3.2410 \(\int \frac{(a+\frac{b}{\sqrt [3]{x}})^2}{x^3} \, dx\)

Optimal. Leaf size=34 \[ -\frac{a^2}{2 x^2}-\frac{6 a b}{7 x^{7/3}}-\frac{3 b^2}{8 x^{8/3}} \]

[Out]

(-3*b^2)/(8*x^(8/3)) - (6*a*b)/(7*x^(7/3)) - a^2/(2*x^2)

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Rubi [A]  time = 0.0173652, 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} \[ -\frac{a^2}{2 x^2}-\frac{6 a b}{7 x^{7/3}}-\frac{3 b^2}{8 x^{8/3}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^(1/3))^2/x^3,x]

[Out]

(-3*b^2)/(8*x^(8/3)) - (6*a*b)/(7*x^(7/3)) - a^2/(2*x^2)

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^3} \, dx &=\int \frac{\left (b+a \sqrt [3]{x}\right )^2}{x^{11/3}} \, dx\\ &=3 \operatorname{Subst}\left (\int \frac{(b+a x)^2}{x^9} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{b^2}{x^9}+\frac{2 a b}{x^8}+\frac{a^2}{x^7}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{3 b^2}{8 x^{8/3}}-\frac{6 a b}{7 x^{7/3}}-\frac{a^2}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0202941, size = 34, normalized size = 1. \[ -\frac{a^2}{2 x^2}-\frac{6 a b}{7 x^{7/3}}-\frac{3 b^2}{8 x^{8/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^(1/3))^2/x^3,x]

[Out]

(-3*b^2)/(8*x^(8/3)) - (6*a*b)/(7*x^(7/3)) - a^2/(2*x^2)

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Maple [A]  time = 0.006, size = 25, normalized size = 0.7 \begin{align*} -{\frac{3\,{b}^{2}}{8}{x}^{-{\frac{8}{3}}}}-{\frac{6\,ab}{7}{x}^{-{\frac{7}{3}}}}-{\frac{{a}^{2}}{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/8*b^2/x^(8/3)-6/7*a*b/x^(7/3)-1/2/x^2*a^2

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Maxima [B]  time = 0.977857, size = 131, normalized size = 3.85 \begin{align*} -\frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{8}}{8 \, b^{6}} + \frac{15 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{7} a}{7 \, b^{6}} - \frac{5 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{6} a^{2}}{b^{6}} + \frac{6 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{5} a^{3}}{b^{6}} - \frac{15 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{4} a^{4}}{4 \, b^{6}} + \frac{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{3} a^{5}}{b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/8*(a + b/x^(1/3))^8/b^6 + 15/7*(a + b/x^(1/3))^7*a/b^6 - 5*(a + b/x^(1/3))^6*a^2/b^6 + 6*(a + b/x^(1/3))^5*
a^3/b^6 - 15/4*(a + b/x^(1/3))^4*a^4/b^6 + (a + b/x^(1/3))^3*a^5/b^6

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Fricas [A]  time = 1.50993, size = 76, normalized size = 2.24 \begin{align*} -\frac{28 \, a^{2} x + 48 \, a b x^{\frac{2}{3}} + 21 \, b^{2} x^{\frac{1}{3}}}{56 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/56*(28*a^2*x + 48*a*b*x^(2/3) + 21*b^2*x^(1/3))/x^3

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Sympy [A]  time = 1.49017, size = 32, normalized size = 0.94 \begin{align*} - \frac{a^{2}}{2 x^{2}} - \frac{6 a b}{7 x^{\frac{7}{3}}} - \frac{3 b^{2}}{8 x^{\frac{8}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**2/(2*x**2) - 6*a*b/(7*x**(7/3)) - 3*b**2/(8*x**(8/3))

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Giac [A]  time = 1.17628, size = 35, normalized size = 1.03 \begin{align*} -\frac{28 \, a^{2} x^{\frac{2}{3}} + 48 \, a b x^{\frac{1}{3}} + 21 \, b^{2}}{56 \, x^{\frac{8}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/56*(28*a^2*x^(2/3) + 48*a*b*x^(1/3) + 21*b^2)/x^(8/3)