3.2436 \(\int \frac{1}{\left (a+\frac{b}{\sqrt [3]{x}}\right )^3 x^4} \, dx\)

Optimal. Leaf size=146 \[ -\frac{84 a^6 \log \left (a \sqrt [3]{x}+b\right )}{b^9}+\frac{28 a^6 \log (x)}{b^9}+\frac{21 a^6}{b^8 \left (a \sqrt [3]{x}+b\right )}+\frac{3 a^6}{2 b^7 \left (a \sqrt [3]{x}+b\right )^2}+\frac{63 a^5}{b^8 \sqrt [3]{x}}-\frac{45 a^4}{2 b^7 x^{2/3}}+\frac{10 a^3}{b^6 x}-\frac{9 a^2}{2 b^5 x^{4/3}}+\frac{9 a}{5 b^4 x^{5/3}}-\frac{1}{2 b^3 x^2} \]

[Out]

(3*a^6)/(2*b^7*(b + a*x^(1/3))^2) + (21*a^6)/(b^8*(b + a*x^(1/3))) - 1/(2*b^3*x^
2) + (9*a)/(5*b^4*x^(5/3)) - (9*a^2)/(2*b^5*x^(4/3)) + (10*a^3)/(b^6*x) - (45*a^
4)/(2*b^7*x^(2/3)) + (63*a^5)/(b^8*x^(1/3)) - (84*a^6*Log[b + a*x^(1/3)])/b^9 +
(28*a^6*Log[x])/b^9

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Rubi [A]  time = 0.243702, antiderivative size = 146, 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 \[ -\frac{84 a^6 \log \left (a \sqrt [3]{x}+b\right )}{b^9}+\frac{28 a^6 \log (x)}{b^9}+\frac{21 a^6}{b^8 \left (a \sqrt [3]{x}+b\right )}+\frac{3 a^6}{2 b^7 \left (a \sqrt [3]{x}+b\right )^2}+\frac{63 a^5}{b^8 \sqrt [3]{x}}-\frac{45 a^4}{2 b^7 x^{2/3}}+\frac{10 a^3}{b^6 x}-\frac{9 a^2}{2 b^5 x^{4/3}}+\frac{9 a}{5 b^4 x^{5/3}}-\frac{1}{2 b^3 x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^(1/3))^3*x^4),x]

[Out]

(3*a^6)/(2*b^7*(b + a*x^(1/3))^2) + (21*a^6)/(b^8*(b + a*x^(1/3))) - 1/(2*b^3*x^
2) + (9*a)/(5*b^4*x^(5/3)) - (9*a^2)/(2*b^5*x^(4/3)) + (10*a^3)/(b^6*x) - (45*a^
4)/(2*b^7*x^(2/3)) + (63*a^5)/(b^8*x^(1/3)) - (84*a^6*Log[b + a*x^(1/3)])/b^9 +
(28*a^6*Log[x])/b^9

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Rubi in Sympy [A]  time = 42.6693, size = 148, normalized size = 1.01 \[ \frac{3 a^{6}}{2 b^{7} \left (a \sqrt [3]{x} + b\right )^{2}} + \frac{21 a^{6}}{b^{8} \left (a \sqrt [3]{x} + b\right )} + \frac{84 a^{6} \log{\left (\sqrt [3]{x} \right )}}{b^{9}} - \frac{84 a^{6} \log{\left (a \sqrt [3]{x} + b \right )}}{b^{9}} + \frac{63 a^{5}}{b^{8} \sqrt [3]{x}} - \frac{45 a^{4}}{2 b^{7} x^{\frac{2}{3}}} + \frac{10 a^{3}}{b^{6} x} - \frac{9 a^{2}}{2 b^{5} x^{\frac{4}{3}}} + \frac{9 a}{5 b^{4} x^{\frac{5}{3}}} - \frac{1}{2 b^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**(1/3))**3/x**4,x)

[Out]

3*a**6/(2*b**7*(a*x**(1/3) + b)**2) + 21*a**6/(b**8*(a*x**(1/3) + b)) + 84*a**6*
log(x**(1/3))/b**9 - 84*a**6*log(a*x**(1/3) + b)/b**9 + 63*a**5/(b**8*x**(1/3))
- 45*a**4/(2*b**7*x**(2/3)) + 10*a**3/(b**6*x) - 9*a**2/(2*b**5*x**(4/3)) + 9*a/
(5*b**4*x**(5/3)) - 1/(2*b**3*x**2)

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Mathematica [A]  time = 0.277282, size = 130, normalized size = 0.89 \[ \frac{-840 a^6 \log \left (a \sqrt [3]{x}+b\right )+280 a^6 \log (x)+\frac{b \left (840 a^7 x^{7/3}+1260 a^6 b x^2+280 a^5 b^2 x^{5/3}-70 a^4 b^3 x^{4/3}+28 a^3 b^4 x-14 a^2 b^5 x^{2/3}+8 a b^6 \sqrt [3]{x}-5 b^7\right )}{x^2 \left (a \sqrt [3]{x}+b\right )^2}}{10 b^9} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^(1/3))^3*x^4),x]

[Out]

((b*(-5*b^7 + 8*a*b^6*x^(1/3) - 14*a^2*b^5*x^(2/3) + 28*a^3*b^4*x - 70*a^4*b^3*x
^(4/3) + 280*a^5*b^2*x^(5/3) + 1260*a^6*b*x^2 + 840*a^7*x^(7/3)))/((b + a*x^(1/3
))^2*x^2) - 840*a^6*Log[b + a*x^(1/3)] + 280*a^6*Log[x])/(10*b^9)

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Maple [A]  time = 0.019, size = 123, normalized size = 0.8 \[{\frac{3\,{a}^{6}}{2\,{b}^{7}} \left ( b+a\sqrt [3]{x} \right ) ^{-2}}+21\,{\frac{{a}^{6}}{{b}^{8} \left ( b+a\sqrt [3]{x} \right ) }}-{\frac{1}{2\,{b}^{3}{x}^{2}}}+{\frac{9\,a}{5\,{b}^{4}}{x}^{-{\frac{5}{3}}}}-{\frac{9\,{a}^{2}}{2\,{b}^{5}}{x}^{-{\frac{4}{3}}}}+10\,{\frac{{a}^{3}}{{b}^{6}x}}-{\frac{45\,{a}^{4}}{2\,{b}^{7}}{x}^{-{\frac{2}{3}}}}+63\,{\frac{{a}^{5}}{{b}^{8}\sqrt [3]{x}}}-84\,{\frac{{a}^{6}\ln \left ( b+a\sqrt [3]{x} \right ) }{{b}^{9}}}+28\,{\frac{{a}^{6}\ln \left ( x \right ) }{{b}^{9}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^(1/3))^3/x^4,x)

[Out]

3/2*a^6/b^7/(b+a*x^(1/3))^2+21*a^6/b^8/(b+a*x^(1/3))-1/2/b^3/x^2+9/5*a/b^4/x^(5/
3)-9/2*a^2/b^5/x^(4/3)+10*a^3/b^6/x-45/2*a^4/b^7/x^(2/3)+63*a^5/b^8/x^(1/3)-84*a
^6*ln(b+a*x^(1/3))/b^9+28*a^6*ln(x)/b^9

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Maxima [A]  time = 1.44466, size = 197, normalized size = 1.35 \[ -\frac{84 \, a^{6} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{b^{9}} - \frac{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{6}}{2 \, b^{9}} + \frac{24 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{5} a}{5 \, b^{9}} - \frac{21 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{4} a^{2}}{b^{9}} + \frac{56 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{3} a^{3}}{b^{9}} - \frac{105 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2} a^{4}}{b^{9}} + \frac{168 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} a^{5}}{b^{9}} - \frac{24 \, a^{7}}{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} b^{9}} + \frac{3 \, a^{8}}{2 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2} b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^(1/3))^3*x^4),x, algorithm="maxima")

[Out]

-84*a^6*log(a + b/x^(1/3))/b^9 - 1/2*(a + b/x^(1/3))^6/b^9 + 24/5*(a + b/x^(1/3)
)^5*a/b^9 - 21*(a + b/x^(1/3))^4*a^2/b^9 + 56*(a + b/x^(1/3))^3*a^3/b^9 - 105*(a
+ b/x^(1/3))^2*a^4/b^9 + 168*(a + b/x^(1/3))*a^5/b^9 - 24*a^7/((a + b/x^(1/3))*
b^9) + 3/2*a^8/((a + b/x^(1/3))^2*b^9)

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Fricas [A]  time = 0.238961, size = 243, normalized size = 1.66 \[ \frac{1260 \, a^{6} b^{2} x^{2} + 28 \, a^{3} b^{5} x - 5 \, b^{8} - 840 \,{\left (a^{8} x^{\frac{8}{3}} + 2 \, a^{7} b x^{\frac{7}{3}} + a^{6} b^{2} x^{2}\right )} \log \left (a x^{\frac{1}{3}} + b\right ) + 840 \,{\left (a^{8} x^{\frac{8}{3}} + 2 \, a^{7} b x^{\frac{7}{3}} + a^{6} b^{2} x^{2}\right )} \log \left (x^{\frac{1}{3}}\right ) + 14 \,{\left (20 \, a^{5} b^{3} x - a^{2} b^{6}\right )} x^{\frac{2}{3}} + 2 \,{\left (420 \, a^{7} b x^{2} - 35 \, a^{4} b^{4} x + 4 \, a b^{7}\right )} x^{\frac{1}{3}}}{10 \,{\left (a^{2} b^{9} x^{\frac{8}{3}} + 2 \, a b^{10} x^{\frac{7}{3}} + b^{11} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^(1/3))^3*x^4),x, algorithm="fricas")

[Out]

1/10*(1260*a^6*b^2*x^2 + 28*a^3*b^5*x - 5*b^8 - 840*(a^8*x^(8/3) + 2*a^7*b*x^(7/
3) + a^6*b^2*x^2)*log(a*x^(1/3) + b) + 840*(a^8*x^(8/3) + 2*a^7*b*x^(7/3) + a^6*
b^2*x^2)*log(x^(1/3)) + 14*(20*a^5*b^3*x - a^2*b^6)*x^(2/3) + 2*(420*a^7*b*x^2 -
35*a^4*b^4*x + 4*a*b^7)*x^(1/3))/(a^2*b^9*x^(8/3) + 2*a*b^10*x^(7/3) + b^11*x^2
)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**(1/3))**3/x**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.21801, size = 166, normalized size = 1.14 \[ -\frac{84 \, a^{6}{\rm ln}\left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{b^{9}} + \frac{28 \, a^{6}{\rm ln}\left ({\left | x \right |}\right )}{b^{9}} + \frac{840 \, a^{7} b x^{\frac{7}{3}} + 1260 \, a^{6} b^{2} x^{2} + 280 \, a^{5} b^{3} x^{\frac{5}{3}} - 70 \, a^{4} b^{4} x^{\frac{4}{3}} + 28 \, a^{3} b^{5} x - 14 \, a^{2} b^{6} x^{\frac{2}{3}} + 8 \, a b^{7} x^{\frac{1}{3}} - 5 \, b^{8}}{10 \,{\left (a x^{\frac{1}{3}} + b\right )}^{2} b^{9} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^(1/3))^3*x^4),x, algorithm="giac")

[Out]

-84*a^6*ln(abs(a*x^(1/3) + b))/b^9 + 28*a^6*ln(abs(x))/b^9 + 1/10*(840*a^7*b*x^(
7/3) + 1260*a^6*b^2*x^2 + 280*a^5*b^3*x^(5/3) - 70*a^4*b^4*x^(4/3) + 28*a^3*b^5*
x - 14*a^2*b^6*x^(2/3) + 8*a*b^7*x^(1/3) - 5*b^8)/((a*x^(1/3) + b)^2*b^9*x^2)