3.2430 \(\int \frac{x^2}{\left (a+\frac{b}{\sqrt [3]{x}}\right )^3} \, dx\)

Optimal. Leaf size=171 \[ \frac{3 b^{11}}{2 a^{12} \left (a \sqrt [3]{x}+b\right )^2}-\frac{33 b^{10}}{a^{12} \left (a \sqrt [3]{x}+b\right )}-\frac{165 b^9 \log \left (a \sqrt [3]{x}+b\right )}{a^{12}}+\frac{135 b^8 \sqrt [3]{x}}{a^{11}}-\frac{54 b^7 x^{2/3}}{a^{10}}+\frac{28 b^6 x}{a^9}-\frac{63 b^5 x^{4/3}}{4 a^8}+\frac{9 b^4 x^{5/3}}{a^7}-\frac{5 b^3 x^2}{a^6}+\frac{18 b^2 x^{7/3}}{7 a^5}-\frac{9 b x^{8/3}}{8 a^4}+\frac{x^3}{3 a^3} \]

[Out]

(3*b^11)/(2*a^12*(b + a*x^(1/3))^2) - (33*b^10)/(a^12*(b + a*x^(1/3))) + (135*b^
8*x^(1/3))/a^11 - (54*b^7*x^(2/3))/a^10 + (28*b^6*x)/a^9 - (63*b^5*x^(4/3))/(4*a
^8) + (9*b^4*x^(5/3))/a^7 - (5*b^3*x^2)/a^6 + (18*b^2*x^(7/3))/(7*a^5) - (9*b*x^
(8/3))/(8*a^4) + x^3/(3*a^3) - (165*b^9*Log[b + a*x^(1/3)])/a^12

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Rubi [A]  time = 0.339949, antiderivative size = 171, 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 \[ \frac{3 b^{11}}{2 a^{12} \left (a \sqrt [3]{x}+b\right )^2}-\frac{33 b^{10}}{a^{12} \left (a \sqrt [3]{x}+b\right )}-\frac{165 b^9 \log \left (a \sqrt [3]{x}+b\right )}{a^{12}}+\frac{135 b^8 \sqrt [3]{x}}{a^{11}}-\frac{54 b^7 x^{2/3}}{a^{10}}+\frac{28 b^6 x}{a^9}-\frac{63 b^5 x^{4/3}}{4 a^8}+\frac{9 b^4 x^{5/3}}{a^7}-\frac{5 b^3 x^2}{a^6}+\frac{18 b^2 x^{7/3}}{7 a^5}-\frac{9 b x^{8/3}}{8 a^4}+\frac{x^3}{3 a^3} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b/x^(1/3))^3,x]

[Out]

(3*b^11)/(2*a^12*(b + a*x^(1/3))^2) - (33*b^10)/(a^12*(b + a*x^(1/3))) + (135*b^
8*x^(1/3))/a^11 - (54*b^7*x^(2/3))/a^10 + (28*b^6*x)/a^9 - (63*b^5*x^(4/3))/(4*a
^8) + (9*b^4*x^(5/3))/a^7 - (5*b^3*x^2)/a^6 + (18*b^2*x^(7/3))/(7*a^5) - (9*b*x^
(8/3))/(8*a^4) + x^3/(3*a^3) - (165*b^9*Log[b + a*x^(1/3)])/a^12

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x^{3}}{3 a^{3}} - \frac{9 b x^{\frac{8}{3}}}{8 a^{4}} + \frac{18 b^{2} x^{\frac{7}{3}}}{7 a^{5}} - \frac{5 b^{3} x^{2}}{a^{6}} + \frac{9 b^{4} x^{\frac{5}{3}}}{a^{7}} - \frac{63 b^{5} x^{\frac{4}{3}}}{4 a^{8}} + \frac{28 b^{6} x}{a^{9}} - \frac{108 b^{7} \int ^{\sqrt [3]{x}} x\, dx}{a^{10}} + \frac{135 b^{8} \sqrt [3]{x}}{a^{11}} + \frac{3 b^{11}}{2 a^{12} \left (a \sqrt [3]{x} + b\right )^{2}} - \frac{33 b^{10}}{a^{12} \left (a \sqrt [3]{x} + b\right )} - \frac{165 b^{9} \log{\left (a \sqrt [3]{x} + b \right )}}{a^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3/(3*a**3) - 9*b*x**(8/3)/(8*a**4) + 18*b**2*x**(7/3)/(7*a**5) - 5*b**3*x**2/
a**6 + 9*b**4*x**(5/3)/a**7 - 63*b**5*x**(4/3)/(4*a**8) + 28*b**6*x/a**9 - 108*b
**7*Integral(x, (x, x**(1/3)))/a**10 + 135*b**8*x**(1/3)/a**11 + 3*b**11/(2*a**1
2*(a*x**(1/3) + b)**2) - 33*b**10/(a**12*(a*x**(1/3) + b)) - 165*b**9*log(a*x**(
1/3) + b)/a**12

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Mathematica [A]  time = 0.102834, size = 157, normalized size = 0.92 \[ \frac{56 a^9 x^3-189 a^8 b x^{8/3}+432 a^7 b^2 x^{7/3}-840 a^6 b^3 x^2+1512 a^5 b^4 x^{5/3}-2646 a^4 b^5 x^{4/3}+4704 a^3 b^6 x-9072 a^2 b^7 x^{2/3}+\frac{252 b^{11}}{\left (a \sqrt [3]{x}+b\right )^2}-\frac{5544 b^{10}}{a \sqrt [3]{x}+b}-27720 b^9 \log \left (a \sqrt [3]{x}+b\right )+22680 a b^8 \sqrt [3]{x}}{168 a^{12}} \]

Antiderivative was successfully verified.

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

[Out]

((252*b^11)/(b + a*x^(1/3))^2 - (5544*b^10)/(b + a*x^(1/3)) + 22680*a*b^8*x^(1/3
) - 9072*a^2*b^7*x^(2/3) + 4704*a^3*b^6*x - 2646*a^4*b^5*x^(4/3) + 1512*a^5*b^4*
x^(5/3) - 840*a^6*b^3*x^2 + 432*a^7*b^2*x^(7/3) - 189*a^8*b*x^(8/3) + 56*a^9*x^3
- 27720*b^9*Log[b + a*x^(1/3)])/(168*a^12)

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Maple [A]  time = 0.013, size = 144, normalized size = 0.8 \[{\frac{3\,{b}^{11}}{2\,{a}^{12}} \left ( b+a\sqrt [3]{x} \right ) ^{-2}}-33\,{\frac{{b}^{10}}{{a}^{12} \left ( b+a\sqrt [3]{x} \right ) }}+135\,{\frac{{b}^{8}\sqrt [3]{x}}{{a}^{11}}}-54\,{\frac{{b}^{7}{x}^{2/3}}{{a}^{10}}}+28\,{\frac{{b}^{6}x}{{a}^{9}}}-{\frac{63\,{b}^{5}}{4\,{a}^{8}}{x}^{{\frac{4}{3}}}}+9\,{\frac{{b}^{4}{x}^{5/3}}{{a}^{7}}}-5\,{\frac{{b}^{3}{x}^{2}}{{a}^{6}}}+{\frac{18\,{b}^{2}}{7\,{a}^{5}}{x}^{{\frac{7}{3}}}}-{\frac{9\,b}{8\,{a}^{4}}{x}^{{\frac{8}{3}}}}+{\frac{{x}^{3}}{3\,{a}^{3}}}-165\,{\frac{{b}^{9}\ln \left ( b+a\sqrt [3]{x} \right ) }{{a}^{12}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(a+b/x^(1/3))^3,x)

[Out]

3/2*b^11/a^12/(b+a*x^(1/3))^2-33*b^10/a^12/(b+a*x^(1/3))+135*b^8*x^(1/3)/a^11-54
*b^7*x^(2/3)/a^10+28*b^6*x/a^9-63/4*b^5*x^(4/3)/a^8+9*b^4*x^(5/3)/a^7-5*b^3*x^2/
a^6+18/7*b^2*x^(7/3)/a^5-9/8*b*x^(8/3)/a^4+1/3*x^3/a^3-165*b^9*ln(b+a*x^(1/3))/a
^12

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Maxima [A]  time = 1.43974, size = 225, normalized size = 1.32 \[ \frac{56 \, a^{10} - \frac{77 \, a^{9} b}{x^{\frac{1}{3}}} + \frac{110 \, a^{8} b^{2}}{x^{\frac{2}{3}}} - \frac{165 \, a^{7} b^{3}}{x} + \frac{264 \, a^{6} b^{4}}{x^{\frac{4}{3}}} - \frac{462 \, a^{5} b^{5}}{x^{\frac{5}{3}}} + \frac{924 \, a^{4} b^{6}}{x^{2}} - \frac{2310 \, a^{3} b^{7}}{x^{\frac{7}{3}}} + \frac{9240 \, a^{2} b^{8}}{x^{\frac{8}{3}}} + \frac{41580 \, a b^{9}}{x^{3}} + \frac{27720 \, b^{10}}{x^{\frac{10}{3}}}}{168 \,{\left (\frac{a^{13}}{x^{3}} + \frac{2 \, a^{12} b}{x^{\frac{10}{3}}} + \frac{a^{11} b^{2}}{x^{\frac{11}{3}}}\right )}} - \frac{165 \, b^{9} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{a^{12}} - \frac{55 \, b^{9} \log \left (x\right )}{a^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/168*(56*a^10 - 77*a^9*b/x^(1/3) + 110*a^8*b^2/x^(2/3) - 165*a^7*b^3/x + 264*a^
6*b^4/x^(4/3) - 462*a^5*b^5/x^(5/3) + 924*a^4*b^6/x^2 - 2310*a^3*b^7/x^(7/3) + 9
240*a^2*b^8/x^(8/3) + 41580*a*b^9/x^3 + 27720*b^10/x^(10/3))/(a^13/x^3 + 2*a^12*
b/x^(10/3) + a^11*b^2/x^(11/3)) - 165*b^9*log(a + b/x^(1/3))/a^12 - 55*b^9*log(x
)/a^12

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Fricas [A]  time = 0.230786, size = 243, normalized size = 1.42 \[ \frac{110 \, a^{9} b^{2} x^{3} - 462 \, a^{6} b^{5} x^{2} + 9240 \, a^{3} b^{8} x - 5292 \, b^{11} - 27720 \,{\left (a^{2} b^{9} x^{\frac{2}{3}} + 2 \, a b^{10} x^{\frac{1}{3}} + b^{11}\right )} \log \left (a x^{\frac{1}{3}} + b\right ) +{\left (56 \, a^{11} x^{3} - 165 \, a^{8} b^{3} x^{2} + 924 \, a^{5} b^{6} x + 36288 \, a^{2} b^{9}\right )} x^{\frac{2}{3}} -{\left (77 \, a^{10} b x^{3} - 264 \, a^{7} b^{4} x^{2} + 2310 \, a^{4} b^{7} x - 17136 \, a b^{10}\right )} x^{\frac{1}{3}}}{168 \,{\left (a^{14} x^{\frac{2}{3}} + 2 \, a^{13} b x^{\frac{1}{3}} + a^{12} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/168*(110*a^9*b^2*x^3 - 462*a^6*b^5*x^2 + 9240*a^3*b^8*x - 5292*b^11 - 27720*(a
^2*b^9*x^(2/3) + 2*a*b^10*x^(1/3) + b^11)*log(a*x^(1/3) + b) + (56*a^11*x^3 - 16
5*a^8*b^3*x^2 + 924*a^5*b^6*x + 36288*a^2*b^9)*x^(2/3) - (77*a^10*b*x^3 - 264*a^
7*b^4*x^2 + 2310*a^4*b^7*x - 17136*a*b^10)*x^(1/3))/(a^14*x^(2/3) + 2*a^13*b*x^(
1/3) + a^12*b^2)

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Sympy [A]  time = 21.1259, size = 624, normalized size = 3.65 \[ \begin{cases} \frac{56 a^{11} x^{\frac{11}{3}}}{168 a^{14} x^{\frac{2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac{77 a^{10} b x^{\frac{10}{3}}}{168 a^{14} x^{\frac{2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} + \frac{110 a^{9} b^{2} x^{3}}{168 a^{14} x^{\frac{2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac{165 a^{8} b^{3} x^{\frac{8}{3}}}{168 a^{14} x^{\frac{2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} + \frac{264 a^{7} b^{4} x^{\frac{7}{3}}}{168 a^{14} x^{\frac{2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac{462 a^{6} b^{5} x^{2}}{168 a^{14} x^{\frac{2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} + \frac{924 a^{5} b^{6} x^{\frac{5}{3}}}{168 a^{14} x^{\frac{2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac{2310 a^{4} b^{7} x^{\frac{4}{3}}}{168 a^{14} x^{\frac{2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} + \frac{9240 a^{3} b^{8} x}{168 a^{14} x^{\frac{2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac{27720 a^{2} b^{9} x^{\frac{2}{3}} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{168 a^{14} x^{\frac{2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac{55440 a b^{10} \sqrt [3]{x} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{168 a^{14} x^{\frac{2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac{55440 a b^{10} \sqrt [3]{x}}{168 a^{14} x^{\frac{2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac{27720 b^{11} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{168 a^{14} x^{\frac{2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac{41580 b^{11}}{168 a^{14} x^{\frac{2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4 b^{3}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((56*a**11*x**(11/3)/(168*a**14*x**(2/3) + 336*a**13*b*x**(1/3) + 168*a
**12*b**2) - 77*a**10*b*x**(10/3)/(168*a**14*x**(2/3) + 336*a**13*b*x**(1/3) + 1
68*a**12*b**2) + 110*a**9*b**2*x**3/(168*a**14*x**(2/3) + 336*a**13*b*x**(1/3) +
168*a**12*b**2) - 165*a**8*b**3*x**(8/3)/(168*a**14*x**(2/3) + 336*a**13*b*x**(
1/3) + 168*a**12*b**2) + 264*a**7*b**4*x**(7/3)/(168*a**14*x**(2/3) + 336*a**13*
b*x**(1/3) + 168*a**12*b**2) - 462*a**6*b**5*x**2/(168*a**14*x**(2/3) + 336*a**1
3*b*x**(1/3) + 168*a**12*b**2) + 924*a**5*b**6*x**(5/3)/(168*a**14*x**(2/3) + 33
6*a**13*b*x**(1/3) + 168*a**12*b**2) - 2310*a**4*b**7*x**(4/3)/(168*a**14*x**(2/
3) + 336*a**13*b*x**(1/3) + 168*a**12*b**2) + 9240*a**3*b**8*x/(168*a**14*x**(2/
3) + 336*a**13*b*x**(1/3) + 168*a**12*b**2) - 27720*a**2*b**9*x**(2/3)*log(x**(1
/3) + b/a)/(168*a**14*x**(2/3) + 336*a**13*b*x**(1/3) + 168*a**12*b**2) - 55440*
a*b**10*x**(1/3)*log(x**(1/3) + b/a)/(168*a**14*x**(2/3) + 336*a**13*b*x**(1/3)
+ 168*a**12*b**2) - 55440*a*b**10*x**(1/3)/(168*a**14*x**(2/3) + 336*a**13*b*x**
(1/3) + 168*a**12*b**2) - 27720*b**11*log(x**(1/3) + b/a)/(168*a**14*x**(2/3) +
336*a**13*b*x**(1/3) + 168*a**12*b**2) - 41580*b**11/(168*a**14*x**(2/3) + 336*a
**13*b*x**(1/3) + 168*a**12*b**2), Ne(a, 0)), (x**4/(4*b**3), True))

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GIAC/XCAS [A]  time = 0.223706, size = 196, normalized size = 1.15 \[ -\frac{165 \, b^{9}{\rm ln}\left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{a^{12}} - \frac{3 \,{\left (22 \, a b^{10} x^{\frac{1}{3}} + 21 \, b^{11}\right )}}{2 \,{\left (a x^{\frac{1}{3}} + b\right )}^{2} a^{12}} + \frac{56 \, a^{24} x^{3} - 189 \, a^{23} b x^{\frac{8}{3}} + 432 \, a^{22} b^{2} x^{\frac{7}{3}} - 840 \, a^{21} b^{3} x^{2} + 1512 \, a^{20} b^{4} x^{\frac{5}{3}} - 2646 \, a^{19} b^{5} x^{\frac{4}{3}} + 4704 \, a^{18} b^{6} x - 9072 \, a^{17} b^{7} x^{\frac{2}{3}} + 22680 \, a^{16} b^{8} x^{\frac{1}{3}}}{168 \, a^{27}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-165*b^9*ln(abs(a*x^(1/3) + b))/a^12 - 3/2*(22*a*b^10*x^(1/3) + 21*b^11)/((a*x^(
1/3) + b)^2*a^12) + 1/168*(56*a^24*x^3 - 189*a^23*b*x^(8/3) + 432*a^22*b^2*x^(7/
3) - 840*a^21*b^3*x^2 + 1512*a^20*b^4*x^(5/3) - 2646*a^19*b^5*x^(4/3) + 4704*a^1
8*b^6*x - 9072*a^17*b^7*x^(2/3) + 22680*a^16*b^8*x^(1/3))/a^27