3.2430 \(\int \frac{x}{(a+\frac{b}{\sqrt [3]{x}})^2} \, dx\)

Optimal. Leaf size=113 \[ \frac{15 b^4 x^{2/3}}{2 a^6}+\frac{9 b^2 x^{4/3}}{4 a^4}+\frac{3 b^7}{a^8 \left (a \sqrt [3]{x}+b\right )}-\frac{18 b^5 \sqrt [3]{x}}{a^7}-\frac{4 b^3 x}{a^5}+\frac{21 b^6 \log \left (a \sqrt [3]{x}+b\right )}{a^8}-\frac{6 b x^{5/3}}{5 a^3}+\frac{x^2}{2 a^2} \]

[Out]

(3*b^7)/(a^8*(b + a*x^(1/3))) - (18*b^5*x^(1/3))/a^7 + (15*b^4*x^(2/3))/(2*a^6) - (4*b^3*x)/a^5 + (9*b^2*x^(4/
3))/(4*a^4) - (6*b*x^(5/3))/(5*a^3) + x^2/(2*a^2) + (21*b^6*Log[b + a*x^(1/3)])/a^8

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Rubi [A]  time = 0.085341, antiderivative size = 113, 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} \[ \frac{15 b^4 x^{2/3}}{2 a^6}+\frac{9 b^2 x^{4/3}}{4 a^4}+\frac{3 b^7}{a^8 \left (a \sqrt [3]{x}+b\right )}-\frac{18 b^5 \sqrt [3]{x}}{a^7}-\frac{4 b^3 x}{a^5}+\frac{21 b^6 \log \left (a \sqrt [3]{x}+b\right )}{a^8}-\frac{6 b x^{5/3}}{5 a^3}+\frac{x^2}{2 a^2} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b/x^(1/3))^2,x]

[Out]

(3*b^7)/(a^8*(b + a*x^(1/3))) - (18*b^5*x^(1/3))/a^7 + (15*b^4*x^(2/3))/(2*a^6) - (4*b^3*x)/a^5 + (9*b^2*x^(4/
3))/(4*a^4) - (6*b*x^(5/3))/(5*a^3) + x^2/(2*a^2) + (21*b^6*Log[b + a*x^(1/3)])/a^8

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}{\left (a+\frac{b}{\sqrt [3]{x}}\right )^2} \, dx &=\int \frac{x^{5/3}}{\left (b+a \sqrt [3]{x}\right )^2} \, dx\\ &=3 \operatorname{Subst}\left (\int \frac{x^7}{(b+a x)^2} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (-\frac{6 b^5}{a^7}+\frac{5 b^4 x}{a^6}-\frac{4 b^3 x^2}{a^5}+\frac{3 b^2 x^3}{a^4}-\frac{2 b x^4}{a^3}+\frac{x^5}{a^2}-\frac{b^7}{a^7 (b+a x)^2}+\frac{7 b^6}{a^7 (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3 b^7}{a^8 \left (b+a \sqrt [3]{x}\right )}-\frac{18 b^5 \sqrt [3]{x}}{a^7}+\frac{15 b^4 x^{2/3}}{2 a^6}-\frac{4 b^3 x}{a^5}+\frac{9 b^2 x^{4/3}}{4 a^4}-\frac{6 b x^{5/3}}{5 a^3}+\frac{x^2}{2 a^2}+\frac{21 b^6 \log \left (b+a \sqrt [3]{x}\right )}{a^8}\\ \end{align*}

Mathematica [A]  time = 0.134864, size = 111, normalized size = 0.98 \[ \frac{a \left (45 a^3 b^2 x^{4/3}-80 a^2 b^3 x-24 a^4 b x^{5/3}+10 a^5 x^2+150 a b^4 x^{2/3}-\frac{60 b^6}{a+\frac{b}{\sqrt [3]{x}}}-360 b^5 \sqrt [3]{x}\right )+420 b^6 \log \left (a+\frac{b}{\sqrt [3]{x}}\right )+140 b^6 \log (x)}{20 a^8} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b/x^(1/3))^2,x]

[Out]

(a*((-60*b^6)/(a + b/x^(1/3)) - 360*b^5*x^(1/3) + 150*a*b^4*x^(2/3) - 80*a^2*b^3*x + 45*a^3*b^2*x^(4/3) - 24*a
^4*b*x^(5/3) + 10*a^5*x^2) + 420*b^6*Log[a + b/x^(1/3)] + 140*b^6*Log[x])/(20*a^8)

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Maple [A]  time = 0.007, size = 94, normalized size = 0.8 \begin{align*} 3\,{\frac{{b}^{7}}{{a}^{8} \left ( b+a\sqrt [3]{x} \right ) }}-18\,{\frac{{b}^{5}\sqrt [3]{x}}{{a}^{7}}}+{\frac{15\,{b}^{4}}{2\,{a}^{6}}{x}^{{\frac{2}{3}}}}-4\,{\frac{{b}^{3}x}{{a}^{5}}}+{\frac{9\,{b}^{2}}{4\,{a}^{4}}{x}^{{\frac{4}{3}}}}-{\frac{6\,b}{5\,{a}^{3}}{x}^{{\frac{5}{3}}}}+{\frac{{x}^{2}}{2\,{a}^{2}}}+21\,{\frac{{b}^{6}\ln \left ( b+a\sqrt [3]{x} \right ) }{{a}^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*b^7/a^8/(b+a*x^(1/3))-18*b^5*x^(1/3)/a^7+15/2*b^4*x^(2/3)/a^6-4*b^3*x/a^5+9/4*b^2*x^(4/3)/a^4-6/5*b*x^(5/3)/
a^3+1/2*x^2/a^2+21*b^6*ln(b+a*x^(1/3))/a^8

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Maxima [A]  time = 0.96935, size = 151, normalized size = 1.34 \begin{align*} \frac{10 \, a^{6} - \frac{14 \, a^{5} b}{x^{\frac{1}{3}}} + \frac{21 \, a^{4} b^{2}}{x^{\frac{2}{3}}} - \frac{35 \, a^{3} b^{3}}{x} + \frac{70 \, a^{2} b^{4}}{x^{\frac{4}{3}}} - \frac{210 \, a b^{5}}{x^{\frac{5}{3}}} - \frac{420 \, b^{6}}{x^{2}}}{20 \,{\left (\frac{a^{8}}{x^{2}} + \frac{a^{7} b}{x^{\frac{7}{3}}}\right )}} + \frac{21 \, b^{6} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{a^{8}} + \frac{7 \, b^{6} \log \left (x\right )}{a^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(10*a^6 - 14*a^5*b/x^(1/3) + 21*a^4*b^2/x^(2/3) - 35*a^3*b^3/x + 70*a^2*b^4/x^(4/3) - 210*a*b^5/x^(5/3) -
420*b^6/x^2)/(a^8/x^2 + a^7*b/x^(7/3)) + 21*b^6*log(a + b/x^(1/3))/a^8 + 7*b^6*log(x)/a^8

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Fricas [A]  time = 1.58512, size = 312, normalized size = 2.76 \begin{align*} \frac{10 \, a^{9} x^{3} - 70 \, a^{6} b^{3} x^{2} - 80 \, a^{3} b^{6} x + 60 \, b^{9} + 420 \,{\left (a^{3} b^{6} x + b^{9}\right )} \log \left (a x^{\frac{1}{3}} + b\right ) - 6 \,{\left (4 \, a^{8} b x^{2} - 21 \, a^{5} b^{4} x - 35 \, a^{2} b^{7}\right )} x^{\frac{2}{3}} + 15 \,{\left (3 \, a^{7} b^{2} x^{2} - 21 \, a^{4} b^{5} x - 28 \, a b^{8}\right )} x^{\frac{1}{3}}}{20 \,{\left (a^{11} x + a^{8} b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(10*a^9*x^3 - 70*a^6*b^3*x^2 - 80*a^3*b^6*x + 60*b^9 + 420*(a^3*b^6*x + b^9)*log(a*x^(1/3) + b) - 6*(4*a^
8*b*x^2 - 21*a^5*b^4*x - 35*a^2*b^7)*x^(2/3) + 15*(3*a^7*b^2*x^2 - 21*a^4*b^5*x - 28*a*b^8)*x^(1/3))/(a^11*x +
a^8*b^3)

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Sympy [A]  time = 2.03926, size = 277, normalized size = 2.45 \begin{align*} \begin{cases} \frac{10 a^{7} x^{\frac{7}{3}}}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} - \frac{14 a^{6} b x^{2}}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} + \frac{21 a^{5} b^{2} x^{\frac{5}{3}}}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} - \frac{35 a^{4} b^{3} x^{\frac{4}{3}}}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} + \frac{70 a^{3} b^{4} x}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} - \frac{210 a^{2} b^{5} x^{\frac{2}{3}}}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} + \frac{420 a b^{6} \sqrt [3]{x} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} + \frac{420 b^{7} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} + \frac{420 b^{7}}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} & \text{for}\: a \neq 0 \\\frac{3 x^{\frac{8}{3}}}{8 b^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((10*a**7*x**(7/3)/(20*a**9*x**(1/3) + 20*a**8*b) - 14*a**6*b*x**2/(20*a**9*x**(1/3) + 20*a**8*b) + 2
1*a**5*b**2*x**(5/3)/(20*a**9*x**(1/3) + 20*a**8*b) - 35*a**4*b**3*x**(4/3)/(20*a**9*x**(1/3) + 20*a**8*b) + 7
0*a**3*b**4*x/(20*a**9*x**(1/3) + 20*a**8*b) - 210*a**2*b**5*x**(2/3)/(20*a**9*x**(1/3) + 20*a**8*b) + 420*a*b
**6*x**(1/3)*log(x**(1/3) + b/a)/(20*a**9*x**(1/3) + 20*a**8*b) + 420*b**7*log(x**(1/3) + b/a)/(20*a**9*x**(1/
3) + 20*a**8*b) + 420*b**7/(20*a**9*x**(1/3) + 20*a**8*b), Ne(a, 0)), (3*x**(8/3)/(8*b**2), True))

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Giac [A]  time = 1.14153, size = 135, normalized size = 1.19 \begin{align*} \frac{21 \, b^{6} \log \left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{a^{8}} + \frac{3 \, b^{7}}{{\left (a x^{\frac{1}{3}} + b\right )} a^{8}} + \frac{10 \, a^{10} x^{2} - 24 \, a^{9} b x^{\frac{5}{3}} + 45 \, a^{8} b^{2} x^{\frac{4}{3}} - 80 \, a^{7} b^{3} x + 150 \, a^{6} b^{4} x^{\frac{2}{3}} - 360 \, a^{5} b^{5} x^{\frac{1}{3}}}{20 \, a^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

21*b^6*log(abs(a*x^(1/3) + b))/a^8 + 3*b^7/((a*x^(1/3) + b)*a^8) + 1/20*(10*a^10*x^2 - 24*a^9*b*x^(5/3) + 45*a
^8*b^2*x^(4/3) - 80*a^7*b^3*x + 150*a^6*b^4*x^(2/3) - 360*a^5*b^5*x^(1/3))/a^12