\(\int \frac {x^2}{a+b \sqrt [3]{x}} \, dx\) [2357]

Optimal result

Integrand size = 15, antiderivative size = 124 \[ \int \frac {x^2}{a+b \sqrt [3]{x}} \, dx=-\frac {3 a^7 \sqrt [3]{x}}{b^8}+\frac {3 a^6 x^{2/3}}{2 b^7}-\frac {a^5 x}{b^6}+\frac {3 a^4 x^{4/3}}{4 b^5}-\frac {3 a^3 x^{5/3}}{5 b^4}+\frac {a^2 x^2}{2 b^3}-\frac {3 a x^{7/3}}{7 b^2}+\frac {3 x^{8/3}}{8 b}+\frac {3 a^8 \log \left (a+b \sqrt [3]{x}\right )}{b^9} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int \frac {x^2}{a+b \sqrt [3]{x}} \, dx=\frac {3 a^8 \log \left (a+b \sqrt [3]{x}\right )}{b^9}-\frac {3 a^7 \sqrt [3]{x}}{b^8}+\frac {3 a^6 x^{2/3}}{2 b^7}-\frac {a^5 x}{b^6}+\frac {3 a^4 x^{4/3}}{4 b^5}-\frac {3 a^3 x^{5/3}}{5 b^4}+\frac {a^2 x^2}{2 b^3}-\frac {3 a x^{7/3}}{7 b^2}+\frac {3 x^{8/3}}{8 b} \]

[In]

Int[x^2/(a + b*x^(1/3)),x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.93 \[ \int \frac {x^2}{a+b \sqrt [3]{x}} \, dx=\frac {\sqrt [3]{x} \left (-840 a^7+420 a^6 b \sqrt [3]{x}-280 a^5 b^2 x^{2/3}+210 a^4 b^3 x-168 a^3 b^4 x^{4/3}+140 a^2 b^5 x^{5/3}-120 a b^6 x^2+105 b^7 x^{7/3}\right )}{280 b^8}+\frac {3 a^8 \log \left (a+b \sqrt [3]{x}\right )}{b^9} \]

[In]

Integrate[x^2/(a + b*x^(1/3)),x]

[Out]

(x^(1/3)*(-840*a^7 + 420*a^6*b*x^(1/3) - 280*a^5*b^2*x^(2/3) + 210*a^4*b^3*x - 168*a^3*b^4*x^(4/3) + 140*a^2*b
^5*x^(5/3) - 120*a*b^6*x^2 + 105*b^7*x^(7/3)))/(280*b^8) + (3*a^8*Log[a + b*x^(1/3)])/b^9

Maple [A] (verified)

Time = 3.70 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.80

method	result	size
derivativedivides	\(-\frac {3 \left (-\frac {x^{\frac {8}{3}} b^{7}}{8}+\frac {a \,x^{\frac {7}{3}} b^{6}}{7}-\frac {a^{2} x^{2} b^{5}}{6}+\frac {a^{3} x^{\frac {5}{3}} b^{4}}{5}-\frac {a^{4} x^{\frac {4}{3}} b^{3}}{4}+\frac {a^{5} b^{2} x}{3}-\frac {b \,a^{6} x^{\frac {2}{3}}}{2}+a^{7} x^{\frac {1}{3}}\right )}{b^{8}}+\frac {3 a^{8} \ln \left (a +b \,x^{\frac {1}{3}}\right )}{b^{9}}\)	\(99\)
default	\(-\frac {3 \left (-\frac {x^{\frac {8}{3}} b^{7}}{8}+\frac {a \,x^{\frac {7}{3}} b^{6}}{7}-\frac {a^{2} x^{2} b^{5}}{6}+\frac {a^{3} x^{\frac {5}{3}} b^{4}}{5}-\frac {a^{4} x^{\frac {4}{3}} b^{3}}{4}+\frac {a^{5} b^{2} x}{3}-\frac {b \,a^{6} x^{\frac {2}{3}}}{2}+a^{7} x^{\frac {1}{3}}\right )}{b^{8}}+\frac {3 a^{8} \ln \left (a +b \,x^{\frac {1}{3}}\right )}{b^{9}}\)	\(99\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.81 \[ \int \frac {x^2}{a+b \sqrt [3]{x}} \, dx=\frac {140 \, a^{2} b^{6} x^{2} - 280 \, a^{5} b^{3} x + 840 \, a^{8} \log \left (b x^{\frac {1}{3}} + a\right ) + 21 \, {\left (5 \, b^{8} x^{2} - 8 \, a^{3} b^{5} x + 20 \, a^{6} b^{2}\right )} x^{\frac {2}{3}} - 30 \, {\left (4 \, a b^{7} x^{2} - 7 \, a^{4} b^{4} x + 28 \, a^{7} b\right )} x^{\frac {1}{3}}}{280 \, b^{9}} \]

[In]

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

[Out]

1/280*(140*a^2*b^6*x^2 - 280*a^5*b^3*x + 840*a^8*log(b*x^(1/3) + a) + 21*(5*b^8*x^2 - 8*a^3*b^5*x + 20*a^6*b^2
)*x^(2/3) - 30*(4*a*b^7*x^2 - 7*a^4*b^4*x + 28*a^7*b)*x^(1/3))/b^9

Sympy [F(-1)]

Timed out. \[ \int \frac {x^2}{a+b \sqrt [3]{x}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.18 \[ \int \frac {x^2}{a+b \sqrt [3]{x}} \, dx=\frac {3 \, a^{8} \log \left (b x^{\frac {1}{3}} + a\right )}{b^{9}} + \frac {3 \, {\left (b x^{\frac {1}{3}} + a\right )}^{8}}{8 \, b^{9}} - \frac {24 \, {\left (b x^{\frac {1}{3}} + a\right )}^{7} a}{7 \, b^{9}} + \frac {14 \, {\left (b x^{\frac {1}{3}} + a\right )}^{6} a^{2}}{b^{9}} - \frac {168 \, {\left (b x^{\frac {1}{3}} + a\right )}^{5} a^{3}}{5 \, b^{9}} + \frac {105 \, {\left (b x^{\frac {1}{3}} + a\right )}^{4} a^{4}}{2 \, b^{9}} - \frac {56 \, {\left (b x^{\frac {1}{3}} + a\right )}^{3} a^{5}}{b^{9}} + \frac {42 \, {\left (b x^{\frac {1}{3}} + a\right )}^{2} a^{6}}{b^{9}} - \frac {24 \, {\left (b x^{\frac {1}{3}} + a\right )} a^{7}}{b^{9}} \]

[In]

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

[Out]

3*a^8*log(b*x^(1/3) + a)/b^9 + 3/8*(b*x^(1/3) + a)^8/b^9 - 24/7*(b*x^(1/3) + a)^7*a/b^9 + 14*(b*x^(1/3) + a)^6
*a^2/b^9 - 168/5*(b*x^(1/3) + a)^5*a^3/b^9 + 105/2*(b*x^(1/3) + a)^4*a^4/b^9 - 56*(b*x^(1/3) + a)^3*a^5/b^9 +
42*(b*x^(1/3) + a)^2*a^6/b^9 - 24*(b*x^(1/3) + a)*a^7/b^9

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.81 \[ \int \frac {x^2}{a+b \sqrt [3]{x}} \, dx=\frac {3 \, a^{8} \log \left ({\left | b x^{\frac {1}{3}} + a \right |}\right )}{b^{9}} + \frac {105 \, b^{7} x^{\frac {8}{3}} - 120 \, a b^{6} x^{\frac {7}{3}} + 140 \, a^{2} b^{5} x^{2} - 168 \, a^{3} b^{4} x^{\frac {5}{3}} + 210 \, a^{4} b^{3} x^{\frac {4}{3}} - 280 \, a^{5} b^{2} x + 420 \, a^{6} b x^{\frac {2}{3}} - 840 \, a^{7} x^{\frac {1}{3}}}{280 \, b^{8}} \]

[In]

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

[Out]

3*a^8*log(abs(b*x^(1/3) + a))/b^9 + 1/280*(105*b^7*x^(8/3) - 120*a*b^6*x^(7/3) + 140*a^2*b^5*x^2 - 168*a^3*b^4
*x^(5/3) + 210*a^4*b^3*x^(4/3) - 280*a^5*b^2*x + 420*a^6*b*x^(2/3) - 840*a^7*x^(1/3))/b^8

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.79 \[ \int \frac {x^2}{a+b \sqrt [3]{x}} \, dx=\frac {3\,x^{8/3}}{8\,b}-\frac {a^5\,x}{b^6}-\frac {3\,a\,x^{7/3}}{7\,b^2}+\frac {3\,a^8\,\ln \left (a+b\,x^{1/3}\right )}{b^9}+\frac {a^2\,x^2}{2\,b^3}-\frac {3\,a^3\,x^{5/3}}{5\,b^4}+\frac {3\,a^4\,x^{4/3}}{4\,b^5}+\frac {3\,a^6\,x^{2/3}}{2\,b^7}-\frac {3\,a^7\,x^{1/3}}{b^8} \]

[In]

int(x^2/(a + b*x^(1/3)),x)

[Out]

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