3.2411 \(\int \frac{(a+\frac{b}{\sqrt [3]{x}})^2}{x^4} \, dx\)

Optimal. Leaf size=34 \[ -\frac{a^2}{3 x^3}-\frac{3 a b}{5 x^{10/3}}-\frac{3 b^2}{11 x^{11/3}} \]

[Out]

(-3*b^2)/(11*x^(11/3)) - (3*a*b)/(5*x^(10/3)) - a^2/(3*x^3)

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Rubi [A]  time = 0.0159659, antiderivative size = 34, 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} \[ -\frac{a^2}{3 x^3}-\frac{3 a b}{5 x^{10/3}}-\frac{3 b^2}{11 x^{11/3}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^(1/3))^2/x^4,x]

[Out]

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

Mathematica [A]  time = 0.0223455, size = 34, normalized size = 1. \[ -\frac{a^2}{3 x^3}-\frac{3 a b}{5 x^{10/3}}-\frac{3 b^2}{11 x^{11/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^(1/3))^2/x^4,x]

[Out]

(-3*b^2)/(11*x^(11/3)) - (3*a*b)/(5*x^(10/3)) - a^2/(3*x^3)

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Maple [A]  time = 0.006, size = 25, normalized size = 0.7 \begin{align*} -{\frac{3\,{b}^{2}}{11}{x}^{-{\frac{11}{3}}}}-{\frac{3\,ab}{5}{x}^{-{\frac{10}{3}}}}-{\frac{{a}^{2}}{3\,{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x^(1/3))^2/x^4,x)

[Out]

-3/11*b^2/x^(11/3)-3/5*a*b/x^(10/3)-1/3*a^2/x^3

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Maxima [B]  time = 1.02447, size = 201, normalized size = 5.91 \begin{align*} -\frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{11}}{11 \, b^{9}} + \frac{12 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{10} a}{5 \, b^{9}} - \frac{28 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{9} a^{2}}{3 \, b^{9}} + \frac{21 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{8} a^{3}}{b^{9}} - \frac{30 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{7} a^{4}}{b^{9}} + \frac{28 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{6} a^{5}}{b^{9}} - \frac{84 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{5} a^{6}}{5 \, b^{9}} + \frac{6 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{4} a^{7}}{b^{9}} - \frac{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{3} a^{8}}{b^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^(1/3))^2/x^4,x, algorithm="maxima")

[Out]

-3/11*(a + b/x^(1/3))^11/b^9 + 12/5*(a + b/x^(1/3))^10*a/b^9 - 28/3*(a + b/x^(1/3))^9*a^2/b^9 + 21*(a + b/x^(1
/3))^8*a^3/b^9 - 30*(a + b/x^(1/3))^7*a^4/b^9 + 28*(a + b/x^(1/3))^6*a^5/b^9 - 84/5*(a + b/x^(1/3))^5*a^6/b^9
+ 6*(a + b/x^(1/3))^4*a^7/b^9 - (a + b/x^(1/3))^3*a^8/b^9

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Fricas [A]  time = 1.42835, size = 77, normalized size = 2.26 \begin{align*} -\frac{55 \, a^{2} x + 99 \, a b x^{\frac{2}{3}} + 45 \, b^{2} x^{\frac{1}{3}}}{165 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^(1/3))^2/x^4,x, algorithm="fricas")

[Out]

-1/165*(55*a^2*x + 99*a*b*x^(2/3) + 45*b^2*x^(1/3))/x^4

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Sympy [A]  time = 2.95926, size = 32, normalized size = 0.94 \begin{align*} - \frac{a^{2}}{3 x^{3}} - \frac{3 a b}{5 x^{\frac{10}{3}}} - \frac{3 b^{2}}{11 x^{\frac{11}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**(1/3))**2/x**4,x)

[Out]

-a**2/(3*x**3) - 3*a*b/(5*x**(10/3)) - 3*b**2/(11*x**(11/3))

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Giac [A]  time = 1.20799, size = 35, normalized size = 1.03 \begin{align*} -\frac{55 \, a^{2} x^{\frac{2}{3}} + 99 \, a b x^{\frac{1}{3}} + 45 \, b^{2}}{165 \, x^{\frac{11}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^(1/3))^2/x^4,x, algorithm="giac")

[Out]

-1/165*(55*a^2*x^(2/3) + 99*a*b*x^(1/3) + 45*b^2)/x^(11/3)