5*b^6*x^(5/3) + 264*a^4*b^7*x^(4/3
) - 165*a^3*b^8*x + 110*a^2*b^9*x^(2/3) - 77*a*b^10*x^(1/3) + 56*b^11)/((a*x^(1/3) + b)^2*b^12*x^3)