3.2408 \(\int \frac{(a+\frac{b}{\sqrt [3]{x}})^2}{x} \, dx\)

Optimal. Leaf size=28 \[ a^2 \log (x)-\frac{6 a b}{\sqrt [3]{x}}-\frac{3 b^2}{2 x^{2/3}} \]

[Out]

(-3*b^2)/(2*x^(2/3)) - (6*a*b)/x^(1/3) + a^2*Log[x]

________________________________________________________________________________________

Rubi [A]  time = 0.0154615, antiderivative size = 28, 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} \[ a^2 \log (x)-\frac{6 a b}{\sqrt [3]{x}}-\frac{3 b^2}{2 x^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^(1/3))^2/x,x]

[Out]

(-3*b^2)/(2*x^(2/3)) - (6*a*b)/x^(1/3) + a^2*Log[x]

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

Mathematica [A]  time = 0.0183545, size = 28, normalized size = 1. \[ a^2 \log (x)-\frac{6 a b}{\sqrt [3]{x}}-\frac{3 b^2}{2 x^{2/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^(1/3))^2/x,x]

[Out]

(-3*b^2)/(2*x^(2/3)) - (6*a*b)/x^(1/3) + a^2*Log[x]

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 23, normalized size = 0.8 \begin{align*} -{\frac{3\,{b}^{2}}{2}{x}^{-{\frac{2}{3}}}}-6\,{\frac{ab}{\sqrt [3]{x}}}+{a}^{2}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.988304, size = 30, normalized size = 1.07 \begin{align*} a^{2} \log \left (x\right ) - \frac{6 \, a b}{x^{\frac{1}{3}}} - \frac{3 \, b^{2}}{2 \, x^{\frac{2}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.44463, size = 81, normalized size = 2.89 \begin{align*} \frac{3 \,{\left (2 \, a^{2} x \log \left (x^{\frac{1}{3}}\right ) - 4 \, a b x^{\frac{2}{3}} - b^{2} x^{\frac{1}{3}}\right )}}{2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.477558, size = 27, normalized size = 0.96 \begin{align*} a^{2} \log{\left (x \right )} - \frac{6 a b}{\sqrt [3]{x}} - \frac{3 b^{2}}{2 x^{\frac{2}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*log(x) - 6*a*b/x**(1/3) - 3*b**2/(2*x**(2/3))

________________________________________________________________________________________

Giac [A]  time = 1.19777, size = 32, normalized size = 1.14 \begin{align*} a^{2} \log \left ({\left | x \right |}\right ) - \frac{3 \,{\left (4 \, a b x^{\frac{1}{3}} + b^{2}\right )}}{2 \, x^{\frac{2}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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