3.2358 \(\int \frac{x^3}{\left (a+b \sqrt [3]{x}\right )^2} \, dx\)

Optimal. Leaf size=171 \[ \frac{3 a^{11}}{b^{12} \left (a+b \sqrt [3]{x}\right )}+\frac{33 a^{10} \log \left (a+b \sqrt [3]{x}\right )}{b^{12}}-\frac{30 a^9 \sqrt [3]{x}}{b^{11}}+\frac{27 a^8 x^{2/3}}{2 b^{10}}-\frac{8 a^7 x}{b^9}+\frac{21 a^6 x^{4/3}}{4 b^8}-\frac{18 a^5 x^{5/3}}{5 b^7}+\frac{5 a^4 x^2}{2 b^6}-\frac{12 a^3 x^{7/3}}{7 b^5}+\frac{9 a^2 x^{8/3}}{8 b^4}-\frac{2 a x^3}{3 b^3}+\frac{3 x^{10/3}}{10 b^2} \]

[Out]

(3*a^11)/(b^12*(a + b*x^(1/3))) - (30*a^9*x^(1/3))/b^11 + (27*a^8*x^(2/3))/(2*b^
10) - (8*a^7*x)/b^9 + (21*a^6*x^(4/3))/(4*b^8) - (18*a^5*x^(5/3))/(5*b^7) + (5*a
^4*x^2)/(2*b^6) - (12*a^3*x^(7/3))/(7*b^5) + (9*a^2*x^(8/3))/(8*b^4) - (2*a*x^3)
/(3*b^3) + (3*x^(10/3))/(10*b^2) + (33*a^10*Log[a + b*x^(1/3)])/b^12

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Rubi [A]  time = 0.297123, 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 \[ \frac{3 a^{11}}{b^{12} \left (a+b \sqrt [3]{x}\right )}+\frac{33 a^{10} \log \left (a+b \sqrt [3]{x}\right )}{b^{12}}-\frac{30 a^9 \sqrt [3]{x}}{b^{11}}+\frac{27 a^8 x^{2/3}}{2 b^{10}}-\frac{8 a^7 x}{b^9}+\frac{21 a^6 x^{4/3}}{4 b^8}-\frac{18 a^5 x^{5/3}}{5 b^7}+\frac{5 a^4 x^2}{2 b^6}-\frac{12 a^3 x^{7/3}}{7 b^5}+\frac{9 a^2 x^{8/3}}{8 b^4}-\frac{2 a x^3}{3 b^3}+\frac{3 x^{10/3}}{10 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b*x^(1/3))^2,x]

[Out]

(3*a^11)/(b^12*(a + b*x^(1/3))) - (30*a^9*x^(1/3))/b^11 + (27*a^8*x^(2/3))/(2*b^
10) - (8*a^7*x)/b^9 + (21*a^6*x^(4/3))/(4*b^8) - (18*a^5*x^(5/3))/(5*b^7) + (5*a
^4*x^2)/(2*b^6) - (12*a^3*x^(7/3))/(7*b^5) + (9*a^2*x^(8/3))/(8*b^4) - (2*a*x^3)
/(3*b^3) + (3*x^(10/3))/(10*b^2) + (33*a^10*Log[a + b*x^(1/3)])/b^12

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{3 a^{11}}{b^{12} \left (a + b \sqrt [3]{x}\right )} + \frac{33 a^{10} \log{\left (a + b \sqrt [3]{x} \right )}}{b^{12}} - \frac{30 a^{9} \sqrt [3]{x}}{b^{11}} + \frac{27 a^{8} \int ^{\sqrt [3]{x}} x\, dx}{b^{10}} - \frac{8 a^{7} x}{b^{9}} + \frac{21 a^{6} x^{\frac{4}{3}}}{4 b^{8}} - \frac{18 a^{5} x^{\frac{5}{3}}}{5 b^{7}} + \frac{5 a^{4} x^{2}}{2 b^{6}} - \frac{12 a^{3} x^{\frac{7}{3}}}{7 b^{5}} + \frac{9 a^{2} x^{\frac{8}{3}}}{8 b^{4}} - \frac{2 a x^{3}}{3 b^{3}} + \frac{3 x^{\frac{10}{3}}}{10 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a**11/(b**12*(a + b*x**(1/3))) + 33*a**10*log(a + b*x**(1/3))/b**12 - 30*a**9*
x**(1/3)/b**11 + 27*a**8*Integral(x, (x, x**(1/3)))/b**10 - 8*a**7*x/b**9 + 21*a
**6*x**(4/3)/(4*b**8) - 18*a**5*x**(5/3)/(5*b**7) + 5*a**4*x**2/(2*b**6) - 12*a*
*3*x**(7/3)/(7*b**5) + 9*a**2*x**(8/3)/(8*b**4) - 2*a*x**3/(3*b**3) + 3*x**(10/3
)/(10*b**2)

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Mathematica [A]  time = 0.119952, size = 171, normalized size = 1. \[ \frac{3 a^{11}}{b^{12} \left (a+b \sqrt [3]{x}\right )}+\frac{33 a^{10} \log \left (a+b \sqrt [3]{x}\right )}{b^{12}}-\frac{30 a^9 \sqrt [3]{x}}{b^{11}}+\frac{27 a^8 x^{2/3}}{2 b^{10}}-\frac{8 a^7 x}{b^9}+\frac{21 a^6 x^{4/3}}{4 b^8}-\frac{18 a^5 x^{5/3}}{5 b^7}+\frac{5 a^4 x^2}{2 b^6}-\frac{12 a^3 x^{7/3}}{7 b^5}+\frac{9 a^2 x^{8/3}}{8 b^4}-\frac{2 a x^3}{3 b^3}+\frac{3 x^{10/3}}{10 b^2} \]

Antiderivative was successfully verified.

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

[Out]

(3*a^11)/(b^12*(a + b*x^(1/3))) - (30*a^9*x^(1/3))/b^11 + (27*a^8*x^(2/3))/(2*b^
10) - (8*a^7*x)/b^9 + (21*a^6*x^(4/3))/(4*b^8) - (18*a^5*x^(5/3))/(5*b^7) + (5*a
^4*x^2)/(2*b^6) - (12*a^3*x^(7/3))/(7*b^5) + (9*a^2*x^(8/3))/(8*b^4) - (2*a*x^3)
/(3*b^3) + (3*x^(10/3))/(10*b^2) + (33*a^10*Log[a + b*x^(1/3)])/b^12

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Maple [A]  time = 0.012, size = 138, normalized size = 0.8 \[ 3\,{\frac{{a}^{11}}{{b}^{12} \left ( a+b\sqrt [3]{x} \right ) }}-30\,{\frac{{a}^{9}\sqrt [3]{x}}{{b}^{11}}}+{\frac{27\,{a}^{8}}{2\,{b}^{10}}{x}^{{\frac{2}{3}}}}-8\,{\frac{{a}^{7}x}{{b}^{9}}}+{\frac{21\,{a}^{6}}{4\,{b}^{8}}{x}^{{\frac{4}{3}}}}-{\frac{18\,{a}^{5}}{5\,{b}^{7}}{x}^{{\frac{5}{3}}}}+{\frac{5\,{a}^{4}{x}^{2}}{2\,{b}^{6}}}-{\frac{12\,{a}^{3}}{7\,{b}^{5}}{x}^{{\frac{7}{3}}}}+{\frac{9\,{a}^{2}}{8\,{b}^{4}}{x}^{{\frac{8}{3}}}}-{\frac{2\,a{x}^{3}}{3\,{b}^{3}}}+{\frac{3}{10\,{b}^{2}}{x}^{{\frac{10}{3}}}}+33\,{\frac{{a}^{10}\ln \left ( a+b\sqrt [3]{x} \right ) }{{b}^{12}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(a+b*x^(1/3))^2,x)

[Out]

3*a^11/b^12/(a+b*x^(1/3))-30*a^9*x^(1/3)/b^11+27/2*a^8*x^(2/3)/b^10-8*a^7*x/b^9+
21/4*a^6*x^(4/3)/b^8-18/5*a^5*x^(5/3)/b^7+5/2*a^4*x^2/b^6-12/7*a^3*x^(7/3)/b^5+9
/8*a^2*x^(8/3)/b^4-2/3*a*x^3/b^3+3/10*x^(10/3)/b^2+33*a^10*ln(a+b*x^(1/3))/b^12

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Maxima [A]  time = 1.44281, size = 266, normalized size = 1.56 \[ \frac{33 \, a^{10} \log \left (b x^{\frac{1}{3}} + a\right )}{b^{12}} + \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{10}}{10 \, b^{12}} - \frac{11 \,{\left (b x^{\frac{1}{3}} + a\right )}^{9} a}{3 \, b^{12}} + \frac{165 \,{\left (b x^{\frac{1}{3}} + a\right )}^{8} a^{2}}{8 \, b^{12}} - \frac{495 \,{\left (b x^{\frac{1}{3}} + a\right )}^{7} a^{3}}{7 \, b^{12}} + \frac{165 \,{\left (b x^{\frac{1}{3}} + a\right )}^{6} a^{4}}{b^{12}} - \frac{1386 \,{\left (b x^{\frac{1}{3}} + a\right )}^{5} a^{5}}{5 \, b^{12}} + \frac{693 \,{\left (b x^{\frac{1}{3}} + a\right )}^{4} a^{6}}{2 \, b^{12}} - \frac{330 \,{\left (b x^{\frac{1}{3}} + a\right )}^{3} a^{7}}{b^{12}} + \frac{495 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} a^{8}}{2 \, b^{12}} - \frac{165 \,{\left (b x^{\frac{1}{3}} + a\right )} a^{9}}{b^{12}} + \frac{3 \, a^{11}}{{\left (b x^{\frac{1}{3}} + a\right )} b^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

33*a^10*log(b*x^(1/3) + a)/b^12 + 3/10*(b*x^(1/3) + a)^10/b^12 - 11/3*(b*x^(1/3)
+ a)^9*a/b^12 + 165/8*(b*x^(1/3) + a)^8*a^2/b^12 - 495/7*(b*x^(1/3) + a)^7*a^3/
b^12 + 165*(b*x^(1/3) + a)^6*a^4/b^12 - 1386/5*(b*x^(1/3) + a)^5*a^5/b^12 + 693/
2*(b*x^(1/3) + a)^4*a^6/b^12 - 330*(b*x^(1/3) + a)^3*a^7/b^12 + 495/2*(b*x^(1/3)
+ a)^2*a^8/b^12 - 165*(b*x^(1/3) + a)*a^9/b^12 + 3*a^11/((b*x^(1/3) + a)*b^12)

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Fricas [A]  time = 0.219541, size = 215, normalized size = 1.26 \[ \frac{385 \, a^{2} b^{9} x^{3} - 924 \, a^{5} b^{6} x^{2} + 4620 \, a^{8} b^{3} x + 2520 \, a^{11} + 27720 \,{\left (a^{10} b x^{\frac{1}{3}} + a^{11}\right )} \log \left (b x^{\frac{1}{3}} + a\right ) + 9 \,{\left (28 \, b^{11} x^{3} - 55 \, a^{3} b^{8} x^{2} + 154 \, a^{6} b^{5} x - 1540 \, a^{9} b^{2}\right )} x^{\frac{2}{3}} - 2 \,{\left (154 \, a b^{10} x^{3} - 330 \, a^{4} b^{7} x^{2} + 1155 \, a^{7} b^{4} x + 12600 \, a^{10} b\right )} x^{\frac{1}{3}}}{840 \,{\left (b^{13} x^{\frac{1}{3}} + a b^{12}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/840*(385*a^2*b^9*x^3 - 924*a^5*b^6*x^2 + 4620*a^8*b^3*x + 2520*a^11 + 27720*(a
^10*b*x^(1/3) + a^11)*log(b*x^(1/3) + a) + 9*(28*b^11*x^3 - 55*a^3*b^8*x^2 + 154
*a^6*b^5*x - 1540*a^9*b^2)*x^(2/3) - 2*(154*a*b^10*x^3 - 330*a^4*b^7*x^2 + 1155*
a^7*b^4*x + 12600*a^10*b)*x^(1/3))/(b^13*x^(1/3) + a*b^12)

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Sympy [A]  time = 138.042, size = 444, normalized size = 2.6 \[ \frac{27720 a^{11} x^{\frac{308}{3}} \log{\left (1 + \frac{b \sqrt [3]{x}}{a} \right )}}{840 a b^{12} x^{\frac{308}{3}} + 840 b^{13} x^{103}} + \frac{27720 a^{10} b x^{103} \log{\left (1 + \frac{b \sqrt [3]{x}}{a} \right )}}{840 a b^{12} x^{\frac{308}{3}} + 840 b^{13} x^{103}} - \frac{27720 a^{10} b x^{103}}{840 a b^{12} x^{\frac{308}{3}} + 840 b^{13} x^{103}} - \frac{13860 a^{9} b^{2} x^{\frac{310}{3}}}{840 a b^{12} x^{\frac{308}{3}} + 840 b^{13} x^{103}} + \frac{4620 a^{8} b^{3} x^{\frac{311}{3}}}{840 a b^{12} x^{\frac{308}{3}} + 840 b^{13} x^{103}} - \frac{2310 a^{7} b^{4} x^{104}}{840 a b^{12} x^{\frac{308}{3}} + 840 b^{13} x^{103}} + \frac{1386 a^{6} b^{5} x^{\frac{313}{3}}}{840 a b^{12} x^{\frac{308}{3}} + 840 b^{13} x^{103}} - \frac{924 a^{5} b^{6} x^{\frac{314}{3}}}{840 a b^{12} x^{\frac{308}{3}} + 840 b^{13} x^{103}} + \frac{660 a^{4} b^{7} x^{105}}{840 a b^{12} x^{\frac{308}{3}} + 840 b^{13} x^{103}} - \frac{495 a^{3} b^{8} x^{\frac{316}{3}}}{840 a b^{12} x^{\frac{308}{3}} + 840 b^{13} x^{103}} + \frac{385 a^{2} b^{9} x^{\frac{317}{3}}}{840 a b^{12} x^{\frac{308}{3}} + 840 b^{13} x^{103}} - \frac{308 a b^{10} x^{106}}{840 a b^{12} x^{\frac{308}{3}} + 840 b^{13} x^{103}} + \frac{252 b^{11} x^{\frac{319}{3}}}{840 a b^{12} x^{\frac{308}{3}} + 840 b^{13} x^{103}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27720*a**11*x**(308/3)*log(1 + b*x**(1/3)/a)/(840*a*b**12*x**(308/3) + 840*b**13
*x**103) + 27720*a**10*b*x**103*log(1 + b*x**(1/3)/a)/(840*a*b**12*x**(308/3) +
840*b**13*x**103) - 27720*a**10*b*x**103/(840*a*b**12*x**(308/3) + 840*b**13*x**
103) - 13860*a**9*b**2*x**(310/3)/(840*a*b**12*x**(308/3) + 840*b**13*x**103) +
4620*a**8*b**3*x**(311/3)/(840*a*b**12*x**(308/3) + 840*b**13*x**103) - 2310*a**
7*b**4*x**104/(840*a*b**12*x**(308/3) + 840*b**13*x**103) + 1386*a**6*b**5*x**(3
13/3)/(840*a*b**12*x**(308/3) + 840*b**13*x**103) - 924*a**5*b**6*x**(314/3)/(84
0*a*b**12*x**(308/3) + 840*b**13*x**103) + 660*a**4*b**7*x**105/(840*a*b**12*x**
(308/3) + 840*b**13*x**103) - 495*a**3*b**8*x**(316/3)/(840*a*b**12*x**(308/3) +
840*b**13*x**103) + 385*a**2*b**9*x**(317/3)/(840*a*b**12*x**(308/3) + 840*b**1
3*x**103) - 308*a*b**10*x**106/(840*a*b**12*x**(308/3) + 840*b**13*x**103) + 252
*b**11*x**(319/3)/(840*a*b**12*x**(308/3) + 840*b**13*x**103)

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GIAC/XCAS [A]  time = 0.223925, size = 194, normalized size = 1.13 \[ \frac{33 \, a^{10}{\rm ln}\left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{b^{12}} + \frac{3 \, a^{11}}{{\left (b x^{\frac{1}{3}} + a\right )} b^{12}} + \frac{252 \, b^{18} x^{\frac{10}{3}} - 560 \, a b^{17} x^{3} + 945 \, a^{2} b^{16} x^{\frac{8}{3}} - 1440 \, a^{3} b^{15} x^{\frac{7}{3}} + 2100 \, a^{4} b^{14} x^{2} - 3024 \, a^{5} b^{13} x^{\frac{5}{3}} + 4410 \, a^{6} b^{12} x^{\frac{4}{3}} - 6720 \, a^{7} b^{11} x + 11340 \, a^{8} b^{10} x^{\frac{2}{3}} - 25200 \, a^{9} b^{9} x^{\frac{1}{3}}}{840 \, b^{20}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

33*a^10*ln(abs(b*x^(1/3) + a))/b^12 + 3*a^11/((b*x^(1/3) + a)*b^12) + 1/840*(252
*b^18*x^(10/3) - 560*a*b^17*x^3 + 945*a^2*b^16*x^(8/3) - 1440*a^3*b^15*x^(7/3) +
2100*a^4*b^14*x^2 - 3024*a^5*b^13*x^(5/3) + 4410*a^6*b^12*x^(4/3) - 6720*a^7*b^
11*x + 11340*a^8*b^10*x^(2/3) - 25200*a^9*b^9*x^(1/3))/b^20