3.2427 \(\int \frac{1}{\left (a+\frac{b}{\sqrt [3]{x}}\right )^2 x^3} \, dx\)

Optimal. Leaf size=97 \[ -\frac{15 a^4 \log \left (a \sqrt [3]{x}+b\right )}{b^6}+\frac{5 a^4 \log (x)}{b^6}+\frac{3 a^4}{b^5 \left (a \sqrt [3]{x}+b\right )}+\frac{12 a^3}{b^5 \sqrt [3]{x}}-\frac{9 a^2}{2 b^4 x^{2/3}}+\frac{2 a}{b^3 x}-\frac{3}{4 b^2 x^{4/3}} \]

[Out]

(3*a^4)/(b^5*(b + a*x^(1/3))) - 3/(4*b^2*x^(4/3)) + (2*a)/(b^3*x) - (9*a^2)/(2*b
^4*x^(2/3)) + (12*a^3)/(b^5*x^(1/3)) - (15*a^4*Log[b + a*x^(1/3)])/b^6 + (5*a^4*
Log[x])/b^6

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Rubi [A]  time = 0.157355, antiderivative size = 97, 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 \[ -\frac{15 a^4 \log \left (a \sqrt [3]{x}+b\right )}{b^6}+\frac{5 a^4 \log (x)}{b^6}+\frac{3 a^4}{b^5 \left (a \sqrt [3]{x}+b\right )}+\frac{12 a^3}{b^5 \sqrt [3]{x}}-\frac{9 a^2}{2 b^4 x^{2/3}}+\frac{2 a}{b^3 x}-\frac{3}{4 b^2 x^{4/3}} \]

Antiderivative was successfully verified.

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

[Out]

(3*a^4)/(b^5*(b + a*x^(1/3))) - 3/(4*b^2*x^(4/3)) + (2*a)/(b^3*x) - (9*a^2)/(2*b
^4*x^(2/3)) + (12*a^3)/(b^5*x^(1/3)) - (15*a^4*Log[b + a*x^(1/3)])/b^6 + (5*a^4*
Log[x])/b^6

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Rubi in Sympy [A]  time = 21.6068, size = 99, normalized size = 1.02 \[ \frac{3 a^{4}}{b^{5} \left (a \sqrt [3]{x} + b\right )} + \frac{15 a^{4} \log{\left (\sqrt [3]{x} \right )}}{b^{6}} - \frac{15 a^{4} \log{\left (a \sqrt [3]{x} + b \right )}}{b^{6}} + \frac{12 a^{3}}{b^{5} \sqrt [3]{x}} - \frac{9 a^{2}}{2 b^{4} x^{\frac{2}{3}}} + \frac{2 a}{b^{3} x} - \frac{3}{4 b^{2} x^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a**4/(b**5*(a*x**(1/3) + b)) + 15*a**4*log(x**(1/3))/b**6 - 15*a**4*log(a*x**(
1/3) + b)/b**6 + 12*a**3/(b**5*x**(1/3)) - 9*a**2/(2*b**4*x**(2/3)) + 2*a/(b**3*
x) - 3/(4*b**2*x**(4/3))

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Mathematica [A]  time = 0.229858, size = 89, normalized size = 0.92 \[ \frac{-60 a^4 \log \left (a \sqrt [3]{x}+b\right )+20 a^4 \log (x)+b \left (\frac{12 a^4}{a \sqrt [3]{x}+b}+\frac{48 a^3}{\sqrt [3]{x}}-\frac{18 a^2 b}{x^{2/3}}+\frac{8 a b^2}{x}-\frac{3 b^3}{x^{4/3}}\right )}{4 b^6} \]

Antiderivative was successfully verified.

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

[Out]

(b*((12*a^4)/(b + a*x^(1/3)) - (3*b^3)/x^(4/3) + (8*a*b^2)/x - (18*a^2*b)/x^(2/3
) + (48*a^3)/x^(1/3)) - 60*a^4*Log[b + a*x^(1/3)] + 20*a^4*Log[x])/(4*b^6)

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Maple [A]  time = 0.017, size = 84, normalized size = 0.9 \[ 3\,{\frac{{a}^{4}}{{b}^{5} \left ( b+a\sqrt [3]{x} \right ) }}-{\frac{3}{4\,{b}^{2}}{x}^{-{\frac{4}{3}}}}+2\,{\frac{a}{{b}^{3}x}}-{\frac{9\,{a}^{2}}{2\,{b}^{4}}{x}^{-{\frac{2}{3}}}}+12\,{\frac{{a}^{3}}{{b}^{5}\sqrt [3]{x}}}-15\,{\frac{{a}^{4}\ln \left ( b+a\sqrt [3]{x} \right ) }{{b}^{6}}}+5\,{\frac{{a}^{4}\ln \left ( x \right ) }{{b}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^(1/3))^2/x^3,x)

[Out]

3*a^4/b^5/(b+a*x^(1/3))-3/4/b^2/x^(4/3)+2*a/b^3/x-9/2*a^2/b^4/x^(2/3)+12*a^3/b^5
/x^(1/3)-15*a^4*ln(b+a*x^(1/3))/b^6+5*a^4*ln(x)/b^6

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Maxima [A]  time = 1.43851, size = 128, normalized size = 1.32 \[ -\frac{15 \, a^{4} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{b^{6}} - \frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{4}}{4 \, b^{6}} + \frac{5 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{3} a}{b^{6}} - \frac{15 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2} a^{2}}{b^{6}} + \frac{30 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} a^{3}}{b^{6}} - \frac{3 \, a^{5}}{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15*a^4*log(a + b/x^(1/3))/b^6 - 3/4*(a + b/x^(1/3))^4/b^6 + 5*(a + b/x^(1/3))^3
*a/b^6 - 15*(a + b/x^(1/3))^2*a^2/b^6 + 30*(a + b/x^(1/3))*a^3/b^6 - 3*a^5/((a +
b/x^(1/3))*b^6)

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Fricas [A]  time = 0.237325, size = 151, normalized size = 1.56 \[ \frac{30 \, a^{3} b^{2} x - 10 \, a^{2} b^{3} x^{\frac{2}{3}} - 3 \, b^{5} - 60 \,{\left (a^{5} x^{\frac{5}{3}} + a^{4} b x^{\frac{4}{3}}\right )} \log \left (a x^{\frac{1}{3}} + b\right ) + 60 \,{\left (a^{5} x^{\frac{5}{3}} + a^{4} b x^{\frac{4}{3}}\right )} \log \left (x^{\frac{1}{3}}\right ) + 5 \,{\left (12 \, a^{4} b x + a b^{4}\right )} x^{\frac{1}{3}}}{4 \,{\left (a b^{6} x^{\frac{5}{3}} + b^{7} x^{\frac{4}{3}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(30*a^3*b^2*x - 10*a^2*b^3*x^(2/3) - 3*b^5 - 60*(a^5*x^(5/3) + a^4*b*x^(4/3)
)*log(a*x^(1/3) + b) + 60*(a^5*x^(5/3) + a^4*b*x^(4/3))*log(x^(1/3)) + 5*(12*a^4
*b*x + a*b^4)*x^(1/3))/(a*b^6*x^(5/3) + b^7*x^(4/3))

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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**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.217641, size = 122, normalized size = 1.26 \[ -\frac{15 \, a^{4}{\rm ln}\left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{b^{6}} + \frac{5 \, a^{4}{\rm ln}\left ({\left | x \right |}\right )}{b^{6}} + \frac{60 \, a^{4} b x^{\frac{4}{3}} + 30 \, a^{3} b^{2} x - 10 \, a^{2} b^{3} x^{\frac{2}{3}} + 5 \, a b^{4} x^{\frac{1}{3}} - 3 \, b^{5}}{4 \,{\left (a x^{\frac{1}{3}} + b\right )} b^{6} x^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15*a^4*ln(abs(a*x^(1/3) + b))/b^6 + 5*a^4*ln(abs(x))/b^6 + 1/4*(60*a^4*b*x^(4/3
) + 30*a^3*b^2*x - 10*a^2*b^3*x^(2/3) + 5*a*b^4*x^(1/3) - 3*b^5)/((a*x^(1/3) + b
)*b^6*x^(4/3))