3.2366 \(\int \frac{x^3}{\left (a+b \sqrt [3]{x}\right )^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.325596, antiderivative size = 171, 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 \[ \frac{3 a^{11}}{2 b^{12} \left (a+b \sqrt [3]{x}\right )^2}-\frac{33 a^{10}}{b^{12} \left (a+b \sqrt [3]{x}\right )}-\frac{165 a^9 \log \left (a+b \sqrt [3]{x}\right )}{b^{12}}+\frac{135 a^8 \sqrt [3]{x}}{b^{11}}-\frac{54 a^7 x^{2/3}}{b^{10}}+\frac{28 a^6 x}{b^9}-\frac{63 a^5 x^{4/3}}{4 b^8}+\frac{9 a^4 x^{5/3}}{b^7}-\frac{5 a^3 x^2}{b^6}+\frac{18 a^2 x^{7/3}}{7 b^5}-\frac{9 a x^{8/3}}{8 b^4}+\frac{x^3}{3 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b*x^(1/3))^3,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.102513, size = 157, normalized size = 0.92 \[ \frac{\frac{252 a^{11}}{\left (a+b \sqrt [3]{x}\right )^2}-\frac{5544 a^{10}}{a+b \sqrt [3]{x}}-27720 a^9 \log \left (a+b \sqrt [3]{x}\right )+22680 a^8 b \sqrt [3]{x}-9072 a^7 b^2 x^{2/3}+4704 a^6 b^3 x-2646 a^5 b^4 x^{4/3}+1512 a^4 b^5 x^{5/3}-840 a^3 b^6 x^2+432 a^2 b^7 x^{7/3}-189 a b^8 x^{8/3}+56 b^9 x^3}{168 b^{12}} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(a+b*x^(1/3))^3,x)

[Out]

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

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Maxima [A]  time = 1.44412, size = 266, normalized size = 1.56 \[ -\frac{165 \, a^{9} \log \left (b x^{\frac{1}{3}} + a\right )}{b^{12}} + \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{9}}{3 \, b^{12}} - \frac{33 \,{\left (b x^{\frac{1}{3}} + a\right )}^{8} a}{8 \, b^{12}} + \frac{165 \,{\left (b x^{\frac{1}{3}} + a\right )}^{7} a^{2}}{7 \, b^{12}} - \frac{165 \,{\left (b x^{\frac{1}{3}} + a\right )}^{6} a^{3}}{2 \, b^{12}} + \frac{198 \,{\left (b x^{\frac{1}{3}} + a\right )}^{5} a^{4}}{b^{12}} - \frac{693 \,{\left (b x^{\frac{1}{3}} + a\right )}^{4} a^{5}}{2 \, b^{12}} + \frac{462 \,{\left (b x^{\frac{1}{3}} + a\right )}^{3} a^{6}}{b^{12}} - \frac{495 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} a^{7}}{b^{12}} + \frac{495 \,{\left (b x^{\frac{1}{3}} + a\right )} a^{8}}{b^{12}} - \frac{33 \, a^{10}}{{\left (b x^{\frac{1}{3}} + a\right )} b^{12}} + \frac{3 \, a^{11}}{2 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} b^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-165*a^9*log(b*x^(1/3) + a)/b^12 + 1/3*(b*x^(1/3) + a)^9/b^12 - 33/8*(b*x^(1/3)
+ a)^8*a/b^12 + 165/7*(b*x^(1/3) + a)^7*a^2/b^12 - 165/2*(b*x^(1/3) + a)^6*a^3/b
^12 + 198*(b*x^(1/3) + a)^5*a^4/b^12 - 693/2*(b*x^(1/3) + a)^4*a^5/b^12 + 462*(b
*x^(1/3) + a)^3*a^6/b^12 - 495*(b*x^(1/3) + a)^2*a^7/b^12 + 495*(b*x^(1/3) + a)*
a^8/b^12 - 33*a^10/((b*x^(1/3) + a)*b^12) + 3/2*a^11/((b*x^(1/3) + a)^2*b^12)

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Fricas [A]  time = 0.234337, size = 243, normalized size = 1.42 \[ \frac{110 \, a^{2} b^{9} x^{3} - 462 \, a^{5} b^{6} x^{2} + 9240 \, a^{8} b^{3} x - 5292 \, a^{11} - 27720 \,{\left (a^{9} b^{2} x^{\frac{2}{3}} + 2 \, a^{10} b x^{\frac{1}{3}} + a^{11}\right )} \log \left (b x^{\frac{1}{3}} + a\right ) +{\left (56 \, b^{11} x^{3} - 165 \, a^{3} b^{8} x^{2} + 924 \, a^{6} b^{5} x + 36288 \, a^{9} b^{2}\right )} x^{\frac{2}{3}} -{\left (77 \, a b^{10} x^{3} - 264 \, a^{4} b^{7} x^{2} + 2310 \, a^{7} b^{4} x - 17136 \, a^{10} b\right )} x^{\frac{1}{3}}}{168 \,{\left (b^{14} x^{\frac{2}{3}} + 2 \, a b^{13} x^{\frac{1}{3}} + a^{2} b^{12}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 21.6122, size = 624, normalized size = 3.65 \[ \begin{cases} - \frac{27720 a^{11} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac{2}{3}}} - \frac{41580 a^{11}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac{2}{3}}} - \frac{55440 a^{10} b \sqrt [3]{x} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac{2}{3}}} - \frac{55440 a^{10} b \sqrt [3]{x}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac{2}{3}}} - \frac{27720 a^{9} b^{2} x^{\frac{2}{3}} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac{2}{3}}} + \frac{9240 a^{8} b^{3} x}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac{2}{3}}} - \frac{2310 a^{7} b^{4} x^{\frac{4}{3}}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac{2}{3}}} + \frac{924 a^{6} b^{5} x^{\frac{5}{3}}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac{2}{3}}} - \frac{462 a^{5} b^{6} x^{2}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac{2}{3}}} + \frac{264 a^{4} b^{7} x^{\frac{7}{3}}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac{2}{3}}} - \frac{165 a^{3} b^{8} x^{\frac{8}{3}}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac{2}{3}}} + \frac{110 a^{2} b^{9} x^{3}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac{2}{3}}} - \frac{77 a b^{10} x^{\frac{10}{3}}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac{2}{3}}} + \frac{56 b^{11} x^{\frac{11}{3}}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac{2}{3}}} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 a^{3}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.224032, size = 196, normalized size = 1.15 \[ -\frac{165 \, a^{9}{\rm ln}\left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{b^{12}} - \frac{3 \,{\left (22 \, a^{10} b x^{\frac{1}{3}} + 21 \, a^{11}\right )}}{2 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} b^{12}} + \frac{56 \, b^{24} x^{3} - 189 \, a b^{23} x^{\frac{8}{3}} + 432 \, a^{2} b^{22} x^{\frac{7}{3}} - 840 \, a^{3} b^{21} x^{2} + 1512 \, a^{4} b^{20} x^{\frac{5}{3}} - 2646 \, a^{5} b^{19} x^{\frac{4}{3}} + 4704 \, a^{6} b^{18} x - 9072 \, a^{7} b^{17} x^{\frac{2}{3}} + 22680 \, a^{8} b^{16} x^{\frac{1}{3}}}{168 \, b^{27}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-165*a^9*ln(abs(b*x^(1/3) + a))/b^12 - 3/2*(22*a^10*b*x^(1/3) + 21*a^11)/((b*x^(
1/3) + a)^2*b^12) + 1/168*(56*b^24*x^3 - 189*a*b^23*x^(8/3) + 432*a^2*b^22*x^(7/
3) - 840*a^3*b^21*x^2 + 1512*a^4*b^20*x^(5/3) - 2646*a^5*b^19*x^(4/3) + 4704*a^6
*b^18*x - 9072*a^7*b^17*x^(2/3) + 22680*a^8*b^16*x^(1/3))/b^27