3.2428 \(\int \frac{1}{\left (a+\frac{b}{\sqrt [3]{x}}\right )^2 x^4} \, dx\)

Optimal. Leaf size=133 \[ \frac{24 a^7 \log \left (a \sqrt [3]{x}+b\right )}{b^9}-\frac{8 a^7 \log (x)}{b^9}-\frac{3 a^7}{b^8 \left (a \sqrt [3]{x}+b\right )}-\frac{21 a^6}{b^8 \sqrt [3]{x}}+\frac{9 a^5}{b^7 x^{2/3}}-\frac{5 a^4}{b^6 x}+\frac{3 a^3}{b^5 x^{4/3}}-\frac{9 a^2}{5 b^4 x^{5/3}}+\frac{a}{b^3 x^2}-\frac{3}{7 b^2 x^{7/3}} \]

[Out]

(-3*a^7)/(b^8*(b + a*x^(1/3))) - 3/(7*b^2*x^(7/3)) + a/(b^3*x^2) - (9*a^2)/(5*b^
4*x^(5/3)) + (3*a^3)/(b^5*x^(4/3)) - (5*a^4)/(b^6*x) + (9*a^5)/(b^7*x^(2/3)) - (
21*a^6)/(b^8*x^(1/3)) + (24*a^7*Log[b + a*x^(1/3)])/b^9 - (8*a^7*Log[x])/b^9

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Rubi [A]  time = 0.221285, antiderivative size = 133, 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 \[ \frac{24 a^7 \log \left (a \sqrt [3]{x}+b\right )}{b^9}-\frac{8 a^7 \log (x)}{b^9}-\frac{3 a^7}{b^8 \left (a \sqrt [3]{x}+b\right )}-\frac{21 a^6}{b^8 \sqrt [3]{x}}+\frac{9 a^5}{b^7 x^{2/3}}-\frac{5 a^4}{b^6 x}+\frac{3 a^3}{b^5 x^{4/3}}-\frac{9 a^2}{5 b^4 x^{5/3}}+\frac{a}{b^3 x^2}-\frac{3}{7 b^2 x^{7/3}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^(1/3))^2*x^4),x]

[Out]

(-3*a^7)/(b^8*(b + a*x^(1/3))) - 3/(7*b^2*x^(7/3)) + a/(b^3*x^2) - (9*a^2)/(5*b^
4*x^(5/3)) + (3*a^3)/(b^5*x^(4/3)) - (5*a^4)/(b^6*x) + (9*a^5)/(b^7*x^(2/3)) - (
21*a^6)/(b^8*x^(1/3)) + (24*a^7*Log[b + a*x^(1/3)])/b^9 - (8*a^7*Log[x])/b^9

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Rubi in Sympy [A]  time = 32.7418, size = 136, normalized size = 1.02 \[ - \frac{3 a^{7}}{b^{8} \left (a \sqrt [3]{x} + b\right )} - \frac{24 a^{7} \log{\left (\sqrt [3]{x} \right )}}{b^{9}} + \frac{24 a^{7} \log{\left (a \sqrt [3]{x} + b \right )}}{b^{9}} - \frac{21 a^{6}}{b^{8} \sqrt [3]{x}} + \frac{9 a^{5}}{b^{7} x^{\frac{2}{3}}} - \frac{5 a^{4}}{b^{6} x} + \frac{3 a^{3}}{b^{5} x^{\frac{4}{3}}} - \frac{9 a^{2}}{5 b^{4} x^{\frac{5}{3}}} + \frac{a}{b^{3} x^{2}} - \frac{3}{7 b^{2} x^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**(1/3))**2/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.359899, size = 132, normalized size = 0.99 \[ -\frac{-840 a^7 \log \left (a \sqrt [3]{x}+b\right )+280 a^7 \log (x)+\frac{b \left (840 a^7 x^{7/3}+420 a^6 b x^2-140 a^5 b^2 x^{5/3}+70 a^4 b^3 x^{4/3}-42 a^3 b^4 x+28 a^2 b^5 x^{2/3}-20 a b^6 \sqrt [3]{x}+15 b^7\right )}{x^{7/3} \left (a \sqrt [3]{x}+b\right )}}{35 b^9} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^(1/3))^2*x^4),x]

[Out]

-((b*(15*b^7 - 20*a*b^6*x^(1/3) + 28*a^2*b^5*x^(2/3) - 42*a^3*b^4*x + 70*a^4*b^3
*x^(4/3) - 140*a^5*b^2*x^(5/3) + 420*a^6*b*x^2 + 840*a^7*x^(7/3)))/((b + a*x^(1/
3))*x^(7/3)) - 840*a^7*Log[b + a*x^(1/3)] + 280*a^7*Log[x])/(35*b^9)

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Maple [A]  time = 0.02, size = 116, normalized size = 0.9 \[ -3\,{\frac{{a}^{7}}{{b}^{8} \left ( b+a\sqrt [3]{x} \right ) }}-{\frac{3}{7\,{b}^{2}}{x}^{-{\frac{7}{3}}}}+{\frac{a}{{b}^{3}{x}^{2}}}-{\frac{9\,{a}^{2}}{5\,{b}^{4}}{x}^{-{\frac{5}{3}}}}+3\,{\frac{{a}^{3}}{{b}^{5}{x}^{4/3}}}-5\,{\frac{{a}^{4}}{{b}^{6}x}}+9\,{\frac{{a}^{5}}{{b}^{7}{x}^{2/3}}}-21\,{\frac{{a}^{6}}{{b}^{8}\sqrt [3]{x}}}+24\,{\frac{{a}^{7}\ln \left ( b+a\sqrt [3]{x} \right ) }{{b}^{9}}}-8\,{\frac{{a}^{7}\ln \left ( x \right ) }{{b}^{9}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^(1/3))^2/x^4,x)

[Out]

-3*a^7/b^8/(b+a*x^(1/3))-3/7/b^2/x^(7/3)+a/b^3/x^2-9/5*a^2/b^4/x^(5/3)+3*a^3/b^5
/x^(4/3)-5*a^4/b^6/x+9*a^5/b^7/x^(2/3)-21*a^6/b^8/x^(1/3)+24*a^7*ln(b+a*x^(1/3))
/b^9-8*a^7*ln(x)/b^9

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Maxima [A]  time = 1.44979, size = 197, normalized size = 1.48 \[ \frac{24 \, a^{7} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{b^{9}} - \frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{7}}{7 \, b^{9}} + \frac{4 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{6} a}{b^{9}} - \frac{84 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{5} a^{2}}{5 \, b^{9}} + \frac{42 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{4} a^{3}}{b^{9}} - \frac{70 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{3} a^{4}}{b^{9}} + \frac{84 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2} a^{5}}{b^{9}} - \frac{84 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} a^{6}}{b^{9}} + \frac{3 \, a^{8}}{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^(1/3))^2*x^4),x, algorithm="maxima")

[Out]

24*a^7*log(a + b/x^(1/3))/b^9 - 3/7*(a + b/x^(1/3))^7/b^9 + 4*(a + b/x^(1/3))^6*
a/b^9 - 84/5*(a + b/x^(1/3))^5*a^2/b^9 + 42*(a + b/x^(1/3))^4*a^3/b^9 - 70*(a +
b/x^(1/3))^3*a^4/b^9 + 84*(a + b/x^(1/3))^2*a^5/b^9 - 84*(a + b/x^(1/3))*a^6/b^9
+ 3*a^8/((a + b/x^(1/3))*b^9)

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Fricas [A]  time = 0.238848, size = 198, normalized size = 1.49 \[ -\frac{420 \, a^{6} b^{2} x^{2} - 42 \, a^{3} b^{5} x + 15 \, b^{8} - 840 \,{\left (a^{8} x^{\frac{8}{3}} + a^{7} b x^{\frac{7}{3}}\right )} \log \left (a x^{\frac{1}{3}} + b\right ) + 840 \,{\left (a^{8} x^{\frac{8}{3}} + a^{7} b x^{\frac{7}{3}}\right )} \log \left (x^{\frac{1}{3}}\right ) - 28 \,{\left (5 \, a^{5} b^{3} x - a^{2} b^{6}\right )} x^{\frac{2}{3}} + 10 \,{\left (84 \, a^{7} b x^{2} + 7 \, a^{4} b^{4} x - 2 \, a b^{7}\right )} x^{\frac{1}{3}}}{35 \,{\left (a b^{9} x^{\frac{8}{3}} + b^{10} x^{\frac{7}{3}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^(1/3))^2*x^4),x, algorithm="fricas")

[Out]

-1/35*(420*a^6*b^2*x^2 - 42*a^3*b^5*x + 15*b^8 - 840*(a^8*x^(8/3) + a^7*b*x^(7/3
))*log(a*x^(1/3) + b) + 840*(a^8*x^(8/3) + a^7*b*x^(7/3))*log(x^(1/3)) - 28*(5*a
^5*b^3*x - a^2*b^6)*x^(2/3) + 10*(84*a^7*b*x^2 + 7*a^4*b^4*x - 2*a*b^7)*x^(1/3))
/(a*b^9*x^(8/3) + b^10*x^(7/3))

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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))**2/x**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216691, size = 166, normalized size = 1.25 \[ \frac{24 \, a^{7}{\rm ln}\left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{b^{9}} - \frac{8 \, a^{7}{\rm ln}\left ({\left | x \right |}\right )}{b^{9}} - \frac{840 \, a^{7} b x^{\frac{7}{3}} + 420 \, a^{6} b^{2} x^{2} - 140 \, a^{5} b^{3} x^{\frac{5}{3}} + 70 \, a^{4} b^{4} x^{\frac{4}{3}} - 42 \, a^{3} b^{5} x + 28 \, a^{2} b^{6} x^{\frac{2}{3}} - 20 \, a b^{7} x^{\frac{1}{3}} + 15 \, b^{8}}{35 \,{\left (a x^{\frac{1}{3}} + b\right )} b^{9} x^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^(1/3))^2*x^4),x, algorithm="giac")

[Out]

24*a^7*ln(abs(a*x^(1/3) + b))/b^9 - 8*a^7*ln(abs(x))/b^9 - 1/35*(840*a^7*b*x^(7/
3) + 420*a^6*b^2*x^2 - 140*a^5*b^3*x^(5/3) + 70*a^4*b^4*x^(4/3) - 42*a^3*b^5*x +
28*a^2*b^6*x^(2/3) - 20*a*b^7*x^(1/3) + 15*b^8)/((a*x^(1/3) + b)*b^9*x^(7/3))