3.2367 \(\int \frac{x^2}{\left (a+b \sqrt [3]{x}\right )^3} \, dx\)

Optimal. Leaf size=134 \[ -\frac{3 a^8}{2 b^9 \left (a+b \sqrt [3]{x}\right )^2}+\frac{24 a^7}{b^9 \left (a+b \sqrt [3]{x}\right )}+\frac{84 a^6 \log \left (a+b \sqrt [3]{x}\right )}{b^9}-\frac{63 a^5 \sqrt [3]{x}}{b^8}+\frac{45 a^4 x^{2/3}}{2 b^7}-\frac{10 a^3 x}{b^6}+\frac{9 a^2 x^{4/3}}{2 b^5}-\frac{9 a x^{5/3}}{5 b^4}+\frac{x^2}{2 b^3} \]

[Out]

(-3*a^8)/(2*b^9*(a + b*x^(1/3))^2) + (24*a^7)/(b^9*(a + b*x^(1/3))) - (63*a^5*x^
(1/3))/b^8 + (45*a^4*x^(2/3))/(2*b^7) - (10*a^3*x)/b^6 + (9*a^2*x^(4/3))/(2*b^5)
- (9*a*x^(5/3))/(5*b^4) + x^2/(2*b^3) + (84*a^6*Log[a + b*x^(1/3)])/b^9

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Rubi [A]  time = 0.225814, antiderivative size = 134, 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 \[ -\frac{3 a^8}{2 b^9 \left (a+b \sqrt [3]{x}\right )^2}+\frac{24 a^7}{b^9 \left (a+b \sqrt [3]{x}\right )}+\frac{84 a^6 \log \left (a+b \sqrt [3]{x}\right )}{b^9}-\frac{63 a^5 \sqrt [3]{x}}{b^8}+\frac{45 a^4 x^{2/3}}{2 b^7}-\frac{10 a^3 x}{b^6}+\frac{9 a^2 x^{4/3}}{2 b^5}-\frac{9 a x^{5/3}}{5 b^4}+\frac{x^2}{2 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b*x^(1/3))^3,x]

[Out]

(-3*a^8)/(2*b^9*(a + b*x^(1/3))^2) + (24*a^7)/(b^9*(a + b*x^(1/3))) - (63*a^5*x^
(1/3))/b^8 + (45*a^4*x^(2/3))/(2*b^7) - (10*a^3*x)/b^6 + (9*a^2*x^(4/3))/(2*b^5)
- (9*a*x^(5/3))/(5*b^4) + x^2/(2*b^3) + (84*a^6*Log[a + b*x^(1/3)])/b^9

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{3 a^{8}}{2 b^{9} \left (a + b \sqrt [3]{x}\right )^{2}} + \frac{24 a^{7}}{b^{9} \left (a + b \sqrt [3]{x}\right )} + \frac{84 a^{6} \log{\left (a + b \sqrt [3]{x} \right )}}{b^{9}} - \frac{63 a^{5} \sqrt [3]{x}}{b^{8}} + \frac{45 a^{4} \int ^{\sqrt [3]{x}} x\, dx}{b^{7}} - \frac{10 a^{3} x}{b^{6}} + \frac{9 a^{2} x^{\frac{4}{3}}}{2 b^{5}} - \frac{9 a x^{\frac{5}{3}}}{5 b^{4}} + \frac{x^{2}}{2 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*a**8/(2*b**9*(a + b*x**(1/3))**2) + 24*a**7/(b**9*(a + b*x**(1/3))) + 84*a**6
*log(a + b*x**(1/3))/b**9 - 63*a**5*x**(1/3)/b**8 + 45*a**4*Integral(x, (x, x**(
1/3)))/b**7 - 10*a**3*x/b**6 + 9*a**2*x**(4/3)/(2*b**5) - 9*a*x**(5/3)/(5*b**4)
+ x**2/(2*b**3)

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Mathematica [A]  time = 0.08003, size = 120, normalized size = 0.9 \[ \frac{-\frac{15 a^8}{\left (a+b \sqrt [3]{x}\right )^2}+\frac{240 a^7}{a+b \sqrt [3]{x}}+840 a^6 \log \left (a+b \sqrt [3]{x}\right )-630 a^5 b \sqrt [3]{x}+225 a^4 b^2 x^{2/3}-100 a^3 b^3 x+45 a^2 b^4 x^{4/3}-18 a b^5 x^{5/3}+5 b^6 x^2}{10 b^9} \]

Antiderivative was successfully verified.

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

[Out]

((-15*a^8)/(a + b*x^(1/3))^2 + (240*a^7)/(a + b*x^(1/3)) - 630*a^5*b*x^(1/3) + 2
25*a^4*b^2*x^(2/3) - 100*a^3*b^3*x + 45*a^2*b^4*x^(4/3) - 18*a*b^5*x^(5/3) + 5*b
^6*x^2 + 840*a^6*Log[a + b*x^(1/3)])/(10*b^9)

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Maple [A]  time = 0.013, size = 111, normalized size = 0.8 \[ -{\frac{3\,{a}^{8}}{2\,{b}^{9}} \left ( a+b\sqrt [3]{x} \right ) ^{-2}}+24\,{\frac{{a}^{7}}{{b}^{9} \left ( a+b\sqrt [3]{x} \right ) }}-63\,{\frac{{a}^{5}\sqrt [3]{x}}{{b}^{8}}}+{\frac{45\,{a}^{4}}{2\,{b}^{7}}{x}^{{\frac{2}{3}}}}-10\,{\frac{{a}^{3}x}{{b}^{6}}}+{\frac{9\,{a}^{2}}{2\,{b}^{5}}{x}^{{\frac{4}{3}}}}-{\frac{9\,a}{5\,{b}^{4}}{x}^{{\frac{5}{3}}}}+{\frac{{x}^{2}}{2\,{b}^{3}}}+84\,{\frac{{a}^{6}\ln \left ( a+b\sqrt [3]{x} \right ) }{{b}^{9}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(a+b*x^(1/3))^3,x)

[Out]

-3/2*a^8/b^9/(a+b*x^(1/3))^2+24*a^7/b^9/(a+b*x^(1/3))-63*a^5*x^(1/3)/b^8+45/2*a^
4*x^(2/3)/b^7-10*a^3*x/b^6+9/2*a^2*x^(4/3)/b^5-9/5*a*x^(5/3)/b^4+1/2*x^2/b^3+84*
a^6*ln(a+b*x^(1/3))/b^9

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Maxima [A]  time = 1.45562, size = 197, normalized size = 1.47 \[ \frac{84 \, a^{6} \log \left (b x^{\frac{1}{3}} + a\right )}{b^{9}} + \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{6}}{2 \, b^{9}} - \frac{24 \,{\left (b x^{\frac{1}{3}} + a\right )}^{5} a}{5 \, b^{9}} + \frac{21 \,{\left (b x^{\frac{1}{3}} + a\right )}^{4} a^{2}}{b^{9}} - \frac{56 \,{\left (b x^{\frac{1}{3}} + a\right )}^{3} a^{3}}{b^{9}} + \frac{105 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} a^{4}}{b^{9}} - \frac{168 \,{\left (b x^{\frac{1}{3}} + a\right )} a^{5}}{b^{9}} + \frac{24 \, a^{7}}{{\left (b x^{\frac{1}{3}} + a\right )} b^{9}} - \frac{3 \, a^{8}}{2 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

84*a^6*log(b*x^(1/3) + a)/b^9 + 1/2*(b*x^(1/3) + a)^6/b^9 - 24/5*(b*x^(1/3) + a)
^5*a/b^9 + 21*(b*x^(1/3) + a)^4*a^2/b^9 - 56*(b*x^(1/3) + a)^3*a^3/b^9 + 105*(b*
x^(1/3) + a)^2*a^4/b^9 - 168*(b*x^(1/3) + a)*a^5/b^9 + 24*a^7/((b*x^(1/3) + a)*b
^9) - 3/2*a^8/((b*x^(1/3) + a)^2*b^9)

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Fricas [A]  time = 0.226016, size = 198, normalized size = 1.48 \[ \frac{14 \, a^{2} b^{6} x^{2} - 280 \, a^{5} b^{3} x + 225 \, a^{8} + 840 \,{\left (a^{6} b^{2} x^{\frac{2}{3}} + 2 \, a^{7} b x^{\frac{1}{3}} + a^{8}\right )} \log \left (b x^{\frac{1}{3}} + a\right ) +{\left (5 \, b^{8} x^{2} - 28 \, a^{3} b^{5} x - 1035 \, a^{6} b^{2}\right )} x^{\frac{2}{3}} - 2 \,{\left (4 \, a b^{7} x^{2} - 35 \, a^{4} b^{4} x + 195 \, a^{7} b\right )} x^{\frac{1}{3}}}{10 \,{\left (b^{11} x^{\frac{2}{3}} + 2 \, a b^{10} x^{\frac{1}{3}} + a^{2} b^{9}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*(14*a^2*b^6*x^2 - 280*a^5*b^3*x + 225*a^8 + 840*(a^6*b^2*x^(2/3) + 2*a^7*b*
x^(1/3) + a^8)*log(b*x^(1/3) + a) + (5*b^8*x^2 - 28*a^3*b^5*x - 1035*a^6*b^2)*x^
(2/3) - 2*(4*a*b^7*x^2 - 35*a^4*b^4*x + 195*a^7*b)*x^(1/3))/(b^11*x^(2/3) + 2*a*
b^10*x^(1/3) + a^2*b^9)

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Sympy [A]  time = 7.33795, size = 493, normalized size = 3.68 \[ \begin{cases} \frac{840 a^{8} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} + \frac{1260 a^{8}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} + \frac{1680 a^{7} b \sqrt [3]{x} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} + \frac{1680 a^{7} b \sqrt [3]{x}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} + \frac{840 a^{6} b^{2} x^{\frac{2}{3}} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} - \frac{280 a^{5} b^{3} x}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} + \frac{70 a^{4} b^{4} x^{\frac{4}{3}}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} - \frac{28 a^{3} b^{5} x^{\frac{5}{3}}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} + \frac{14 a^{2} b^{6} x^{2}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} - \frac{8 a b^{7} x^{\frac{7}{3}}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} + \frac{5 b^{8} x^{\frac{8}{3}}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} & \text{for}\: b \neq 0 \\\frac{x^{3}}{3 a^{3}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((840*a**8*log(a/b + x**(1/3))/(10*a**2*b**9 + 20*a*b**10*x**(1/3) + 10
*b**11*x**(2/3)) + 1260*a**8/(10*a**2*b**9 + 20*a*b**10*x**(1/3) + 10*b**11*x**(
2/3)) + 1680*a**7*b*x**(1/3)*log(a/b + x**(1/3))/(10*a**2*b**9 + 20*a*b**10*x**(
1/3) + 10*b**11*x**(2/3)) + 1680*a**7*b*x**(1/3)/(10*a**2*b**9 + 20*a*b**10*x**(
1/3) + 10*b**11*x**(2/3)) + 840*a**6*b**2*x**(2/3)*log(a/b + x**(1/3))/(10*a**2*
b**9 + 20*a*b**10*x**(1/3) + 10*b**11*x**(2/3)) - 280*a**5*b**3*x/(10*a**2*b**9
+ 20*a*b**10*x**(1/3) + 10*b**11*x**(2/3)) + 70*a**4*b**4*x**(4/3)/(10*a**2*b**9
+ 20*a*b**10*x**(1/3) + 10*b**11*x**(2/3)) - 28*a**3*b**5*x**(5/3)/(10*a**2*b**
9 + 20*a*b**10*x**(1/3) + 10*b**11*x**(2/3)) + 14*a**2*b**6*x**2/(10*a**2*b**9 +
20*a*b**10*x**(1/3) + 10*b**11*x**(2/3)) - 8*a*b**7*x**(7/3)/(10*a**2*b**9 + 20
*a*b**10*x**(1/3) + 10*b**11*x**(2/3)) + 5*b**8*x**(8/3)/(10*a**2*b**9 + 20*a*b*
*10*x**(1/3) + 10*b**11*x**(2/3)), Ne(b, 0)), (x**3/(3*a**3), True))

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GIAC/XCAS [A]  time = 0.223357, size = 151, normalized size = 1.13 \[ \frac{84 \, a^{6}{\rm ln}\left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{b^{9}} + \frac{3 \,{\left (16 \, a^{7} b x^{\frac{1}{3}} + 15 \, a^{8}\right )}}{2 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} b^{9}} + \frac{5 \, b^{15} x^{2} - 18 \, a b^{14} x^{\frac{5}{3}} + 45 \, a^{2} b^{13} x^{\frac{4}{3}} - 100 \, a^{3} b^{12} x + 225 \, a^{4} b^{11} x^{\frac{2}{3}} - 630 \, a^{5} b^{10} x^{\frac{1}{3}}}{10 \, b^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

84*a^6*ln(abs(b*x^(1/3) + a))/b^9 + 3/2*(16*a^7*b*x^(1/3) + 15*a^8)/((b*x^(1/3)
+ a)^2*b^9) + 1/10*(5*b^15*x^2 - 18*a*b^14*x^(5/3) + 45*a^2*b^13*x^(4/3) - 100*a
^3*b^12*x + 225*a^4*b^11*x^(2/3) - 630*a^5*b^10*x^(1/3))/b^18