3.2358 \(\int \frac{x}{a+b \sqrt [3]{x}} \, dx\)

Optimal. Leaf size=80 \[ -\frac{3 a^3 x^{2/3}}{2 b^4}+\frac{3 a^4 \sqrt [3]{x}}{b^5}+\frac{a^2 x}{b^3}-\frac{3 a^5 \log \left (a+b \sqrt [3]{x}\right )}{b^6}-\frac{3 a x^{4/3}}{4 b^2}+\frac{3 x^{5/3}}{5 b} \]

[Out]

(3*a^4*x^(1/3))/b^5 - (3*a^3*x^(2/3))/(2*b^4) + (a^2*x)/b^3 - (3*a*x^(4/3))/(4*b^2) + (3*x^(5/3))/(5*b) - (3*a
^5*Log[a + b*x^(1/3)])/b^6

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Rubi [A]  time = 0.0460705, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ -\frac{3 a^3 x^{2/3}}{2 b^4}+\frac{3 a^4 \sqrt [3]{x}}{b^5}+\frac{a^2 x}{b^3}-\frac{3 a^5 \log \left (a+b \sqrt [3]{x}\right )}{b^6}-\frac{3 a x^{4/3}}{4 b^2}+\frac{3 x^{5/3}}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x^(1/3)),x]

[Out]

(3*a^4*x^(1/3))/b^5 - (3*a^3*x^(2/3))/(2*b^4) + (a^2*x)/b^3 - (3*a*x^(4/3))/(4*b^2) + (3*x^(5/3))/(5*b) - (3*a
^5*Log[a + b*x^(1/3)])/b^6

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

Mathematica [A]  time = 0.0356736, size = 77, normalized size = 0.96 \[ \frac{-30 a^3 b^2 x^{2/3}+20 a^2 b^3 x+60 a^4 b \sqrt [3]{x}-60 a^5 \log \left (a+b \sqrt [3]{x}\right )-15 a b^4 x^{4/3}+12 b^5 x^{5/3}}{20 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x^(1/3)),x]

[Out]

(60*a^4*b*x^(1/3) - 30*a^3*b^2*x^(2/3) + 20*a^2*b^3*x - 15*a*b^4*x^(4/3) + 12*b^5*x^(5/3) - 60*a^5*Log[a + b*x
^(1/3)])/(20*b^6)

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Maple [A]  time = 0., size = 65, normalized size = 0.8 \begin{align*} 3\,{\frac{{a}^{4}\sqrt [3]{x}}{{b}^{5}}}-{\frac{3\,{a}^{3}}{2\,{b}^{4}}{x}^{{\frac{2}{3}}}}+{\frac{{a}^{2}x}{{b}^{3}}}-{\frac{3\,a}{4\,{b}^{2}}{x}^{{\frac{4}{3}}}}+{\frac{3}{5\,b}{x}^{{\frac{5}{3}}}}-3\,{\frac{{a}^{5}\ln \left ( a+b\sqrt [3]{x} \right ) }{{b}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.972349, size = 128, normalized size = 1.6 \begin{align*} -\frac{3 \, a^{5} \log \left (b x^{\frac{1}{3}} + a\right )}{b^{6}} + \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{5}}{5 \, b^{6}} - \frac{15 \,{\left (b x^{\frac{1}{3}} + a\right )}^{4} a}{4 \, b^{6}} + \frac{10 \,{\left (b x^{\frac{1}{3}} + a\right )}^{3} a^{2}}{b^{6}} - \frac{15 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} a^{3}}{b^{6}} + \frac{15 \,{\left (b x^{\frac{1}{3}} + a\right )} a^{4}}{b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*a^5*log(b*x^(1/3) + a)/b^6 + 3/5*(b*x^(1/3) + a)^5/b^6 - 15/4*(b*x^(1/3) + a)^4*a/b^6 + 10*(b*x^(1/3) + a)^
3*a^2/b^6 - 15*(b*x^(1/3) + a)^2*a^3/b^6 + 15*(b*x^(1/3) + a)*a^4/b^6

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Fricas [A]  time = 1.50553, size = 162, normalized size = 2.02 \begin{align*} \frac{20 \, a^{2} b^{3} x - 60 \, a^{5} \log \left (b x^{\frac{1}{3}} + a\right ) + 6 \,{\left (2 \, b^{5} x - 5 \, a^{3} b^{2}\right )} x^{\frac{2}{3}} - 15 \,{\left (a b^{4} x - 4 \, a^{4} b\right )} x^{\frac{1}{3}}}{20 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(20*a^2*b^3*x - 60*a^5*log(b*x^(1/3) + a) + 6*(2*b^5*x - 5*a^3*b^2)*x^(2/3) - 15*(a*b^4*x - 4*a^4*b)*x^(1
/3))/b^6

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Sympy [A]  time = 9.98872, size = 80, normalized size = 1. \begin{align*} - \frac{3 a^{5} \log{\left (1 + \frac{b \sqrt [3]{x}}{a} \right )}}{b^{6}} + \frac{3 a^{4} \sqrt [3]{x}}{b^{5}} - \frac{3 a^{3} x^{\frac{2}{3}}}{2 b^{4}} + \frac{a^{2} x}{b^{3}} - \frac{3 a x^{\frac{4}{3}}}{4 b^{2}} + \frac{3 x^{\frac{5}{3}}}{5 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*a**5*log(1 + b*x**(1/3)/a)/b**6 + 3*a**4*x**(1/3)/b**5 - 3*a**3*x**(2/3)/(2*b**4) + a**2*x/b**3 - 3*a*x**(4
/3)/(4*b**2) + 3*x**(5/3)/(5*b)

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Giac [A]  time = 1.17946, size = 90, normalized size = 1.12 \begin{align*} -\frac{3 \, a^{5} \log \left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{b^{6}} + \frac{12 \, b^{4} x^{\frac{5}{3}} - 15 \, a b^{3} x^{\frac{4}{3}} + 20 \, a^{2} b^{2} x - 30 \, a^{3} b x^{\frac{2}{3}} + 60 \, a^{4} x^{\frac{1}{3}}}{20 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*a^5*log(abs(b*x^(1/3) + a))/b^6 + 1/20*(12*b^4*x^(5/3) - 15*a*b^3*x^(4/3) + 20*a^2*b^2*x - 30*a^3*b*x^(2/3)
+ 60*a^4*x^(1/3))/b^5