3.2442 \(\int \frac{1}{(a+\frac{b}{\sqrt [3]{x}})^3 x^3} \, dx\)

Optimal. Leaf size=103 \[ -\frac{12 a^3}{b^5 \left (a \sqrt [3]{x}+b\right )}-\frac{3 a^3}{2 b^4 \left (a \sqrt [3]{x}+b\right )^2}-\frac{18 a^2}{b^5 \sqrt [3]{x}}+\frac{30 a^3 \log \left (a \sqrt [3]{x}+b\right )}{b^6}-\frac{10 a^3 \log (x)}{b^6}+\frac{9 a}{2 b^4 x^{2/3}}-\frac{1}{b^3 x} \]

[Out]

(-3*a^3)/(2*b^4*(b + a*x^(1/3))^2) - (12*a^3)/(b^5*(b + a*x^(1/3))) - 1/(b^3*x) + (9*a)/(2*b^4*x^(2/3)) - (18*
a^2)/(b^5*x^(1/3)) + (30*a^3*Log[b + a*x^(1/3)])/b^6 - (10*a^3*Log[x])/b^6

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Rubi [A]  time = 0.0684681, antiderivative size = 103, 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} \[ -\frac{12 a^3}{b^5 \left (a \sqrt [3]{x}+b\right )}-\frac{3 a^3}{2 b^4 \left (a \sqrt [3]{x}+b\right )^2}-\frac{18 a^2}{b^5 \sqrt [3]{x}}+\frac{30 a^3 \log \left (a \sqrt [3]{x}+b\right )}{b^6}-\frac{10 a^3 \log (x)}{b^6}+\frac{9 a}{2 b^4 x^{2/3}}-\frac{1}{b^3 x} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^(1/3))^3*x^3),x]

[Out]

(-3*a^3)/(2*b^4*(b + a*x^(1/3))^2) - (12*a^3)/(b^5*(b + a*x^(1/3))) - 1/(b^3*x) + (9*a)/(2*b^4*x^(2/3)) - (18*
a^2)/(b^5*x^(1/3)) + (30*a^3*Log[b + a*x^(1/3)])/b^6 - (10*a^3*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 )^3 x^3} \, dx &=\int \frac{1}{\left (b+a \sqrt [3]{x}\right )^3 x^2} \, dx\\ &=3 \operatorname{Subst}\left (\int \frac{1}{x^4 (b+a x)^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{1}{b^3 x^4}-\frac{3 a}{b^4 x^3}+\frac{6 a^2}{b^5 x^2}-\frac{10 a^3}{b^6 x}+\frac{a^4}{b^4 (b+a x)^3}+\frac{4 a^4}{b^5 (b+a x)^2}+\frac{10 a^4}{b^6 (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{3 a^3}{2 b^4 \left (b+a \sqrt [3]{x}\right )^2}-\frac{12 a^3}{b^5 \left (b+a \sqrt [3]{x}\right )}-\frac{1}{b^3 x}+\frac{9 a}{2 b^4 x^{2/3}}-\frac{18 a^2}{b^5 \sqrt [3]{x}}+\frac{30 a^3 \log \left (b+a \sqrt [3]{x}\right )}{b^6}-\frac{10 a^3 \log (x)}{b^6}\\ \end{align*}

Mathematica [A]  time = 0.142595, size = 93, normalized size = 0.9 \[ -\frac{\frac{b \left (20 a^2 b^2 x^{2/3}+90 a^3 b x+60 a^4 x^{4/3}-5 a b^3 \sqrt [3]{x}+2 b^4\right )}{x \left (a \sqrt [3]{x}+b\right )^2}-60 a^3 \log \left (a \sqrt [3]{x}+b\right )+20 a^3 \log (x)}{2 b^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*(2*b^4 - 5*a*b^3*x^(1/3) + 20*a^2*b^2*x^(2/3) + 90*a^3*b*x + 60*a^4*x^(4/3)))/((b + a*x^(1/3))^2*x) - 60*
a^3*Log[b + a*x^(1/3)] + 20*a^3*Log[x])/(2*b^6)

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Maple [A]  time = 0.012, size = 90, normalized size = 0.9 \begin{align*} -{\frac{3\,{a}^{3}}{2\,{b}^{4}} \left ( b+a\sqrt [3]{x} \right ) ^{-2}}-12\,{\frac{{a}^{3}}{{b}^{5} \left ( b+a\sqrt [3]{x} \right ) }}-{\frac{1}{{b}^{3}x}}+{\frac{9\,a}{2\,{b}^{4}}{x}^{-{\frac{2}{3}}}}-18\,{\frac{{a}^{2}}{{b}^{5}\sqrt [3]{x}}}+30\,{\frac{{a}^{3}\ln \left ( b+a\sqrt [3]{x} \right ) }{{b}^{6}}}-10\,{\frac{{a}^{3}\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))^3/x^3,x)

[Out]

-3/2*a^3/b^4/(b+a*x^(1/3))^2-12*a^3/b^5/(b+a*x^(1/3))-1/b^3/x+9/2*a/b^4/x^(2/3)-18*a^2/b^5/x^(1/3)+30*a^3*ln(b
+a*x^(1/3))/b^6-10*a^3*ln(x)/b^6

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Maxima [A]  time = 0.982002, size = 128, normalized size = 1.24 \begin{align*} \frac{30 \, a^{3} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{b^{6}} - \frac{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{3}}{b^{6}} + \frac{15 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2} a}{2 \, b^{6}} - \frac{30 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} a^{2}}{b^{6}} + \frac{15 \, a^{4}}{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} b^{6}} - \frac{3 \, a^{5}}{2 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2} b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.57148, size = 421, normalized size = 4.09 \begin{align*} -\frac{20 \, a^{6} b^{3} x^{2} + 31 \, a^{3} b^{6} x + 2 \, b^{9} - 60 \,{\left (a^{9} x^{3} + 2 \, a^{6} b^{3} x^{2} + a^{3} b^{6} x\right )} \log \left (a x^{\frac{1}{3}} + b\right ) + 60 \,{\left (a^{9} x^{3} + 2 \, a^{6} b^{3} x^{2} + a^{3} b^{6} x\right )} \log \left (x^{\frac{1}{3}}\right ) + 3 \,{\left (20 \, a^{8} b x^{2} + 35 \, a^{5} b^{4} x + 12 \, a^{2} b^{7}\right )} x^{\frac{2}{3}} - 3 \,{\left (10 \, a^{7} b^{2} x^{2} + 16 \, a^{4} b^{5} x + 3 \, a b^{8}\right )} x^{\frac{1}{3}}}{2 \,{\left (a^{6} b^{6} x^{3} + 2 \, a^{3} b^{9} x^{2} + b^{12} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(20*a^6*b^3*x^2 + 31*a^3*b^6*x + 2*b^9 - 60*(a^9*x^3 + 2*a^6*b^3*x^2 + a^3*b^6*x)*log(a*x^(1/3) + b) + 60
*(a^9*x^3 + 2*a^6*b^3*x^2 + a^3*b^6*x)*log(x^(1/3)) + 3*(20*a^8*b*x^2 + 35*a^5*b^4*x + 12*a^2*b^7)*x^(2/3) - 3
*(10*a^7*b^2*x^2 + 16*a^4*b^5*x + 3*a*b^8)*x^(1/3))/(a^6*b^6*x^3 + 2*a^3*b^9*x^2 + b^12*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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 [A]  time = 1.17221, size = 122, normalized size = 1.18 \begin{align*} \frac{30 \, a^{3} \log \left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{b^{6}} - \frac{10 \, a^{3} \log \left ({\left | x \right |}\right )}{b^{6}} - \frac{60 \, a^{4} b x^{\frac{4}{3}} + 90 \, a^{3} b^{2} x + 20 \, a^{2} b^{3} x^{\frac{2}{3}} - 5 \, a b^{4} x^{\frac{1}{3}} + 2 \, b^{5}}{2 \,{\left (a x^{\frac{1}{3}} + b\right )}^{2} b^{6} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

30*a^3*log(abs(a*x^(1/3) + b))/b^6 - 10*a^3*log(abs(x))/b^6 - 1/2*(60*a^4*b*x^(4/3) + 90*a^3*b^2*x + 20*a^2*b^
3*x^(2/3) - 5*a*b^4*x^(1/3) + 2*b^5)/((a*x^(1/3) + b)^2*b^6*x)