3.2422 \(\int \frac{x^2}{\left (a+\frac{b}{\sqrt [3]{x}}\right )^2} \, dx\)

Optimal. Leaf size=150 \[ -\frac{3 b^{10}}{a^{11} \left (a \sqrt [3]{x}+b\right )}-\frac{30 b^9 \log \left (a \sqrt [3]{x}+b\right )}{a^{11}}+\frac{27 b^8 \sqrt [3]{x}}{a^{10}}-\frac{12 b^7 x^{2/3}}{a^9}+\frac{7 b^6 x}{a^8}-\frac{9 b^5 x^{4/3}}{2 a^7}+\frac{3 b^4 x^{5/3}}{a^6}-\frac{2 b^3 x^2}{a^5}+\frac{9 b^2 x^{7/3}}{7 a^4}-\frac{3 b x^{8/3}}{4 a^3}+\frac{x^3}{3 a^2} \]

[Out]

(-3*b^10)/(a^11*(b + a*x^(1/3))) + (27*b^8*x^(1/3))/a^10 - (12*b^7*x^(2/3))/a^9
+ (7*b^6*x)/a^8 - (9*b^5*x^(4/3))/(2*a^7) + (3*b^4*x^(5/3))/a^6 - (2*b^3*x^2)/a^
5 + (9*b^2*x^(7/3))/(7*a^4) - (3*b*x^(8/3))/(4*a^3) + x^3/(3*a^2) - (30*b^9*Log[
b + a*x^(1/3)])/a^11

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Rubi [A]  time = 0.281659, antiderivative size = 150, 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 \[ -\frac{3 b^{10}}{a^{11} \left (a \sqrt [3]{x}+b\right )}-\frac{30 b^9 \log \left (a \sqrt [3]{x}+b\right )}{a^{11}}+\frac{27 b^8 \sqrt [3]{x}}{a^{10}}-\frac{12 b^7 x^{2/3}}{a^9}+\frac{7 b^6 x}{a^8}-\frac{9 b^5 x^{4/3}}{2 a^7}+\frac{3 b^4 x^{5/3}}{a^6}-\frac{2 b^3 x^2}{a^5}+\frac{9 b^2 x^{7/3}}{7 a^4}-\frac{3 b x^{8/3}}{4 a^3}+\frac{x^3}{3 a^2} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b/x^(1/3))^2,x]

[Out]

(-3*b^10)/(a^11*(b + a*x^(1/3))) + (27*b^8*x^(1/3))/a^10 - (12*b^7*x^(2/3))/a^9
+ (7*b^6*x)/a^8 - (9*b^5*x^(4/3))/(2*a^7) + (3*b^4*x^(5/3))/a^6 - (2*b^3*x^2)/a^
5 + (9*b^2*x^(7/3))/(7*a^4) - (3*b*x^(8/3))/(4*a^3) + x^3/(3*a^2) - (30*b^9*Log[
b + a*x^(1/3)])/a^11

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x^{3}}{3 a^{2}} - \frac{3 b x^{\frac{8}{3}}}{4 a^{3}} + \frac{9 b^{2} x^{\frac{7}{3}}}{7 a^{4}} - \frac{2 b^{3} x^{2}}{a^{5}} + \frac{3 b^{4} x^{\frac{5}{3}}}{a^{6}} - \frac{9 b^{5} x^{\frac{4}{3}}}{2 a^{7}} + \frac{7 b^{6} x}{a^{8}} - \frac{24 b^{7} \int ^{\sqrt [3]{x}} x\, dx}{a^{9}} + \frac{27 b^{8} \sqrt [3]{x}}{a^{10}} - \frac{3 b^{10}}{a^{11} \left (a \sqrt [3]{x} + b\right )} - \frac{30 b^{9} \log{\left (a \sqrt [3]{x} + b \right )}}{a^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3/(3*a**2) - 3*b*x**(8/3)/(4*a**3) + 9*b**2*x**(7/3)/(7*a**4) - 2*b**3*x**2/a
**5 + 3*b**4*x**(5/3)/a**6 - 9*b**5*x**(4/3)/(2*a**7) + 7*b**6*x/a**8 - 24*b**7*
Integral(x, (x, x**(1/3)))/a**9 + 27*b**8*x**(1/3)/a**10 - 3*b**10/(a**11*(a*x**
(1/3) + b)) - 30*b**9*log(a*x**(1/3) + b)/a**11

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Mathematica [A]  time = 0.104455, size = 150, normalized size = 1. \[ -\frac{3 b^{10}}{a^{11} \left (a \sqrt [3]{x}+b\right )}-\frac{30 b^9 \log \left (a \sqrt [3]{x}+b\right )}{a^{11}}+\frac{27 b^8 \sqrt [3]{x}}{a^{10}}-\frac{12 b^7 x^{2/3}}{a^9}+\frac{7 b^6 x}{a^8}-\frac{9 b^5 x^{4/3}}{2 a^7}+\frac{3 b^4 x^{5/3}}{a^6}-\frac{2 b^3 x^2}{a^5}+\frac{9 b^2 x^{7/3}}{7 a^4}-\frac{3 b x^{8/3}}{4 a^3}+\frac{x^3}{3 a^2} \]

Antiderivative was successfully verified.

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

[Out]

(-3*b^10)/(a^11*(b + a*x^(1/3))) + (27*b^8*x^(1/3))/a^10 - (12*b^7*x^(2/3))/a^9
+ (7*b^6*x)/a^8 - (9*b^5*x^(4/3))/(2*a^7) + (3*b^4*x^(5/3))/a^6 - (2*b^3*x^2)/a^
5 + (9*b^2*x^(7/3))/(7*a^4) - (3*b*x^(8/3))/(4*a^3) + x^3/(3*a^2) - (30*b^9*Log[
b + a*x^(1/3)])/a^11

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Maple [A]  time = 0.012, size = 127, normalized size = 0.9 \[ -3\,{\frac{{b}^{10}}{{a}^{11} \left ( b+a\sqrt [3]{x} \right ) }}+27\,{\frac{{b}^{8}\sqrt [3]{x}}{{a}^{10}}}-12\,{\frac{{b}^{7}{x}^{2/3}}{{a}^{9}}}+7\,{\frac{{b}^{6}x}{{a}^{8}}}-{\frac{9\,{b}^{5}}{2\,{a}^{7}}{x}^{{\frac{4}{3}}}}+3\,{\frac{{b}^{4}{x}^{5/3}}{{a}^{6}}}-2\,{\frac{{b}^{3}{x}^{2}}{{a}^{5}}}+{\frac{9\,{b}^{2}}{7\,{a}^{4}}{x}^{{\frac{7}{3}}}}-{\frac{3\,b}{4\,{a}^{3}}{x}^{{\frac{8}{3}}}}+{\frac{{x}^{3}}{3\,{a}^{2}}}-30\,{\frac{{b}^{9}\ln \left ( b+a\sqrt [3]{x} \right ) }{{a}^{11}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(a+b/x^(1/3))^2,x)

[Out]

-3*b^10/a^11/(b+a*x^(1/3))+27*b^8*x^(1/3)/a^10-12*b^7*x^(2/3)/a^9+7*b^6*x/a^8-9/
2*b^5*x^(4/3)/a^7+3*b^4*x^(5/3)/a^6-2*b^3*x^2/a^5+9/7*b^2*x^(7/3)/a^4-3/4*b*x^(8
/3)/a^3+1/3*x^3/a^2-30*b^9*ln(b+a*x^(1/3))/a^11

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Maxima [A]  time = 1.44173, size = 196, normalized size = 1.31 \[ \frac{28 \, a^{9} - \frac{35 \, a^{8} b}{x^{\frac{1}{3}}} + \frac{45 \, a^{7} b^{2}}{x^{\frac{2}{3}}} - \frac{60 \, a^{6} b^{3}}{x} + \frac{84 \, a^{5} b^{4}}{x^{\frac{4}{3}}} - \frac{126 \, a^{4} b^{5}}{x^{\frac{5}{3}}} + \frac{210 \, a^{3} b^{6}}{x^{2}} - \frac{420 \, a^{2} b^{7}}{x^{\frac{7}{3}}} + \frac{1260 \, a b^{8}}{x^{\frac{8}{3}}} + \frac{2520 \, b^{9}}{x^{3}}}{84 \,{\left (\frac{a^{11}}{x^{3}} + \frac{a^{10} b}{x^{\frac{10}{3}}}\right )}} - \frac{30 \, b^{9} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{a^{11}} - \frac{10 \, b^{9} \log \left (x\right )}{a^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/84*(28*a^9 - 35*a^8*b/x^(1/3) + 45*a^7*b^2/x^(2/3) - 60*a^6*b^3/x + 84*a^5*b^4
/x^(4/3) - 126*a^4*b^5/x^(5/3) + 210*a^3*b^6/x^2 - 420*a^2*b^7/x^(7/3) + 1260*a*
b^8/x^(8/3) + 2520*b^9/x^3)/(a^11/x^3 + a^10*b/x^(10/3)) - 30*b^9*log(a + b/x^(1
/3))/a^11 - 10*b^9*log(x)/a^11

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Fricas [A]  time = 0.22898, size = 200, normalized size = 1.33 \[ -\frac{35 \, a^{9} b x^{3} - 84 \, a^{6} b^{4} x^{2} + 420 \, a^{3} b^{7} x + 252 \, b^{10} + 2520 \,{\left (a b^{9} x^{\frac{1}{3}} + b^{10}\right )} \log \left (a x^{\frac{1}{3}} + b\right ) - 9 \,{\left (5 \, a^{8} b^{2} x^{2} - 14 \, a^{5} b^{5} x + 140 \, a^{2} b^{8}\right )} x^{\frac{2}{3}} - 2 \,{\left (14 \, a^{10} x^{3} - 30 \, a^{7} b^{3} x^{2} + 105 \, a^{4} b^{6} x + 1134 \, a b^{9}\right )} x^{\frac{1}{3}}}{84 \,{\left (a^{12} x^{\frac{1}{3}} + a^{11} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/84*(35*a^9*b*x^3 - 84*a^6*b^4*x^2 + 420*a^3*b^7*x + 252*b^10 + 2520*(a*b^9*x^
(1/3) + b^10)*log(a*x^(1/3) + b) - 9*(5*a^8*b^2*x^2 - 14*a^5*b^5*x + 140*a^2*b^8
)*x^(2/3) - 2*(14*a^10*x^3 - 30*a^7*b^3*x^2 + 105*a^4*b^6*x + 1134*a*b^9)*x^(1/3
))/(a^12*x^(1/3) + a^11*b)

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Sympy [A]  time = 23.5188, size = 367, normalized size = 2.45 \[ \begin{cases} \frac{28 a^{10} x^{\frac{10}{3}}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac{35 a^{9} b x^{3}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} + \frac{45 a^{8} b^{2} x^{\frac{8}{3}}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac{60 a^{7} b^{3} x^{\frac{7}{3}}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} + \frac{84 a^{6} b^{4} x^{2}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac{126 a^{5} b^{5} x^{\frac{5}{3}}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} + \frac{210 a^{4} b^{6} x^{\frac{4}{3}}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac{420 a^{3} b^{7} x}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} + \frac{1260 a^{2} b^{8} x^{\frac{2}{3}}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac{2520 a b^{9} \sqrt [3]{x} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac{2520 b^{10} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac{2520 b^{10}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} & \text{for}\: a \neq 0 \\\frac{3 x^{\frac{11}{3}}}{11 b^{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((28*a**10*x**(10/3)/(84*a**12*x**(1/3) + 84*a**11*b) - 35*a**9*b*x**3/
(84*a**12*x**(1/3) + 84*a**11*b) + 45*a**8*b**2*x**(8/3)/(84*a**12*x**(1/3) + 84
*a**11*b) - 60*a**7*b**3*x**(7/3)/(84*a**12*x**(1/3) + 84*a**11*b) + 84*a**6*b**
4*x**2/(84*a**12*x**(1/3) + 84*a**11*b) - 126*a**5*b**5*x**(5/3)/(84*a**12*x**(1
/3) + 84*a**11*b) + 210*a**4*b**6*x**(4/3)/(84*a**12*x**(1/3) + 84*a**11*b) - 42
0*a**3*b**7*x/(84*a**12*x**(1/3) + 84*a**11*b) + 1260*a**2*b**8*x**(2/3)/(84*a**
12*x**(1/3) + 84*a**11*b) - 2520*a*b**9*x**(1/3)*log(x**(1/3) + b/a)/(84*a**12*x
**(1/3) + 84*a**11*b) - 2520*b**10*log(x**(1/3) + b/a)/(84*a**12*x**(1/3) + 84*a
**11*b) - 2520*b**10/(84*a**12*x**(1/3) + 84*a**11*b), Ne(a, 0)), (3*x**(11/3)/(
11*b**2), True))

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GIAC/XCAS [A]  time = 0.214604, size = 180, normalized size = 1.2 \[ -\frac{30 \, b^{9}{\rm ln}\left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{a^{11}} - \frac{3 \, b^{10}}{{\left (a x^{\frac{1}{3}} + b\right )} a^{11}} + \frac{28 \, a^{16} x^{3} - 63 \, a^{15} b x^{\frac{8}{3}} + 108 \, a^{14} b^{2} x^{\frac{7}{3}} - 168 \, a^{13} b^{3} x^{2} + 252 \, a^{12} b^{4} x^{\frac{5}{3}} - 378 \, a^{11} b^{5} x^{\frac{4}{3}} + 588 \, a^{10} b^{6} x - 1008 \, a^{9} b^{7} x^{\frac{2}{3}} + 2268 \, a^{8} b^{8} x^{\frac{1}{3}}}{84 \, a^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-30*b^9*ln(abs(a*x^(1/3) + b))/a^11 - 3*b^10/((a*x^(1/3) + b)*a^11) + 1/84*(28*a
^16*x^3 - 63*a^15*b*x^(8/3) + 108*a^14*b^2*x^(7/3) - 168*a^13*b^3*x^2 + 252*a^12
*b^4*x^(5/3) - 378*a^11*b^5*x^(4/3) + 588*a^10*b^6*x - 1008*a^9*b^7*x^(2/3) + 22
68*a^8*b^8*x^(1/3))/a^18