Integrand size = 15, antiderivative size = 134 \[ \int \frac {x^2}{\left (a+b \sqrt [3]{x}\right )^3} \, dx=-\frac {3 a^8}{2 b^9 \left (a+b \sqrt [3]{x}\right )^2}+\frac {24 a^7}{b^9 \left (a+b \sqrt [3]{x}\right )}-\frac {63 a^5 \sqrt [3]{x}}{b^8}+\frac {45 a^4 x^{2/3}}{2 b^7}-\frac {10 a^3 x}{b^6}+\frac {9 a^2 x^{4/3}}{2 b^5}-\frac {9 a x^{5/3}}{5 b^4}+\frac {x^2}{2 b^3}+\frac {84 a^6 \log \left (a+b \sqrt [3]{x}\right )}{b^9} \]
-3/2*a^8/b^9/(a+b*x^(1/3))^2+24*a^7/b^9/(a+b*x^(1/3))-63*a^5*x^(1/3)/b^8+4 5/2*a^4*x^(2/3)/b^7-10*a^3*x/b^6+9/2*a^2*x^(4/3)/b^5-9/5*a*x^(5/3)/b^4+1/2 *x^2/b^3+84*a^6*ln(a+b*x^(1/3))/b^9
Time = 0.09 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\left (a+b \sqrt [3]{x}\right )^3} \, dx=\frac {225 a^8-390 a^7 b \sqrt [3]{x}-1035 a^6 b^2 x^{2/3}-280 a^5 b^3 x+70 a^4 b^4 x^{4/3}-28 a^3 b^5 x^{5/3}+14 a^2 b^6 x^2-8 a b^7 x^{7/3}+5 b^8 x^{8/3}}{10 b^9 \left (a+b \sqrt [3]{x}\right )^2}+\frac {84 a^6 \log \left (a+b \sqrt [3]{x}\right )}{b^9} \]
(225*a^8 - 390*a^7*b*x^(1/3) - 1035*a^6*b^2*x^(2/3) - 280*a^5*b^3*x + 70*a ^4*b^4*x^(4/3) - 28*a^3*b^5*x^(5/3) + 14*a^2*b^6*x^2 - 8*a*b^7*x^(7/3) + 5 *b^8*x^(8/3))/(10*b^9*(a + b*x^(1/3))^2) + (84*a^6*Log[a + b*x^(1/3)])/b^9
Time = 0.29 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {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.
\(\displaystyle \int \frac {x^2}{\left (a+b \sqrt [3]{x}\right )^3} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 3 \int \frac {x^{8/3}}{\left (a+b \sqrt [3]{x}\right )^3}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle 3 \int \left (\frac {a^8}{b^8 \left (a+b \sqrt [3]{x}\right )^3}-\frac {8 a^7}{b^8 \left (a+b \sqrt [3]{x}\right )^2}+\frac {28 a^6}{b^8 \left (a+b \sqrt [3]{x}\right )}-\frac {21 a^5}{b^8}+\frac {15 \sqrt [3]{x} a^4}{b^7}-\frac {10 x^{2/3} a^3}{b^6}+\frac {6 x a^2}{b^5}-\frac {3 x^{4/3} a}{b^4}+\frac {x^{5/3}}{b^3}\right )d\sqrt [3]{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \left (-\frac {a^8}{2 b^9 \left (a+b \sqrt [3]{x}\right )^2}+\frac {8 a^7}{b^9 \left (a+b \sqrt [3]{x}\right )}+\frac {28 a^6 \log \left (a+b \sqrt [3]{x}\right )}{b^9}-\frac {21 a^5 \sqrt [3]{x}}{b^8}+\frac {15 a^4 x^{2/3}}{2 b^7}-\frac {10 a^3 x}{3 b^6}+\frac {3 a^2 x^{4/3}}{2 b^5}-\frac {3 a x^{5/3}}{5 b^4}+\frac {x^2}{6 b^3}\right )\) |
3*(-1/2*a^8/(b^9*(a + b*x^(1/3))^2) + (8*a^7)/(b^9*(a + b*x^(1/3))) - (21* a^5*x^(1/3))/b^8 + (15*a^4*x^(2/3))/(2*b^7) - (10*a^3*x)/(3*b^6) + (3*a^2* x^(4/3))/(2*b^5) - (3*a*x^(5/3))/(5*b^4) + x^2/(6*b^3) + (28*a^6*Log[a + b *x^(1/3)])/b^9)
3.24.74.3.1 Defintions of rubi rules used
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] :> 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 = 12.85 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(-\frac {3 \left (-\frac {b^{5} x^{2}}{6}+\frac {3 a \,x^{\frac {5}{3}} b^{4}}{5}-\frac {3 a^{2} x^{\frac {4}{3}} b^{3}}{2}+\frac {10 a^{3} b^{2} x}{3}-\frac {15 a^{4} x^{\frac {2}{3}} b}{2}+21 a^{5} x^{\frac {1}{3}}\right )}{b^{8}}+\frac {84 a^{6} \ln \left (a +b \,x^{\frac {1}{3}}\right )}{b^{9}}-\frac {3 a^{8}}{2 b^{9} \left (a +b \,x^{\frac {1}{3}}\right )^{2}}+\frac {24 a^{7}}{b^{9} \left (a +b \,x^{\frac {1}{3}}\right )}\) | \(112\) |
default | \(-\frac {3 \left (-\frac {b^{5} x^{2}}{6}+\frac {3 a \,x^{\frac {5}{3}} b^{4}}{5}-\frac {3 a^{2} x^{\frac {4}{3}} b^{3}}{2}+\frac {10 a^{3} b^{2} x}{3}-\frac {15 a^{4} x^{\frac {2}{3}} b}{2}+21 a^{5} x^{\frac {1}{3}}\right )}{b^{8}}+\frac {84 a^{6} \ln \left (a +b \,x^{\frac {1}{3}}\right )}{b^{9}}-\frac {3 a^{8}}{2 b^{9} \left (a +b \,x^{\frac {1}{3}}\right )^{2}}+\frac {24 a^{7}}{b^{9} \left (a +b \,x^{\frac {1}{3}}\right )}\) | \(112\) |
-3/b^8*(-1/6*b^5*x^2+3/5*a*x^(5/3)*b^4-3/2*a^2*x^(4/3)*b^3+10/3*a^3*b^2*x- 15/2*a^4*x^(2/3)*b+21*a^5*x^(1/3))+84*a^6*ln(a+b*x^(1/3))/b^9-3/2*a^8/b^9/ (a+b*x^(1/3))^2+24*a^7/b^9/(a+b*x^(1/3))
Time = 0.28 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.43 \[ \int \frac {x^2}{\left (a+b \sqrt [3]{x}\right )^3} \, dx=\frac {5 \, b^{12} x^{4} - 90 \, a^{3} b^{9} x^{3} - 195 \, a^{6} b^{6} x^{2} + 170 \, a^{9} b^{3} x + 225 \, a^{12} + 840 \, {\left (a^{6} b^{6} x^{2} + 2 \, a^{9} b^{3} x + a^{12}\right )} \log \left (b x^{\frac {1}{3}} + a\right ) - 3 \, {\left (6 \, a b^{11} x^{3} - 63 \, a^{4} b^{8} x^{2} - 224 \, a^{7} b^{5} x - 140 \, a^{10} b^{2}\right )} x^{\frac {2}{3}} + 15 \, {\left (3 \, a^{2} b^{10} x^{3} - 36 \, a^{5} b^{7} x^{2} - 98 \, a^{8} b^{4} x - 56 \, a^{11} b\right )} x^{\frac {1}{3}}}{10 \, {\left (b^{15} x^{2} + 2 \, a^{3} b^{12} x + a^{6} b^{9}\right )}} \]
1/10*(5*b^12*x^4 - 90*a^3*b^9*x^3 - 195*a^6*b^6*x^2 + 170*a^9*b^3*x + 225* a^12 + 840*(a^6*b^6*x^2 + 2*a^9*b^3*x + a^12)*log(b*x^(1/3) + a) - 3*(6*a* b^11*x^3 - 63*a^4*b^8*x^2 - 224*a^7*b^5*x - 140*a^10*b^2)*x^(2/3) + 15*(3* a^2*b^10*x^3 - 36*a^5*b^7*x^2 - 98*a^8*b^4*x - 56*a^11*b)*x^(1/3))/(b^15*x ^2 + 2*a^3*b^12*x + a^6*b^9)
Leaf count of result is larger than twice the leaf count of optimal. 493 vs. \(2 (131) = 262\).
Time = 0.56 (sec) , antiderivative size = 493, normalized size of antiderivative = 3.68 \[ \int \frac {x^2}{\left (a+b \sqrt [3]{x}\right )^3} \, dx=\begin {cases} \frac {840 a^{8} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac {2}{3}}} + \frac {1260 a^{8}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac {2}{3}}} + \frac {1680 a^{7} b \sqrt [3]{x} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac {2}{3}}} + \frac {1680 a^{7} b \sqrt [3]{x}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac {2}{3}}} + \frac {840 a^{6} b^{2} x^{\frac {2}{3}} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac {2}{3}}} - \frac {280 a^{5} b^{3} x}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac {2}{3}}} + \frac {70 a^{4} b^{4} x^{\frac {4}{3}}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac {2}{3}}} - \frac {28 a^{3} b^{5} x^{\frac {5}{3}}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac {2}{3}}} + \frac {14 a^{2} b^{6} x^{2}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac {2}{3}}} - \frac {8 a b^{7} x^{\frac {7}{3}}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac {2}{3}}} + \frac {5 b^{8} x^{\frac {8}{3}}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac {2}{3}}} & \text {for}\: b \neq 0 \\\frac {x^{3}}{3 a^{3}} & \text {otherwise} \end {cases} \]
Piecewise((840*a**8*log(a/b + x**(1/3))/(10*a**2*b**9 + 20*a*b**10*x**(1/3 ) + 10*b**11*x**(2/3)) + 1260*a**8/(10*a**2*b**9 + 20*a*b**10*x**(1/3) + 1 0*b**11*x**(2/3)) + 1680*a**7*b*x**(1/3)*log(a/b + x**(1/3))/(10*a**2*b**9 + 20*a*b**10*x**(1/3) + 10*b**11*x**(2/3)) + 1680*a**7*b*x**(1/3)/(10*a** 2*b**9 + 20*a*b**10*x**(1/3) + 10*b**11*x**(2/3)) + 840*a**6*b**2*x**(2/3) *log(a/b + x**(1/3))/(10*a**2*b**9 + 20*a*b**10*x**(1/3) + 10*b**11*x**(2/ 3)) - 280*a**5*b**3*x/(10*a**2*b**9 + 20*a*b**10*x**(1/3) + 10*b**11*x**(2 /3)) + 70*a**4*b**4*x**(4/3)/(10*a**2*b**9 + 20*a*b**10*x**(1/3) + 10*b**1 1*x**(2/3)) - 28*a**3*b**5*x**(5/3)/(10*a**2*b**9 + 20*a*b**10*x**(1/3) + 10*b**11*x**(2/3)) + 14*a**2*b**6*x**2/(10*a**2*b**9 + 20*a*b**10*x**(1/3) + 10*b**11*x**(2/3)) - 8*a*b**7*x**(7/3)/(10*a**2*b**9 + 20*a*b**10*x**(1 /3) + 10*b**11*x**(2/3)) + 5*b**8*x**(8/3)/(10*a**2*b**9 + 20*a*b**10*x**( 1/3) + 10*b**11*x**(2/3)), Ne(b, 0)), (x**3/(3*a**3), True))
Time = 0.20 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.09 \[ \int \frac {x^2}{\left (a+b \sqrt [3]{x}\right )^3} \, dx=\frac {84 \, a^{6} \log \left (b x^{\frac {1}{3}} + a\right )}{b^{9}} + \frac {{\left (b x^{\frac {1}{3}} + a\right )}^{6}}{2 \, b^{9}} - \frac {24 \, {\left (b x^{\frac {1}{3}} + a\right )}^{5} a}{5 \, b^{9}} + \frac {21 \, {\left (b x^{\frac {1}{3}} + a\right )}^{4} a^{2}}{b^{9}} - \frac {56 \, {\left (b x^{\frac {1}{3}} + a\right )}^{3} a^{3}}{b^{9}} + \frac {105 \, {\left (b x^{\frac {1}{3}} + a\right )}^{2} a^{4}}{b^{9}} - \frac {168 \, {\left (b x^{\frac {1}{3}} + a\right )} a^{5}}{b^{9}} + \frac {24 \, a^{7}}{{\left (b x^{\frac {1}{3}} + a\right )} b^{9}} - \frac {3 \, a^{8}}{2 \, {\left (b x^{\frac {1}{3}} + a\right )}^{2} b^{9}} \]
84*a^6*log(b*x^(1/3) + a)/b^9 + 1/2*(b*x^(1/3) + a)^6/b^9 - 24/5*(b*x^(1/3 ) + a)^5*a/b^9 + 21*(b*x^(1/3) + a)^4*a^2/b^9 - 56*(b*x^(1/3) + a)^3*a^3/b ^9 + 105*(b*x^(1/3) + a)^2*a^4/b^9 - 168*(b*x^(1/3) + a)*a^5/b^9 + 24*a^7/ ((b*x^(1/3) + a)*b^9) - 3/2*a^8/((b*x^(1/3) + a)^2*b^9)
Time = 0.27 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.84 \[ \int \frac {x^2}{\left (a+b \sqrt [3]{x}\right )^3} \, dx=\frac {84 \, a^{6} \log \left ({\left | b x^{\frac {1}{3}} + a \right |}\right )}{b^{9}} + \frac {3 \, {\left (16 \, a^{7} b x^{\frac {1}{3}} + 15 \, a^{8}\right )}}{2 \, {\left (b x^{\frac {1}{3}} + a\right )}^{2} b^{9}} + \frac {5 \, b^{15} x^{2} - 18 \, a b^{14} x^{\frac {5}{3}} + 45 \, a^{2} b^{13} x^{\frac {4}{3}} - 100 \, a^{3} b^{12} x + 225 \, a^{4} b^{11} x^{\frac {2}{3}} - 630 \, a^{5} b^{10} x^{\frac {1}{3}}}{10 \, b^{18}} \]
84*a^6*log(abs(b*x^(1/3) + a))/b^9 + 3/2*(16*a^7*b*x^(1/3) + 15*a^8)/((b*x ^(1/3) + a)^2*b^9) + 1/10*(5*b^15*x^2 - 18*a*b^14*x^(5/3) + 45*a^2*b^13*x^ (4/3) - 100*a^3*b^12*x + 225*a^4*b^11*x^(2/3) - 630*a^5*b^10*x^(1/3))/b^18
Time = 0.05 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.90 \[ \int \frac {x^2}{\left (a+b \sqrt [3]{x}\right )^3} \, dx=\frac {\frac {45\,a^8}{2\,b}+24\,a^7\,x^{1/3}}{a^2\,b^8+b^{10}\,x^{2/3}+2\,a\,b^9\,x^{1/3}}+\frac {x^2}{2\,b^3}-\frac {10\,a^3\,x}{b^6}-\frac {9\,a\,x^{5/3}}{5\,b^4}+\frac {84\,a^6\,\ln \left (a+b\,x^{1/3}\right )}{b^9}+\frac {9\,a^2\,x^{4/3}}{2\,b^5}+\frac {45\,a^4\,x^{2/3}}{2\,b^7}-\frac {63\,a^5\,x^{1/3}}{b^8} \]