3.2429 \(\int \frac{1}{\left (a+\frac{b}{\sqrt [3]{x}}\right )^2 x^5} \, dx\)

Optimal. Leaf size=183 \[ -\frac{33 a^{10} \log \left (a \sqrt [3]{x}+b\right )}{b^{12}}+\frac{11 a^{10} \log (x)}{b^{12}}+\frac{3 a^{10}}{b^{11} \left (a \sqrt [3]{x}+b\right )}+\frac{30 a^9}{b^{11} \sqrt [3]{x}}-\frac{27 a^8}{2 b^{10} x^{2/3}}+\frac{8 a^7}{b^9 x}-\frac{21 a^6}{4 b^8 x^{4/3}}+\frac{18 a^5}{5 b^7 x^{5/3}}-\frac{5 a^4}{2 b^6 x^2}+\frac{12 a^3}{7 b^5 x^{7/3}}-\frac{9 a^2}{8 b^4 x^{8/3}}+\frac{2 a}{3 b^3 x^3}-\frac{3}{10 b^2 x^{10/3}} \]

[Out]

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

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Rubi [A]  time = 0.30437, antiderivative size = 183, 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 \[ -\frac{33 a^{10} \log \left (a \sqrt [3]{x}+b\right )}{b^{12}}+\frac{11 a^{10} \log (x)}{b^{12}}+\frac{3 a^{10}}{b^{11} \left (a \sqrt [3]{x}+b\right )}+\frac{30 a^9}{b^{11} \sqrt [3]{x}}-\frac{27 a^8}{2 b^{10} x^{2/3}}+\frac{8 a^7}{b^9 x}-\frac{21 a^6}{4 b^8 x^{4/3}}+\frac{18 a^5}{5 b^7 x^{5/3}}-\frac{5 a^4}{2 b^6 x^2}+\frac{12 a^3}{7 b^5 x^{7/3}}-\frac{9 a^2}{8 b^4 x^{8/3}}+\frac{2 a}{3 b^3 x^3}-\frac{3}{10 b^2 x^{10/3}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^(1/3))^2*x^5),x]

[Out]

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

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Rubi in Sympy [A]  time = 59.1396, size = 187, normalized size = 1.02 \[ \frac{3 a^{10}}{b^{11} \left (a \sqrt [3]{x} + b\right )} + \frac{33 a^{10} \log{\left (\sqrt [3]{x} \right )}}{b^{12}} - \frac{33 a^{10} \log{\left (a \sqrt [3]{x} + b \right )}}{b^{12}} + \frac{30 a^{9}}{b^{11} \sqrt [3]{x}} - \frac{27 a^{8}}{2 b^{10} x^{\frac{2}{3}}} + \frac{8 a^{7}}{b^{9} x} - \frac{21 a^{6}}{4 b^{8} x^{\frac{4}{3}}} + \frac{18 a^{5}}{5 b^{7} x^{\frac{5}{3}}} - \frac{5 a^{4}}{2 b^{6} x^{2}} + \frac{12 a^{3}}{7 b^{5} x^{\frac{7}{3}}} - \frac{9 a^{2}}{8 b^{4} x^{\frac{8}{3}}} + \frac{2 a}{3 b^{3} x^{3}} - \frac{3}{10 b^{2} x^{\frac{10}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**(1/3))**2/x**5,x)

[Out]

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

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Mathematica [A]  time = 0.45466, size = 169, normalized size = 0.92 \[ \frac{-27720 a^{10} \log \left (a \sqrt [3]{x}+b\right )+9240 a^{10} \log (x)+\frac{b \left (27720 a^{10} x^{10/3}+13860 a^9 b x^3-4620 a^8 b^2 x^{8/3}+2310 a^7 b^3 x^{7/3}-1386 a^6 b^4 x^2+924 a^5 b^5 x^{5/3}-660 a^4 b^6 x^{4/3}+495 a^3 b^7 x-385 a^2 b^8 x^{2/3}+308 a b^9 \sqrt [3]{x}-252 b^{10}\right )}{x^{10/3} \left (a \sqrt [3]{x}+b\right )}}{840 b^{12}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^(1/3))^2*x^5),x]

[Out]

((b*(-252*b^10 + 308*a*b^9*x^(1/3) - 385*a^2*b^8*x^(2/3) + 495*a^3*b^7*x - 660*a
^4*b^6*x^(4/3) + 924*a^5*b^5*x^(5/3) - 1386*a^6*b^4*x^2 + 2310*a^7*b^3*x^(7/3) -
4620*a^8*b^2*x^(8/3) + 13860*a^9*b*x^3 + 27720*a^10*x^(10/3)))/((b + a*x^(1/3))
*x^(10/3)) - 27720*a^10*Log[b + a*x^(1/3)] + 9240*a^10*Log[x])/(840*b^12)

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Maple [A]  time = 0.021, size = 150, normalized size = 0.8 \[ 3\,{\frac{{a}^{10}}{{b}^{11} \left ( b+a\sqrt [3]{x} \right ) }}-{\frac{3}{10\,{b}^{2}}{x}^{-{\frac{10}{3}}}}+{\frac{2\,a}{3\,{b}^{3}{x}^{3}}}-{\frac{9\,{a}^{2}}{8\,{b}^{4}}{x}^{-{\frac{8}{3}}}}+{\frac{12\,{a}^{3}}{7\,{b}^{5}}{x}^{-{\frac{7}{3}}}}-{\frac{5\,{a}^{4}}{2\,{b}^{6}{x}^{2}}}+{\frac{18\,{a}^{5}}{5\,{b}^{7}}{x}^{-{\frac{5}{3}}}}-{\frac{21\,{a}^{6}}{4\,{b}^{8}}{x}^{-{\frac{4}{3}}}}+8\,{\frac{{a}^{7}}{{b}^{9}x}}-{\frac{27\,{a}^{8}}{2\,{b}^{10}}{x}^{-{\frac{2}{3}}}}+30\,{\frac{{a}^{9}}{{b}^{11}\sqrt [3]{x}}}-33\,{\frac{{a}^{10}\ln \left ( b+a\sqrt [3]{x} \right ) }{{b}^{12}}}+11\,{\frac{{a}^{10}\ln \left ( x \right ) }{{b}^{12}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^(1/3))^2/x^5,x)

[Out]

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

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Maxima [A]  time = 1.43217, size = 266, normalized size = 1.45 \[ -\frac{33 \, a^{10} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{b^{12}} - \frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{10}}{10 \, b^{12}} + \frac{11 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{9} a}{3 \, b^{12}} - \frac{165 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{8} a^{2}}{8 \, b^{12}} + \frac{495 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{7} a^{3}}{7 \, b^{12}} - \frac{165 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{6} a^{4}}{b^{12}} + \frac{1386 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{5} a^{5}}{5 \, b^{12}} - \frac{693 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{4} a^{6}}{2 \, b^{12}} + \frac{330 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{3} a^{7}}{b^{12}} - \frac{495 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2} a^{8}}{2 \, b^{12}} + \frac{165 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} a^{9}}{b^{12}} - \frac{3 \, a^{11}}{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} b^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^(1/3))^2*x^5),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.239893, size = 243, normalized size = 1.33 \[ \frac{13860 \, a^{9} b^{2} x^{3} - 1386 \, a^{6} b^{5} x^{2} + 495 \, a^{3} b^{8} x - 252 \, b^{11} - 27720 \,{\left (a^{11} x^{\frac{11}{3}} + a^{10} b x^{\frac{10}{3}}\right )} \log \left (a x^{\frac{1}{3}} + b\right ) + 27720 \,{\left (a^{11} x^{\frac{11}{3}} + a^{10} b x^{\frac{10}{3}}\right )} \log \left (x^{\frac{1}{3}}\right ) - 77 \,{\left (60 \, a^{8} b^{3} x^{2} - 12 \, a^{5} b^{6} x + 5 \, a^{2} b^{9}\right )} x^{\frac{2}{3}} + 22 \,{\left (1260 \, a^{10} b x^{3} + 105 \, a^{7} b^{4} x^{2} - 30 \, a^{4} b^{7} x + 14 \, a b^{10}\right )} x^{\frac{1}{3}}}{840 \,{\left (a b^{12} x^{\frac{11}{3}} + b^{13} x^{\frac{10}{3}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^(1/3))^2*x^5),x, algorithm="fricas")

[Out]

1/840*(13860*a^9*b^2*x^3 - 1386*a^6*b^5*x^2 + 495*a^3*b^8*x - 252*b^11 - 27720*(
a^11*x^(11/3) + a^10*b*x^(10/3))*log(a*x^(1/3) + b) + 27720*(a^11*x^(11/3) + a^1
0*b*x^(10/3))*log(x^(1/3)) - 77*(60*a^8*b^3*x^2 - 12*a^5*b^6*x + 5*a^2*b^9)*x^(2
/3) + 22*(1260*a^10*b*x^3 + 105*a^7*b^4*x^2 - 30*a^4*b^7*x + 14*a*b^10)*x^(1/3))
/(a*b^12*x^(11/3) + b^13*x^(10/3))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**(1/3))**2/x**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.218323, size = 211, normalized size = 1.15 \[ -\frac{33 \, a^{10}{\rm ln}\left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{b^{12}} + \frac{11 \, a^{10}{\rm ln}\left ({\left | x \right |}\right )}{b^{12}} + \frac{27720 \, a^{10} b x^{\frac{10}{3}} + 13860 \, a^{9} b^{2} x^{3} - 4620 \, a^{8} b^{3} x^{\frac{8}{3}} + 2310 \, a^{7} b^{4} x^{\frac{7}{3}} - 1386 \, a^{6} b^{5} x^{2} + 924 \, a^{5} b^{6} x^{\frac{5}{3}} - 660 \, a^{4} b^{7} x^{\frac{4}{3}} + 495 \, a^{3} b^{8} x - 385 \, a^{2} b^{9} x^{\frac{2}{3}} + 308 \, a b^{10} x^{\frac{1}{3}} - 252 \, b^{11}}{840 \,{\left (a x^{\frac{1}{3}} + b\right )} b^{12} x^{\frac{10}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^(1/3))^2*x^5),x, algorithm="giac")

[Out]

-33*a^10*ln(abs(a*x^(1/3) + b))/b^12 + 11*a^10*ln(abs(x))/b^12 + 1/840*(27720*a^
10*b*x^(10/3) + 13860*a^9*b^2*x^3 - 4620*a^8*b^3*x^(8/3) + 2310*a^7*b^4*x^(7/3)
- 1386*a^6*b^5*x^2 + 924*a^5*b^6*x^(5/3) - 660*a^4*b^7*x^(4/3) + 495*a^3*b^8*x -
385*a^2*b^9*x^(2/3) + 308*a*b^10*x^(1/3) - 252*b^11)/((a*x^(1/3) + b)*b^12*x^(1
0/3))