3.2428 \(\int \frac{1}{(a+\frac{b}{\sqrt [3]{x}}) x^5} \, dx\)

Optimal. Leaf size=179 \[ \frac{3 a^9}{2 b^{10} x^{2/3}}+\frac{3 a^7}{4 b^8 x^{4/3}}-\frac{3 a^6}{5 b^7 x^{5/3}}+\frac{a^5}{2 b^6 x^2}-\frac{3 a^4}{7 b^5 x^{7/3}}+\frac{3 a^3}{8 b^4 x^{8/3}}-\frac{a^2}{3 b^3 x^3}-\frac{3 a^{10}}{b^{11} \sqrt [3]{x}}-\frac{a^8}{b^9 x}+\frac{3 a^{11} \log \left (a \sqrt [3]{x}+b\right )}{b^{12}}-\frac{a^{11} \log (x)}{b^{12}}+\frac{3 a}{10 b^2 x^{10/3}}-\frac{3}{11 b x^{11/3}} \]

[Out]

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

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Rubi [A]  time = 0.0939117, antiderivative size = 179, 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, Rules used = {263, 266, 44} \[ \frac{3 a^9}{2 b^{10} x^{2/3}}+\frac{3 a^7}{4 b^8 x^{4/3}}-\frac{3 a^6}{5 b^7 x^{5/3}}+\frac{a^5}{2 b^6 x^2}-\frac{3 a^4}{7 b^5 x^{7/3}}+\frac{3 a^3}{8 b^4 x^{8/3}}-\frac{a^2}{3 b^3 x^3}-\frac{3 a^{10}}{b^{11} \sqrt [3]{x}}-\frac{a^8}{b^9 x}+\frac{3 a^{11} \log \left (a \sqrt [3]{x}+b\right )}{b^{12}}-\frac{a^{11} \log (x)}{b^{12}}+\frac{3 a}{10 b^2 x^{10/3}}-\frac{3}{11 b x^{11/3}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^(1/3))*x^5),x]

[Out]

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

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{\left (a+\frac{b}{\sqrt [3]{x}}\right ) x^5} \, dx &=\int \frac{1}{\left (b+a \sqrt [3]{x}\right ) x^{14/3}} \, dx\\ &=3 \operatorname{Subst}\left (\int \frac{1}{x^{12} (b+a x)} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{1}{b x^{12}}-\frac{a}{b^2 x^{11}}+\frac{a^2}{b^3 x^{10}}-\frac{a^3}{b^4 x^9}+\frac{a^4}{b^5 x^8}-\frac{a^5}{b^6 x^7}+\frac{a^6}{b^7 x^6}-\frac{a^7}{b^8 x^5}+\frac{a^8}{b^9 x^4}-\frac{a^9}{b^{10} x^3}+\frac{a^{10}}{b^{11} x^2}-\frac{a^{11}}{b^{12} x}+\frac{a^{12}}{b^{12} (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{3}{11 b x^{11/3}}+\frac{3 a}{10 b^2 x^{10/3}}-\frac{a^2}{3 b^3 x^3}+\frac{3 a^3}{8 b^4 x^{8/3}}-\frac{3 a^4}{7 b^5 x^{7/3}}+\frac{a^5}{2 b^6 x^2}-\frac{3 a^6}{5 b^7 x^{5/3}}+\frac{3 a^7}{4 b^8 x^{4/3}}-\frac{a^8}{b^9 x}+\frac{3 a^9}{2 b^{10} x^{2/3}}-\frac{3 a^{10}}{b^{11} \sqrt [3]{x}}+\frac{3 a^{11} \log \left (b+a \sqrt [3]{x}\right )}{b^{12}}-\frac{a^{11} \log (x)}{b^{12}}\\ \end{align*}

Mathematica [A]  time = 0.168314, size = 158, normalized size = 0.88 \[ \frac{\frac{b \left (-9240 a^8 b^2 x^{8/3}+6930 a^7 b^3 x^{7/3}-5544 a^6 b^4 x^2+4620 a^5 b^5 x^{5/3}-3960 a^4 b^6 x^{4/3}-3080 a^2 b^8 x^{2/3}+3465 a^3 b^7 x+13860 a^9 b x^3-27720 a^{10} x^{10/3}+2772 a b^9 \sqrt [3]{x}-2520 b^{10}\right )}{x^{11/3}}+27720 a^{11} \log \left (a \sqrt [3]{x}+b\right )-9240 a^{11} \log (x)}{9240 b^{12}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^(1/3))*x^5),x]

[Out]

((b*(-2520*b^10 + 2772*a*b^9*x^(1/3) - 3080*a^2*b^8*x^(2/3) + 3465*a^3*b^7*x - 3960*a^4*b^6*x^(4/3) + 4620*a^5
*b^5*x^(5/3) - 5544*a^6*b^4*x^2 + 6930*a^7*b^3*x^(7/3) - 9240*a^8*b^2*x^(8/3) + 13860*a^9*b*x^3 - 27720*a^10*x
^(10/3)))/x^(11/3) + 27720*a^11*Log[b + a*x^(1/3)] - 9240*a^11*Log[x])/(9240*b^12)

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Maple [A]  time = 0.012, size = 144, normalized size = 0.8 \begin{align*} -{\frac{3}{11\,b}{x}^{-{\frac{11}{3}}}}+{\frac{3\,a}{10\,{b}^{2}}{x}^{-{\frac{10}{3}}}}-{\frac{{a}^{2}}{3\,{b}^{3}{x}^{3}}}+{\frac{3\,{a}^{3}}{8\,{b}^{4}}{x}^{-{\frac{8}{3}}}}-{\frac{3\,{a}^{4}}{7\,{b}^{5}}{x}^{-{\frac{7}{3}}}}+{\frac{{a}^{5}}{2\,{b}^{6}{x}^{2}}}-{\frac{3\,{a}^{6}}{5\,{b}^{7}}{x}^{-{\frac{5}{3}}}}+{\frac{3\,{a}^{7}}{4\,{b}^{8}}{x}^{-{\frac{4}{3}}}}-{\frac{{a}^{8}}{{b}^{9}x}}+{\frac{3\,{a}^{9}}{2\,{b}^{10}}{x}^{-{\frac{2}{3}}}}-3\,{\frac{{a}^{10}}{{b}^{11}\sqrt [3]{x}}}+3\,{\frac{{a}^{11}\ln \left ( b+a\sqrt [3]{x} \right ) }{{b}^{12}}}-{\frac{{a}^{11}\ln \left ( x \right ) }{{b}^{12}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x^(1/3))/x^5,x)

[Out]

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

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Maxima [A]  time = 0.97146, size = 266, normalized size = 1.49 \begin{align*} \frac{3 \, a^{11} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{b^{12}} - \frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{11}}{11 \, b^{12}} + \frac{33 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{10} a}{10 \, b^{12}} - \frac{55 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{9} a^{2}}{3 \, b^{12}} + \frac{495 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{8} a^{3}}{8 \, b^{12}} - \frac{990 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{7} a^{4}}{7 \, b^{12}} + \frac{231 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{6} a^{5}}{b^{12}} - \frac{1386 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{5} a^{6}}{5 \, b^{12}} + \frac{495 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{4} a^{7}}{2 \, b^{12}} - \frac{165 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{3} a^{8}}{b^{12}} + \frac{165 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2} a^{9}}{2 \, b^{12}} - \frac{33 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} a^{10}}{b^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^(1/3))/x^5,x, algorithm="maxima")

[Out]

3*a^11*log(a + b/x^(1/3))/b^12 - 3/11*(a + b/x^(1/3))^11/b^12 + 33/10*(a + b/x^(1/3))^10*a/b^12 - 55/3*(a + b/
x^(1/3))^9*a^2/b^12 + 495/8*(a + b/x^(1/3))^8*a^3/b^12 - 990/7*(a + b/x^(1/3))^7*a^4/b^12 + 231*(a + b/x^(1/3)
)^6*a^5/b^12 - 1386/5*(a + b/x^(1/3))^5*a^6/b^12 + 495/2*(a + b/x^(1/3))^4*a^7/b^12 - 165*(a + b/x^(1/3))^3*a^
8/b^12 + 165/2*(a + b/x^(1/3))^2*a^9/b^12 - 33*(a + b/x^(1/3))*a^10/b^12

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Fricas [A]  time = 1.50655, size = 389, normalized size = 2.17 \begin{align*} \frac{27720 \, a^{11} x^{4} \log \left (a x^{\frac{1}{3}} + b\right ) - 27720 \, a^{11} x^{4} \log \left (x^{\frac{1}{3}}\right ) - 9240 \, a^{8} b^{3} x^{3} + 4620 \, a^{5} b^{6} x^{2} - 3080 \, a^{2} b^{9} x - 198 \,{\left (140 \, a^{10} b x^{3} - 35 \, a^{7} b^{4} x^{2} + 20 \, a^{4} b^{7} x - 14 \, a b^{10}\right )} x^{\frac{2}{3}} + 63 \,{\left (220 \, a^{9} b^{2} x^{3} - 88 \, a^{6} b^{5} x^{2} + 55 \, a^{3} b^{8} x - 40 \, b^{11}\right )} x^{\frac{1}{3}}}{9240 \, b^{12} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^(1/3))/x^5,x, algorithm="fricas")

[Out]

1/9240*(27720*a^11*x^4*log(a*x^(1/3) + b) - 27720*a^11*x^4*log(x^(1/3)) - 9240*a^8*b^3*x^3 + 4620*a^5*b^6*x^2
- 3080*a^2*b^9*x - 198*(140*a^10*b*x^3 - 35*a^7*b^4*x^2 + 20*a^4*b^7*x - 14*a*b^10)*x^(2/3) + 63*(220*a^9*b^2*
x^3 - 88*a^6*b^5*x^2 + 55*a^3*b^8*x - 40*b^11)*x^(1/3))/(b^12*x^4)

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Sympy [A]  time = 83.7714, size = 201, normalized size = 1.12 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{11}{3}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{1}{4 a x^{4}} & \text{for}\: b = 0 \\- \frac{3}{11 b x^{\frac{11}{3}}} & \text{for}\: a = 0 \\- \frac{a^{11} \log{\left (x \right )}}{b^{12}} + \frac{3 a^{11} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{b^{12}} - \frac{3 a^{10}}{b^{11} \sqrt [3]{x}} + \frac{3 a^{9}}{2 b^{10} x^{\frac{2}{3}}} - \frac{a^{8}}{b^{9} x} + \frac{3 a^{7}}{4 b^{8} x^{\frac{4}{3}}} - \frac{3 a^{6}}{5 b^{7} x^{\frac{5}{3}}} + \frac{a^{5}}{2 b^{6} x^{2}} - \frac{3 a^{4}}{7 b^{5} x^{\frac{7}{3}}} + \frac{3 a^{3}}{8 b^{4} x^{\frac{8}{3}}} - \frac{a^{2}}{3 b^{3} x^{3}} + \frac{3 a}{10 b^{2} x^{\frac{10}{3}}} - \frac{3}{11 b x^{\frac{11}{3}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**(1/3))/x**5,x)

[Out]

Piecewise((zoo/x**(11/3), Eq(a, 0) & Eq(b, 0)), (-1/(4*a*x**4), Eq(b, 0)), (-3/(11*b*x**(11/3)), Eq(a, 0)), (-
a**11*log(x)/b**12 + 3*a**11*log(x**(1/3) + b/a)/b**12 - 3*a**10/(b**11*x**(1/3)) + 3*a**9/(2*b**10*x**(2/3))
- a**8/(b**9*x) + 3*a**7/(4*b**8*x**(4/3)) - 3*a**6/(5*b**7*x**(5/3)) + a**5/(2*b**6*x**2) - 3*a**4/(7*b**5*x*
*(7/3)) + 3*a**3/(8*b**4*x**(8/3)) - a**2/(3*b**3*x**3) + 3*a/(10*b**2*x**(10/3)) - 3/(11*b*x**(11/3)), True))

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Giac [A]  time = 1.18851, size = 198, normalized size = 1.11 \begin{align*} \frac{3 \, a^{11} \log \left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{b^{12}} - \frac{a^{11} \log \left ({\left | x \right |}\right )}{b^{12}} - \frac{27720 \, a^{10} b x^{\frac{10}{3}} - 13860 \, a^{9} b^{2} x^{3} + 9240 \, a^{8} b^{3} x^{\frac{8}{3}} - 6930 \, a^{7} b^{4} x^{\frac{7}{3}} + 5544 \, a^{6} b^{5} x^{2} - 4620 \, a^{5} b^{6} x^{\frac{5}{3}} + 3960 \, a^{4} b^{7} x^{\frac{4}{3}} - 3465 \, a^{3} b^{8} x + 3080 \, a^{2} b^{9} x^{\frac{2}{3}} - 2772 \, a b^{10} x^{\frac{1}{3}} + 2520 \, b^{11}}{9240 \, b^{12} x^{\frac{11}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^(1/3))/x^5,x, algorithm="giac")

[Out]

3*a^11*log(abs(a*x^(1/3) + b))/b^12 - a^11*log(abs(x))/b^12 - 1/9240*(27720*a^10*b*x^(10/3) - 13860*a^9*b^2*x^
3 + 9240*a^8*b^3*x^(8/3) - 6930*a^7*b^4*x^(7/3) + 5544*a^6*b^5*x^2 - 4620*a^5*b^6*x^(5/3) + 3960*a^4*b^7*x^(4/
3) - 3465*a^3*b^8*x + 3080*a^2*b^9*x^(2/3) - 2772*a*b^10*x^(1/3) + 2520*b^11)/(b^12*x^(11/3))