3.2420 \(\int \frac{(a+\frac{b}{\sqrt [3]{x}})^3}{x^4} \, dx\)

Optimal. Leaf size=47 \[ -\frac{9 a^2 b}{10 x^{10/3}}-\frac{a^3}{3 x^3}-\frac{9 a b^2}{11 x^{11/3}}-\frac{b^3}{4 x^4} \]

[Out]

-b^3/(4*x^4) - (9*a*b^2)/(11*x^(11/3)) - (9*a^2*b)/(10*x^(10/3)) - a^3/(3*x^3)

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Rubi [A]  time = 0.0214049, antiderivative size = 47, 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, 43} \[ -\frac{9 a^2 b}{10 x^{10/3}}-\frac{a^3}{3 x^3}-\frac{9 a b^2}{11 x^{11/3}}-\frac{b^3}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^(1/3))^3/x^4,x]

[Out]

-b^3/(4*x^4) - (9*a*b^2)/(11*x^(11/3)) - (9*a^2*b)/(10*x^(10/3)) - a^3/(3*x^3)

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 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\left (a+\frac{b}{\sqrt [3]{x}}\right )^3}{x^4} \, dx &=\int \frac{\left (b+a \sqrt [3]{x}\right )^3}{x^5} \, dx\\ &=3 \operatorname{Subst}\left (\int \frac{(b+a x)^3}{x^{13}} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{b^3}{x^{13}}+\frac{3 a b^2}{x^{12}}+\frac{3 a^2 b}{x^{11}}+\frac{a^3}{x^{10}}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{b^3}{4 x^4}-\frac{9 a b^2}{11 x^{11/3}}-\frac{9 a^2 b}{10 x^{10/3}}-\frac{a^3}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0161741, size = 41, normalized size = 0.87 \[ -\frac{594 a^2 b x^{2/3}+220 a^3 x+540 a b^2 \sqrt [3]{x}+165 b^3}{660 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^(1/3))^3/x^4,x]

[Out]

-(165*b^3 + 540*a*b^2*x^(1/3) + 594*a^2*b*x^(2/3) + 220*a^3*x)/(660*x^4)

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Maple [A]  time = 0.004, size = 36, normalized size = 0.8 \begin{align*} -{\frac{{b}^{3}}{4\,{x}^{4}}}-{\frac{9\,{b}^{2}a}{11}{x}^{-{\frac{11}{3}}}}-{\frac{9\,b{a}^{2}}{10}{x}^{-{\frac{10}{3}}}}-{\frac{{a}^{3}}{3\,{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x^(1/3))^3/x^4,x)

[Out]

-1/4*b^3/x^4-9/11*a*b^2/x^(11/3)-9/10*a^2*b/x^(10/3)-1/3*a^3/x^3

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Maxima [B]  time = 0.956527, size = 201, normalized size = 4.28 \begin{align*} -\frac{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{12}}{4 \, b^{9}} + \frac{24 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{11} a}{11 \, b^{9}} - \frac{42 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{10} a^{2}}{5 \, b^{9}} + \frac{56 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{9} a^{3}}{3 \, b^{9}} - \frac{105 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{8} a^{4}}{4 \, b^{9}} + \frac{24 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{7} a^{5}}{b^{9}} - \frac{14 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{6} a^{6}}{b^{9}} + \frac{24 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{5} a^{7}}{5 \, b^{9}} - \frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{4} a^{8}}{4 \, b^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(a + b/x^(1/3))^12/b^9 + 24/11*(a + b/x^(1/3))^11*a/b^9 - 42/5*(a + b/x^(1/3))^10*a^2/b^9 + 56/3*(a + b/x
^(1/3))^9*a^3/b^9 - 105/4*(a + b/x^(1/3))^8*a^4/b^9 + 24*(a + b/x^(1/3))^7*a^5/b^9 - 14*(a + b/x^(1/3))^6*a^6/
b^9 + 24/5*(a + b/x^(1/3))^5*a^7/b^9 - 3/4*(a + b/x^(1/3))^4*a^8/b^9

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Fricas [A]  time = 1.59053, size = 100, normalized size = 2.13 \begin{align*} -\frac{220 \, a^{3} x + 594 \, a^{2} b x^{\frac{2}{3}} + 540 \, a b^{2} x^{\frac{1}{3}} + 165 \, b^{3}}{660 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/660*(220*a^3*x + 594*a^2*b*x^(2/3) + 540*a*b^2*x^(1/3) + 165*b^3)/x^4

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Sympy [A]  time = 5.37162, size = 44, normalized size = 0.94 \begin{align*} - \frac{a^{3}}{3 x^{3}} - \frac{9 a^{2} b}{10 x^{\frac{10}{3}}} - \frac{9 a b^{2}}{11 x^{\frac{11}{3}}} - \frac{b^{3}}{4 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**(1/3))**3/x**4,x)

[Out]

-a**3/(3*x**3) - 9*a**2*b/(10*x**(10/3)) - 9*a*b**2/(11*x**(11/3)) - b**3/(4*x**4)

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Giac [A]  time = 1.14757, size = 47, normalized size = 1. \begin{align*} -\frac{220 \, a^{3} x + 594 \, a^{2} b x^{\frac{2}{3}} + 540 \, a b^{2} x^{\frac{1}{3}} + 165 \, b^{3}}{660 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/660*(220*a^3*x + 594*a^2*b*x^(2/3) + 540*a*b^2*x^(1/3) + 165*b^3)/x^4