3.2426 \(\int \frac{1}{(a+\frac{b}{\sqrt [3]{x}}) x^3} \, dx\)

Optimal. Leaf size=93 \[ \frac{3 a^3}{2 b^4 x^{2/3}}-\frac{3 a^4}{b^5 \sqrt [3]{x}}-\frac{a^2}{b^3 x}+\frac{3 a^5 \log \left (a \sqrt [3]{x}+b\right )}{b^6}-\frac{a^5 \log (x)}{b^6}+\frac{3 a}{4 b^2 x^{4/3}}-\frac{3}{5 b x^{5/3}} \]

[Out]

-3/(5*b*x^(5/3)) + (3*a)/(4*b^2*x^(4/3)) - a^2/(b^3*x) + (3*a^3)/(2*b^4*x^(2/3)) - (3*a^4)/(b^5*x^(1/3)) + (3*
a^5*Log[b + a*x^(1/3)])/b^6 - (a^5*Log[x])/b^6

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Rubi [A]  time = 0.0514041, antiderivative size = 93, 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} \[ \frac{3 a^3}{2 b^4 x^{2/3}}-\frac{3 a^4}{b^5 \sqrt [3]{x}}-\frac{a^2}{b^3 x}+\frac{3 a^5 \log \left (a \sqrt [3]{x}+b\right )}{b^6}-\frac{a^5 \log (x)}{b^6}+\frac{3 a}{4 b^2 x^{4/3}}-\frac{3}{5 b x^{5/3}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^(1/3))*x^3),x]

[Out]

-3/(5*b*x^(5/3)) + (3*a)/(4*b^2*x^(4/3)) - a^2/(b^3*x) + (3*a^3)/(2*b^4*x^(2/3)) - (3*a^4)/(b^5*x^(1/3)) + (3*
a^5*Log[b + a*x^(1/3)])/b^6 - (a^5*Log[x])/b^6

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

Mathematica [A]  time = 0.0782991, size = 84, normalized size = 0.9 \[ -\frac{\frac{b \left (20 a^2 b^2 x^{2/3}-30 a^3 b x+60 a^4 x^{4/3}-15 a b^3 \sqrt [3]{x}+12 b^4\right )}{x^{5/3}}-60 a^5 \log \left (a \sqrt [3]{x}+b\right )+20 a^5 \log (x)}{20 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^(1/3))*x^3),x]

[Out]

-((b*(12*b^4 - 15*a*b^3*x^(1/3) + 20*a^2*b^2*x^(2/3) - 30*a^3*b*x + 60*a^4*x^(4/3)))/x^(5/3) - 60*a^5*Log[b +
a*x^(1/3)] + 20*a^5*Log[x])/(20*b^6)

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Maple [A]  time = 0.008, size = 78, normalized size = 0.8 \begin{align*} -{\frac{3}{5\,b}{x}^{-{\frac{5}{3}}}}+{\frac{3\,a}{4\,{b}^{2}}{x}^{-{\frac{4}{3}}}}-{\frac{{a}^{2}}{{b}^{3}x}}+{\frac{3\,{a}^{3}}{2\,{b}^{4}}{x}^{-{\frac{2}{3}}}}-3\,{\frac{{a}^{4}}{{b}^{5}\sqrt [3]{x}}}+3\,{\frac{{a}^{5}\ln \left ( b+a\sqrt [3]{x} \right ) }{{b}^{6}}}-{\frac{{a}^{5}\ln \left ( x \right ) }{{b}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/5/b/x^(5/3)+3/4*a/b^2/x^(4/3)-a^2/b^3/x+3/2*a^3/b^4/x^(2/3)-3*a^4/b^5/x^(1/3)+3*a^5*ln(b+a*x^(1/3))/b^6-a^5
*ln(x)/b^6

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Maxima [A]  time = 0.97827, size = 128, normalized size = 1.38 \begin{align*} \frac{3 \, a^{5} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{b^{6}} - \frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{5}}{5 \, b^{6}} + \frac{15 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{4} a}{4 \, b^{6}} - \frac{10 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{3} a^{2}}{b^{6}} + \frac{15 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2} a^{3}}{b^{6}} - \frac{15 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} a^{4}}{b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*a^5*log(a + b/x^(1/3))/b^6 - 3/5*(a + b/x^(1/3))^5/b^6 + 15/4*(a + b/x^(1/3))^4*a/b^6 - 10*(a + b/x^(1/3))^3
*a^2/b^6 + 15*(a + b/x^(1/3))^2*a^3/b^6 - 15*(a + b/x^(1/3))*a^4/b^6

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Fricas [A]  time = 1.54642, size = 211, normalized size = 2.27 \begin{align*} \frac{60 \, a^{5} x^{2} \log \left (a x^{\frac{1}{3}} + b\right ) - 60 \, a^{5} x^{2} \log \left (x^{\frac{1}{3}}\right ) - 20 \, a^{2} b^{3} x - 15 \,{\left (4 \, a^{4} b x - a b^{4}\right )} x^{\frac{2}{3}} + 6 \,{\left (5 \, a^{3} b^{2} x - 2 \, b^{5}\right )} x^{\frac{1}{3}}}{20 \, b^{6} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(60*a^5*x^2*log(a*x^(1/3) + b) - 60*a^5*x^2*log(x^(1/3)) - 20*a^2*b^3*x - 15*(4*a^4*b*x - a*b^4)*x^(2/3)
+ 6*(5*a^3*b^2*x - 2*b^5)*x^(1/3))/(b^6*x^2)

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Sympy [A]  time = 5.9519, size = 116, normalized size = 1.25 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{5}{3}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{3}{5 b x^{\frac{5}{3}}} & \text{for}\: a = 0 \\- \frac{1}{2 a x^{2}} & \text{for}\: b = 0 \\- \frac{a^{5} \log{\left (x \right )}}{b^{6}} + \frac{3 a^{5} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{b^{6}} - \frac{3 a^{4}}{b^{5} \sqrt [3]{x}} + \frac{3 a^{3}}{2 b^{4} x^{\frac{2}{3}}} - \frac{a^{2}}{b^{3} x} + \frac{3 a}{4 b^{2} x^{\frac{4}{3}}} - \frac{3}{5 b x^{\frac{5}{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**3,x)

[Out]

Piecewise((zoo/x**(5/3), Eq(a, 0) & Eq(b, 0)), (-3/(5*b*x**(5/3)), Eq(a, 0)), (-1/(2*a*x**2), Eq(b, 0)), (-a**
5*log(x)/b**6 + 3*a**5*log(x**(1/3) + b/a)/b**6 - 3*a**4/(b**5*x**(1/3)) + 3*a**3/(2*b**4*x**(2/3)) - a**2/(b*
*3*x) + 3*a/(4*b**2*x**(4/3)) - 3/(5*b*x**(5/3)), True))

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Giac [A]  time = 1.23034, size = 109, normalized size = 1.17 \begin{align*} \frac{3 \, a^{5} \log \left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{b^{6}} - \frac{a^{5} \log \left ({\left | x \right |}\right )}{b^{6}} - \frac{60 \, a^{4} b x^{\frac{4}{3}} - 30 \, a^{3} b^{2} x + 20 \, a^{2} b^{3} x^{\frac{2}{3}} - 15 \, a b^{4} x^{\frac{1}{3}} + 12 \, b^{5}}{20 \, b^{6} x^{\frac{5}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*a^5*log(abs(a*x^(1/3) + b))/b^6 - a^5*log(abs(x))/b^6 - 1/20*(60*a^4*b*x^(4/3) - 30*a^3*b^2*x + 20*a^2*b^3*x
^(2/3) - 15*a*b^4*x^(1/3) + 12*b^5)/(b^6*x^(5/3))