3.2372 \(\int \frac{1}{\left (a+b \sqrt [3]{x}\right )^3 x^3} \, dx\)

Optimal. Leaf size=146 \[ -\frac{84 b^6 \log \left (a+b \sqrt [3]{x}\right )}{a^9}+\frac{28 b^6 \log (x)}{a^9}+\frac{21 b^6}{a^8 \left (a+b \sqrt [3]{x}\right )}+\frac{63 b^5}{a^8 \sqrt [3]{x}}+\frac{3 b^6}{2 a^7 \left (a+b \sqrt [3]{x}\right )^2}-\frac{45 b^4}{2 a^7 x^{2/3}}+\frac{10 b^3}{a^6 x}-\frac{9 b^2}{2 a^5 x^{4/3}}+\frac{9 b}{5 a^4 x^{5/3}}-\frac{1}{2 a^3 x^2} \]

[Out]

(3*b^6)/(2*a^7*(a + b*x^(1/3))^2) + (21*b^6)/(a^8*(a + b*x^(1/3))) - 1/(2*a^3*x^
2) + (9*b)/(5*a^4*x^(5/3)) - (9*b^2)/(2*a^5*x^(4/3)) + (10*b^3)/(a^6*x) - (45*b^
4)/(2*a^7*x^(2/3)) + (63*b^5)/(a^8*x^(1/3)) - (84*b^6*Log[a + b*x^(1/3)])/a^9 +
(28*b^6*Log[x])/a^9

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Rubi [A]  time = 0.232653, antiderivative size = 146, 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 \[ -\frac{84 b^6 \log \left (a+b \sqrt [3]{x}\right )}{a^9}+\frac{28 b^6 \log (x)}{a^9}+\frac{21 b^6}{a^8 \left (a+b \sqrt [3]{x}\right )}+\frac{63 b^5}{a^8 \sqrt [3]{x}}+\frac{3 b^6}{2 a^7 \left (a+b \sqrt [3]{x}\right )^2}-\frac{45 b^4}{2 a^7 x^{2/3}}+\frac{10 b^3}{a^6 x}-\frac{9 b^2}{2 a^5 x^{4/3}}+\frac{9 b}{5 a^4 x^{5/3}}-\frac{1}{2 a^3 x^2} \]

Antiderivative was successfully verified.

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

[Out]

(3*b^6)/(2*a^7*(a + b*x^(1/3))^2) + (21*b^6)/(a^8*(a + b*x^(1/3))) - 1/(2*a^3*x^
2) + (9*b)/(5*a^4*x^(5/3)) - (9*b^2)/(2*a^5*x^(4/3)) + (10*b^3)/(a^6*x) - (45*b^
4)/(2*a^7*x^(2/3)) + (63*b^5)/(a^8*x^(1/3)) - (84*b^6*Log[a + b*x^(1/3)])/a^9 +
(28*b^6*Log[x])/a^9

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Rubi in Sympy [A]  time = 39.3296, size = 148, normalized size = 1.01 \[ - \frac{1}{2 a^{3} x^{2}} + \frac{9 b}{5 a^{4} x^{\frac{5}{3}}} - \frac{9 b^{2}}{2 a^{5} x^{\frac{4}{3}}} + \frac{10 b^{3}}{a^{6} x} + \frac{3 b^{6}}{2 a^{7} \left (a + b \sqrt [3]{x}\right )^{2}} - \frac{45 b^{4}}{2 a^{7} x^{\frac{2}{3}}} + \frac{21 b^{6}}{a^{8} \left (a + b \sqrt [3]{x}\right )} + \frac{63 b^{5}}{a^{8} \sqrt [3]{x}} + \frac{84 b^{6} \log{\left (\sqrt [3]{x} \right )}}{a^{9}} - \frac{84 b^{6} \log{\left (a + b \sqrt [3]{x} \right )}}{a^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(2*a**3*x**2) + 9*b/(5*a**4*x**(5/3)) - 9*b**2/(2*a**5*x**(4/3)) + 10*b**3/(a
**6*x) + 3*b**6/(2*a**7*(a + b*x**(1/3))**2) - 45*b**4/(2*a**7*x**(2/3)) + 21*b*
*6/(a**8*(a + b*x**(1/3))) + 63*b**5/(a**8*x**(1/3)) + 84*b**6*log(x**(1/3))/a**
9 - 84*b**6*log(a + b*x**(1/3))/a**9

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Mathematica [A]  time = 0.279978, size = 130, normalized size = 0.89 \[ \frac{\frac{a \left (-5 a^7+8 a^6 b \sqrt [3]{x}-14 a^5 b^2 x^{2/3}+28 a^4 b^3 x-70 a^3 b^4 x^{4/3}+280 a^2 b^5 x^{5/3}+1260 a b^6 x^2+840 b^7 x^{7/3}\right )}{x^2 \left (a+b \sqrt [3]{x}\right )^2}-840 b^6 \log \left (a+b \sqrt [3]{x}\right )+280 b^6 \log (x)}{10 a^9} \]

Antiderivative was successfully verified.

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

[Out]

((a*(-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 + b*x^(1/3
))^2*x^2) - 840*b^6*Log[a + b*x^(1/3)] + 280*b^6*Log[x])/(10*a^9)

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Maple [A]  time = 0.003, size = 123, normalized size = 0.8 \[{\frac{3\,{b}^{6}}{2\,{a}^{7}} \left ( a+b\sqrt [3]{x} \right ) ^{-2}}+21\,{\frac{{b}^{6}}{{a}^{8} \left ( a+b\sqrt [3]{x} \right ) }}-{\frac{1}{2\,{x}^{2}{a}^{3}}}+{\frac{9\,b}{5\,{a}^{4}}{x}^{-{\frac{5}{3}}}}-{\frac{9\,{b}^{2}}{2\,{a}^{5}}{x}^{-{\frac{4}{3}}}}+10\,{\frac{{b}^{3}}{{a}^{6}x}}-{\frac{45\,{b}^{4}}{2\,{a}^{7}}{x}^{-{\frac{2}{3}}}}+63\,{\frac{{b}^{5}}{{a}^{8}\sqrt [3]{x}}}-84\,{\frac{{b}^{6}\ln \left ( a+b\sqrt [3]{x} \right ) }{{a}^{9}}}+28\,{\frac{{b}^{6}\ln \left ( x \right ) }{{a}^{9}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*x^(1/3))^3/x^3,x)

[Out]

3/2*b^6/a^7/(a+b*x^(1/3))^2+21*b^6/a^8/(a+b*x^(1/3))-1/2/x^2/a^3+9/5*b/a^4/x^(5/
3)-9/2*b^2/a^5/x^(4/3)+10*b^3/a^6/x-45/2*b^4/a^7/x^(2/3)+63*b^5/a^8/x^(1/3)-84*b
^6*ln(a+b*x^(1/3))/a^9+28*b^6*ln(x)/a^9

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Maxima [A]  time = 1.45677, size = 178, normalized size = 1.22 \[ \frac{840 \, b^{7} x^{\frac{7}{3}} + 1260 \, a b^{6} x^{2} + 280 \, a^{2} b^{5} x^{\frac{5}{3}} - 70 \, a^{3} b^{4} x^{\frac{4}{3}} + 28 \, a^{4} b^{3} x - 14 \, a^{5} b^{2} x^{\frac{2}{3}} + 8 \, a^{6} b x^{\frac{1}{3}} - 5 \, a^{7}}{10 \,{\left (a^{8} b^{2} x^{\frac{8}{3}} + 2 \, a^{9} b x^{\frac{7}{3}} + a^{10} x^{2}\right )}} - \frac{84 \, b^{6} \log \left (b x^{\frac{1}{3}} + a\right )}{a^{9}} + \frac{28 \, b^{6} \log \left (x\right )}{a^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.229895, size = 243, normalized size = 1.66 \[ \frac{1260 \, a^{2} b^{6} x^{2} + 28 \, a^{5} b^{3} x - 5 \, a^{8} - 840 \,{\left (b^{8} x^{\frac{8}{3}} + 2 \, a b^{7} x^{\frac{7}{3}} + a^{2} b^{6} x^{2}\right )} \log \left (b x^{\frac{1}{3}} + a\right ) + 840 \,{\left (b^{8} x^{\frac{8}{3}} + 2 \, a b^{7} x^{\frac{7}{3}} + a^{2} b^{6} x^{2}\right )} \log \left (x^{\frac{1}{3}}\right ) + 14 \,{\left (20 \, a^{3} b^{5} x - a^{6} b^{2}\right )} x^{\frac{2}{3}} + 2 \,{\left (420 \, a b^{7} x^{2} - 35 \, a^{4} b^{4} x + 4 \, a^{7} b\right )} x^{\frac{1}{3}}}{10 \,{\left (a^{9} b^{2} x^{\frac{8}{3}} + 2 \, a^{10} b x^{\frac{7}{3}} + a^{11} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*(1260*a^2*b^6*x^2 + 28*a^5*b^3*x - 5*a^8 - 840*(b^8*x^(8/3) + 2*a*b^7*x^(7/
3) + a^2*b^6*x^2)*log(b*x^(1/3) + a) + 840*(b^8*x^(8/3) + 2*a*b^7*x^(7/3) + a^2*
b^6*x^2)*log(x^(1/3)) + 14*(20*a^3*b^5*x - a^6*b^2)*x^(2/3) + 2*(420*a*b^7*x^2 -
35*a^4*b^4*x + 4*a^7*b)*x^(1/3))/(a^9*b^2*x^(8/3) + 2*a^10*b*x^(7/3) + a^11*x^2
)

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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))**3/x**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.223216, size = 166, normalized size = 1.14 \[ -\frac{84 \, b^{6}{\rm ln}\left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{a^{9}} + \frac{28 \, b^{6}{\rm ln}\left ({\left | x \right |}\right )}{a^{9}} + \frac{840 \, a b^{7} x^{\frac{7}{3}} + 1260 \, a^{2} b^{6} x^{2} + 280 \, a^{3} b^{5} x^{\frac{5}{3}} - 70 \, a^{4} b^{4} x^{\frac{4}{3}} + 28 \, a^{5} b^{3} x - 14 \, a^{6} b^{2} x^{\frac{2}{3}} + 8 \, a^{7} b x^{\frac{1}{3}} - 5 \, a^{8}}{10 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} a^{9} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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