3.2438 \(\int \frac{x}{(a+\frac{b}{\sqrt [3]{x}})^3} \, dx\)

Optimal. Leaf size=134 \[ \frac{45 b^4 x^{2/3}}{2 a^7}+\frac{9 b^2 x^{4/3}}{2 a^5}-\frac{3 b^8}{2 a^9 \left (a \sqrt [3]{x}+b\right )^2}+\frac{24 b^7}{a^9 \left (a \sqrt [3]{x}+b\right )}-\frac{63 b^5 \sqrt [3]{x}}{a^8}-\frac{10 b^3 x}{a^6}+\frac{84 b^6 \log \left (a \sqrt [3]{x}+b\right )}{a^9}-\frac{9 b x^{5/3}}{5 a^4}+\frac{x^2}{2 a^3} \]

[Out]

(-3*b^8)/(2*a^9*(b + a*x^(1/3))^2) + (24*b^7)/(a^9*(b + a*x^(1/3))) - (63*b^5*x^(1/3))/a^8 + (45*b^4*x^(2/3))/
(2*a^7) - (10*b^3*x)/a^6 + (9*b^2*x^(4/3))/(2*a^5) - (9*b*x^(5/3))/(5*a^4) + x^2/(2*a^3) + (84*b^6*Log[b + a*x
^(1/3)])/a^9

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Rubi [A]  time = 0.105557, antiderivative size = 134, 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} \[ \frac{45 b^4 x^{2/3}}{2 a^7}+\frac{9 b^2 x^{4/3}}{2 a^5}-\frac{3 b^8}{2 a^9 \left (a \sqrt [3]{x}+b\right )^2}+\frac{24 b^7}{a^9 \left (a \sqrt [3]{x}+b\right )}-\frac{63 b^5 \sqrt [3]{x}}{a^8}-\frac{10 b^3 x}{a^6}+\frac{84 b^6 \log \left (a \sqrt [3]{x}+b\right )}{a^9}-\frac{9 b x^{5/3}}{5 a^4}+\frac{x^2}{2 a^3} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b/x^(1/3))^3,x]

[Out]

(-3*b^8)/(2*a^9*(b + a*x^(1/3))^2) + (24*b^7)/(a^9*(b + a*x^(1/3))) - (63*b^5*x^(1/3))/a^8 + (45*b^4*x^(2/3))/
(2*a^7) - (10*b^3*x)/a^6 + (9*b^2*x^(4/3))/(2*a^5) - (9*b*x^(5/3))/(5*a^4) + x^2/(2*a^3) + (84*b^6*Log[b + a*x
^(1/3)])/a^9

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 )^3} \, dx &=\int \frac{x^2}{\left (b+a \sqrt [3]{x}\right )^3} \, dx\\ &=3 \operatorname{Subst}\left (\int \frac{x^8}{(b+a x)^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (-\frac{21 b^5}{a^8}+\frac{15 b^4 x}{a^7}-\frac{10 b^3 x^2}{a^6}+\frac{6 b^2 x^3}{a^5}-\frac{3 b x^4}{a^4}+\frac{x^5}{a^3}+\frac{b^8}{a^8 (b+a x)^3}-\frac{8 b^7}{a^8 (b+a x)^2}+\frac{28 b^6}{a^8 (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{3 b^8}{2 a^9 \left (b+a \sqrt [3]{x}\right )^2}+\frac{24 b^7}{a^9 \left (b+a \sqrt [3]{x}\right )}-\frac{63 b^5 \sqrt [3]{x}}{a^8}+\frac{45 b^4 x^{2/3}}{2 a^7}-\frac{10 b^3 x}{a^6}+\frac{9 b^2 x^{4/3}}{2 a^5}-\frac{9 b x^{5/3}}{5 a^4}+\frac{x^2}{2 a^3}+\frac{84 b^6 \log \left (b+a \sqrt [3]{x}\right )}{a^9}\\ \end{align*}

Mathematica [A]  time = 0.152537, size = 132, normalized size = 0.99 \[ \frac{\frac{a \sqrt [3]{x} \left (14 a^5 b^2 x^{5/3}-28 a^4 b^3 x^{4/3}-280 a^2 b^5 x^{2/3}+70 a^3 b^4 x-8 a^6 b x^2+5 a^7 x^{7/3}-1260 a b^6 \sqrt [3]{x}-840 b^7\right )}{\left (a \sqrt [3]{x}+b\right )^2}+840 b^6 \log \left (a+\frac{b}{\sqrt [3]{x}}\right )+280 b^6 \log (x)}{10 a^9} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b/x^(1/3))^3,x]

[Out]

((a*x^(1/3)*(-840*b^7 - 1260*a*b^6*x^(1/3) - 280*a^2*b^5*x^(2/3) + 70*a^3*b^4*x - 28*a^4*b^3*x^(4/3) + 14*a^5*
b^2*x^(5/3) - 8*a^6*b*x^2 + 5*a^7*x^(7/3)))/(b + a*x^(1/3))^2 + 840*b^6*Log[a + b/x^(1/3)] + 280*b^6*Log[x])/(
10*a^9)

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Maple [A]  time = 0.008, size = 111, normalized size = 0.8 \begin{align*} -{\frac{3\,{b}^{8}}{2\,{a}^{9}} \left ( b+a\sqrt [3]{x} \right ) ^{-2}}+24\,{\frac{{b}^{7}}{{a}^{9} \left ( b+a\sqrt [3]{x} \right ) }}-63\,{\frac{{b}^{5}\sqrt [3]{x}}{{a}^{8}}}+{\frac{45\,{b}^{4}}{2\,{a}^{7}}{x}^{{\frac{2}{3}}}}-10\,{\frac{{b}^{3}x}{{a}^{6}}}+{\frac{9\,{b}^{2}}{2\,{a}^{5}}{x}^{{\frac{4}{3}}}}-{\frac{9\,b}{5\,{a}^{4}}{x}^{{\frac{5}{3}}}}+{\frac{{x}^{2}}{2\,{a}^{3}}}+84\,{\frac{{b}^{6}\ln \left ( b+a\sqrt [3]{x} \right ) }{{a}^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*b^8/a^9/(b+a*x^(1/3))^2+24*b^7/a^9/(b+a*x^(1/3))-63*b^5*x^(1/3)/a^8+45/2*b^4*x^(2/3)/a^7-10*b^3*x/a^6+9/2
*b^2*x^(4/3)/a^5-9/5*b*x^(5/3)/a^4+1/2*x^2/a^3+84*b^6*ln(b+a*x^(1/3))/a^9

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Maxima [A]  time = 1.05029, size = 181, normalized size = 1.35 \begin{align*} \frac{5 \, a^{7} - \frac{8 \, a^{6} b}{x^{\frac{1}{3}}} + \frac{14 \, a^{5} b^{2}}{x^{\frac{2}{3}}} - \frac{28 \, a^{4} b^{3}}{x} + \frac{70 \, a^{3} b^{4}}{x^{\frac{4}{3}}} - \frac{280 \, a^{2} b^{5}}{x^{\frac{5}{3}}} - \frac{1260 \, a b^{6}}{x^{2}} - \frac{840 \, b^{7}}{x^{\frac{7}{3}}}}{10 \,{\left (\frac{a^{10}}{x^{2}} + \frac{2 \, a^{9} b}{x^{\frac{7}{3}}} + \frac{a^{8} b^{2}}{x^{\frac{8}{3}}}\right )}} + \frac{84 \, b^{6} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{a^{9}} + \frac{28 \, b^{6} \log \left (x\right )}{a^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(5*a^7 - 8*a^6*b/x^(1/3) + 14*a^5*b^2/x^(2/3) - 28*a^4*b^3/x + 70*a^3*b^4/x^(4/3) - 280*a^2*b^5/x^(5/3) -
1260*a*b^6/x^2 - 840*b^7/x^(7/3))/(a^10/x^2 + 2*a^9*b/x^(7/3) + a^8*b^2/x^(8/3)) + 84*b^6*log(a + b/x^(1/3))/
a^9 + 28*b^6*log(x)/a^9

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Fricas [A]  time = 1.47062, size = 440, normalized size = 3.28 \begin{align*} \frac{5 \, a^{12} x^{4} - 90 \, a^{9} b^{3} x^{3} - 195 \, a^{6} b^{6} x^{2} + 170 \, a^{3} b^{9} x + 225 \, b^{12} + 840 \,{\left (a^{6} b^{6} x^{2} + 2 \, a^{3} b^{9} x + b^{12}\right )} \log \left (a x^{\frac{1}{3}} + b\right ) - 3 \,{\left (6 \, a^{11} b x^{3} - 63 \, a^{8} b^{4} x^{2} - 224 \, a^{5} b^{7} x - 140 \, a^{2} b^{10}\right )} x^{\frac{2}{3}} + 15 \,{\left (3 \, a^{10} b^{2} x^{3} - 36 \, a^{7} b^{5} x^{2} - 98 \, a^{4} b^{8} x - 56 \, a b^{11}\right )} x^{\frac{1}{3}}}{10 \,{\left (a^{15} x^{2} + 2 \, a^{12} b^{3} x + a^{9} b^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(5*a^12*x^4 - 90*a^9*b^3*x^3 - 195*a^6*b^6*x^2 + 170*a^3*b^9*x + 225*b^12 + 840*(a^6*b^6*x^2 + 2*a^3*b^9*
x + b^12)*log(a*x^(1/3) + b) - 3*(6*a^11*b*x^3 - 63*a^8*b^4*x^2 - 224*a^5*b^7*x - 140*a^2*b^10)*x^(2/3) + 15*(
3*a^10*b^2*x^3 - 36*a^7*b^5*x^2 - 98*a^4*b^8*x - 56*a*b^11)*x^(1/3))/(a^15*x^2 + 2*a^12*b^3*x + a^9*b^6)

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Sympy [A]  time = 2.86735, size = 493, normalized size = 3.68 \begin{align*} \begin{cases} \frac{5 a^{8} x^{\frac{8}{3}}}{10 a^{11} x^{\frac{2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} - \frac{8 a^{7} b x^{\frac{7}{3}}}{10 a^{11} x^{\frac{2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} + \frac{14 a^{6} b^{2} x^{2}}{10 a^{11} x^{\frac{2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} - \frac{28 a^{5} b^{3} x^{\frac{5}{3}}}{10 a^{11} x^{\frac{2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} + \frac{70 a^{4} b^{4} x^{\frac{4}{3}}}{10 a^{11} x^{\frac{2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} - \frac{280 a^{3} b^{5} x}{10 a^{11} x^{\frac{2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} + \frac{840 a^{2} b^{6} x^{\frac{2}{3}} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{10 a^{11} x^{\frac{2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} + \frac{1680 a b^{7} \sqrt [3]{x} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{10 a^{11} x^{\frac{2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} + \frac{1680 a b^{7} \sqrt [3]{x}}{10 a^{11} x^{\frac{2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} + \frac{840 b^{8} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{10 a^{11} x^{\frac{2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} + \frac{1260 b^{8}}{10 a^{11} x^{\frac{2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} & \text{for}\: a \neq 0 \\\frac{x^{3}}{3 b^{3}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((5*a**8*x**(8/3)/(10*a**11*x**(2/3) + 20*a**10*b*x**(1/3) + 10*a**9*b**2) - 8*a**7*b*x**(7/3)/(10*a*
*11*x**(2/3) + 20*a**10*b*x**(1/3) + 10*a**9*b**2) + 14*a**6*b**2*x**2/(10*a**11*x**(2/3) + 20*a**10*b*x**(1/3
) + 10*a**9*b**2) - 28*a**5*b**3*x**(5/3)/(10*a**11*x**(2/3) + 20*a**10*b*x**(1/3) + 10*a**9*b**2) + 70*a**4*b
**4*x**(4/3)/(10*a**11*x**(2/3) + 20*a**10*b*x**(1/3) + 10*a**9*b**2) - 280*a**3*b**5*x/(10*a**11*x**(2/3) + 2
0*a**10*b*x**(1/3) + 10*a**9*b**2) + 840*a**2*b**6*x**(2/3)*log(x**(1/3) + b/a)/(10*a**11*x**(2/3) + 20*a**10*
b*x**(1/3) + 10*a**9*b**2) + 1680*a*b**7*x**(1/3)*log(x**(1/3) + b/a)/(10*a**11*x**(2/3) + 20*a**10*b*x**(1/3)
+ 10*a**9*b**2) + 1680*a*b**7*x**(1/3)/(10*a**11*x**(2/3) + 20*a**10*b*x**(1/3) + 10*a**9*b**2) + 840*b**8*lo
g(x**(1/3) + b/a)/(10*a**11*x**(2/3) + 20*a**10*b*x**(1/3) + 10*a**9*b**2) + 1260*b**8/(10*a**11*x**(2/3) + 20
*a**10*b*x**(1/3) + 10*a**9*b**2), Ne(a, 0)), (x**3/(3*b**3), True))

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Giac [A]  time = 1.2136, size = 151, normalized size = 1.13 \begin{align*} \frac{84 \, b^{6} \log \left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{a^{9}} + \frac{3 \,{\left (16 \, a b^{7} x^{\frac{1}{3}} + 15 \, b^{8}\right )}}{2 \,{\left (a x^{\frac{1}{3}} + b\right )}^{2} a^{9}} + \frac{5 \, a^{15} x^{2} - 18 \, a^{14} b x^{\frac{5}{3}} + 45 \, a^{13} b^{2} x^{\frac{4}{3}} - 100 \, a^{12} b^{3} x + 225 \, a^{11} b^{4} x^{\frac{2}{3}} - 630 \, a^{10} b^{5} x^{\frac{1}{3}}}{10 \, a^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

84*b^6*log(abs(a*x^(1/3) + b))/a^9 + 3/2*(16*a*b^7*x^(1/3) + 15*b^8)/((a*x^(1/3) + b)^2*a^9) + 1/10*(5*a^15*x^
2 - 18*a^14*b*x^(5/3) + 45*a^13*b^2*x^(4/3) - 100*a^12*b^3*x + 225*a^11*b^4*x^(2/3) - 630*a^10*b^5*x^(1/3))/a^
18