3.2308 \(\int \frac{(a+b \sqrt [3]{x})^3}{x} \, dx\)

Optimal. Leaf size=36 \[ 9 a^2 b \sqrt [3]{x}+a^3 \log (x)+\frac{9}{2} a b^2 x^{2/3}+b^3 x \]

[Out]

9*a^2*b*x^(1/3) + (9*a*b^2*x^(2/3))/2 + b^3*x + a^3*Log[x]

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Rubi [A]  time = 0.0167985, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ 9 a^2 b \sqrt [3]{x}+a^3 \log (x)+\frac{9}{2} a b^2 x^{2/3}+b^3 x \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^3/x,x]

[Out]

9*a^2*b*x^(1/3) + (9*a*b^2*x^(2/3))/2 + b^3*x + a^3*Log[x]

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

Mathematica [A]  time = 0.0128372, size = 36, normalized size = 1. \[ 9 a^2 b \sqrt [3]{x}+a^3 \log (x)+\frac{9}{2} a b^2 x^{2/3}+b^3 x \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^3/x,x]

[Out]

9*a^2*b*x^(1/3) + (9*a*b^2*x^(2/3))/2 + b^3*x + a^3*Log[x]

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Maple [A]  time = 0.004, size = 31, normalized size = 0.9 \begin{align*} 9\,{a}^{2}b\sqrt [3]{x}+{\frac{9\,{b}^{2}a}{2}{x}^{{\frac{2}{3}}}}+{b}^{3}x+{a}^{3}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9*a^2*b*x^(1/3)+9/2*a*b^2*x^(2/3)+b^3*x+a^3*ln(x)

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Maxima [A]  time = 0.990877, size = 41, normalized size = 1.14 \begin{align*} b^{3} x + a^{3} \log \left (x\right ) + \frac{9}{2} \, a b^{2} x^{\frac{2}{3}} + 9 \, a^{2} b x^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^3*x + a^3*log(x) + 9/2*a*b^2*x^(2/3) + 9*a^2*b*x^(1/3)

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Fricas [A]  time = 1.53227, size = 89, normalized size = 2.47 \begin{align*} b^{3} x + 3 \, a^{3} \log \left (x^{\frac{1}{3}}\right ) + \frac{9}{2} \, a b^{2} x^{\frac{2}{3}} + 9 \, a^{2} b x^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^3*x + 3*a^3*log(x^(1/3)) + 9/2*a*b^2*x^(2/3) + 9*a^2*b*x^(1/3)

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Sympy [A]  time = 0.29268, size = 36, normalized size = 1. \begin{align*} a^{3} \log{\left (x \right )} + 9 a^{2} b \sqrt [3]{x} + \frac{9 a b^{2} x^{\frac{2}{3}}}{2} + b^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*log(x) + 9*a**2*b*x**(1/3) + 9*a*b**2*x**(2/3)/2 + b**3*x

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Giac [A]  time = 1.12399, size = 42, normalized size = 1.17 \begin{align*} b^{3} x + a^{3} \log \left ({\left | x \right |}\right ) + \frac{9}{2} \, a b^{2} x^{\frac{2}{3}} + 9 \, a^{2} b x^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^3*x + a^3*log(abs(x)) + 9/2*a*b^2*x^(2/3) + 9*a^2*b*x^(1/3)