3.2431 \(\int \frac{x}{\left (a+\frac{b}{\sqrt [3]{x}}\right )^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.236058, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{3 b^8}{2 a^9 \left (a \sqrt [3]{x}+b\right )^2}+\frac{24 b^7}{a^9 \left (a \sqrt [3]{x}+b\right )}+\frac{84 b^6 \log \left (a \sqrt [3]{x}+b\right )}{a^9}-\frac{63 b^5 \sqrt [3]{x}}{a^8}+\frac{45 b^4 x^{2/3}}{2 a^7}-\frac{10 b^3 x}{a^6}+\frac{9 b^2 x^{4/3}}{2 a^5}-\frac{9 b x^{5/3}}{5 a^4}+\frac{x^2}{2 a^3} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b/x^(1/3))^3,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0741221, size = 120, normalized size = 0.9 \[ \frac{5 a^6 x^2-18 a^5 b x^{5/3}+45 a^4 b^2 x^{4/3}-100 a^3 b^3 x+225 a^2 b^4 x^{2/3}-\frac{15 b^8}{\left (a \sqrt [3]{x}+b\right )^2}+\frac{240 b^7}{a \sqrt [3]{x}+b}+840 b^6 \log \left (a \sqrt [3]{x}+b\right )-630 a b^5 \sqrt [3]{x}}{10 a^9} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 111, normalized size = 0.8 \[ -{\frac{3\,{b}^{8}}{2\,{a}^{9}} \left ( b+a\sqrt [3]{x} \right ) ^{-2}}+24\,{\frac{{b}^{7}}{{a}^{9} \left ( b+a\sqrt [3]{x} \right ) }}-63\,{\frac{{b}^{5}\sqrt [3]{x}}{{a}^{8}}}+{\frac{45\,{b}^{4}}{2\,{a}^{7}}{x}^{{\frac{2}{3}}}}-10\,{\frac{{b}^{3}x}{{a}^{6}}}+{\frac{9\,{b}^{2}}{2\,{a}^{5}}{x}^{{\frac{4}{3}}}}-{\frac{9\,b}{5\,{a}^{4}}{x}^{{\frac{5}{3}}}}+{\frac{{x}^{2}}{2\,{a}^{3}}}+84\,{\frac{{b}^{6}\ln \left ( b+a\sqrt [3]{x} \right ) }{{a}^{9}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b/x^(1/3))^3,x)

[Out]

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

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Maxima [A]  time = 1.46607, size = 181, normalized size = 1.35 \[ \frac{5 \, a^{7} - \frac{8 \, a^{6} b}{x^{\frac{1}{3}}} + \frac{14 \, a^{5} b^{2}}{x^{\frac{2}{3}}} - \frac{28 \, a^{4} b^{3}}{x} + \frac{70 \, a^{3} b^{4}}{x^{\frac{4}{3}}} - \frac{280 \, a^{2} b^{5}}{x^{\frac{5}{3}}} - \frac{1260 \, a b^{6}}{x^{2}} - \frac{840 \, b^{7}}{x^{\frac{7}{3}}}}{10 \,{\left (\frac{a^{10}}{x^{2}} + \frac{2 \, a^{9} b}{x^{\frac{7}{3}}} + \frac{a^{8} b^{2}}{x^{\frac{8}{3}}}\right )}} + \frac{84 \, b^{6} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{a^{9}} + \frac{28 \, b^{6} \log \left (x\right )}{a^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*(5*a^7 - 8*a^6*b/x^(1/3) + 14*a^5*b^2/x^(2/3) - 28*a^4*b^3/x + 70*a^3*b^4/x
^(4/3) - 280*a^2*b^5/x^(5/3) - 1260*a*b^6/x^2 - 840*b^7/x^(7/3))/(a^10/x^2 + 2*a
^9*b/x^(7/3) + a^8*b^2/x^(8/3)) + 84*b^6*log(a + b/x^(1/3))/a^9 + 28*b^6*log(x)/
a^9

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Fricas [A]  time = 0.227002, size = 198, normalized size = 1.48 \[ \frac{14 \, a^{6} b^{2} x^{2} - 280 \, a^{3} b^{5} x + 225 \, b^{8} + 840 \,{\left (a^{2} b^{6} x^{\frac{2}{3}} + 2 \, a b^{7} x^{\frac{1}{3}} + b^{8}\right )} \log \left (a x^{\frac{1}{3}} + b\right ) +{\left (5 \, a^{8} x^{2} - 28 \, a^{5} b^{3} x - 1035 \, a^{2} b^{6}\right )} x^{\frac{2}{3}} - 2 \,{\left (4 \, a^{7} b x^{2} - 35 \, a^{4} b^{4} x + 195 \, a b^{7}\right )} x^{\frac{1}{3}}}{10 \,{\left (a^{11} x^{\frac{2}{3}} + 2 \, a^{10} b x^{\frac{1}{3}} + a^{9} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 62.6536, size = 1069, normalized size = 7.98 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*a**9*x**60/(10*a**12*x**58 + 30*a**11*b*x**(173/3) + 30*a**10*b**2*x**(172/3)
+ 10*a**9*b**3*x**57) - 3*a**8*b*x**(179/3)/(10*a**12*x**58 + 30*a**11*b*x**(173
/3) + 30*a**10*b**2*x**(172/3) + 10*a**9*b**3*x**57) + 6*a**7*b**2*x**(178/3)/(1
0*a**12*x**58 + 30*a**11*b*x**(173/3) + 30*a**10*b**2*x**(172/3) + 10*a**9*b**3*
x**57) - 14*a**6*b**3*x**59/(10*a**12*x**58 + 30*a**11*b*x**(173/3) + 30*a**10*b
**2*x**(172/3) + 10*a**9*b**3*x**57) + 42*a**5*b**4*x**(176/3)/(10*a**12*x**58 +
30*a**11*b*x**(173/3) + 30*a**10*b**2*x**(172/3) + 10*a**9*b**3*x**57) - 210*a*
*4*b**5*x**(175/3)/(10*a**12*x**58 + 30*a**11*b*x**(173/3) + 30*a**10*b**2*x**(1
72/3) + 10*a**9*b**3*x**57) - 840*a**3*b**6*x**58*log(b/(a*x**(1/3)))/(10*a**12*
x**58 + 30*a**11*b*x**(173/3) + 30*a**10*b**2*x**(172/3) + 10*a**9*b**3*x**57) +
840*a**3*b**6*x**58*log(1 + b/(a*x**(1/3)))/(10*a**12*x**58 + 30*a**11*b*x**(17
3/3) + 30*a**10*b**2*x**(172/3) + 10*a**9*b**3*x**57) - 1540*a**3*b**6*x**58/(10
*a**12*x**58 + 30*a**11*b*x**(173/3) + 30*a**10*b**2*x**(172/3) + 10*a**9*b**3*x
**57) - 2520*a**2*b**7*x**(173/3)*log(b/(a*x**(1/3)))/(10*a**12*x**58 + 30*a**11
*b*x**(173/3) + 30*a**10*b**2*x**(172/3) + 10*a**9*b**3*x**57) + 2520*a**2*b**7*
x**(173/3)*log(1 + b/(a*x**(1/3)))/(10*a**12*x**58 + 30*a**11*b*x**(173/3) + 30*
a**10*b**2*x**(172/3) + 10*a**9*b**3*x**57) - 2100*a**2*b**7*x**(173/3)/(10*a**1
2*x**58 + 30*a**11*b*x**(173/3) + 30*a**10*b**2*x**(172/3) + 10*a**9*b**3*x**57)
- 2520*a*b**8*x**(172/3)*log(b/(a*x**(1/3)))/(10*a**12*x**58 + 30*a**11*b*x**(1
73/3) + 30*a**10*b**2*x**(172/3) + 10*a**9*b**3*x**57) + 2520*a*b**8*x**(172/3)*
log(1 + b/(a*x**(1/3)))/(10*a**12*x**58 + 30*a**11*b*x**(173/3) + 30*a**10*b**2*
x**(172/3) + 10*a**9*b**3*x**57) - 840*a*b**8*x**(172/3)/(10*a**12*x**58 + 30*a*
*11*b*x**(173/3) + 30*a**10*b**2*x**(172/3) + 10*a**9*b**3*x**57) - 840*b**9*x**
57*log(b/(a*x**(1/3)))/(10*a**12*x**58 + 30*a**11*b*x**(173/3) + 30*a**10*b**2*x
**(172/3) + 10*a**9*b**3*x**57) + 840*b**9*x**57*log(1 + b/(a*x**(1/3)))/(10*a**
12*x**58 + 30*a**11*b*x**(173/3) + 30*a**10*b**2*x**(172/3) + 10*a**9*b**3*x**57
)

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GIAC/XCAS [A]  time = 0.21799, size = 151, normalized size = 1.13 \[ \frac{84 \, b^{6}{\rm ln}\left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{a^{9}} + \frac{3 \,{\left (16 \, a b^{7} x^{\frac{1}{3}} + 15 \, b^{8}\right )}}{2 \,{\left (a x^{\frac{1}{3}} + b\right )}^{2} a^{9}} + \frac{5 \, a^{15} x^{2} - 18 \, a^{14} b x^{\frac{5}{3}} + 45 \, a^{13} b^{2} x^{\frac{4}{3}} - 100 \, a^{12} b^{3} x + 225 \, a^{11} b^{4} x^{\frac{2}{3}} - 630 \, a^{10} b^{5} x^{\frac{1}{3}}}{10 \, a^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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