3.2356 \(\int \frac{1}{\left (a+b \sqrt [3]{x}\right ) x^4} \, dx\)

Optimal. Leaf size=149 \[ \frac{3 b^9 \log \left (a+b \sqrt [3]{x}\right )}{a^{10}}-\frac{b^9 \log (x)}{a^{10}}-\frac{3 b^8}{a^9 \sqrt [3]{x}}+\frac{3 b^7}{2 a^8 x^{2/3}}-\frac{b^6}{a^7 x}+\frac{3 b^5}{4 a^6 x^{4/3}}-\frac{3 b^4}{5 a^5 x^{5/3}}+\frac{b^3}{2 a^4 x^2}-\frac{3 b^2}{7 a^3 x^{7/3}}+\frac{3 b}{8 a^2 x^{8/3}}-\frac{1}{3 a x^3} \]

[Out]

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

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Rubi [A]  time = 0.181564, antiderivative size = 149, 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 \[ \frac{3 b^9 \log \left (a+b \sqrt [3]{x}\right )}{a^{10}}-\frac{b^9 \log (x)}{a^{10}}-\frac{3 b^8}{a^9 \sqrt [3]{x}}+\frac{3 b^7}{2 a^8 x^{2/3}}-\frac{b^6}{a^7 x}+\frac{3 b^5}{4 a^6 x^{4/3}}-\frac{3 b^4}{5 a^5 x^{5/3}}+\frac{b^3}{2 a^4 x^2}-\frac{3 b^2}{7 a^3 x^{7/3}}+\frac{3 b}{8 a^2 x^{8/3}}-\frac{1}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x^(1/3))*x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 28.153, size = 150, normalized size = 1.01 \[ - \frac{1}{3 a x^{3}} + \frac{3 b}{8 a^{2} x^{\frac{8}{3}}} - \frac{3 b^{2}}{7 a^{3} x^{\frac{7}{3}}} + \frac{b^{3}}{2 a^{4} x^{2}} - \frac{3 b^{4}}{5 a^{5} x^{\frac{5}{3}}} + \frac{3 b^{5}}{4 a^{6} x^{\frac{4}{3}}} - \frac{b^{6}}{a^{7} x} + \frac{3 b^{7}}{2 a^{8} x^{\frac{2}{3}}} - \frac{3 b^{8}}{a^{9} \sqrt [3]{x}} - \frac{3 b^{9} \log{\left (\sqrt [3]{x} \right )}}{a^{10}} + \frac{3 b^{9} \log{\left (a + b \sqrt [3]{x} \right )}}{a^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0913375, size = 132, normalized size = 0.89 \[ \frac{\frac{a \left (-280 a^8+315 a^7 b \sqrt [3]{x}-360 a^6 b^2 x^{2/3}+420 a^5 b^3 x-504 a^4 b^4 x^{4/3}+630 a^3 b^5 x^{5/3}-840 a^2 b^6 x^2+1260 a b^7 x^{7/3}-2520 b^8 x^{8/3}\right )}{x^3}+2520 b^9 \log \left (a+b \sqrt [3]{x}\right )-840 b^9 \log (x)}{840 a^{10}} \]

Antiderivative was successfully verified.

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

[Out]

((a*(-280*a^8 + 315*a^7*b*x^(1/3) - 360*a^6*b^2*x^(2/3) + 420*a^5*b^3*x - 504*a^
4*b^4*x^(4/3) + 630*a^3*b^5*x^(5/3) - 840*a^2*b^6*x^2 + 1260*a*b^7*x^(7/3) - 252
0*b^8*x^(8/3)))/x^3 + 2520*b^9*Log[a + b*x^(1/3)] - 840*b^9*Log[x])/(840*a^10)

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Maple [A]  time = 0.017, size = 122, normalized size = 0.8 \[ -{\frac{1}{3\,a{x}^{3}}}+{\frac{3\,b}{8\,{a}^{2}}{x}^{-{\frac{8}{3}}}}-{\frac{3\,{b}^{2}}{7\,{a}^{3}}{x}^{-{\frac{7}{3}}}}+{\frac{{b}^{3}}{2\,{a}^{4}{x}^{2}}}-{\frac{3\,{b}^{4}}{5\,{a}^{5}}{x}^{-{\frac{5}{3}}}}+{\frac{3\,{b}^{5}}{4\,{a}^{6}}{x}^{-{\frac{4}{3}}}}-{\frac{{b}^{6}}{{a}^{7}x}}+{\frac{3\,{b}^{7}}{2\,{a}^{8}}{x}^{-{\frac{2}{3}}}}-3\,{\frac{{b}^{8}}{{a}^{9}\sqrt [3]{x}}}+3\,{\frac{{b}^{9}\ln \left ( a+b\sqrt [3]{x} \right ) }{{a}^{10}}}-{\frac{{b}^{9}\ln \left ( x \right ) }{{a}^{10}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*x^(1/3))/x^4,x)

[Out]

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

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Maxima [A]  time = 1.44351, size = 162, normalized size = 1.09 \[ \frac{3 \, b^{9} \log \left (b x^{\frac{1}{3}} + a\right )}{a^{10}} - \frac{b^{9} \log \left (x\right )}{a^{10}} - \frac{2520 \, b^{8} x^{\frac{8}{3}} - 1260 \, a b^{7} x^{\frac{7}{3}} + 840 \, a^{2} b^{6} x^{2} - 630 \, a^{3} b^{5} x^{\frac{5}{3}} + 504 \, a^{4} b^{4} x^{\frac{4}{3}} - 420 \, a^{5} b^{3} x + 360 \, a^{6} b^{2} x^{\frac{2}{3}} - 315 \, a^{7} b x^{\frac{1}{3}} + 280 \, a^{8}}{840 \, a^{9} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*b^9*log(b*x^(1/3) + a)/a^10 - b^9*log(x)/a^10 - 1/840*(2520*b^8*x^(8/3) - 1260
*a*b^7*x^(7/3) + 840*a^2*b^6*x^2 - 630*a^3*b^5*x^(5/3) + 504*a^4*b^4*x^(4/3) - 4
20*a^5*b^3*x + 360*a^6*b^2*x^(2/3) - 315*a^7*b*x^(1/3) + 280*a^8)/(a^9*x^3)

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Fricas [A]  time = 0.227347, size = 170, normalized size = 1.14 \[ \frac{2520 \, b^{9} x^{3} \log \left (b x^{\frac{1}{3}} + a\right ) - 2520 \, b^{9} x^{3} \log \left (x^{\frac{1}{3}}\right ) - 840 \, a^{3} b^{6} x^{2} + 420 \, a^{6} b^{3} x - 280 \, a^{9} - 90 \,{\left (28 \, a b^{8} x^{2} - 7 \, a^{4} b^{5} x + 4 \, a^{7} b^{2}\right )} x^{\frac{2}{3}} + 63 \,{\left (20 \, a^{2} b^{7} x^{2} - 8 \, a^{5} b^{4} x + 5 \, a^{8} b\right )} x^{\frac{1}{3}}}{840 \, a^{10} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/840*(2520*b^9*x^3*log(b*x^(1/3) + a) - 2520*b^9*x^3*log(x^(1/3)) - 840*a^3*b^6
*x^2 + 420*a^6*b^3*x - 280*a^9 - 90*(28*a*b^8*x^2 - 7*a^4*b^5*x + 4*a^7*b^2)*x^(
2/3) + 63*(20*a^2*b^7*x^2 - 8*a^5*b^4*x + 5*a^8*b)*x^(1/3))/(a^10*x^3)

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Sympy [A]  time = 87.8231, size = 172, normalized size = 1.15 \[ \begin{cases} \frac{\tilde{\infty }}{x^{\frac{10}{3}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{3}{10 b x^{\frac{10}{3}}} & \text{for}\: a = 0 \\- \frac{1}{3 a x^{3}} & \text{for}\: b = 0 \\- \frac{1}{3 a x^{3}} + \frac{3 b}{8 a^{2} x^{\frac{8}{3}}} - \frac{3 b^{2}}{7 a^{3} x^{\frac{7}{3}}} + \frac{b^{3}}{2 a^{4} x^{2}} - \frac{3 b^{4}}{5 a^{5} x^{\frac{5}{3}}} + \frac{3 b^{5}}{4 a^{6} x^{\frac{4}{3}}} - \frac{b^{6}}{a^{7} x} + \frac{3 b^{7}}{2 a^{8} x^{\frac{2}{3}}} - \frac{3 b^{8}}{a^{9} \sqrt [3]{x}} - \frac{b^{9} \log{\left (x \right )}}{a^{10}} + \frac{3 b^{9} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{a^{10}} & \text{otherwise} \end{cases} \]

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**(10/3), Eq(a, 0) & Eq(b, 0)), (-3/(10*b*x**(10/3)), Eq(a, 0)),
(-1/(3*a*x**3), Eq(b, 0)), (-1/(3*a*x**3) + 3*b/(8*a**2*x**(8/3)) - 3*b**2/(7*a
**3*x**(7/3)) + b**3/(2*a**4*x**2) - 3*b**4/(5*a**5*x**(5/3)) + 3*b**5/(4*a**6*x
**(4/3)) - b**6/(a**7*x) + 3*b**7/(2*a**8*x**(2/3)) - 3*b**8/(a**9*x**(1/3)) - b
**9*log(x)/a**10 + 3*b**9*log(a/b + x**(1/3))/a**10, True))

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GIAC/XCAS [A]  time = 0.218496, size = 169, normalized size = 1.13 \[ \frac{3 \, b^{9}{\rm ln}\left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{a^{10}} - \frac{b^{9}{\rm ln}\left ({\left | x \right |}\right )}{a^{10}} - \frac{2520 \, a b^{8} x^{\frac{8}{3}} - 1260 \, a^{2} b^{7} x^{\frac{7}{3}} + 840 \, a^{3} b^{6} x^{2} - 630 \, a^{4} b^{5} x^{\frac{5}{3}} + 504 \, a^{5} b^{4} x^{\frac{4}{3}} - 420 \, a^{6} b^{3} x + 360 \, a^{7} b^{2} x^{\frac{2}{3}} - 315 \, a^{8} b x^{\frac{1}{3}} + 280 \, a^{9}}{840 \, a^{10} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*b^9*ln(abs(b*x^(1/3) + a))/a^10 - b^9*ln(abs(x))/a^10 - 1/840*(2520*a*b^8*x^(8
/3) - 1260*a^2*b^7*x^(7/3) + 840*a^3*b^6*x^2 - 630*a^4*b^5*x^(5/3) + 504*a^5*b^4
*x^(4/3) - 420*a^6*b^3*x + 360*a^7*b^2*x^(2/3) - 315*a^8*b*x^(1/3) + 280*a^9)/(a
^10*x^3)