3.2429 \(\int \frac{x^2}{(a+\frac{b}{\sqrt [3]{x}})^2} \, dx\)

Optimal. Leaf size=150 \[ -\frac{12 b^7 x^{2/3}}{a^9}-\frac{9 b^5 x^{4/3}}{2 a^7}+\frac{3 b^4 x^{5/3}}{a^6}-\frac{2 b^3 x^2}{a^5}+\frac{9 b^2 x^{7/3}}{7 a^4}-\frac{3 b^{10}}{a^{11} \left (a \sqrt [3]{x}+b\right )}+\frac{27 b^8 \sqrt [3]{x}}{a^{10}}+\frac{7 b^6 x}{a^8}-\frac{30 b^9 \log \left (a \sqrt [3]{x}+b\right )}{a^{11}}-\frac{3 b x^{8/3}}{4 a^3}+\frac{x^3}{3 a^2} \]

[Out]

(-3*b^10)/(a^11*(b + a*x^(1/3))) + (27*b^8*x^(1/3))/a^10 - (12*b^7*x^(2/3))/a^9 + (7*b^6*x)/a^8 - (9*b^5*x^(4/
3))/(2*a^7) + (3*b^4*x^(5/3))/a^6 - (2*b^3*x^2)/a^5 + (9*b^2*x^(7/3))/(7*a^4) - (3*b*x^(8/3))/(4*a^3) + x^3/(3
*a^2) - (30*b^9*Log[b + a*x^(1/3)])/a^11

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Rubi [A]  time = 0.131766, antiderivative size = 150, 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} \[ -\frac{12 b^7 x^{2/3}}{a^9}-\frac{9 b^5 x^{4/3}}{2 a^7}+\frac{3 b^4 x^{5/3}}{a^6}-\frac{2 b^3 x^2}{a^5}+\frac{9 b^2 x^{7/3}}{7 a^4}-\frac{3 b^{10}}{a^{11} \left (a \sqrt [3]{x}+b\right )}+\frac{27 b^8 \sqrt [3]{x}}{a^{10}}+\frac{7 b^6 x}{a^8}-\frac{30 b^9 \log \left (a \sqrt [3]{x}+b\right )}{a^{11}}-\frac{3 b x^{8/3}}{4 a^3}+\frac{x^3}{3 a^2} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b/x^(1/3))^2,x]

[Out]

(-3*b^10)/(a^11*(b + a*x^(1/3))) + (27*b^8*x^(1/3))/a^10 - (12*b^7*x^(2/3))/a^9 + (7*b^6*x)/a^8 - (9*b^5*x^(4/
3))/(2*a^7) + (3*b^4*x^(5/3))/a^6 - (2*b^3*x^2)/a^5 + (9*b^2*x^(7/3))/(7*a^4) - (3*b*x^(8/3))/(4*a^3) + x^3/(3
*a^2) - (30*b^9*Log[b + a*x^(1/3)])/a^11

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}{\left (a+\frac{b}{\sqrt [3]{x}}\right )^2} \, dx &=\int \frac{x^{8/3}}{\left (b+a \sqrt [3]{x}\right )^2} \, dx\\ &=3 \operatorname{Subst}\left (\int \frac{x^{10}}{(b+a x)^2} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{9 b^8}{a^{10}}-\frac{8 b^7 x}{a^9}+\frac{7 b^6 x^2}{a^8}-\frac{6 b^5 x^3}{a^7}+\frac{5 b^4 x^4}{a^6}-\frac{4 b^3 x^5}{a^5}+\frac{3 b^2 x^6}{a^4}-\frac{2 b x^7}{a^3}+\frac{x^8}{a^2}+\frac{b^{10}}{a^{10} (b+a x)^2}-\frac{10 b^9}{a^{10} (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{3 b^{10}}{a^{11} \left (b+a \sqrt [3]{x}\right )}+\frac{27 b^8 \sqrt [3]{x}}{a^{10}}-\frac{12 b^7 x^{2/3}}{a^9}+\frac{7 b^6 x}{a^8}-\frac{9 b^5 x^{4/3}}{2 a^7}+\frac{3 b^4 x^{5/3}}{a^6}-\frac{2 b^3 x^2}{a^5}+\frac{9 b^2 x^{7/3}}{7 a^4}-\frac{3 b x^{8/3}}{4 a^3}+\frac{x^3}{3 a^2}-\frac{30 b^9 \log \left (b+a \sqrt [3]{x}\right )}{a^{11}}\\ \end{align*}

Mathematica [A]  time = 0.183681, size = 148, normalized size = 0.99 \[ \frac{a \left (-378 a^3 b^5 x^{4/3}+252 a^4 b^4 x^{5/3}-168 a^5 b^3 x^2+108 a^6 b^2 x^{7/3}+588 a^2 b^6 x-63 a^7 b x^{8/3}+28 a^8 x^3-1008 a b^7 x^{2/3}+\frac{252 b^9}{a+\frac{b}{\sqrt [3]{x}}}+2268 b^8 \sqrt [3]{x}\right )-2520 b^9 \log \left (a+\frac{b}{\sqrt [3]{x}}\right )-840 b^9 \log (x)}{84 a^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b/x^(1/3))^2,x]

[Out]

(a*((252*b^9)/(a + b/x^(1/3)) + 2268*b^8*x^(1/3) - 1008*a*b^7*x^(2/3) + 588*a^2*b^6*x - 378*a^3*b^5*x^(4/3) +
252*a^4*b^4*x^(5/3) - 168*a^5*b^3*x^2 + 108*a^6*b^2*x^(7/3) - 63*a^7*b*x^(8/3) + 28*a^8*x^3) - 2520*b^9*Log[a
+ b/x^(1/3)] - 840*b^9*Log[x])/(84*a^11)

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Maple [A]  time = 0.008, size = 127, normalized size = 0.9 \begin{align*} -3\,{\frac{{b}^{10}}{{a}^{11} \left ( b+a\sqrt [3]{x} \right ) }}+27\,{\frac{{b}^{8}\sqrt [3]{x}}{{a}^{10}}}-12\,{\frac{{b}^{7}{x}^{2/3}}{{a}^{9}}}+7\,{\frac{{b}^{6}x}{{a}^{8}}}-{\frac{9\,{b}^{5}}{2\,{a}^{7}}{x}^{{\frac{4}{3}}}}+3\,{\frac{{b}^{4}{x}^{5/3}}{{a}^{6}}}-2\,{\frac{{b}^{3}{x}^{2}}{{a}^{5}}}+{\frac{9\,{b}^{2}}{7\,{a}^{4}}{x}^{{\frac{7}{3}}}}-{\frac{3\,b}{4\,{a}^{3}}{x}^{{\frac{8}{3}}}}+{\frac{{x}^{3}}{3\,{a}^{2}}}-30\,{\frac{{b}^{9}\ln \left ( b+a\sqrt [3]{x} \right ) }{{a}^{11}}} \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*b^10/a^11/(b+a*x^(1/3))+27*b^8*x^(1/3)/a^10-12*b^7*x^(2/3)/a^9+7*b^6*x/a^8-9/2*b^5*x^(4/3)/a^7+3*b^4*x^(5/3
)/a^6-2*b^3*x^2/a^5+9/7*b^2*x^(7/3)/a^4-3/4*b*x^(8/3)/a^3+1/3*x^3/a^2-30*b^9*ln(b+a*x^(1/3))/a^11

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Maxima [A]  time = 1.0255, size = 196, normalized size = 1.31 \begin{align*} \frac{28 \, a^{9} - \frac{35 \, a^{8} b}{x^{\frac{1}{3}}} + \frac{45 \, a^{7} b^{2}}{x^{\frac{2}{3}}} - \frac{60 \, a^{6} b^{3}}{x} + \frac{84 \, a^{5} b^{4}}{x^{\frac{4}{3}}} - \frac{126 \, a^{4} b^{5}}{x^{\frac{5}{3}}} + \frac{210 \, a^{3} b^{6}}{x^{2}} - \frac{420 \, a^{2} b^{7}}{x^{\frac{7}{3}}} + \frac{1260 \, a b^{8}}{x^{\frac{8}{3}}} + \frac{2520 \, b^{9}}{x^{3}}}{84 \,{\left (\frac{a^{11}}{x^{3}} + \frac{a^{10} b}{x^{\frac{10}{3}}}\right )}} - \frac{30 \, b^{9} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{a^{11}} - \frac{10 \, b^{9} \log \left (x\right )}{a^{11}} \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]

1/84*(28*a^9 - 35*a^8*b/x^(1/3) + 45*a^7*b^2/x^(2/3) - 60*a^6*b^3/x + 84*a^5*b^4/x^(4/3) - 126*a^4*b^5/x^(5/3)
+ 210*a^3*b^6/x^2 - 420*a^2*b^7/x^(7/3) + 1260*a*b^8/x^(8/3) + 2520*b^9/x^3)/(a^11/x^3 + a^10*b/x^(10/3)) - 3
0*b^9*log(a + b/x^(1/3))/a^11 - 10*b^9*log(x)/a^11

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Fricas [A]  time = 1.56666, size = 398, normalized size = 2.65 \begin{align*} \frac{28 \, a^{12} x^{4} - 140 \, a^{9} b^{3} x^{3} + 420 \, a^{6} b^{6} x^{2} + 588 \, a^{3} b^{9} x - 252 \, b^{12} - 2520 \,{\left (a^{3} b^{9} x + b^{12}\right )} \log \left (a x^{\frac{1}{3}} + b\right ) - 63 \,{\left (a^{11} b x^{3} - 3 \, a^{8} b^{4} x^{2} + 12 \, a^{5} b^{7} x + 20 \, a^{2} b^{10}\right )} x^{\frac{2}{3}} + 18 \,{\left (6 \, a^{10} b^{2} x^{3} - 15 \, a^{7} b^{5} x^{2} + 105 \, a^{4} b^{8} x + 140 \, a b^{11}\right )} x^{\frac{1}{3}}}{84 \,{\left (a^{14} x + a^{11} b^{3}\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]

1/84*(28*a^12*x^4 - 140*a^9*b^3*x^3 + 420*a^6*b^6*x^2 + 588*a^3*b^9*x - 252*b^12 - 2520*(a^3*b^9*x + b^12)*log
(a*x^(1/3) + b) - 63*(a^11*b*x^3 - 3*a^8*b^4*x^2 + 12*a^5*b^7*x + 20*a^2*b^10)*x^(2/3) + 18*(6*a^10*b^2*x^3 -
15*a^7*b^5*x^2 + 105*a^4*b^8*x + 140*a*b^11)*x^(1/3))/(a^14*x + a^11*b^3)

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Sympy [A]  time = 6.14106, size = 374, normalized size = 2.49 \begin{align*} \begin{cases} \frac{28 a^{10} x^{\frac{10}{3}}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac{35 a^{9} b x^{3}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} + \frac{45 a^{8} b^{2} x^{\frac{8}{3}}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac{60 a^{7} b^{3} x^{\frac{7}{3}}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} + \frac{84 a^{6} b^{4} x^{2}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac{126 a^{5} b^{5} x^{\frac{5}{3}}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} + \frac{210 a^{4} b^{6} x^{\frac{4}{3}}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac{420 a^{3} b^{7} x}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} + \frac{1260 a^{2} b^{8} x^{\frac{2}{3}}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac{2520 a b^{9} \sqrt [3]{x} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} + \frac{2520 a b^{9} \sqrt [3]{x}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac{2520 b^{10} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} & \text{for}\: a \neq 0 \\\frac{3 x^{\frac{11}{3}}}{11 b^{2}} & \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))**2,x)

[Out]

Piecewise((28*a**10*x**(10/3)/(84*a**12*x**(1/3) + 84*a**11*b) - 35*a**9*b*x**3/(84*a**12*x**(1/3) + 84*a**11*
b) + 45*a**8*b**2*x**(8/3)/(84*a**12*x**(1/3) + 84*a**11*b) - 60*a**7*b**3*x**(7/3)/(84*a**12*x**(1/3) + 84*a*
*11*b) + 84*a**6*b**4*x**2/(84*a**12*x**(1/3) + 84*a**11*b) - 126*a**5*b**5*x**(5/3)/(84*a**12*x**(1/3) + 84*a
**11*b) + 210*a**4*b**6*x**(4/3)/(84*a**12*x**(1/3) + 84*a**11*b) - 420*a**3*b**7*x/(84*a**12*x**(1/3) + 84*a*
*11*b) + 1260*a**2*b**8*x**(2/3)/(84*a**12*x**(1/3) + 84*a**11*b) - 2520*a*b**9*x**(1/3)*log(x**(1/3) + b/a)/(
84*a**12*x**(1/3) + 84*a**11*b) + 2520*a*b**9*x**(1/3)/(84*a**12*x**(1/3) + 84*a**11*b) - 2520*b**10*log(x**(1
/3) + b/a)/(84*a**12*x**(1/3) + 84*a**11*b), Ne(a, 0)), (3*x**(11/3)/(11*b**2), True))

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Giac [A]  time = 1.21109, size = 180, normalized size = 1.2 \begin{align*} -\frac{30 \, b^{9} \log \left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{a^{11}} - \frac{3 \, b^{10}}{{\left (a x^{\frac{1}{3}} + b\right )} a^{11}} + \frac{28 \, a^{16} x^{3} - 63 \, a^{15} b x^{\frac{8}{3}} + 108 \, a^{14} b^{2} x^{\frac{7}{3}} - 168 \, a^{13} b^{3} x^{2} + 252 \, a^{12} b^{4} x^{\frac{5}{3}} - 378 \, a^{11} b^{5} x^{\frac{4}{3}} + 588 \, a^{10} b^{6} x - 1008 \, a^{9} b^{7} x^{\frac{2}{3}} + 2268 \, a^{8} b^{8} x^{\frac{1}{3}}}{84 \, a^{18}} \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]

-30*b^9*log(abs(a*x^(1/3) + b))/a^11 - 3*b^10/((a*x^(1/3) + b)*a^11) + 1/84*(28*a^16*x^3 - 63*a^15*b*x^(8/3) +
108*a^14*b^2*x^(7/3) - 168*a^13*b^3*x^2 + 252*a^12*b^4*x^(5/3) - 378*a^11*b^5*x^(4/3) + 588*a^10*b^6*x - 1008
*a^9*b^7*x^(2/3) + 2268*a^8*b^8*x^(1/3))/a^18