3.2423 \(\int \frac{x}{\left (a+\frac{b}{\sqrt [3]{x}}\right )^2} \, dx\)

Optimal. Leaf size=113 \[ \frac{3 b^7}{a^8 \left (a \sqrt [3]{x}+b\right )}+\frac{21 b^6 \log \left (a \sqrt [3]{x}+b\right )}{a^8}-\frac{18 b^5 \sqrt [3]{x}}{a^7}+\frac{15 b^4 x^{2/3}}{2 a^6}-\frac{4 b^3 x}{a^5}+\frac{9 b^2 x^{4/3}}{4 a^4}-\frac{6 b x^{5/3}}{5 a^3}+\frac{x^2}{2 a^2} \]

[Out]

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

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Rubi [A]  time = 0.194628, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{3 b^7}{a^8 \left (a \sqrt [3]{x}+b\right )}+\frac{21 b^6 \log \left (a \sqrt [3]{x}+b\right )}{a^8}-\frac{18 b^5 \sqrt [3]{x}}{a^7}+\frac{15 b^4 x^{2/3}}{2 a^6}-\frac{4 b^3 x}{a^5}+\frac{9 b^2 x^{4/3}}{4 a^4}-\frac{6 b x^{5/3}}{5 a^3}+\frac{x^2}{2 a^2} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b/x^(1/3))^2,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x^{2}}{2 a^{2}} - \frac{6 b x^{\frac{5}{3}}}{5 a^{3}} + \frac{9 b^{2} x^{\frac{4}{3}}}{4 a^{4}} - \frac{4 b^{3} x}{a^{5}} + \frac{15 b^{4} \int ^{\sqrt [3]{x}} x\, dx}{a^{6}} - \frac{18 b^{5} \sqrt [3]{x}}{a^{7}} + \frac{3 b^{7}}{a^{8} \left (a \sqrt [3]{x} + b\right )} + \frac{21 b^{6} \log{\left (a \sqrt [3]{x} + b \right )}}{a^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0605347, size = 104, normalized size = 0.92 \[ \frac{10 a^6 x^2-24 a^5 b x^{5/3}+45 a^4 b^2 x^{4/3}-80 a^3 b^3 x+150 a^2 b^4 x^{2/3}+\frac{60 b^7}{a \sqrt [3]{x}+b}+420 b^6 \log \left (a \sqrt [3]{x}+b\right )-360 a b^5 \sqrt [3]{x}}{20 a^8} \]

Antiderivative was successfully verified.

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

[Out]

((60*b^7)/(b + a*x^(1/3)) - 360*a*b^5*x^(1/3) + 150*a^2*b^4*x^(2/3) - 80*a^3*b^3
*x + 45*a^4*b^2*x^(4/3) - 24*a^5*b*x^(5/3) + 10*a^6*x^2 + 420*b^6*Log[b + a*x^(1
/3)])/(20*a^8)

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Maple [A]  time = 0.011, size = 94, normalized size = 0.8 \[ 3\,{\frac{{b}^{7}}{{a}^{8} \left ( b+a\sqrt [3]{x} \right ) }}-18\,{\frac{{b}^{5}\sqrt [3]{x}}{{a}^{7}}}+{\frac{15\,{b}^{4}}{2\,{a}^{6}}{x}^{{\frac{2}{3}}}}-4\,{\frac{{b}^{3}x}{{a}^{5}}}+{\frac{9\,{b}^{2}}{4\,{a}^{4}}{x}^{{\frac{4}{3}}}}-{\frac{6\,b}{5\,{a}^{3}}{x}^{{\frac{5}{3}}}}+{\frac{{x}^{2}}{2\,{a}^{2}}}+21\,{\frac{{b}^{6}\ln \left ( b+a\sqrt [3]{x} \right ) }{{a}^{8}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b/x^(1/3))^2,x)

[Out]

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

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Maxima [A]  time = 1.42656, size = 151, normalized size = 1.34 \[ \frac{10 \, a^{6} - \frac{14 \, a^{5} b}{x^{\frac{1}{3}}} + \frac{21 \, a^{4} b^{2}}{x^{\frac{2}{3}}} - \frac{35 \, a^{3} b^{3}}{x} + \frac{70 \, a^{2} b^{4}}{x^{\frac{4}{3}}} - \frac{210 \, a b^{5}}{x^{\frac{5}{3}}} - \frac{420 \, b^{6}}{x^{2}}}{20 \,{\left (\frac{a^{8}}{x^{2}} + \frac{a^{7} b}{x^{\frac{7}{3}}}\right )}} + \frac{21 \, b^{6} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{a^{8}} + \frac{7 \, b^{6} \log \left (x\right )}{a^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/20*(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^8/x^2 + a^7*b/x^(7/3)) + 21*b^6*l
og(a + b/x^(1/3))/a^8 + 7*b^6*log(x)/a^8

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Fricas [A]  time = 0.230294, size = 154, normalized size = 1.36 \[ -\frac{14 \, a^{6} b x^{2} - 70 \, a^{3} b^{4} x - 60 \, b^{7} - 420 \,{\left (a b^{6} x^{\frac{1}{3}} + b^{7}\right )} \log \left (a x^{\frac{1}{3}} + b\right ) - 21 \,{\left (a^{5} b^{2} x - 10 \, a^{2} b^{5}\right )} x^{\frac{2}{3}} - 5 \,{\left (2 \, a^{7} x^{2} - 7 \, a^{4} b^{3} x - 72 \, a b^{6}\right )} x^{\frac{1}{3}}}{20 \,{\left (a^{9} x^{\frac{1}{3}} + a^{8} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*(14*a^6*b*x^2 - 70*a^3*b^4*x - 60*b^7 - 420*(a*b^6*x^(1/3) + b^7)*log(a*x^
(1/3) + b) - 21*(a^5*b^2*x - 10*a^2*b^5)*x^(2/3) - 5*(2*a^7*x^2 - 7*a^4*b^3*x -
72*a*b^6)*x^(1/3))/(a^9*x^(1/3) + a^8*b)

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Sympy [A]  time = 48.0661, size = 415, normalized size = 3.67 \[ \frac{10 a^{7} x^{\frac{137}{3}}}{20 a^{9} x^{\frac{131}{3}} + 20 a^{8} b x^{\frac{130}{3}}} - \frac{14 a^{6} b x^{\frac{136}{3}}}{20 a^{9} x^{\frac{131}{3}} + 20 a^{8} b x^{\frac{130}{3}}} + \frac{21 a^{5} b^{2} x^{45}}{20 a^{9} x^{\frac{131}{3}} + 20 a^{8} b x^{\frac{130}{3}}} - \frac{35 a^{4} b^{3} x^{\frac{134}{3}}}{20 a^{9} x^{\frac{131}{3}} + 20 a^{8} b x^{\frac{130}{3}}} + \frac{70 a^{3} b^{4} x^{\frac{133}{3}}}{20 a^{9} x^{\frac{131}{3}} + 20 a^{8} b x^{\frac{130}{3}}} - \frac{210 a^{2} b^{5} x^{44}}{20 a^{9} x^{\frac{131}{3}} + 20 a^{8} b x^{\frac{130}{3}}} - \frac{420 a b^{6} x^{\frac{131}{3}} \log{\left (\frac{b}{a \sqrt [3]{x}} \right )}}{20 a^{9} x^{\frac{131}{3}} + 20 a^{8} b x^{\frac{130}{3}}} + \frac{420 a b^{6} x^{\frac{131}{3}} \log{\left (1 + \frac{b}{a \sqrt [3]{x}} \right )}}{20 a^{9} x^{\frac{131}{3}} + 20 a^{8} b x^{\frac{130}{3}}} - \frac{420 a b^{6} x^{\frac{131}{3}}}{20 a^{9} x^{\frac{131}{3}} + 20 a^{8} b x^{\frac{130}{3}}} - \frac{420 b^{7} x^{\frac{130}{3}} \log{\left (\frac{b}{a \sqrt [3]{x}} \right )}}{20 a^{9} x^{\frac{131}{3}} + 20 a^{8} b x^{\frac{130}{3}}} + \frac{420 b^{7} x^{\frac{130}{3}} \log{\left (1 + \frac{b}{a \sqrt [3]{x}} \right )}}{20 a^{9} x^{\frac{131}{3}} + 20 a^{8} b x^{\frac{130}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

10*a**7*x**(137/3)/(20*a**9*x**(131/3) + 20*a**8*b*x**(130/3)) - 14*a**6*b*x**(1
36/3)/(20*a**9*x**(131/3) + 20*a**8*b*x**(130/3)) + 21*a**5*b**2*x**45/(20*a**9*
x**(131/3) + 20*a**8*b*x**(130/3)) - 35*a**4*b**3*x**(134/3)/(20*a**9*x**(131/3)
+ 20*a**8*b*x**(130/3)) + 70*a**3*b**4*x**(133/3)/(20*a**9*x**(131/3) + 20*a**8
*b*x**(130/3)) - 210*a**2*b**5*x**44/(20*a**9*x**(131/3) + 20*a**8*b*x**(130/3))
- 420*a*b**6*x**(131/3)*log(b/(a*x**(1/3)))/(20*a**9*x**(131/3) + 20*a**8*b*x**
(130/3)) + 420*a*b**6*x**(131/3)*log(1 + b/(a*x**(1/3)))/(20*a**9*x**(131/3) + 2
0*a**8*b*x**(130/3)) - 420*a*b**6*x**(131/3)/(20*a**9*x**(131/3) + 20*a**8*b*x**
(130/3)) - 420*b**7*x**(130/3)*log(b/(a*x**(1/3)))/(20*a**9*x**(131/3) + 20*a**8
*b*x**(130/3)) + 420*b**7*x**(130/3)*log(1 + b/(a*x**(1/3)))/(20*a**9*x**(131/3)
+ 20*a**8*b*x**(130/3))

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GIAC/XCAS [A]  time = 0.215663, size = 135, normalized size = 1.19 \[ \frac{21 \, b^{6}{\rm ln}\left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{a^{8}} + \frac{3 \, b^{7}}{{\left (a x^{\frac{1}{3}} + b\right )} a^{8}} + \frac{10 \, a^{10} x^{2} - 24 \, a^{9} b x^{\frac{5}{3}} + 45 \, a^{8} b^{2} x^{\frac{4}{3}} - 80 \, a^{7} b^{3} x + 150 \, a^{6} b^{4} x^{\frac{2}{3}} - 360 \, a^{5} b^{5} x^{\frac{1}{3}}}{20 \, a^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

21*b^6*ln(abs(a*x^(1/3) + b))/a^8 + 3*b^7/((a*x^(1/3) + b)*a^8) + 1/20*(10*a^10*
x^2 - 24*a^9*b*x^(5/3) + 45*a^8*b^2*x^(4/3) - 80*a^7*b^3*x + 150*a^6*b^4*x^(2/3)
- 360*a^5*b^5*x^(1/3))/a^12