3.2366 \(\int \frac{x^2}{(a+b \sqrt [3]{x})^2} \, dx\)

Optimal. Leaf size=122 \[ -\frac{9 a^5 x^{2/3}}{b^7}-\frac{3 a^3 x^{4/3}}{b^5}+\frac{9 a^2 x^{5/3}}{5 b^4}-\frac{3 a^8}{b^9 \left (a+b \sqrt [3]{x}\right )}+\frac{21 a^6 \sqrt [3]{x}}{b^8}+\frac{5 a^4 x}{b^6}-\frac{24 a^7 \log \left (a+b \sqrt [3]{x}\right )}{b^9}-\frac{a x^2}{b^3}+\frac{3 x^{7/3}}{7 b^2} \]

[Out]

(-3*a^8)/(b^9*(a + b*x^(1/3))) + (21*a^6*x^(1/3))/b^8 - (9*a^5*x^(2/3))/b^7 + (5*a^4*x)/b^6 - (3*a^3*x^(4/3))/
b^5 + (9*a^2*x^(5/3))/(5*b^4) - (a*x^2)/b^3 + (3*x^(7/3))/(7*b^2) - (24*a^7*Log[a + b*x^(1/3)])/b^9

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Rubi [A]  time = 0.0885099, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac{9 a^5 x^{2/3}}{b^7}-\frac{3 a^3 x^{4/3}}{b^5}+\frac{9 a^2 x^{5/3}}{5 b^4}-\frac{3 a^8}{b^9 \left (a+b \sqrt [3]{x}\right )}+\frac{21 a^6 \sqrt [3]{x}}{b^8}+\frac{5 a^4 x}{b^6}-\frac{24 a^7 \log \left (a+b \sqrt [3]{x}\right )}{b^9}-\frac{a x^2}{b^3}+\frac{3 x^{7/3}}{7 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*x^(1/3))^2,x]

[Out]

(-3*a^8)/(b^9*(a + b*x^(1/3))) + (21*a^6*x^(1/3))/b^8 - (9*a^5*x^(2/3))/b^7 + (5*a^4*x)/b^6 - (3*a^3*x^(4/3))/
b^5 + (9*a^2*x^(5/3))/(5*b^4) - (a*x^2)/b^3 + (3*x^(7/3))/(7*b^2) - (24*a^7*Log[a + b*x^(1/3)])/b^9

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

Mathematica [A]  time = 0.0972802, size = 122, normalized size = 1. \[ -\frac{9 a^5 x^{2/3}}{b^7}-\frac{3 a^3 x^{4/3}}{b^5}+\frac{9 a^2 x^{5/3}}{5 b^4}-\frac{3 a^8}{b^9 \left (a+b \sqrt [3]{x}\right )}+\frac{21 a^6 \sqrt [3]{x}}{b^8}+\frac{5 a^4 x}{b^6}-\frac{24 a^7 \log \left (a+b \sqrt [3]{x}\right )}{b^9}-\frac{a x^2}{b^3}+\frac{3 x^{7/3}}{7 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*x^(1/3))^2,x]

[Out]

(-3*a^8)/(b^9*(a + b*x^(1/3))) + (21*a^6*x^(1/3))/b^8 - (9*a^5*x^(2/3))/b^7 + (5*a^4*x)/b^6 - (3*a^3*x^(4/3))/
b^5 + (9*a^2*x^(5/3))/(5*b^4) - (a*x^2)/b^3 + (3*x^(7/3))/(7*b^2) - (24*a^7*Log[a + b*x^(1/3)])/b^9

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Maple [A]  time = 0.007, size = 105, normalized size = 0.9 \begin{align*} -3\,{\frac{{a}^{8}}{{b}^{9} \left ( a+b\sqrt [3]{x} \right ) }}+21\,{\frac{{a}^{6}\sqrt [3]{x}}{{b}^{8}}}-9\,{\frac{{a}^{5}{x}^{2/3}}{{b}^{7}}}+5\,{\frac{{a}^{4}x}{{b}^{6}}}-3\,{\frac{{a}^{3}{x}^{4/3}}{{b}^{5}}}+{\frac{9\,{a}^{2}}{5\,{b}^{4}}{x}^{{\frac{5}{3}}}}-{\frac{a{x}^{2}}{{b}^{3}}}+{\frac{3}{7\,{b}^{2}}{x}^{{\frac{7}{3}}}}-24\,{\frac{{a}^{7}\ln \left ( a+b\sqrt [3]{x} \right ) }{{b}^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*a^8/b^9/(a+b*x^(1/3))+21*a^6*x^(1/3)/b^8-9*a^5*x^(2/3)/b^7+5*a^4*x/b^6-3*a^3*x^(4/3)/b^5+9/5*a^2*x^(5/3)/b^
4-a*x^2/b^3+3/7*x^(7/3)/b^2-24*a^7*ln(a+b*x^(1/3))/b^9

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Maxima [A]  time = 0.980532, size = 197, normalized size = 1.61 \begin{align*} -\frac{24 \, a^{7} \log \left (b x^{\frac{1}{3}} + a\right )}{b^{9}} + \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{7}}{7 \, b^{9}} - \frac{4 \,{\left (b x^{\frac{1}{3}} + a\right )}^{6} a}{b^{9}} + \frac{84 \,{\left (b x^{\frac{1}{3}} + a\right )}^{5} a^{2}}{5 \, b^{9}} - \frac{42 \,{\left (b x^{\frac{1}{3}} + a\right )}^{4} a^{3}}{b^{9}} + \frac{70 \,{\left (b x^{\frac{1}{3}} + a\right )}^{3} a^{4}}{b^{9}} - \frac{84 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} a^{5}}{b^{9}} + \frac{84 \,{\left (b x^{\frac{1}{3}} + a\right )} a^{6}}{b^{9}} - \frac{3 \, a^{8}}{{\left (b x^{\frac{1}{3}} + a\right )} b^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-24*a^7*log(b*x^(1/3) + a)/b^9 + 3/7*(b*x^(1/3) + a)^7/b^9 - 4*(b*x^(1/3) + a)^6*a/b^9 + 84/5*(b*x^(1/3) + a)^
5*a^2/b^9 - 42*(b*x^(1/3) + a)^4*a^3/b^9 + 70*(b*x^(1/3) + a)^3*a^4/b^9 - 84*(b*x^(1/3) + a)^2*a^5/b^9 + 84*(b
*x^(1/3) + a)*a^6/b^9 - 3*a^8/((b*x^(1/3) + a)*b^9)

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Fricas [A]  time = 1.49412, size = 342, normalized size = 2.8 \begin{align*} -\frac{35 \, a b^{9} x^{3} - 140 \, a^{4} b^{6} x^{2} - 175 \, a^{7} b^{3} x + 105 \, a^{10} + 840 \,{\left (a^{7} b^{3} x + a^{10}\right )} \log \left (b x^{\frac{1}{3}} + a\right ) - 21 \,{\left (3 \, a^{2} b^{8} x^{2} - 12 \, a^{5} b^{5} x - 20 \, a^{8} b^{2}\right )} x^{\frac{2}{3}} - 15 \,{\left (b^{10} x^{3} - 6 \, a^{3} b^{7} x^{2} + 42 \, a^{6} b^{4} x + 56 \, a^{9} b\right )} x^{\frac{1}{3}}}{35 \,{\left (b^{12} x + a^{3} b^{9}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/35*(35*a*b^9*x^3 - 140*a^4*b^6*x^2 - 175*a^7*b^3*x + 105*a^10 + 840*(a^7*b^3*x + a^10)*log(b*x^(1/3) + a) -
21*(3*a^2*b^8*x^2 - 12*a^5*b^5*x - 20*a^8*b^2)*x^(2/3) - 15*(b^10*x^3 - 6*a^3*b^7*x^2 + 42*a^6*b^4*x + 56*a^9
*b)*x^(1/3))/(b^12*x + a^3*b^9)

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Sympy [B]  time = 20.672, size = 343, normalized size = 2.81 \begin{align*} - \frac{840 a^{8} x^{\frac{176}{3}} \log{\left (1 + \frac{b \sqrt [3]{x}}{a} \right )}}{35 a b^{9} x^{\frac{176}{3}} + 35 b^{10} x^{59}} - \frac{840 a^{7} b x^{59} \log{\left (1 + \frac{b \sqrt [3]{x}}{a} \right )}}{35 a b^{9} x^{\frac{176}{3}} + 35 b^{10} x^{59}} + \frac{840 a^{7} b x^{59}}{35 a b^{9} x^{\frac{176}{3}} + 35 b^{10} x^{59}} + \frac{420 a^{6} b^{2} x^{\frac{178}{3}}}{35 a b^{9} x^{\frac{176}{3}} + 35 b^{10} x^{59}} - \frac{140 a^{5} b^{3} x^{\frac{179}{3}}}{35 a b^{9} x^{\frac{176}{3}} + 35 b^{10} x^{59}} + \frac{70 a^{4} b^{4} x^{60}}{35 a b^{9} x^{\frac{176}{3}} + 35 b^{10} x^{59}} - \frac{42 a^{3} b^{5} x^{\frac{181}{3}}}{35 a b^{9} x^{\frac{176}{3}} + 35 b^{10} x^{59}} + \frac{28 a^{2} b^{6} x^{\frac{182}{3}}}{35 a b^{9} x^{\frac{176}{3}} + 35 b^{10} x^{59}} - \frac{20 a b^{7} x^{61}}{35 a b^{9} x^{\frac{176}{3}} + 35 b^{10} x^{59}} + \frac{15 b^{8} x^{\frac{184}{3}}}{35 a b^{9} x^{\frac{176}{3}} + 35 b^{10} x^{59}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-840*a**8*x**(176/3)*log(1 + b*x**(1/3)/a)/(35*a*b**9*x**(176/3) + 35*b**10*x**59) - 840*a**7*b*x**59*log(1 +
b*x**(1/3)/a)/(35*a*b**9*x**(176/3) + 35*b**10*x**59) + 840*a**7*b*x**59/(35*a*b**9*x**(176/3) + 35*b**10*x**5
9) + 420*a**6*b**2*x**(178/3)/(35*a*b**9*x**(176/3) + 35*b**10*x**59) - 140*a**5*b**3*x**(179/3)/(35*a*b**9*x*
*(176/3) + 35*b**10*x**59) + 70*a**4*b**4*x**60/(35*a*b**9*x**(176/3) + 35*b**10*x**59) - 42*a**3*b**5*x**(181
/3)/(35*a*b**9*x**(176/3) + 35*b**10*x**59) + 28*a**2*b**6*x**(182/3)/(35*a*b**9*x**(176/3) + 35*b**10*x**59)
- 20*a*b**7*x**61/(35*a*b**9*x**(176/3) + 35*b**10*x**59) + 15*b**8*x**(184/3)/(35*a*b**9*x**(176/3) + 35*b**1
0*x**59)

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Giac [A]  time = 1.12792, size = 150, normalized size = 1.23 \begin{align*} -\frac{24 \, a^{7} \log \left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{b^{9}} - \frac{3 \, a^{8}}{{\left (b x^{\frac{1}{3}} + a\right )} b^{9}} + \frac{15 \, b^{12} x^{\frac{7}{3}} - 35 \, a b^{11} x^{2} + 63 \, a^{2} b^{10} x^{\frac{5}{3}} - 105 \, a^{3} b^{9} x^{\frac{4}{3}} + 175 \, a^{4} b^{8} x - 315 \, a^{5} b^{7} x^{\frac{2}{3}} + 735 \, a^{6} b^{6} x^{\frac{1}{3}}}{35 \, b^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-24*a^7*log(abs(b*x^(1/3) + a))/b^9 - 3*a^8/((b*x^(1/3) + a)*b^9) + 1/35*(15*b^12*x^(7/3) - 35*a*b^11*x^2 + 63
*a^2*b^10*x^(5/3) - 105*a^3*b^9*x^(4/3) + 175*a^4*b^8*x - 315*a^5*b^7*x^(2/3) + 735*a^6*b^6*x^(1/3))/b^14