Integrand size = 15, antiderivative size = 150 \[ \int \frac {x^2}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^2} \, dx=-\frac {3 b^{10}}{a^{11} \left (b+a \sqrt [3]{x}\right )}+\frac {27 b^8 \sqrt [3]{x}}{a^{10}}-\frac {12 b^7 x^{2/3}}{a^9}+\frac {7 b^6 x}{a^8}-\frac {9 b^5 x^{4/3}}{2 a^7}+\frac {3 b^4 x^{5/3}}{a^6}-\frac {2 b^3 x^2}{a^5}+\frac {9 b^2 x^{7/3}}{7 a^4}-\frac {3 b x^{8/3}}{4 a^3}+\frac {x^3}{3 a^2}-\frac {30 b^9 \log \left (b+a \sqrt [3]{x}\right )}{a^{11}} \] Output:
-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
Time = 0.08 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.05 \[ \int \frac {x^2}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^2} \, dx=\frac {-252 b^{10}+2268 a b^9 \sqrt [3]{x}+1260 a^2 b^8 x^{2/3}-420 a^3 b^7 x+210 a^4 b^6 x^{4/3}-126 a^5 b^5 x^{5/3}+84 a^6 b^4 x^2-60 a^7 b^3 x^{7/3}+45 a^8 b^2 x^{8/3}-35 a^9 b x^3+28 a^{10} x^{10/3}}{84 a^{11} \left (b+a \sqrt [3]{x}\right )}-\frac {30 b^9 \log \left (b+a \sqrt [3]{x}\right )}{a^{11}} \] Input:
Integrate[x^2/(a + b/x^(1/3))^2,x]
Output:
(-252*b^10 + 2268*a*b^9*x^(1/3) + 1260*a^2*b^8*x^(2/3) - 420*a^3*b^7*x + 2 10*a^4*b^6*x^(4/3) - 126*a^5*b^5*x^(5/3) + 84*a^6*b^4*x^2 - 60*a^7*b^3*x^( 7/3) + 45*a^8*b^2*x^(8/3) - 35*a^9*b*x^3 + 28*a^10*x^(10/3))/(84*a^11*(b + a*x^(1/3))) - (30*b^9*Log[b + a*x^(1/3)])/a^11
Time = 0.52 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {795, 798, 49, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^2}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^2} \, dx\) |
\(\Big \downarrow \) 795 |
\(\displaystyle \int \frac {x^{8/3}}{\left (a \sqrt [3]{x}+b\right )^2}dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 3 \int \frac {x^{10/3}}{\left (\sqrt [3]{x} a+b\right )^2}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle 3 \int \left (\frac {b^{10}}{a^{10} \left (\sqrt [3]{x} a+b\right )^2}-\frac {10 b^9}{a^{10} \left (\sqrt [3]{x} a+b\right )}+\frac {9 b^8}{a^{10}}-\frac {8 \sqrt [3]{x} b^7}{a^9}+\frac {7 x^{2/3} b^6}{a^8}-\frac {6 x b^5}{a^7}+\frac {5 x^{4/3} b^4}{a^6}-\frac {4 x^{5/3} b^3}{a^5}+\frac {3 x^2 b^2}{a^4}-\frac {2 x^{7/3} b}{a^3}+\frac {x^{8/3}}{a^2}\right )d\sqrt [3]{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \left (-\frac {b^{10}}{a^{11} \left (a \sqrt [3]{x}+b\right )}-\frac {10 b^9 \log \left (a \sqrt [3]{x}+b\right )}{a^{11}}+\frac {9 b^8 \sqrt [3]{x}}{a^{10}}-\frac {4 b^7 x^{2/3}}{a^9}+\frac {7 b^6 x}{3 a^8}-\frac {3 b^5 x^{4/3}}{2 a^7}+\frac {b^4 x^{5/3}}{a^6}-\frac {2 b^3 x^2}{3 a^5}+\frac {3 b^2 x^{7/3}}{7 a^4}-\frac {b x^{8/3}}{4 a^3}+\frac {x^3}{9 a^2}\right )\) |
Input:
Int[x^2/(a + b/x^(1/3))^2,x]
Output:
3*(-(b^10/(a^11*(b + a*x^(1/3)))) + (9*b^8*x^(1/3))/a^10 - (4*b^7*x^(2/3)) /a^9 + (7*b^6*x)/(3*a^8) - (3*b^5*x^(4/3))/(2*a^7) + (b^4*x^(5/3))/a^6 - ( 2*b^3*x^2)/(3*a^5) + (3*b^2*x^(7/3))/(7*a^4) - (b*x^(8/3))/(4*a^3) + x^3/( 9*a^2) - (10*b^9*Log[b + a*x^(1/3)])/a^11)
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}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
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]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[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]]
Time = 0.57 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(\frac {\frac {a^{8} x^{3}}{3}-\frac {3 b \,x^{\frac {8}{3}} a^{7}}{4}+\frac {9 b^{2} x^{\frac {7}{3}} a^{6}}{7}-2 a^{5} b^{3} x^{2}+3 x^{\frac {5}{3}} a^{4} b^{4}-\frac {9 b^{5} x^{\frac {4}{3}} a^{3}}{2}+7 a^{2} b^{6} x -12 a \,b^{7} x^{\frac {2}{3}}+27 b^{8} x^{\frac {1}{3}}}{a^{10}}-\frac {30 b^{9} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{a^{11}}-\frac {3 b^{10}}{a^{11} \left (b +a \,x^{\frac {1}{3}}\right )}\) | \(127\) |
default | \(\frac {\frac {a^{8} x^{3}}{3}-\frac {3 b \,x^{\frac {8}{3}} a^{7}}{4}+\frac {9 b^{2} x^{\frac {7}{3}} a^{6}}{7}-2 a^{5} b^{3} x^{2}+3 x^{\frac {5}{3}} a^{4} b^{4}-\frac {9 b^{5} x^{\frac {4}{3}} a^{3}}{2}+7 a^{2} b^{6} x -12 a \,b^{7} x^{\frac {2}{3}}+27 b^{8} x^{\frac {1}{3}}}{a^{10}}-\frac {30 b^{9} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{a^{11}}-\frac {3 b^{10}}{a^{11} \left (b +a \,x^{\frac {1}{3}}\right )}\) | \(127\) |
Input:
int(x^2/(a+b/x^(1/3))^2,x,method=_RETURNVERBOSE)
Output:
3/a^10*(1/9*a^8*x^3-1/4*b*x^(8/3)*a^7+3/7*b^2*x^(7/3)*a^6-2/3*a^5*b^3*x^2+ x^(5/3)*a^4*b^4-3/2*b^5*x^(4/3)*a^3+7/3*a^2*b^6*x-4*a*b^7*x^(2/3)+9*b^8*x^ (1/3))-30*b^9*ln(b+a*x^(1/3))/a^11-3*b^10/a^11/(b+a*x^(1/3))
Time = 0.08 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.13 \[ \int \frac {x^2}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^2} \, dx=\frac {28 \, a^{12} x^{4} - 140 \, a^{9} b^{3} x^{3} + 420 \, a^{6} b^{6} x^{2} + 588 \, a^{3} b^{9} x - 252 \, b^{12} - 2520 \, {\left (a^{3} b^{9} x + b^{12}\right )} \log \left (a x^{\frac {1}{3}} + b\right ) - 63 \, {\left (a^{11} b x^{3} - 3 \, a^{8} b^{4} x^{2} + 12 \, a^{5} b^{7} x + 20 \, a^{2} b^{10}\right )} x^{\frac {2}{3}} + 18 \, {\left (6 \, a^{10} b^{2} x^{3} - 15 \, a^{7} b^{5} x^{2} + 105 \, a^{4} b^{8} x + 140 \, a b^{11}\right )} x^{\frac {1}{3}}}{84 \, {\left (a^{14} x + a^{11} b^{3}\right )}} \] Input:
integrate(x^2/(a+b/x^(1/3))^2,x, algorithm="fricas")
Output:
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 - 25 2*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)
Leaf count of result is larger than twice the leaf count of optimal. 367 vs. \(2 (150) = 300\).
Time = 0.95 (sec) , antiderivative size = 367, normalized size of antiderivative = 2.45 \[ \int \frac {x^2}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^2} \, dx=\begin {cases} \frac {28 a^{10} x^{\frac {10}{3}}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac {35 a^{9} b x^{3}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} + \frac {45 a^{8} b^{2} x^{\frac {8}{3}}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac {60 a^{7} b^{3} x^{\frac {7}{3}}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} + \frac {84 a^{6} b^{4} x^{2}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac {126 a^{5} b^{5} x^{\frac {5}{3}}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} + \frac {210 a^{4} b^{6} x^{\frac {4}{3}}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac {420 a^{3} b^{7} x}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} + \frac {1260 a^{2} b^{8} x^{\frac {2}{3}}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac {2520 a b^{9} \sqrt [3]{x} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac {2520 b^{10} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac {2520 b^{10}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} & \text {for}\: a \neq 0 \\\frac {3 x^{\frac {11}{3}}}{11 b^{2}} & \text {otherwise} \end {cases} \] Input:
integrate(x**2/(a+b/x**(1/3))**2,x)
Output:
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**1 1*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))
Time = 0.04 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.97 \[ \int \frac {x^2}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^2} \, dx=\frac {28 \, a^{9} - \frac {35 \, a^{8} b}{x^{\frac {1}{3}}} + \frac {45 \, a^{7} b^{2}}{x^{\frac {2}{3}}} - \frac {60 \, a^{6} b^{3}}{x} + \frac {84 \, a^{5} b^{4}}{x^{\frac {4}{3}}} - \frac {126 \, a^{4} b^{5}}{x^{\frac {5}{3}}} + \frac {210 \, a^{3} b^{6}}{x^{2}} - \frac {420 \, a^{2} b^{7}}{x^{\frac {7}{3}}} + \frac {1260 \, a b^{8}}{x^{\frac {8}{3}}} + \frac {2520 \, b^{9}}{x^{3}}}{84 \, {\left (\frac {a^{11}}{x^{3}} + \frac {a^{10} b}{x^{\frac {10}{3}}}\right )}} - \frac {30 \, b^{9} \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{a^{11}} - \frac {10 \, b^{9} \log \left (x\right )}{a^{11}} \] Input:
integrate(x^2/(a+b/x^(1/3))^2,x, algorithm="maxima")
Output:
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)) - 30* b^9*log(a + b/x^(1/3))/a^11 - 10*b^9*log(x)/a^11
Time = 0.12 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.89 \[ \int \frac {x^2}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^2} \, dx=-\frac {30 \, b^{9} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{a^{11}} - \frac {3 \, b^{10}}{{\left (a x^{\frac {1}{3}} + b\right )} a^{11}} + \frac {28 \, a^{16} x^{3} - 63 \, a^{15} b x^{\frac {8}{3}} + 108 \, a^{14} b^{2} x^{\frac {7}{3}} - 168 \, a^{13} b^{3} x^{2} + 252 \, a^{12} b^{4} x^{\frac {5}{3}} - 378 \, a^{11} b^{5} x^{\frac {4}{3}} + 588 \, a^{10} b^{6} x - 1008 \, a^{9} b^{7} x^{\frac {2}{3}} + 2268 \, a^{8} b^{8} x^{\frac {1}{3}}}{84 \, a^{18}} \] Input:
integrate(x^2/(a+b/x^(1/3))^2,x, algorithm="giac")
Output:
-30*b^9*log(abs(a*x^(1/3) + b))/a^11 - 3*b^10/((a*x^(1/3) + b)*a^11) + 1/8 4*(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
Time = 0.40 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.88 \[ \int \frac {x^2}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^2} \, dx=\frac {x^3}{3\,a^2}-\frac {3\,b^{10}}{a\,\left (a^{10}\,b+a^{11}\,x^{1/3}\right )}-\frac {3\,b\,x^{8/3}}{4\,a^3}+\frac {7\,b^6\,x}{a^8}-\frac {30\,b^9\,\ln \left (b+a\,x^{1/3}\right )}{a^{11}}-\frac {2\,b^3\,x^2}{a^5}+\frac {9\,b^2\,x^{7/3}}{7\,a^4}+\frac {3\,b^4\,x^{5/3}}{a^6}-\frac {9\,b^5\,x^{4/3}}{2\,a^7}-\frac {12\,b^7\,x^{2/3}}{a^9}+\frac {27\,b^8\,x^{1/3}}{a^{10}} \] Input:
int(x^2/(a + b/x^(1/3))^2,x)
Output:
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) - (1 2*b^7*x^(2/3))/a^9 + (27*b^8*x^(1/3))/a^10
Time = 0.21 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.97 \[ \int \frac {x^2}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^2} \, dx=\frac {45 x^{\frac {8}{3}} a^{8} b^{2}-126 x^{\frac {5}{3}} a^{5} b^{5}+1260 x^{\frac {2}{3}} a^{2} b^{8}-2520 x^{\frac {1}{3}} \mathrm {log}\left (x^{\frac {1}{3}} a +b \right ) a \,b^{9}+28 x^{\frac {10}{3}} a^{10}-60 x^{\frac {7}{3}} a^{7} b^{3}+210 x^{\frac {4}{3}} a^{4} b^{6}+2520 x^{\frac {1}{3}} a \,b^{9}-2520 \,\mathrm {log}\left (x^{\frac {1}{3}} a +b \right ) b^{10}-35 a^{9} b \,x^{3}+84 a^{6} b^{4} x^{2}-420 a^{3} b^{7} x}{84 a^{11} \left (x^{\frac {1}{3}} a +b \right )} \] Input:
int(x^2/(a+b/x^(1/3))^2,x)
Output:
(45*x**(2/3)*a**8*b**2*x**2 - 126*x**(2/3)*a**5*b**5*x + 1260*x**(2/3)*a** 2*b**8 - 2520*x**(1/3)*log(x**(1/3)*a + b)*a*b**9 + 28*x**(1/3)*a**10*x**3 - 60*x**(1/3)*a**7*b**3*x**2 + 210*x**(1/3)*a**4*b**6*x + 2520*x**(1/3)*a *b**9 - 2520*log(x**(1/3)*a + b)*b**10 - 35*a**9*b*x**3 + 84*a**6*b**4*x** 2 - 420*a**3*b**7*x)/(84*a**11*(x**(1/3)*a + b))