\(\int \frac {x^2}{a+\frac {b}{\sqrt [3]{x}}} \, dx\) [2421]

Optimal result

Integrand size = 15, antiderivative size = 136 \[ \int \frac {x^2}{a+\frac {b}{\sqrt [3]{x}}} \, dx=\frac {3 b^8 \sqrt [3]{x}}{a^9}-\frac {3 b^7 x^{2/3}}{2 a^8}+\frac {b^6 x}{a^7}-\frac {3 b^5 x^{4/3}}{4 a^6}+\frac {3 b^4 x^{5/3}}{5 a^5}-\frac {b^3 x^2}{2 a^4}+\frac {3 b^2 x^{7/3}}{7 a^3}-\frac {3 b x^{8/3}}{8 a^2}+\frac {x^3}{3 a}-\frac {3 b^9 \log \left (b+a \sqrt [3]{x}\right )}{a^{10}} \]

[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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {269, 272, 45} \[ \int \frac {x^2}{a+\frac {b}{\sqrt [3]{x}}} \, dx=-\frac {3 b^9 \log \left (a \sqrt [3]{x}+b\right )}{a^{10}}+\frac {3 b^8 \sqrt [3]{x}}{a^9}-\frac {3 b^7 x^{2/3}}{2 a^8}+\frac {b^6 x}{a^7}-\frac {3 b^5 x^{4/3}}{4 a^6}+\frac {3 b^4 x^{5/3}}{5 a^5}-\frac {b^3 x^2}{2 a^4}+\frac {3 b^2 x^{7/3}}{7 a^3}-\frac {3 b x^{8/3}}{8 a^2}+\frac {x^3}{3 a} \]

[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 45

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])

Rule 269

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 272

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]]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^{7/3}}{b+a \sqrt [3]{x}} \, dx \\ & = 3 \text {Subst}\left (\int \frac {x^9}{b+a x} \, dx,x,\sqrt [3]{x}\right ) \\ & = 3 \text {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] (verified)

Time = 0.11 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.93 \[ \int \frac {x^2}{a+\frac {b}{\sqrt [3]{x}}} \, dx=\frac {2520 b^8 \sqrt [3]{x}-1260 a b^7 x^{2/3}+840 a^2 b^6 x-630 a^3 b^5 x^{4/3}+504 a^4 b^4 x^{5/3}-420 a^5 b^3 x^2+360 a^6 b^2 x^{7/3}-315 a^7 b x^{8/3}+280 a^8 x^3}{840 a^9}-\frac {3 b^9 \log \left (b+a \sqrt [3]{x}\right )}{a^{10}} \]

[In]

Integrate[x^2/(a + b/x^(1/3)),x]

[Out]

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

Maple [A] (verified)

Time = 6.17 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.81

method	result	size
derivativedivides	\(\frac {\frac {x^{3} a^{8}}{3}-\frac {3 b \,x^{\frac {8}{3}} a^{7}}{8}+\frac {3 b^{2} x^{\frac {7}{3}} a^{6}}{7}-\frac {a^{5} b^{3} x^{2}}{2}+\frac {3 b^{4} x^{\frac {5}{3}} a^{4}}{5}-\frac {3 b^{5} x^{\frac {4}{3}} a^{3}}{4}+a^{2} b^{6} x -\frac {3 a \,b^{7} x^{\frac {2}{3}}}{2}+3 b^{8} x^{\frac {1}{3}}}{a^{9}}-\frac {3 b^{9} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{a^{10}}\)	\(110\)
default	\(\frac {\frac {x^{3} a^{8}}{3}-\frac {3 b \,x^{\frac {8}{3}} a^{7}}{8}+\frac {3 b^{2} x^{\frac {7}{3}} a^{6}}{7}-\frac {a^{5} b^{3} x^{2}}{2}+\frac {3 b^{4} x^{\frac {5}{3}} a^{4}}{5}-\frac {3 b^{5} x^{\frac {4}{3}} a^{3}}{4}+a^{2} b^{6} x -\frac {3 a \,b^{7} x^{\frac {2}{3}}}{2}+3 b^{8} x^{\frac {1}{3}}}{a^{9}}-\frac {3 b^{9} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{a^{10}}\)	\(110\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.82 \[ \int \frac {x^2}{a+\frac {b}{\sqrt [3]{x}}} \, dx=\frac {280 \, a^{9} x^{3} - 420 \, a^{6} b^{3} x^{2} + 840 \, a^{3} b^{6} x - 2520 \, b^{9} \log \left (a x^{\frac {1}{3}} + b\right ) - 63 \, {\left (5 \, a^{8} b x^{2} - 8 \, a^{5} b^{4} x + 20 \, a^{2} b^{7}\right )} x^{\frac {2}{3}} + 90 \, {\left (4 \, a^{7} b^{2} x^{2} - 7 \, a^{4} b^{5} x + 28 \, a b^{8}\right )} x^{\frac {1}{3}}}{840 \, a^{10}} \]

[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

Sympy [A] (verification not implemented)

Time = 0.54 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.05 \[ \int \frac {x^2}{a+\frac {b}{\sqrt [3]{x}}} \, dx=\begin {cases} \frac {x^{3}}{3 a} - \frac {3 b x^{\frac {8}{3}}}{8 a^{2}} + \frac {3 b^{2} x^{\frac {7}{3}}}{7 a^{3}} - \frac {b^{3} x^{2}}{2 a^{4}} + \frac {3 b^{4} x^{\frac {5}{3}}}{5 a^{5}} - \frac {3 b^{5} x^{\frac {4}{3}}}{4 a^{6}} + \frac {b^{6} x}{a^{7}} - \frac {3 b^{7} x^{\frac {2}{3}}}{2 a^{8}} + \frac {3 b^{8} \sqrt [3]{x}}{a^{9}} - \frac {3 b^{9} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{a^{10}} & \text {for}\: a \neq 0 \\\frac {3 x^{\frac {10}{3}}}{10 b} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.90 \[ \int \frac {x^2}{a+\frac {b}{\sqrt [3]{x}}} \, dx=-\frac {3 \, b^{9} \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{a^{10}} - \frac {b^{9} \log \left (x\right )}{a^{10}} + \frac {{\left (280 \, a^{8} - \frac {315 \, a^{7} b}{x^{\frac {1}{3}}} + \frac {360 \, a^{6} b^{2}}{x^{\frac {2}{3}}} - \frac {420 \, a^{5} b^{3}}{x} + \frac {504 \, a^{4} b^{4}}{x^{\frac {4}{3}}} - \frac {630 \, a^{3} b^{5}}{x^{\frac {5}{3}}} + \frac {840 \, a^{2} b^{6}}{x^{2}} - \frac {1260 \, a b^{7}}{x^{\frac {7}{3}}} + \frac {2520 \, b^{8}}{x^{\frac {8}{3}}}\right )} x^{3}}{840 \, a^{9}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.82 \[ \int \frac {x^2}{a+\frac {b}{\sqrt [3]{x}}} \, dx=-\frac {3 \, b^{9} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{a^{10}} + \frac {280 \, a^{8} x^{3} - 315 \, a^{7} b x^{\frac {8}{3}} + 360 \, a^{6} b^{2} x^{\frac {7}{3}} - 420 \, a^{5} b^{3} x^{2} + 504 \, a^{4} b^{4} x^{\frac {5}{3}} - 630 \, a^{3} b^{5} x^{\frac {4}{3}} + 840 \, a^{2} b^{6} x - 1260 \, a b^{7} x^{\frac {2}{3}} + 2520 \, b^{8} x^{\frac {1}{3}}}{840 \, a^{9}} \]

[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

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.79 \[ \int \frac {x^2}{a+\frac {b}{\sqrt [3]{x}}} \, dx=\frac {x^3}{3\,a}-\frac {3\,b\,x^{8/3}}{8\,a^2}+\frac {b^6\,x}{a^7}-\frac {3\,b^9\,\ln \left (b+a\,x^{1/3}\right )}{a^{10}}-\frac {b^3\,x^2}{2\,a^4}+\frac {3\,b^2\,x^{7/3}}{7\,a^3}+\frac {3\,b^4\,x^{5/3}}{5\,a^5}-\frac {3\,b^5\,x^{4/3}}{4\,a^6}-\frac {3\,b^7\,x^{2/3}}{2\,a^8}+\frac {3\,b^8\,x^{1/3}}{a^9} \]

[In]

int(x^2/(a + b/x^(1/3)),x)

[Out]

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