3.2373 \(\int \frac{1}{\left (a+b \sqrt [3]{x}\right )^3 x^4} \, dx\)

Optimal. Leaf size=183 \[ \frac{165 b^9 \log \left (a+b \sqrt [3]{x}\right )}{a^{12}}-\frac{55 b^9 \log (x)}{a^{12}}-\frac{30 b^9}{a^{11} \left (a+b \sqrt [3]{x}\right )}-\frac{135 b^8}{a^{11} \sqrt [3]{x}}-\frac{3 b^9}{2 a^{10} \left (a+b \sqrt [3]{x}\right )^2}+\frac{54 b^7}{a^{10} x^{2/3}}-\frac{28 b^6}{a^9 x}+\frac{63 b^5}{4 a^8 x^{4/3}}-\frac{9 b^4}{a^7 x^{5/3}}+\frac{5 b^3}{a^6 x^2}-\frac{18 b^2}{7 a^5 x^{7/3}}+\frac{9 b}{8 a^4 x^{8/3}}-\frac{1}{3 a^3 x^3} \]

[Out]

(-3*b^9)/(2*a^10*(a + b*x^(1/3))^2) - (30*b^9)/(a^11*(a + b*x^(1/3))) - 1/(3*a^3
*x^3) + (9*b)/(8*a^4*x^(8/3)) - (18*b^2)/(7*a^5*x^(7/3)) + (5*b^3)/(a^6*x^2) - (
9*b^4)/(a^7*x^(5/3)) + (63*b^5)/(4*a^8*x^(4/3)) - (28*b^6)/(a^9*x) + (54*b^7)/(a
^10*x^(2/3)) - (135*b^8)/(a^11*x^(1/3)) + (165*b^9*Log[a + b*x^(1/3)])/a^12 - (5
5*b^9*Log[x])/a^12

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Rubi [A]  time = 0.317381, antiderivative size = 183, 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 \[ \frac{165 b^9 \log \left (a+b \sqrt [3]{x}\right )}{a^{12}}-\frac{55 b^9 \log (x)}{a^{12}}-\frac{30 b^9}{a^{11} \left (a+b \sqrt [3]{x}\right )}-\frac{135 b^8}{a^{11} \sqrt [3]{x}}-\frac{3 b^9}{2 a^{10} \left (a+b \sqrt [3]{x}\right )^2}+\frac{54 b^7}{a^{10} x^{2/3}}-\frac{28 b^6}{a^9 x}+\frac{63 b^5}{4 a^8 x^{4/3}}-\frac{9 b^4}{a^7 x^{5/3}}+\frac{5 b^3}{a^6 x^2}-\frac{18 b^2}{7 a^5 x^{7/3}}+\frac{9 b}{8 a^4 x^{8/3}}-\frac{1}{3 a^3 x^3} \]

Antiderivative was successfully verified.

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

[Out]

(-3*b^9)/(2*a^10*(a + b*x^(1/3))^2) - (30*b^9)/(a^11*(a + b*x^(1/3))) - 1/(3*a^3
*x^3) + (9*b)/(8*a^4*x^(8/3)) - (18*b^2)/(7*a^5*x^(7/3)) + (5*b^3)/(a^6*x^2) - (
9*b^4)/(a^7*x^(5/3)) + (63*b^5)/(4*a^8*x^(4/3)) - (28*b^6)/(a^9*x) + (54*b^7)/(a
^10*x^(2/3)) - (135*b^8)/(a^11*x^(1/3)) + (165*b^9*Log[a + b*x^(1/3)])/a^12 - (5
5*b^9*Log[x])/a^12

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.419691, size = 167, normalized size = 0.91 \[ -\frac{\frac{a \left (56 a^{10}-77 a^9 b \sqrt [3]{x}+110 a^8 b^2 x^{2/3}-165 a^7 b^3 x+264 a^6 b^4 x^{4/3}-462 a^5 b^5 x^{5/3}+924 a^4 b^6 x^2-2310 a^3 b^7 x^{7/3}+9240 a^2 b^8 x^{8/3}+41580 a b^9 x^3+27720 b^{10} x^{10/3}\right )}{x^3 \left (a+b \sqrt [3]{x}\right )^2}-27720 b^9 \log \left (a+b \sqrt [3]{x}\right )+9240 b^9 \log (x)}{168 a^{12}} \]

Antiderivative was successfully verified.

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

[Out]

-((a*(56*a^10 - 77*a^9*b*x^(1/3) + 110*a^8*b^2*x^(2/3) - 165*a^7*b^3*x + 264*a^6
*b^4*x^(4/3) - 462*a^5*b^5*x^(5/3) + 924*a^4*b^6*x^2 - 2310*a^3*b^7*x^(7/3) + 92
40*a^2*b^8*x^(8/3) + 41580*a*b^9*x^3 + 27720*b^10*x^(10/3)))/((a + b*x^(1/3))^2*
x^3) - 27720*b^9*Log[a + b*x^(1/3)] + 9240*b^9*Log[x])/(168*a^12)

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Maple [A]  time = 0.021, size = 156, normalized size = 0.9 \[ -{\frac{3\,{b}^{9}}{2\,{a}^{10}} \left ( a+b\sqrt [3]{x} \right ) ^{-2}}-30\,{\frac{{b}^{9}}{{a}^{11} \left ( a+b\sqrt [3]{x} \right ) }}-{\frac{1}{3\,{a}^{3}{x}^{3}}}+{\frac{9\,b}{8\,{a}^{4}}{x}^{-{\frac{8}{3}}}}-{\frac{18\,{b}^{2}}{7\,{a}^{5}}{x}^{-{\frac{7}{3}}}}+5\,{\frac{{b}^{3}}{{a}^{6}{x}^{2}}}-9\,{\frac{{b}^{4}}{{a}^{7}{x}^{5/3}}}+{\frac{63\,{b}^{5}}{4\,{a}^{8}}{x}^{-{\frac{4}{3}}}}-28\,{\frac{{b}^{6}}{{a}^{9}x}}+54\,{\frac{{b}^{7}}{{a}^{10}{x}^{2/3}}}-135\,{\frac{{b}^{8}}{{a}^{11}\sqrt [3]{x}}}+165\,{\frac{{b}^{9}\ln \left ( a+b\sqrt [3]{x} \right ) }{{a}^{12}}}-55\,{\frac{{b}^{9}\ln \left ( x \right ) }{{a}^{12}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*x^(1/3))^3/x^4,x)

[Out]

-3/2*b^9/a^10/(a+b*x^(1/3))^2-30*b^9/a^11/(a+b*x^(1/3))-1/3/a^3/x^3+9/8*b/a^4/x^
(8/3)-18/7*b^2/a^5/x^(7/3)+5*b^3/a^6/x^2-9*b^4/a^7/x^(5/3)+63/4*b^5/a^8/x^(4/3)-
28*b^6/a^9/x+54*b^7/a^10/x^(2/3)-135*b^8/a^11/x^(1/3)+165*b^9*ln(a+b*x^(1/3))/a^
12-55*b^9*ln(x)/a^12

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Maxima [A]  time = 1.45634, size = 223, normalized size = 1.22 \[ -\frac{27720 \, b^{10} x^{\frac{10}{3}} + 41580 \, a b^{9} x^{3} + 9240 \, a^{2} b^{8} x^{\frac{8}{3}} - 2310 \, a^{3} b^{7} x^{\frac{7}{3}} + 924 \, a^{4} b^{6} x^{2} - 462 \, a^{5} b^{5} x^{\frac{5}{3}} + 264 \, a^{6} b^{4} x^{\frac{4}{3}} - 165 \, a^{7} b^{3} x + 110 \, a^{8} b^{2} x^{\frac{2}{3}} - 77 \, a^{9} b x^{\frac{1}{3}} + 56 \, a^{10}}{168 \,{\left (a^{11} b^{2} x^{\frac{11}{3}} + 2 \, a^{12} b x^{\frac{10}{3}} + a^{13} x^{3}\right )}} + \frac{165 \, b^{9} \log \left (b x^{\frac{1}{3}} + a\right )}{a^{12}} - \frac{55 \, b^{9} \log \left (x\right )}{a^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/168*(27720*b^10*x^(10/3) + 41580*a*b^9*x^3 + 9240*a^2*b^8*x^(8/3) - 2310*a^3*
b^7*x^(7/3) + 924*a^4*b^6*x^2 - 462*a^5*b^5*x^(5/3) + 264*a^6*b^4*x^(4/3) - 165*
a^7*b^3*x + 110*a^8*b^2*x^(2/3) - 77*a^9*b*x^(1/3) + 56*a^10)/(a^11*b^2*x^(11/3)
+ 2*a^12*b*x^(10/3) + a^13*x^3) + 165*b^9*log(b*x^(1/3) + a)/a^12 - 55*b^9*log(
x)/a^12

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Fricas [A]  time = 0.23256, size = 288, normalized size = 1.57 \[ -\frac{41580 \, a^{2} b^{9} x^{3} + 924 \, a^{5} b^{6} x^{2} - 165 \, a^{8} b^{3} x + 56 \, a^{11} - 27720 \,{\left (b^{11} x^{\frac{11}{3}} + 2 \, a b^{10} x^{\frac{10}{3}} + a^{2} b^{9} x^{3}\right )} \log \left (b x^{\frac{1}{3}} + a\right ) + 27720 \,{\left (b^{11} x^{\frac{11}{3}} + 2 \, a b^{10} x^{\frac{10}{3}} + a^{2} b^{9} x^{3}\right )} \log \left (x^{\frac{1}{3}}\right ) + 22 \,{\left (420 \, a^{3} b^{8} x^{2} - 21 \, a^{6} b^{5} x + 5 \, a^{9} b^{2}\right )} x^{\frac{2}{3}} + 11 \,{\left (2520 \, a b^{10} x^{3} - 210 \, a^{4} b^{7} x^{2} + 24 \, a^{7} b^{4} x - 7 \, a^{10} b\right )} x^{\frac{1}{3}}}{168 \,{\left (a^{12} b^{2} x^{\frac{11}{3}} + 2 \, a^{13} b x^{\frac{10}{3}} + a^{14} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/168*(41580*a^2*b^9*x^3 + 924*a^5*b^6*x^2 - 165*a^8*b^3*x + 56*a^11 - 27720*(b
^11*x^(11/3) + 2*a*b^10*x^(10/3) + a^2*b^9*x^3)*log(b*x^(1/3) + a) + 27720*(b^11
*x^(11/3) + 2*a*b^10*x^(10/3) + a^2*b^9*x^3)*log(x^(1/3)) + 22*(420*a^3*b^8*x^2
- 21*a^6*b^5*x + 5*a^9*b^2)*x^(2/3) + 11*(2520*a*b^10*x^3 - 210*a^4*b^7*x^2 + 24
*a^7*b^4*x - 7*a^10*b)*x^(1/3))/(a^12*b^2*x^(11/3) + 2*a^13*b*x^(10/3) + a^14*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))**3/x**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.224199, size = 211, normalized size = 1.15 \[ \frac{165 \, b^{9}{\rm ln}\left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{a^{12}} - \frac{55 \, b^{9}{\rm ln}\left ({\left | x \right |}\right )}{a^{12}} - \frac{27720 \, a b^{10} x^{\frac{10}{3}} + 41580 \, a^{2} b^{9} x^{3} + 9240 \, a^{3} b^{8} x^{\frac{8}{3}} - 2310 \, a^{4} b^{7} x^{\frac{7}{3}} + 924 \, a^{5} b^{6} x^{2} - 462 \, a^{6} b^{5} x^{\frac{5}{3}} + 264 \, a^{7} b^{4} x^{\frac{4}{3}} - 165 \, a^{8} b^{3} x + 110 \, a^{9} b^{2} x^{\frac{2}{3}} - 77 \, a^{10} b x^{\frac{1}{3}} + 56 \, a^{11}}{168 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} a^{12} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

165*b^9*ln(abs(b*x^(1/3) + a))/a^12 - 55*b^9*ln(abs(x))/a^12 - 1/168*(27720*a*b^
10*x^(10/3) + 41580*a^2*b^9*x^3 + 9240*a^3*b^8*x^(8/3) - 2310*a^4*b^7*x^(7/3) +
924*a^5*b^6*x^2 - 462*a^6*b^5*x^(5/3) + 264*a^7*b^4*x^(4/3) - 165*a^8*b^3*x + 11
0*a^9*b^2*x^(2/3) - 77*a^10*b*x^(1/3) + 56*a^11)/((b*x^(1/3) + a)^2*a^12*x^3)