3.2365 \(\int \frac{1}{\left (a+b \sqrt [3]{x}\right )^2 x^4} \, dx\)

Optimal. Leaf size=162 \[ \frac{30 b^9 \log \left (a+b \sqrt [3]{x}\right )}{a^{11}}-\frac{10 b^9 \log (x)}{a^{11}}-\frac{3 b^9}{a^{10} \left (a+b \sqrt [3]{x}\right )}-\frac{27 b^8}{a^{10} \sqrt [3]{x}}+\frac{12 b^7}{a^9 x^{2/3}}-\frac{7 b^6}{a^8 x}+\frac{9 b^5}{2 a^7 x^{4/3}}-\frac{3 b^4}{a^6 x^{5/3}}+\frac{2 b^3}{a^5 x^2}-\frac{9 b^2}{7 a^4 x^{7/3}}+\frac{3 b}{4 a^3 x^{8/3}}-\frac{1}{3 a^2 x^3} \]

[Out]

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

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Rubi [A]  time = 0.260817, antiderivative size = 162, 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 \[ \frac{30 b^9 \log \left (a+b \sqrt [3]{x}\right )}{a^{11}}-\frac{10 b^9 \log (x)}{a^{11}}-\frac{3 b^9}{a^{10} \left (a+b \sqrt [3]{x}\right )}-\frac{27 b^8}{a^{10} \sqrt [3]{x}}+\frac{12 b^7}{a^9 x^{2/3}}-\frac{7 b^6}{a^8 x}+\frac{9 b^5}{2 a^7 x^{4/3}}-\frac{3 b^4}{a^6 x^{5/3}}+\frac{2 b^3}{a^5 x^2}-\frac{9 b^2}{7 a^4 x^{7/3}}+\frac{3 b}{4 a^3 x^{8/3}}-\frac{1}{3 a^2 x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x^(1/3))^2*x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 45.4411, size = 167, normalized size = 1.03 \[ - \frac{1}{3 a^{2} x^{3}} + \frac{3 b}{4 a^{3} x^{\frac{8}{3}}} - \frac{9 b^{2}}{7 a^{4} x^{\frac{7}{3}}} + \frac{2 b^{3}}{a^{5} x^{2}} - \frac{3 b^{4}}{a^{6} x^{\frac{5}{3}}} + \frac{9 b^{5}}{2 a^{7} x^{\frac{4}{3}}} - \frac{7 b^{6}}{a^{8} x} + \frac{12 b^{7}}{a^{9} x^{\frac{2}{3}}} - \frac{3 b^{9}}{a^{10} \left (a + b \sqrt [3]{x}\right )} - \frac{27 b^{8}}{a^{10} \sqrt [3]{x}} - \frac{30 b^{9} \log{\left (\sqrt [3]{x} \right )}}{a^{11}} + \frac{30 b^{9} \log{\left (a + b \sqrt [3]{x} \right )}}{a^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b*x**(1/3))**2/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.355602, size = 154, normalized size = 0.95 \[ -\frac{\frac{a \left (28 a^9-35 a^8 b \sqrt [3]{x}+45 a^7 b^2 x^{2/3}-60 a^6 b^3 x+84 a^5 b^4 x^{4/3}-126 a^4 b^5 x^{5/3}+210 a^3 b^6 x^2-420 a^2 b^7 x^{7/3}+1260 a b^8 x^{8/3}+2520 b^9 x^3\right )}{x^3 \left (a+b \sqrt [3]{x}\right )}-2520 b^9 \log \left (a+b \sqrt [3]{x}\right )+840 b^9 \log (x)}{84 a^{11}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x^(1/3))^2*x^4),x]

[Out]

-((a*(28*a^9 - 35*a^8*b*x^(1/3) + 45*a^7*b^2*x^(2/3) - 60*a^6*b^3*x + 84*a^5*b^4
*x^(4/3) - 126*a^4*b^5*x^(5/3) + 210*a^3*b^6*x^2 - 420*a^2*b^7*x^(7/3) + 1260*a*
b^8*x^(8/3) + 2520*b^9*x^3))/((a + b*x^(1/3))*x^3) - 2520*b^9*Log[a + b*x^(1/3)]
+ 840*b^9*Log[x])/(84*a^11)

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Maple [A]  time = 0.022, size = 139, normalized size = 0.9 \[ -3\,{\frac{{b}^{9}}{{a}^{10} \left ( a+b\sqrt [3]{x} \right ) }}-{\frac{1}{3\,{x}^{3}{a}^{2}}}+{\frac{3\,b}{4\,{a}^{3}}{x}^{-{\frac{8}{3}}}}-{\frac{9\,{b}^{2}}{7\,{a}^{4}}{x}^{-{\frac{7}{3}}}}+2\,{\frac{{b}^{3}}{{a}^{5}{x}^{2}}}-3\,{\frac{{b}^{4}}{{a}^{6}{x}^{5/3}}}+{\frac{9\,{b}^{5}}{2\,{a}^{7}}{x}^{-{\frac{4}{3}}}}-7\,{\frac{{b}^{6}}{{a}^{8}x}}+12\,{\frac{{b}^{7}}{{a}^{9}{x}^{2/3}}}-27\,{\frac{{b}^{8}}{{a}^{10}\sqrt [3]{x}}}+30\,{\frac{{b}^{9}\ln \left ( a+b\sqrt [3]{x} \right ) }{{a}^{11}}}-10\,{\frac{{b}^{9}\ln \left ( x \right ) }{{a}^{11}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*x^(1/3))^2/x^4,x)

[Out]

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

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Maxima [A]  time = 1.44067, size = 193, normalized size = 1.19 \[ -\frac{2520 \, b^{9} x^{3} + 1260 \, a b^{8} x^{\frac{8}{3}} - 420 \, a^{2} b^{7} x^{\frac{7}{3}} + 210 \, a^{3} b^{6} x^{2} - 126 \, a^{4} b^{5} x^{\frac{5}{3}} + 84 \, a^{5} b^{4} x^{\frac{4}{3}} - 60 \, a^{6} b^{3} x + 45 \, a^{7} b^{2} x^{\frac{2}{3}} - 35 \, a^{8} b x^{\frac{1}{3}} + 28 \, a^{9}}{84 \,{\left (a^{10} b x^{\frac{10}{3}} + a^{11} x^{3}\right )}} + \frac{30 \, b^{9} \log \left (b x^{\frac{1}{3}} + a\right )}{a^{11}} - \frac{10 \, b^{9} \log \left (x\right )}{a^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^(1/3) + a)^2*x^4),x, algorithm="maxima")

[Out]

-1/84*(2520*b^9*x^3 + 1260*a*b^8*x^(8/3) - 420*a^2*b^7*x^(7/3) + 210*a^3*b^6*x^2
- 126*a^4*b^5*x^(5/3) + 84*a^5*b^4*x^(4/3) - 60*a^6*b^3*x + 45*a^7*b^2*x^(2/3)
- 35*a^8*b*x^(1/3) + 28*a^9)/(a^10*b*x^(10/3) + a^11*x^3) + 30*b^9*log(b*x^(1/3)
+ a)/a^11 - 10*b^9*log(x)/a^11

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Fricas [A]  time = 0.228253, size = 228, normalized size = 1.41 \[ -\frac{2520 \, a b^{9} x^{3} + 210 \, a^{4} b^{6} x^{2} - 60 \, a^{7} b^{3} x + 28 \, a^{10} - 2520 \,{\left (b^{10} x^{\frac{10}{3}} + a b^{9} x^{3}\right )} \log \left (b x^{\frac{1}{3}} + a\right ) + 2520 \,{\left (b^{10} x^{\frac{10}{3}} + a b^{9} x^{3}\right )} \log \left (x^{\frac{1}{3}}\right ) + 9 \,{\left (140 \, a^{2} b^{8} x^{2} - 14 \, a^{5} b^{5} x + 5 \, a^{8} b^{2}\right )} x^{\frac{2}{3}} - 7 \,{\left (60 \, a^{3} b^{7} x^{2} - 12 \, a^{6} b^{4} x + 5 \, a^{9} b\right )} x^{\frac{1}{3}}}{84 \,{\left (a^{11} b x^{\frac{10}{3}} + a^{12} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^(1/3) + a)^2*x^4),x, algorithm="fricas")

[Out]

-1/84*(2520*a*b^9*x^3 + 210*a^4*b^6*x^2 - 60*a^7*b^3*x + 28*a^10 - 2520*(b^10*x^
(10/3) + a*b^9*x^3)*log(b*x^(1/3) + a) + 2520*(b^10*x^(10/3) + a*b^9*x^3)*log(x^
(1/3)) + 9*(140*a^2*b^8*x^2 - 14*a^5*b^5*x + 5*a^8*b^2)*x^(2/3) - 7*(60*a^3*b^7*
x^2 - 12*a^6*b^4*x + 5*a^9*b)*x^(1/3))/(a^11*b*x^(10/3) + a^12*x^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**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.22353, size = 196, normalized size = 1.21 \[ \frac{30 \, b^{9}{\rm ln}\left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{a^{11}} - \frac{10 \, b^{9}{\rm ln}\left ({\left | x \right |}\right )}{a^{11}} - \frac{2520 \, a b^{9} x^{3} + 1260 \, a^{2} b^{8} x^{\frac{8}{3}} - 420 \, a^{3} b^{7} x^{\frac{7}{3}} + 210 \, a^{4} b^{6} x^{2} - 126 \, a^{5} b^{5} x^{\frac{5}{3}} + 84 \, a^{6} b^{4} x^{\frac{4}{3}} - 60 \, a^{7} b^{3} x + 45 \, a^{8} b^{2} x^{\frac{2}{3}} - 35 \, a^{9} b x^{\frac{1}{3}} + 28 \, a^{10}}{84 \,{\left (b x^{\frac{1}{3}} + a\right )} a^{11} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^(1/3) + a)^2*x^4),x, algorithm="giac")

[Out]

30*b^9*ln(abs(b*x^(1/3) + a))/a^11 - 10*b^9*ln(abs(x))/a^11 - 1/84*(2520*a*b^9*x
^3 + 1260*a^2*b^8*x^(8/3) - 420*a^3*b^7*x^(7/3) + 210*a^4*b^6*x^2 - 126*a^5*b^5*
x^(5/3) + 84*a^6*b^4*x^(4/3) - 60*a^7*b^3*x + 45*a^8*b^2*x^(2/3) - 35*a^9*b*x^(1
/3) + 28*a^10)/((b*x^(1/3) + a)*a^11*x^3)