3.2421 \(\int \frac{x^2}{a+\frac{b}{\sqrt [3]{x}}} \, dx\)

Optimal. Leaf size=136 \[ -\frac{3 b^7 x^{2/3}}{2 a^8}-\frac{3 b^5 x^{4/3}}{4 a^6}+\frac{3 b^4 x^{5/3}}{5 a^5}-\frac{b^3 x^2}{2 a^4}+\frac{3 b^2 x^{7/3}}{7 a^3}+\frac{3 b^8 \sqrt [3]{x}}{a^9}+\frac{b^6 x}{a^7}-\frac{3 b^9 \log \left (a \sqrt [3]{x}+b\right )}{a^{10}}-\frac{3 b x^{8/3}}{8 a^2}+\frac{x^3}{3 a} \]

[Out]

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

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Rubi [A]  time = 0.0892607, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {263, 266, 43} \[ -\frac{3 b^7 x^{2/3}}{2 a^8}-\frac{3 b^5 x^{4/3}}{4 a^6}+\frac{3 b^4 x^{5/3}}{5 a^5}-\frac{b^3 x^2}{2 a^4}+\frac{3 b^2 x^{7/3}}{7 a^3}+\frac{3 b^8 \sqrt [3]{x}}{a^9}+\frac{b^6 x}{a^7}-\frac{3 b^9 \log \left (a \sqrt [3]{x}+b\right )}{a^{10}}-\frac{3 b x^{8/3}}{8 a^2}+\frac{x^3}{3 a} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b/x^(1/3)),x]

[Out]

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

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

Mathematica [A]  time = 0.0952866, size = 146, normalized size = 1.07 \[ -\frac{3 b^7 x^{2/3}}{2 a^8}-\frac{3 b^5 x^{4/3}}{4 a^6}+\frac{3 b^4 x^{5/3}}{5 a^5}-\frac{b^3 x^2}{2 a^4}+\frac{3 b^2 x^{7/3}}{7 a^3}+\frac{3 b^8 \sqrt [3]{x}}{a^9}+\frac{b^6 x}{a^7}-\frac{3 b^9 \log \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^{10}}-\frac{b^9 \log (x)}{a^{10}}-\frac{3 b x^{8/3}}{8 a^2}+\frac{x^3}{3 a} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b/x^(1/3)),x]

[Out]

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

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Maple [A]  time = 0.004, size = 109, normalized size = 0.8 \begin{align*} 3\,{\frac{{b}^{8}\sqrt [3]{x}}{{a}^{9}}}-{\frac{3\,{b}^{7}}{2\,{a}^{8}}{x}^{{\frac{2}{3}}}}+{\frac{{b}^{6}x}{{a}^{7}}}-{\frac{3\,{b}^{5}}{4\,{a}^{6}}{x}^{{\frac{4}{3}}}}+{\frac{3\,{b}^{4}}{5\,{a}^{5}}{x}^{{\frac{5}{3}}}}-{\frac{{b}^{3}{x}^{2}}{2\,{a}^{4}}}+{\frac{3\,{b}^{2}}{7\,{a}^{3}}{x}^{{\frac{7}{3}}}}-{\frac{3\,b}{8\,{a}^{2}}{x}^{{\frac{8}{3}}}}+{\frac{{x}^{3}}{3\,a}}-3\,{\frac{{b}^{9}\ln \left ( b+a\sqrt [3]{x} \right ) }{{a}^{10}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0163, size = 165, normalized size = 1.21 \begin{align*} -\frac{3 \, b^{9} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{a^{10}} - \frac{b^{9} \log \left (x\right )}{a^{10}} + \frac{{\left (280 \, a^{8} - \frac{315 \, a^{7} b}{x^{\frac{1}{3}}} + \frac{360 \, a^{6} b^{2}}{x^{\frac{2}{3}}} - \frac{420 \, a^{5} b^{3}}{x} + \frac{504 \, a^{4} b^{4}}{x^{\frac{4}{3}}} - \frac{630 \, a^{3} b^{5}}{x^{\frac{5}{3}}} + \frac{840 \, a^{2} b^{6}}{x^{2}} - \frac{1260 \, a b^{7}}{x^{\frac{7}{3}}} + \frac{2520 \, b^{8}}{x^{\frac{8}{3}}}\right )} x^{3}}{840 \, a^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*b^9*log(a + b/x^(1/3))/a^10 - b^9*log(x)/a^10 + 1/840*(280*a^8 - 315*a^7*b/x^(1/3) + 360*a^6*b^2/x^(2/3) -
420*a^5*b^3/x + 504*a^4*b^4/x^(4/3) - 630*a^3*b^5/x^(5/3) + 840*a^2*b^6/x^2 - 1260*a*b^7/x^(7/3) + 2520*b^8/x^
(8/3))*x^3/a^9

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Fricas [A]  time = 1.4833, size = 267, normalized size = 1.96 \begin{align*} \frac{280 \, a^{9} x^{3} - 420 \, a^{6} b^{3} x^{2} + 840 \, a^{3} b^{6} x - 2520 \, b^{9} \log \left (a x^{\frac{1}{3}} + b\right ) - 63 \,{\left (5 \, a^{8} b x^{2} - 8 \, a^{5} b^{4} x + 20 \, a^{2} b^{7}\right )} x^{\frac{2}{3}} + 90 \,{\left (4 \, a^{7} b^{2} x^{2} - 7 \, a^{4} b^{5} x + 28 \, a b^{8}\right )} x^{\frac{1}{3}}}{840 \, a^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/840*(280*a^9*x^3 - 420*a^6*b^3*x^2 + 840*a^3*b^6*x - 2520*b^9*log(a*x^(1/3) + b) - 63*(5*a^8*b*x^2 - 8*a^5*b
^4*x + 20*a^2*b^7)*x^(2/3) + 90*(4*a^7*b^2*x^2 - 7*a^4*b^5*x + 28*a*b^8)*x^(1/3))/a^10

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Sympy [A]  time = 5.4587, size = 143, normalized size = 1.05 \begin{align*} \begin{cases} \frac{x^{3}}{3 a} - \frac{3 b x^{\frac{8}{3}}}{8 a^{2}} + \frac{3 b^{2} x^{\frac{7}{3}}}{7 a^{3}} - \frac{b^{3} x^{2}}{2 a^{4}} + \frac{3 b^{4} x^{\frac{5}{3}}}{5 a^{5}} - \frac{3 b^{5} x^{\frac{4}{3}}}{4 a^{6}} + \frac{b^{6} x}{a^{7}} - \frac{3 b^{7} x^{\frac{2}{3}}}{2 a^{8}} + \frac{3 b^{8} \sqrt [3]{x}}{a^{9}} - \frac{3 b^{9} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{a^{10}} & \text{for}\: a \neq 0 \\\frac{3 x^{\frac{10}{3}}}{10 b} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.20175, size = 150, normalized size = 1.1 \begin{align*} -\frac{3 \, b^{9} \log \left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{a^{10}} + \frac{280 \, a^{8} x^{3} - 315 \, a^{7} b x^{\frac{8}{3}} + 360 \, a^{6} b^{2} x^{\frac{7}{3}} - 420 \, a^{5} b^{3} x^{2} + 504 \, a^{4} b^{4} x^{\frac{5}{3}} - 630 \, a^{3} b^{5} x^{\frac{4}{3}} + 840 \, a^{2} b^{6} x - 1260 \, a b^{7} x^{\frac{2}{3}} + 2520 \, b^{8} x^{\frac{1}{3}}}{840 \, a^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*b^9*log(abs(a*x^(1/3) + b))/a^10 + 1/840*(280*a^8*x^3 - 315*a^7*b*x^(8/3) + 360*a^6*b^2*x^(7/3) - 420*a^5*b
^3*x^2 + 504*a^4*b^4*x^(5/3) - 630*a^3*b^5*x^(4/3) + 840*a^2*b^6*x - 1260*a*b^7*x^(2/3) + 2520*b^8*x^(1/3))/a^
9