3.2364 \(\int \frac{1}{\left (a+b \sqrt [3]{x}\right )^2 x^3} \, dx\)

Optimal. Leaf size=125 \[ -\frac{21 b^6 \log \left (a+b \sqrt [3]{x}\right )}{a^8}+\frac{7 b^6 \log (x)}{a^8}+\frac{3 b^6}{a^7 \left (a+b \sqrt [3]{x}\right )}+\frac{18 b^5}{a^7 \sqrt [3]{x}}-\frac{15 b^4}{2 a^6 x^{2/3}}+\frac{4 b^3}{a^5 x}-\frac{9 b^2}{4 a^4 x^{4/3}}+\frac{6 b}{5 a^3 x^{5/3}}-\frac{1}{2 a^2 x^2} \]

[Out]

(3*b^6)/(a^7*(a + b*x^(1/3))) - 1/(2*a^2*x^2) + (6*b)/(5*a^3*x^(5/3)) - (9*b^2)/
(4*a^4*x^(4/3)) + (4*b^3)/(a^5*x) - (15*b^4)/(2*a^6*x^(2/3)) + (18*b^5)/(a^7*x^(
1/3)) - (21*b^6*Log[a + b*x^(1/3)])/a^8 + (7*b^6*Log[x])/a^8

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Rubi [A]  time = 0.188413, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{21 b^6 \log \left (a+b \sqrt [3]{x}\right )}{a^8}+\frac{7 b^6 \log (x)}{a^8}+\frac{3 b^6}{a^7 \left (a+b \sqrt [3]{x}\right )}+\frac{18 b^5}{a^7 \sqrt [3]{x}}-\frac{15 b^4}{2 a^6 x^{2/3}}+\frac{4 b^3}{a^5 x}-\frac{9 b^2}{4 a^4 x^{4/3}}+\frac{6 b}{5 a^3 x^{5/3}}-\frac{1}{2 a^2 x^2} \]

Antiderivative was successfully verified.

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

[Out]

(3*b^6)/(a^7*(a + b*x^(1/3))) - 1/(2*a^2*x^2) + (6*b)/(5*a^3*x^(5/3)) - (9*b^2)/
(4*a^4*x^(4/3)) + (4*b^3)/(a^5*x) - (15*b^4)/(2*a^6*x^(2/3)) + (18*b^5)/(a^7*x^(
1/3)) - (21*b^6*Log[a + b*x^(1/3)])/a^8 + (7*b^6*Log[x])/a^8

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Rubi in Sympy [A]  time = 27.422, size = 128, normalized size = 1.02 \[ - \frac{1}{2 a^{2} x^{2}} + \frac{6 b}{5 a^{3} x^{\frac{5}{3}}} - \frac{9 b^{2}}{4 a^{4} x^{\frac{4}{3}}} + \frac{4 b^{3}}{a^{5} x} - \frac{15 b^{4}}{2 a^{6} x^{\frac{2}{3}}} + \frac{3 b^{6}}{a^{7} \left (a + b \sqrt [3]{x}\right )} + \frac{18 b^{5}}{a^{7} \sqrt [3]{x}} + \frac{21 b^{6} \log{\left (\sqrt [3]{x} \right )}}{a^{8}} - \frac{21 b^{6} \log{\left (a + b \sqrt [3]{x} \right )}}{a^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(2*a**2*x**2) + 6*b/(5*a**3*x**(5/3)) - 9*b**2/(4*a**4*x**(4/3)) + 4*b**3/(a*
*5*x) - 15*b**4/(2*a**6*x**(2/3)) + 3*b**6/(a**7*(a + b*x**(1/3))) + 18*b**5/(a*
*7*x**(1/3)) + 21*b**6*log(x**(1/3))/a**8 - 21*b**6*log(a + b*x**(1/3))/a**8

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Mathematica [A]  time = 0.237039, size = 117, normalized size = 0.94 \[ \frac{\frac{a \left (-10 a^6+14 a^5 b \sqrt [3]{x}-21 a^4 b^2 x^{2/3}+35 a^3 b^3 x-70 a^2 b^4 x^{4/3}+210 a b^5 x^{5/3}+420 b^6 x^2\right )}{x^2 \left (a+b \sqrt [3]{x}\right )}-420 b^6 \log \left (a+b \sqrt [3]{x}\right )+140 b^6 \log (x)}{20 a^8} \]

Antiderivative was successfully verified.

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

[Out]

((a*(-10*a^6 + 14*a^5*b*x^(1/3) - 21*a^4*b^2*x^(2/3) + 35*a^3*b^3*x - 70*a^2*b^4
*x^(4/3) + 210*a*b^5*x^(5/3) + 420*b^6*x^2))/((a + b*x^(1/3))*x^2) - 420*b^6*Log
[a + b*x^(1/3)] + 140*b^6*Log[x])/(20*a^8)

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Maple [A]  time = 0.003, size = 106, normalized size = 0.9 \[ 3\,{\frac{{b}^{6}}{{a}^{7} \left ( a+b\sqrt [3]{x} \right ) }}-{\frac{1}{2\,{a}^{2}{x}^{2}}}+{\frac{6\,b}{5\,{a}^{3}}{x}^{-{\frac{5}{3}}}}-{\frac{9\,{b}^{2}}{4\,{a}^{4}}{x}^{-{\frac{4}{3}}}}+4\,{\frac{{b}^{3}}{x{a}^{5}}}-{\frac{15\,{b}^{4}}{2\,{a}^{6}}{x}^{-{\frac{2}{3}}}}+18\,{\frac{{b}^{5}}{{a}^{7}\sqrt [3]{x}}}-21\,{\frac{{b}^{6}\ln \left ( a+b\sqrt [3]{x} \right ) }{{a}^{8}}}+7\,{\frac{{b}^{6}\ln \left ( x \right ) }{{a}^{8}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*x^(1/3))^2/x^3,x)

[Out]

3*b^6/a^7/(a+b*x^(1/3))-1/2/a^2/x^2+6/5*b/a^3/x^(5/3)-9/4*b^2/a^4/x^(4/3)+4*b^3/
x/a^5-15/2*b^4/a^6/x^(2/3)+18*b^5/a^7/x^(1/3)-21*b^6*ln(a+b*x^(1/3))/a^8+7*b^6*l
n(x)/a^8

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Maxima [A]  time = 1.44637, size = 149, normalized size = 1.19 \[ \frac{420 \, b^{6} x^{2} + 210 \, a b^{5} x^{\frac{5}{3}} - 70 \, a^{2} b^{4} x^{\frac{4}{3}} + 35 \, a^{3} b^{3} x - 21 \, a^{4} b^{2} x^{\frac{2}{3}} + 14 \, a^{5} b x^{\frac{1}{3}} - 10 \, a^{6}}{20 \,{\left (a^{7} b x^{\frac{7}{3}} + a^{8} x^{2}\right )}} - \frac{21 \, b^{6} \log \left (b x^{\frac{1}{3}} + a\right )}{a^{8}} + \frac{7 \, b^{6} \log \left (x\right )}{a^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/20*(420*b^6*x^2 + 210*a*b^5*x^(5/3) - 70*a^2*b^4*x^(4/3) + 35*a^3*b^3*x - 21*a
^4*b^2*x^(2/3) + 14*a^5*b*x^(1/3) - 10*a^6)/(a^7*b*x^(7/3) + a^8*x^2) - 21*b^6*l
og(b*x^(1/3) + a)/a^8 + 7*b^6*log(x)/a^8

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Fricas [A]  time = 0.228479, size = 184, normalized size = 1.47 \[ \frac{420 \, a b^{6} x^{2} + 35 \, a^{4} b^{3} x - 10 \, a^{7} - 420 \,{\left (b^{7} x^{\frac{7}{3}} + a b^{6} x^{2}\right )} \log \left (b x^{\frac{1}{3}} + a\right ) + 420 \,{\left (b^{7} x^{\frac{7}{3}} + a b^{6} x^{2}\right )} \log \left (x^{\frac{1}{3}}\right ) + 21 \,{\left (10 \, a^{2} b^{5} x - a^{5} b^{2}\right )} x^{\frac{2}{3}} - 14 \,{\left (5 \, a^{3} b^{4} x - a^{6} b\right )} x^{\frac{1}{3}}}{20 \,{\left (a^{8} b x^{\frac{7}{3}} + a^{9} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/20*(420*a*b^6*x^2 + 35*a^4*b^3*x - 10*a^7 - 420*(b^7*x^(7/3) + a*b^6*x^2)*log(
b*x^(1/3) + a) + 420*(b^7*x^(7/3) + a*b^6*x^2)*log(x^(1/3)) + 21*(10*a^2*b^5*x -
a^5*b^2)*x^(2/3) - 14*(5*a^3*b^4*x - a^6*b)*x^(1/3))/(a^8*b*x^(7/3) + a^9*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))**2/x**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.218207, size = 151, normalized size = 1.21 \[ -\frac{21 \, b^{6}{\rm ln}\left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{a^{8}} + \frac{7 \, b^{6}{\rm ln}\left ({\left | x \right |}\right )}{a^{8}} + \frac{420 \, a b^{6} x^{2} + 210 \, a^{2} b^{5} x^{\frac{5}{3}} - 70 \, a^{3} b^{4} x^{\frac{4}{3}} + 35 \, a^{4} b^{3} x - 21 \, a^{5} b^{2} x^{\frac{2}{3}} + 14 \, a^{6} b x^{\frac{1}{3}} - 10 \, a^{7}}{20 \,{\left (b x^{\frac{1}{3}} + a\right )} a^{8} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-21*b^6*ln(abs(b*x^(1/3) + a))/a^8 + 7*b^6*ln(abs(x))/a^8 + 1/20*(420*a*b^6*x^2
+ 210*a^2*b^5*x^(5/3) - 70*a^3*b^4*x^(4/3) + 35*a^4*b^3*x - 21*a^5*b^2*x^(2/3) +
14*a^6*b*x^(1/3) - 10*a^7)/((b*x^(1/3) + a)*a^8*x^2)