3.2422 \(\int \frac{x}{a+\frac{b}{\sqrt [3]{x}}} \, dx\)

Optimal. Leaf size=94 \[ \frac{3 b^4 x^{2/3}}{2 a^5}+\frac{3 b^2 x^{4/3}}{4 a^3}-\frac{3 b^5 \sqrt [3]{x}}{a^6}-\frac{b^3 x}{a^4}+\frac{3 b^6 \log \left (a \sqrt [3]{x}+b\right )}{a^7}-\frac{3 b x^{5/3}}{5 a^2}+\frac{x^2}{2 a} \]

[Out]

(-3*b^5*x^(1/3))/a^6 + (3*b^4*x^(2/3))/(2*a^5) - (b^3*x)/a^4 + (3*b^2*x^(4/3))/(4*a^3) - (3*b*x^(5/3))/(5*a^2)
+ x^2/(2*a) + (3*b^6*Log[b + a*x^(1/3)])/a^7

________________________________________________________________________________________

Rubi [A]  time = 0.0535049, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {263, 266, 43} \[ \frac{3 b^4 x^{2/3}}{2 a^5}+\frac{3 b^2 x^{4/3}}{4 a^3}-\frac{3 b^5 \sqrt [3]{x}}{a^6}-\frac{b^3 x}{a^4}+\frac{3 b^6 \log \left (a \sqrt [3]{x}+b\right )}{a^7}-\frac{3 b x^{5/3}}{5 a^2}+\frac{x^2}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b/x^(1/3)),x]

[Out]

(-3*b^5*x^(1/3))/a^6 + (3*b^4*x^(2/3))/(2*a^5) - (b^3*x)/a^4 + (3*b^2*x^(4/3))/(4*a^3) - (3*b*x^(5/3))/(5*a^2)
+ x^2/(2*a) + (3*b^6*Log[b + a*x^(1/3)])/a^7

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{x}{a+\frac{b}{\sqrt [3]{x}}} \, dx &=\int \frac{x^{4/3}}{b+a \sqrt [3]{x}} \, dx\\ &=3 \operatorname{Subst}\left (\int \frac{x^6}{b+a x} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (-\frac{b^5}{a^6}+\frac{b^4 x}{a^5}-\frac{b^3 x^2}{a^4}+\frac{b^2 x^3}{a^3}-\frac{b x^4}{a^2}+\frac{x^5}{a}+\frac{b^6}{a^6 (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{3 b^5 \sqrt [3]{x}}{a^6}+\frac{3 b^4 x^{2/3}}{2 a^5}-\frac{b^3 x}{a^4}+\frac{3 b^2 x^{4/3}}{4 a^3}-\frac{3 b x^{5/3}}{5 a^2}+\frac{x^2}{2 a}+\frac{3 b^6 \log \left (b+a \sqrt [3]{x}\right )}{a^7}\\ \end{align*}

Mathematica [A]  time = 0.0387983, size = 95, normalized size = 1.01 \[ \frac{15 a^4 b^2 x^{4/3}+30 a^2 b^4 x^{2/3}-20 a^3 b^3 x-12 a^5 b x^{5/3}+10 a^6 x^2-60 a b^5 \sqrt [3]{x}+60 b^6 \log \left (a+\frac{b}{\sqrt [3]{x}}\right )+20 b^6 \log (x)}{20 a^7} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b/x^(1/3)),x]

[Out]

(-60*a*b^5*x^(1/3) + 30*a^2*b^4*x^(2/3) - 20*a^3*b^3*x + 15*a^4*b^2*x^(4/3) - 12*a^5*b*x^(5/3) + 10*a^6*x^2 +
60*b^6*Log[a + b/x^(1/3)] + 20*b^6*Log[x])/(20*a^7)

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 77, normalized size = 0.8 \begin{align*} -3\,{\frac{{b}^{5}\sqrt [3]{x}}{{a}^{6}}}+{\frac{3\,{b}^{4}}{2\,{a}^{5}}{x}^{{\frac{2}{3}}}}-{\frac{{b}^{3}x}{{a}^{4}}}+{\frac{3\,{b}^{2}}{4\,{a}^{3}}{x}^{{\frac{4}{3}}}}-{\frac{3\,b}{5\,{a}^{2}}{x}^{{\frac{5}{3}}}}+{\frac{{x}^{2}}{2\,a}}+3\,{\frac{{b}^{6}\ln \left ( b+a\sqrt [3]{x} \right ) }{{a}^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.982674, size = 119, normalized size = 1.27 \begin{align*} \frac{3 \, b^{6} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{a^{7}} + \frac{b^{6} \log \left (x\right )}{a^{7}} + \frac{{\left (10 \, a^{5} - \frac{12 \, a^{4} b}{x^{\frac{1}{3}}} + \frac{15 \, a^{3} b^{2}}{x^{\frac{2}{3}}} - \frac{20 \, a^{2} b^{3}}{x} + \frac{30 \, a b^{4}}{x^{\frac{4}{3}}} - \frac{60 \, b^{5}}{x^{\frac{5}{3}}}\right )} x^{2}}{20 \, a^{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*b^6*log(a + b/x^(1/3))/a^7 + b^6*log(x)/a^7 + 1/20*(10*a^5 - 12*a^4*b/x^(1/3) + 15*a^3*b^2/x^(2/3) - 20*a^2*
b^3/x + 30*a*b^4/x^(4/3) - 60*b^5/x^(5/3))*x^2/a^6

________________________________________________________________________________________

Fricas [A]  time = 1.43823, size = 185, normalized size = 1.97 \begin{align*} \frac{10 \, a^{6} x^{2} - 20 \, a^{3} b^{3} x + 60 \, b^{6} \log \left (a x^{\frac{1}{3}} + b\right ) - 6 \,{\left (2 \, a^{5} b x - 5 \, a^{2} b^{4}\right )} x^{\frac{2}{3}} + 15 \,{\left (a^{4} b^{2} x - 4 \, a b^{5}\right )} x^{\frac{1}{3}}}{20 \, a^{7}} \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*(10*a^6*x^2 - 20*a^3*b^3*x + 60*b^6*log(a*x^(1/3) + b) - 6*(2*a^5*b*x - 5*a^2*b^4)*x^(2/3) + 15*(a^4*b^2*
x - 4*a*b^5)*x^(1/3))/a^7

________________________________________________________________________________________

Sympy [A]  time = 1.56533, size = 100, normalized size = 1.06 \begin{align*} \begin{cases} \frac{x^{2}}{2 a} - \frac{3 b x^{\frac{5}{3}}}{5 a^{2}} + \frac{3 b^{2} x^{\frac{4}{3}}}{4 a^{3}} - \frac{b^{3} x}{a^{4}} + \frac{3 b^{4} x^{\frac{2}{3}}}{2 a^{5}} - \frac{3 b^{5} \sqrt [3]{x}}{a^{6}} + \frac{3 b^{6} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{a^{7}} & \text{for}\: a \neq 0 \\\frac{3 x^{\frac{7}{3}}}{7 b} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**2/(2*a) - 3*b*x**(5/3)/(5*a**2) + 3*b**2*x**(4/3)/(4*a**3) - b**3*x/a**4 + 3*b**4*x**(2/3)/(2*a*
*5) - 3*b**5*x**(1/3)/a**6 + 3*b**6*log(x**(1/3) + b/a)/a**7, Ne(a, 0)), (3*x**(7/3)/(7*b), True))

________________________________________________________________________________________

Giac [A]  time = 1.22304, size = 105, normalized size = 1.12 \begin{align*} \frac{3 \, b^{6} \log \left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{a^{7}} + \frac{10 \, a^{5} x^{2} - 12 \, a^{4} b x^{\frac{5}{3}} + 15 \, a^{3} b^{2} x^{\frac{4}{3}} - 20 \, a^{2} b^{3} x + 30 \, a b^{4} x^{\frac{2}{3}} - 60 \, b^{5} x^{\frac{1}{3}}}{20 \, a^{6}} \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*b^6*log(abs(a*x^(1/3) + b))/a^7 + 1/20*(10*a^5*x^2 - 12*a^4*b*x^(5/3) + 15*a^3*b^2*x^(4/3) - 20*a^2*b^3*x +
30*a*b^4*x^(2/3) - 60*b^5*x^(1/3))/a^6