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

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

[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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Rubi [A]  time = 0.13, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 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])

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]]

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*}

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Mathematica [A]  time = 0.20, size = 148, normalized size = 0.99 \[ \frac {a \left (28 a^8 x^3-63 a^7 b x^{8/3}+108 a^6 b^2 x^{7/3}-168 a^5 b^3 x^2+252 a^4 b^4 x^{5/3}-378 a^3 b^5 x^{4/3}+588 a^2 b^6 x+\frac {252 b^9}{a+\frac {b}{\sqrt [3]{x}}}-1008 a b^7 x^{2/3}+2268 b^8 \sqrt [3]{x}\right )-2520 b^9 \log \left (a+\frac {b}{\sqrt [3]{x}}\right )-840 b^9 \log (x)}{84 a^{11}} \]

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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fricas [A]  time = 0.66, size = 169, normalized size = 1.13 \[ \frac {28 \, a^{12} x^{4} - 140 \, a^{9} b^{3} x^{3} + 420 \, a^{6} b^{6} x^{2} + 588 \, a^{3} b^{9} x - 252 \, b^{12} - 2520 \, {\left (a^{3} b^{9} x + b^{12}\right )} \log \left (a x^{\frac {1}{3}} + b\right ) - 63 \, {\left (a^{11} b x^{3} - 3 \, a^{8} b^{4} x^{2} + 12 \, a^{5} b^{7} x + 20 \, a^{2} b^{10}\right )} x^{\frac {2}{3}} + 18 \, {\left (6 \, a^{10} b^{2} x^{3} - 15 \, a^{7} b^{5} x^{2} + 105 \, a^{4} b^{8} x + 140 \, a b^{11}\right )} x^{\frac {1}{3}}}{84 \, {\left (a^{14} x + a^{11} b^{3}\right )}} \]

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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giac [A]  time = 0.17, size = 133, normalized size = 0.89 \[ -\frac {30 \, b^{9} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{a^{11}} - \frac {3 \, b^{10}}{{\left (a x^{\frac {1}{3}} + b\right )} a^{11}} + \frac {28 \, a^{16} x^{3} - 63 \, a^{15} b x^{\frac {8}{3}} + 108 \, a^{14} b^{2} x^{\frac {7}{3}} - 168 \, a^{13} b^{3} x^{2} + 252 \, a^{12} b^{4} x^{\frac {5}{3}} - 378 \, a^{11} b^{5} x^{\frac {4}{3}} + 588 \, a^{10} b^{6} x - 1008 \, a^{9} b^{7} x^{\frac {2}{3}} + 2268 \, a^{8} b^{8} x^{\frac {1}{3}}}{84 \, a^{18}} \]

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

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maple [A]  time = 0.01, size = 127, normalized size = 0.85 \[ \frac {x^{3}}{3 a^{2}}-\frac {3 b \,x^{\frac {8}{3}}}{4 a^{3}}+\frac {9 b^{2} x^{\frac {7}{3}}}{7 a^{4}}-\frac {2 b^{3} x^{2}}{a^{5}}+\frac {3 b^{4} x^{\frac {5}{3}}}{a^{6}}-\frac {9 b^{5} x^{\frac {4}{3}}}{2 a^{7}}+\frac {7 b^{6} x}{a^{8}}-\frac {3 b^{10}}{\left (a \,x^{\frac {1}{3}}+b \right ) a^{11}}-\frac {30 b^{9} \ln \left (a \,x^{\frac {1}{3}}+b \right )}{a^{11}}-\frac {12 b^{7} x^{\frac {2}{3}}}{a^{9}}+\frac {27 b^{8} x^{\frac {1}{3}}}{a^{10}} \]

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/(a*x^(1/3)+b)+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/a^5*b^3*x^2+9/7*b^2*x^(7/3)/a^4-3/4*b*x^(8/3)/a^3+1/3/a^2*x^3-30*b^9*ln(a*x^(1/3)+b)/a^11

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maxima [A]  time = 0.48, size = 145, normalized size = 0.97 \[ \frac {28 \, a^{9} - \frac {35 \, a^{8} b}{x^{\frac {1}{3}}} + \frac {45 \, a^{7} b^{2}}{x^{\frac {2}{3}}} - \frac {60 \, a^{6} b^{3}}{x} + \frac {84 \, a^{5} b^{4}}{x^{\frac {4}{3}}} - \frac {126 \, a^{4} b^{5}}{x^{\frac {5}{3}}} + \frac {210 \, a^{3} b^{6}}{x^{2}} - \frac {420 \, a^{2} b^{7}}{x^{\frac {7}{3}}} + \frac {1260 \, a b^{8}}{x^{\frac {8}{3}}} + \frac {2520 \, b^{9}}{x^{3}}}{84 \, {\left (\frac {a^{11}}{x^{3}} + \frac {a^{10} b}{x^{\frac {10}{3}}}\right )}} - \frac {30 \, b^{9} \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{a^{11}} - \frac {10 \, b^{9} \log \relax (x)}{a^{11}} \]

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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mupad [B]  time = 0.07, size = 132, normalized size = 0.88 \[ \frac {x^3}{3\,a^2}-\frac {3\,b^{10}}{a\,\left (a^{10}\,b+a^{11}\,x^{1/3}\right )}-\frac {3\,b\,x^{8/3}}{4\,a^3}+\frac {7\,b^6\,x}{a^8}-\frac {30\,b^9\,\ln \left (b+a\,x^{1/3}\right )}{a^{11}}-\frac {2\,b^3\,x^2}{a^5}+\frac {9\,b^2\,x^{7/3}}{7\,a^4}+\frac {3\,b^4\,x^{5/3}}{a^6}-\frac {9\,b^5\,x^{4/3}}{2\,a^7}-\frac {12\,b^7\,x^{2/3}}{a^9}+\frac {27\,b^8\,x^{1/3}}{a^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a + b/x^(1/3))^2,x)

[Out]

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

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sympy [A]  time = 7.05, size = 367, normalized size = 2.45 \[ \begin {cases} \frac {28 a^{10} x^{\frac {10}{3}}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac {35 a^{9} b x^{3}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} + \frac {45 a^{8} b^{2} x^{\frac {8}{3}}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac {60 a^{7} b^{3} x^{\frac {7}{3}}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} + \frac {84 a^{6} b^{4} x^{2}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac {126 a^{5} b^{5} x^{\frac {5}{3}}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} + \frac {210 a^{4} b^{6} x^{\frac {4}{3}}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac {420 a^{3} b^{7} x}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} + \frac {1260 a^{2} b^{8} x^{\frac {2}{3}}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac {2520 a b^{9} \sqrt [3]{x} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac {2520 b^{10} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac {2520 b^{10}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} & \text {for}\: a \neq 0 \\\frac {3 x^{\frac {11}{3}}}{11 b^{2}} & \text {otherwise} \end {cases} \]

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*b**10*log(x**(1/3) + b/a)/(84*a**12*x**(1/3) + 84*a**11*b) - 2520*b**10
/(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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