3.2421 \(\int \frac{1}{\left (a+\frac{b}{\sqrt [3]{x}}\right ) x^5} \, dx\)

Optimal. Leaf size=179 \[ \frac{3 a^{11} \log \left (a \sqrt [3]{x}+b\right )}{b^{12}}-\frac{a^{11} \log (x)}{b^{12}}-\frac{3 a^{10}}{b^{11} \sqrt [3]{x}}+\frac{3 a^9}{2 b^{10} x^{2/3}}-\frac{a^8}{b^9 x}+\frac{3 a^7}{4 b^8 x^{4/3}}-\frac{3 a^6}{5 b^7 x^{5/3}}+\frac{a^5}{2 b^6 x^2}-\frac{3 a^4}{7 b^5 x^{7/3}}+\frac{3 a^3}{8 b^4 x^{8/3}}-\frac{a^2}{3 b^3 x^3}+\frac{3 a}{10 b^2 x^{10/3}}-\frac{3}{11 b x^{11/3}} \]

[Out]

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

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Rubi [A]  time = 0.2345, antiderivative size = 179, 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 \[ \frac{3 a^{11} \log \left (a \sqrt [3]{x}+b\right )}{b^{12}}-\frac{a^{11} \log (x)}{b^{12}}-\frac{3 a^{10}}{b^{11} \sqrt [3]{x}}+\frac{3 a^9}{2 b^{10} x^{2/3}}-\frac{a^8}{b^9 x}+\frac{3 a^7}{4 b^8 x^{4/3}}-\frac{3 a^6}{5 b^7 x^{5/3}}+\frac{a^5}{2 b^6 x^2}-\frac{3 a^4}{7 b^5 x^{7/3}}+\frac{3 a^3}{8 b^4 x^{8/3}}-\frac{a^2}{3 b^3 x^3}+\frac{3 a}{10 b^2 x^{10/3}}-\frac{3}{11 b x^{11/3}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^(1/3))*x^5),x]

[Out]

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

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Rubi in Sympy [A]  time = 35.9582, size = 178, normalized size = 0.99 \[ - \frac{3 a^{11} \log{\left (\sqrt [3]{x} \right )}}{b^{12}} + \frac{3 a^{11} \log{\left (a \sqrt [3]{x} + b \right )}}{b^{12}} - \frac{3 a^{10}}{b^{11} \sqrt [3]{x}} + \frac{3 a^{9}}{2 b^{10} x^{\frac{2}{3}}} - \frac{a^{8}}{b^{9} x} + \frac{3 a^{7}}{4 b^{8} x^{\frac{4}{3}}} - \frac{3 a^{6}}{5 b^{7} x^{\frac{5}{3}}} + \frac{a^{5}}{2 b^{6} x^{2}} - \frac{3 a^{4}}{7 b^{5} x^{\frac{7}{3}}} + \frac{3 a^{3}}{8 b^{4} x^{\frac{8}{3}}} - \frac{a^{2}}{3 b^{3} x^{3}} + \frac{3 a}{10 b^{2} x^{\frac{10}{3}}} - \frac{3}{11 b x^{\frac{11}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.331807, size = 158, normalized size = 0.88 \[ \frac{27720 a^{11} \log \left (a \sqrt [3]{x}+b\right )-9240 a^{11} \log (x)+\frac{b \left (-27720 a^{10} x^{10/3}+13860 a^9 b x^3-9240 a^8 b^2 x^{8/3}+6930 a^7 b^3 x^{7/3}-5544 a^6 b^4 x^2+4620 a^5 b^5 x^{5/3}-3960 a^4 b^6 x^{4/3}+3465 a^3 b^7 x-3080 a^2 b^8 x^{2/3}+2772 a b^9 \sqrt [3]{x}-2520 b^{10}\right )}{x^{11/3}}}{9240 b^{12}} \]

Antiderivative was successfully verified.

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

[Out]

((b*(-2520*b^10 + 2772*a*b^9*x^(1/3) - 3080*a^2*b^8*x^(2/3) + 3465*a^3*b^7*x - 3
960*a^4*b^6*x^(4/3) + 4620*a^5*b^5*x^(5/3) - 5544*a^6*b^4*x^2 + 6930*a^7*b^3*x^(
7/3) - 9240*a^8*b^2*x^(8/3) + 13860*a^9*b*x^3 - 27720*a^10*x^(10/3)))/x^(11/3) +
27720*a^11*Log[b + a*x^(1/3)] - 9240*a^11*Log[x])/(9240*b^12)

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Maple [A]  time = 0.016, size = 144, normalized size = 0.8 \[ -{\frac{3}{11\,b}{x}^{-{\frac{11}{3}}}}+{\frac{3\,a}{10\,{b}^{2}}{x}^{-{\frac{10}{3}}}}-{\frac{{a}^{2}}{3\,{b}^{3}{x}^{3}}}+{\frac{3\,{a}^{3}}{8\,{b}^{4}}{x}^{-{\frac{8}{3}}}}-{\frac{3\,{a}^{4}}{7\,{b}^{5}}{x}^{-{\frac{7}{3}}}}+{\frac{{a}^{5}}{2\,{b}^{6}{x}^{2}}}-{\frac{3\,{a}^{6}}{5\,{b}^{7}}{x}^{-{\frac{5}{3}}}}+{\frac{3\,{a}^{7}}{4\,{b}^{8}}{x}^{-{\frac{4}{3}}}}-{\frac{{a}^{8}}{{b}^{9}x}}+{\frac{3\,{a}^{9}}{2\,{b}^{10}}{x}^{-{\frac{2}{3}}}}-3\,{\frac{{a}^{10}}{{b}^{11}\sqrt [3]{x}}}+3\,{\frac{{a}^{11}\ln \left ( b+a\sqrt [3]{x} \right ) }{{b}^{12}}}-{\frac{{a}^{11}\ln \left ( x \right ) }{{b}^{12}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^(1/3))/x^5,x)

[Out]

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

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Maxima [A]  time = 1.43824, size = 266, normalized size = 1.49 \[ \frac{3 \, a^{11} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{b^{12}} - \frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{11}}{11 \, b^{12}} + \frac{33 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{10} a}{10 \, b^{12}} - \frac{55 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{9} a^{2}}{3 \, b^{12}} + \frac{495 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{8} a^{3}}{8 \, b^{12}} - \frac{990 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{7} a^{4}}{7 \, b^{12}} + \frac{231 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{6} a^{5}}{b^{12}} - \frac{1386 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{5} a^{6}}{5 \, b^{12}} + \frac{495 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{4} a^{7}}{2 \, b^{12}} - \frac{165 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{3} a^{8}}{b^{12}} + \frac{165 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2} a^{9}}{2 \, b^{12}} - \frac{33 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} a^{10}}{b^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a^11*log(a + b/x^(1/3))/b^12 - 3/11*(a + b/x^(1/3))^11/b^12 + 33/10*(a + b/x^(
1/3))^10*a/b^12 - 55/3*(a + b/x^(1/3))^9*a^2/b^12 + 495/8*(a + b/x^(1/3))^8*a^3/
b^12 - 990/7*(a + b/x^(1/3))^7*a^4/b^12 + 231*(a + b/x^(1/3))^6*a^5/b^12 - 1386/
5*(a + b/x^(1/3))^5*a^6/b^12 + 495/2*(a + b/x^(1/3))^4*a^7/b^12 - 165*(a + b/x^(
1/3))^3*a^8/b^12 + 165/2*(a + b/x^(1/3))^2*a^9/b^12 - 33*(a + b/x^(1/3))*a^10/b^
12

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Fricas [A]  time = 0.236319, size = 200, normalized size = 1.12 \[ \frac{27720 \, a^{11} x^{\frac{11}{3}} \log \left (a x^{\frac{1}{3}} + b\right ) - 27720 \, a^{11} x^{\frac{11}{3}} \log \left (x^{\frac{1}{3}}\right ) + 13860 \, a^{9} b^{2} x^{3} - 5544 \, a^{6} b^{5} x^{2} + 3465 \, a^{3} b^{8} x - 2520 \, b^{11} - 1540 \,{\left (6 \, a^{8} b^{3} x^{2} - 3 \, a^{5} b^{6} x + 2 \, a^{2} b^{9}\right )} x^{\frac{2}{3}} - 198 \,{\left (140 \, a^{10} b x^{3} - 35 \, a^{7} b^{4} x^{2} + 20 \, a^{4} b^{7} x - 14 \, a b^{10}\right )} x^{\frac{1}{3}}}{9240 \, b^{12} x^{\frac{11}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9240*(27720*a^11*x^(11/3)*log(a*x^(1/3) + b) - 27720*a^11*x^(11/3)*log(x^(1/3)
) + 13860*a^9*b^2*x^3 - 5544*a^6*b^5*x^2 + 3465*a^3*b^8*x - 2520*b^11 - 1540*(6*
a^8*b^3*x^2 - 3*a^5*b^6*x + 2*a^2*b^9)*x^(2/3) - 198*(140*a^10*b*x^3 - 35*a^7*b^
4*x^2 + 20*a^4*b^7*x - 14*a*b^10)*x^(1/3))/(b^12*x^(11/3))

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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))/x**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.217236, size = 198, normalized size = 1.11 \[ \frac{3 \, a^{11}{\rm ln}\left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{b^{12}} - \frac{a^{11}{\rm ln}\left ({\left | x \right |}\right )}{b^{12}} - \frac{27720 \, a^{10} b x^{\frac{10}{3}} - 13860 \, a^{9} b^{2} x^{3} + 9240 \, a^{8} b^{3} x^{\frac{8}{3}} - 6930 \, a^{7} b^{4} x^{\frac{7}{3}} + 5544 \, a^{6} b^{5} x^{2} - 4620 \, a^{5} b^{6} x^{\frac{5}{3}} + 3960 \, a^{4} b^{7} x^{\frac{4}{3}} - 3465 \, a^{3} b^{8} x + 3080 \, a^{2} b^{9} x^{\frac{2}{3}} - 2772 \, a b^{10} x^{\frac{1}{3}} + 2520 \, b^{11}}{9240 \, b^{12} x^{\frac{11}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a^11*ln(abs(a*x^(1/3) + b))/b^12 - a^11*ln(abs(x))/b^12 - 1/9240*(27720*a^10*b
*x^(10/3) - 13860*a^9*b^2*x^3 + 9240*a^8*b^3*x^(8/3) - 6930*a^7*b^4*x^(7/3) + 55
44*a^6*b^5*x^2 - 4620*a^5*b^6*x^(5/3) + 3960*a^4*b^7*x^(4/3) - 3465*a^3*b^8*x +
3080*a^2*b^9*x^(2/3) - 2772*a*b^10*x^(1/3) + 2520*b^11)/(b^12*x^(11/3))