3.2420 \(\int \frac{1}{\left (a+\frac{b}{\sqrt [3]{x}}\right ) x^4} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.181652, antiderivative size = 134, 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 \[ -\frac{3 a^8 \log \left (a \sqrt [3]{x}+b\right )}{b^9}+\frac{a^8 \log (x)}{b^9}+\frac{3 a^7}{b^8 \sqrt [3]{x}}-\frac{3 a^6}{2 b^7 x^{2/3}}+\frac{a^5}{b^6 x}-\frac{3 a^4}{4 b^5 x^{4/3}}+\frac{3 a^3}{5 b^4 x^{5/3}}-\frac{a^2}{2 b^3 x^2}+\frac{3 a}{7 b^2 x^{7/3}}-\frac{3}{8 b x^{8/3}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^(1/3))*x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 26.558, size = 136, normalized size = 1.01 \[ \frac{3 a^{8} \log{\left (\sqrt [3]{x} \right )}}{b^{9}} - \frac{3 a^{8} \log{\left (a \sqrt [3]{x} + b \right )}}{b^{9}} + \frac{3 a^{7}}{b^{8} \sqrt [3]{x}} - \frac{3 a^{6}}{2 b^{7} x^{\frac{2}{3}}} + \frac{a^{5}}{b^{6} x} - \frac{3 a^{4}}{4 b^{5} x^{\frac{4}{3}}} + \frac{3 a^{3}}{5 b^{4} x^{\frac{5}{3}}} - \frac{a^{2}}{2 b^{3} x^{2}} + \frac{3 a}{7 b^{2} x^{\frac{7}{3}}} - \frac{3}{8 b x^{\frac{8}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**(1/3))/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.152259, size = 121, normalized size = 0.9 \[ \frac{-840 a^8 \log \left (a \sqrt [3]{x}+b\right )+280 a^8 \log (x)+\frac{b \left (840 a^7 x^{7/3}-420 a^6 b x^2+280 a^5 b^2 x^{5/3}-210 a^4 b^3 x^{4/3}+168 a^3 b^4 x-140 a^2 b^5 x^{2/3}+120 a b^6 \sqrt [3]{x}-105 b^7\right )}{x^{8/3}}}{280 b^9} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^(1/3))*x^4),x]

[Out]

((b*(-105*b^7 + 120*a*b^6*x^(1/3) - 140*a^2*b^5*x^(2/3) + 168*a^3*b^4*x - 210*a^
4*b^3*x^(4/3) + 280*a^5*b^2*x^(5/3) - 420*a^6*b*x^2 + 840*a^7*x^(7/3)))/x^(8/3)
- 840*a^8*Log[b + a*x^(1/3)] + 280*a^8*Log[x])/(280*b^9)

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Maple [A]  time = 0.016, size = 109, normalized size = 0.8 \[ -{\frac{3}{8\,b}{x}^{-{\frac{8}{3}}}}+{\frac{3\,a}{7\,{b}^{2}}{x}^{-{\frac{7}{3}}}}-{\frac{{a}^{2}}{2\,{b}^{3}{x}^{2}}}+{\frac{3\,{a}^{3}}{5\,{b}^{4}}{x}^{-{\frac{5}{3}}}}-{\frac{3\,{a}^{4}}{4\,{b}^{5}}{x}^{-{\frac{4}{3}}}}+{\frac{{a}^{5}}{{b}^{6}x}}-{\frac{3\,{a}^{6}}{2\,{b}^{7}}{x}^{-{\frac{2}{3}}}}+3\,{\frac{{a}^{7}}{{b}^{8}\sqrt [3]{x}}}-3\,{\frac{{a}^{8}\ln \left ( b+a\sqrt [3]{x} \right ) }{{b}^{9}}}+{\frac{{a}^{8}\ln \left ( x \right ) }{{b}^{9}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^(1/3))/x^4,x)

[Out]

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

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Maxima [A]  time = 1.44244, size = 197, normalized size = 1.47 \[ -\frac{3 \, a^{8} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{b^{9}} - \frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{8}}{8 \, b^{9}} + \frac{24 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{7} a}{7 \, b^{9}} - \frac{14 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{6} a^{2}}{b^{9}} + \frac{168 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{5} a^{3}}{5 \, b^{9}} - \frac{105 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{4} a^{4}}{2 \, b^{9}} + \frac{56 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{3} a^{5}}{b^{9}} - \frac{42 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2} a^{6}}{b^{9}} + \frac{24 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} a^{7}}{b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^(1/3))*x^4),x, algorithm="maxima")

[Out]

-3*a^8*log(a + b/x^(1/3))/b^9 - 3/8*(a + b/x^(1/3))^8/b^9 + 24/7*(a + b/x^(1/3))
^7*a/b^9 - 14*(a + b/x^(1/3))^6*a^2/b^9 + 168/5*(a + b/x^(1/3))^5*a^3/b^9 - 105/
2*(a + b/x^(1/3))^4*a^4/b^9 + 56*(a + b/x^(1/3))^3*a^5/b^9 - 42*(a + b/x^(1/3))^
2*a^6/b^9 + 24*(a + b/x^(1/3))*a^7/b^9

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Fricas [A]  time = 0.237042, size = 155, normalized size = 1.16 \[ -\frac{840 \, a^{8} x^{\frac{8}{3}} \log \left (a x^{\frac{1}{3}} + b\right ) - 840 \, a^{8} x^{\frac{8}{3}} \log \left (x^{\frac{1}{3}}\right ) + 420 \, a^{6} b^{2} x^{2} - 168 \, a^{3} b^{5} x + 105 \, b^{8} - 140 \,{\left (2 \, a^{5} b^{3} x - a^{2} b^{6}\right )} x^{\frac{2}{3}} - 30 \,{\left (28 \, a^{7} b x^{2} - 7 \, a^{4} b^{4} x + 4 \, a b^{7}\right )} x^{\frac{1}{3}}}{280 \, b^{9} x^{\frac{8}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^(1/3))*x^4),x, algorithm="fricas")

[Out]

-1/280*(840*a^8*x^(8/3)*log(a*x^(1/3) + b) - 840*a^8*x^(8/3)*log(x^(1/3)) + 420*
a^6*b^2*x^2 - 168*a^3*b^5*x + 105*b^8 - 140*(2*a^5*b^3*x - a^2*b^6)*x^(2/3) - 30
*(28*a^7*b*x^2 - 7*a^4*b^4*x + 4*a*b^7)*x^(1/3))/(b^9*x^(8/3))

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Sympy [A]  time = 61.8769, size = 158, normalized size = 1.18 \[ \begin{cases} \frac{\tilde{\infty }}{x^{\frac{8}{3}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{3}{8 b x^{\frac{8}{3}}} & \text{for}\: a = 0 \\- \frac{1}{3 a x^{3}} & \text{for}\: b = 0 \\\frac{a^{8} \log{\left (x \right )}}{b^{9}} - \frac{3 a^{8} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{b^{9}} + \frac{3 a^{7}}{b^{8} \sqrt [3]{x}} - \frac{3 a^{6}}{2 b^{7} x^{\frac{2}{3}}} + \frac{a^{5}}{b^{6} x} - \frac{3 a^{4}}{4 b^{5} x^{\frac{4}{3}}} + \frac{3 a^{3}}{5 b^{4} x^{\frac{5}{3}}} - \frac{a^{2}}{2 b^{3} x^{2}} + \frac{3 a}{7 b^{2} x^{\frac{7}{3}}} - \frac{3}{8 b x^{\frac{8}{3}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**(1/3))/x**4,x)

[Out]

Piecewise((zoo/x**(8/3), Eq(a, 0) & Eq(b, 0)), (-3/(8*b*x**(8/3)), Eq(a, 0)), (-
1/(3*a*x**3), Eq(b, 0)), (a**8*log(x)/b**9 - 3*a**8*log(x**(1/3) + b/a)/b**9 + 3
*a**7/(b**8*x**(1/3)) - 3*a**6/(2*b**7*x**(2/3)) + a**5/(b**6*x) - 3*a**4/(4*b**
5*x**(4/3)) + 3*a**3/(5*b**4*x**(5/3)) - a**2/(2*b**3*x**2) + 3*a/(7*b**2*x**(7/
3)) - 3/(8*b*x**(8/3)), True))

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GIAC/XCAS [A]  time = 0.21732, size = 153, normalized size = 1.14 \[ -\frac{3 \, a^{8}{\rm ln}\left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{b^{9}} + \frac{a^{8}{\rm ln}\left ({\left | x \right |}\right )}{b^{9}} + \frac{840 \, a^{7} b x^{\frac{7}{3}} - 420 \, a^{6} b^{2} x^{2} + 280 \, a^{5} b^{3} x^{\frac{5}{3}} - 210 \, a^{4} b^{4} x^{\frac{4}{3}} + 168 \, a^{3} b^{5} x - 140 \, a^{2} b^{6} x^{\frac{2}{3}} + 120 \, a b^{7} x^{\frac{1}{3}} - 105 \, b^{8}}{280 \, b^{9} x^{\frac{8}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^(1/3))*x^4),x, algorithm="giac")

[Out]

-3*a^8*ln(abs(a*x^(1/3) + b))/b^9 + a^8*ln(abs(x))/b^9 + 1/280*(840*a^7*b*x^(7/3
) - 420*a^6*b^2*x^2 + 280*a^5*b^3*x^(5/3) - 210*a^4*b^4*x^(4/3) + 168*a^3*b^5*x
- 140*a^2*b^6*x^(2/3) + 120*a*b^7*x^(1/3) - 105*b^8)/(b^9*x^(8/3))