3.2409 \(\int \frac {(a+\frac {b}{\sqrt [3]{x}})^2}{x^2} \, dx\)

Optimal. Leaf size=32 \[ -\frac {a^2}{x}-\frac {3 a b}{2 x^{4/3}}-\frac {3 b^2}{5 x^{5/3}} \]

[Out]

-3/5*b^2/x^(5/3)-3/2*a*b/x^(4/3)-a^2/x

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Rubi [A]  time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {263, 266, 43} \[ -\frac {a^2}{x}-\frac {3 a b}{2 x^{4/3}}-\frac {3 b^2}{5 x^{5/3}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^(1/3))^2/x^2,x]

[Out]

(-3*b^2)/(5*x^(5/3)) - (3*a*b)/(2*x^(4/3)) - a^2/x

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])

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]]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 32, normalized size = 1.00 \[ -\frac {a^2}{x}-\frac {3 a b}{2 x^{4/3}}-\frac {3 b^2}{5 x^{5/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^(1/3))^2/x^2,x]

[Out]

(-3*b^2)/(5*x^(5/3)) - (3*a*b)/(2*x^(4/3)) - a^2/x

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fricas [A]  time = 0.67, size = 27, normalized size = 0.84 \[ -\frac {10 \, a^{2} x + 15 \, a b x^{\frac {2}{3}} + 6 \, b^{2} x^{\frac {1}{3}}}{10 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.17, size = 26, normalized size = 0.81 \[ -\frac {10 \, a^{2} x^{\frac {2}{3}} + 15 \, a b x^{\frac {1}{3}} + 6 \, b^{2}}{10 \, x^{\frac {5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 25, normalized size = 0.78 \[ -\frac {a^{2}}{x}-\frac {3 a b}{2 x^{\frac {4}{3}}}-\frac {3 b^{2}}{5 x^{\frac {5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/5*b^2/x^(5/3)-3/2*a*b/x^(4/3)-a^2/x

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maxima [A]  time = 0.47, size = 47, normalized size = 1.47 \[ -\frac {3 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{5}}{5 \, b^{3}} + \frac {3 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{4} a}{2 \, b^{3}} - \frac {{\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{3} a^{2}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.05, size = 24, normalized size = 0.75 \[ -\frac {a^2}{x}-\frac {3\,b^2}{5\,x^{5/3}}-\frac {3\,a\,b}{2\,x^{4/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x^(1/3))^2/x^2,x)

[Out]

- a^2/x - (3*b^2)/(5*x^(5/3)) - (3*a*b)/(2*x^(4/3))

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sympy [A]  time = 0.96, size = 29, normalized size = 0.91 \[ - \frac {a^{2}}{x} - \frac {3 a b}{2 x^{\frac {4}{3}}} - \frac {3 b^{2}}{5 x^{\frac {5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**2/x - 3*a*b/(2*x**(4/3)) - 3*b**2/(5*x**(5/3))

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