Integrand size = 15, antiderivative size = 171 \[ \int \frac {x^3}{\left (a+b \sqrt [3]{x}\right )^2} \, dx=\frac {3 a^{11}}{b^{12} \left (a+b \sqrt [3]{x}\right )}-\frac {30 a^9 \sqrt [3]{x}}{b^{11}}+\frac {27 a^8 x^{2/3}}{2 b^{10}}-\frac {8 a^7 x}{b^9}+\frac {21 a^6 x^{4/3}}{4 b^8}-\frac {18 a^5 x^{5/3}}{5 b^7}+\frac {5 a^4 x^2}{2 b^6}-\frac {12 a^3 x^{7/3}}{7 b^5}+\frac {9 a^2 x^{8/3}}{8 b^4}-\frac {2 a x^3}{3 b^3}+\frac {3 x^{10/3}}{10 b^2}+\frac {33 a^{10} \log \left (a+b \sqrt [3]{x}\right )}{b^{12}} \]
3*a^11/b^12/(a+b*x^(1/3))-30*a^9*x^(1/3)/b^11+27/2*a^8*x^(2/3)/b^10-8*a^7* x/b^9+21/4*a^6*x^(4/3)/b^8-18/5*a^5*x^(5/3)/b^7+5/2*a^4*x^2/b^6-12/7*a^3*x ^(7/3)/b^5+9/8*a^2*x^(8/3)/b^4-2/3*a*x^3/b^3+3/10*x^(10/3)/b^2+33*a^10*ln( a+b*x^(1/3))/b^12
Time = 0.12 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00 \[ \int \frac {x^3}{\left (a+b \sqrt [3]{x}\right )^2} \, dx=\frac {2520 a^{11}-25200 a^{10} b \sqrt [3]{x}-13860 a^9 b^2 x^{2/3}+4620 a^8 b^3 x-2310 a^7 b^4 x^{4/3}+1386 a^6 b^5 x^{5/3}-924 a^5 b^6 x^2+660 a^4 b^7 x^{7/3}-495 a^3 b^8 x^{8/3}+385 a^2 b^9 x^3-308 a b^{10} x^{10/3}+252 b^{11} x^{11/3}}{840 b^{12} \left (a+b \sqrt [3]{x}\right )}+\frac {33 a^{10} \log \left (a+b \sqrt [3]{x}\right )}{b^{12}} \]
(2520*a^11 - 25200*a^10*b*x^(1/3) - 13860*a^9*b^2*x^(2/3) + 4620*a^8*b^3*x - 2310*a^7*b^4*x^(4/3) + 1386*a^6*b^5*x^(5/3) - 924*a^5*b^6*x^2 + 660*a^4 *b^7*x^(7/3) - 495*a^3*b^8*x^(8/3) + 385*a^2*b^9*x^3 - 308*a*b^10*x^(10/3) + 252*b^11*x^(11/3))/(840*b^12*(a + b*x^(1/3))) + (33*a^10*Log[a + b*x^(1 /3)])/b^12
Time = 0.33 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.02, 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.
\(\displaystyle \int \frac {x^3}{\left (a+b \sqrt [3]{x}\right )^2} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 3 \int \frac {x^{11/3}}{\left (a+b \sqrt [3]{x}\right )^2}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle 3 \int \left (-\frac {a^{11}}{b^{11} \left (a+b \sqrt [3]{x}\right )^2}+\frac {11 a^{10}}{b^{11} \left (a+b \sqrt [3]{x}\right )}-\frac {10 a^9}{b^{11}}+\frac {9 \sqrt [3]{x} a^8}{b^{10}}-\frac {8 x^{2/3} a^7}{b^9}+\frac {7 x a^6}{b^8}-\frac {6 x^{4/3} a^5}{b^7}+\frac {5 x^{5/3} a^4}{b^6}-\frac {4 x^2 a^3}{b^5}+\frac {3 x^{7/3} a^2}{b^4}-\frac {2 x^{8/3} a}{b^3}+\frac {x^3}{b^2}\right )d\sqrt [3]{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \left (\frac {a^{11}}{b^{12} \left (a+b \sqrt [3]{x}\right )}+\frac {11 a^{10} \log \left (a+b \sqrt [3]{x}\right )}{b^{12}}-\frac {10 a^9 \sqrt [3]{x}}{b^{11}}+\frac {9 a^8 x^{2/3}}{2 b^{10}}-\frac {8 a^7 x}{3 b^9}+\frac {7 a^6 x^{4/3}}{4 b^8}-\frac {6 a^5 x^{5/3}}{5 b^7}+\frac {5 a^4 x^2}{6 b^6}-\frac {4 a^3 x^{7/3}}{7 b^5}+\frac {3 a^2 x^{8/3}}{8 b^4}-\frac {2 a x^3}{9 b^3}+\frac {x^{10/3}}{10 b^2}\right )\) |
3*(a^11/(b^12*(a + b*x^(1/3))) - (10*a^9*x^(1/3))/b^11 + (9*a^8*x^(2/3))/( 2*b^10) - (8*a^7*x)/(3*b^9) + (7*a^6*x^(4/3))/(4*b^8) - (6*a^5*x^(5/3))/(5 *b^7) + (5*a^4*x^2)/(6*b^6) - (4*a^3*x^(7/3))/(7*b^5) + (3*a^2*x^(8/3))/(8 *b^4) - (2*a*x^3)/(9*b^3) + x^(10/3)/(10*b^2) + (11*a^10*Log[a + b*x^(1/3) ])/b^12)
3.24.65.3.1 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 = 3.52 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(-\frac {3 \left (-\frac {x^{\frac {10}{3}} b^{9}}{10}+\frac {2 a \,x^{3} b^{8}}{9}-\frac {3 a^{2} x^{\frac {8}{3}} b^{7}}{8}+\frac {4 a^{3} x^{\frac {7}{3}} b^{6}}{7}-\frac {5 a^{4} x^{2} b^{5}}{6}+\frac {6 a^{5} x^{\frac {5}{3}} b^{4}}{5}-\frac {7 a^{6} x^{\frac {4}{3}} b^{3}}{4}+\frac {8 a^{7} x \,b^{2}}{3}-\frac {9 a^{8} x^{\frac {2}{3}} b}{2}+10 a^{9} x^{\frac {1}{3}}\right )}{b^{11}}+\frac {33 a^{10} \ln \left (a +b \,x^{\frac {1}{3}}\right )}{b^{12}}+\frac {3 a^{11}}{b^{12} \left (a +b \,x^{\frac {1}{3}}\right )}\) | \(139\) |
default | \(-\frac {3 \left (-\frac {x^{\frac {10}{3}} b^{9}}{10}+\frac {2 a \,x^{3} b^{8}}{9}-\frac {3 a^{2} x^{\frac {8}{3}} b^{7}}{8}+\frac {4 a^{3} x^{\frac {7}{3}} b^{6}}{7}-\frac {5 a^{4} x^{2} b^{5}}{6}+\frac {6 a^{5} x^{\frac {5}{3}} b^{4}}{5}-\frac {7 a^{6} x^{\frac {4}{3}} b^{3}}{4}+\frac {8 a^{7} x \,b^{2}}{3}-\frac {9 a^{8} x^{\frac {2}{3}} b}{2}+10 a^{9} x^{\frac {1}{3}}\right )}{b^{11}}+\frac {33 a^{10} \ln \left (a +b \,x^{\frac {1}{3}}\right )}{b^{12}}+\frac {3 a^{11}}{b^{12} \left (a +b \,x^{\frac {1}{3}}\right )}\) | \(139\) |
-3/b^11*(-1/10*x^(10/3)*b^9+2/9*a*x^3*b^8-3/8*a^2*x^(8/3)*b^7+4/7*a^3*x^(7 /3)*b^6-5/6*a^4*x^2*b^5+6/5*a^5*x^(5/3)*b^4-7/4*a^6*x^(4/3)*b^3+8/3*a^7*x* b^2-9/2*a^8*x^(2/3)*b+10*a^9*x^(1/3))+33*a^10*ln(a+b*x^(1/3))/b^12+3*a^11/ b^12/(a+b*x^(1/3))
Time = 0.28 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.06 \[ \int \frac {x^3}{\left (a+b \sqrt [3]{x}\right )^2} \, dx=-\frac {560 \, a b^{12} x^{4} - 1540 \, a^{4} b^{9} x^{3} + 4620 \, a^{7} b^{6} x^{2} + 6720 \, a^{10} b^{3} x - 2520 \, a^{13} - 27720 \, {\left (a^{10} b^{3} x + a^{13}\right )} \log \left (b x^{\frac {1}{3}} + a\right ) - 63 \, {\left (15 \, a^{2} b^{11} x^{3} - 33 \, a^{5} b^{8} x^{2} + 132 \, a^{8} b^{5} x + 220 \, a^{11} b^{2}\right )} x^{\frac {2}{3}} - 18 \, {\left (14 \, b^{13} x^{4} - 66 \, a^{3} b^{10} x^{3} + 165 \, a^{6} b^{7} x^{2} - 1155 \, a^{9} b^{4} x - 1540 \, a^{12} b\right )} x^{\frac {1}{3}}}{840 \, {\left (b^{15} x + a^{3} b^{12}\right )}} \]
-1/840*(560*a*b^12*x^4 - 1540*a^4*b^9*x^3 + 4620*a^7*b^6*x^2 + 6720*a^10*b ^3*x - 2520*a^13 - 27720*(a^10*b^3*x + a^13)*log(b*x^(1/3) + a) - 63*(15*a ^2*b^11*x^3 - 33*a^5*b^8*x^2 + 132*a^8*b^5*x + 220*a^11*b^2)*x^(2/3) - 18* (14*b^13*x^4 - 66*a^3*b^10*x^3 + 165*a^6*b^7*x^2 - 1155*a^9*b^4*x - 1540*a ^12*b)*x^(1/3))/(b^15*x + a^3*b^12)
Timed out. \[ \int \frac {x^3}{\left (a+b \sqrt [3]{x}\right )^2} \, dx=\text {Timed out} \]
Time = 0.20 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.15 \[ \int \frac {x^3}{\left (a+b \sqrt [3]{x}\right )^2} \, dx=\frac {33 \, a^{10} \log \left (b x^{\frac {1}{3}} + a\right )}{b^{12}} + \frac {3 \, {\left (b x^{\frac {1}{3}} + a\right )}^{10}}{10 \, b^{12}} - \frac {11 \, {\left (b x^{\frac {1}{3}} + a\right )}^{9} a}{3 \, b^{12}} + \frac {165 \, {\left (b x^{\frac {1}{3}} + a\right )}^{8} a^{2}}{8 \, b^{12}} - \frac {495 \, {\left (b x^{\frac {1}{3}} + a\right )}^{7} a^{3}}{7 \, b^{12}} + \frac {165 \, {\left (b x^{\frac {1}{3}} + a\right )}^{6} a^{4}}{b^{12}} - \frac {1386 \, {\left (b x^{\frac {1}{3}} + a\right )}^{5} a^{5}}{5 \, b^{12}} + \frac {693 \, {\left (b x^{\frac {1}{3}} + a\right )}^{4} a^{6}}{2 \, b^{12}} - \frac {330 \, {\left (b x^{\frac {1}{3}} + a\right )}^{3} a^{7}}{b^{12}} + \frac {495 \, {\left (b x^{\frac {1}{3}} + a\right )}^{2} a^{8}}{2 \, b^{12}} - \frac {165 \, {\left (b x^{\frac {1}{3}} + a\right )} a^{9}}{b^{12}} + \frac {3 \, a^{11}}{{\left (b x^{\frac {1}{3}} + a\right )} b^{12}} \]
33*a^10*log(b*x^(1/3) + a)/b^12 + 3/10*(b*x^(1/3) + a)^10/b^12 - 11/3*(b*x ^(1/3) + a)^9*a/b^12 + 165/8*(b*x^(1/3) + a)^8*a^2/b^12 - 495/7*(b*x^(1/3) + a)^7*a^3/b^12 + 165*(b*x^(1/3) + a)^6*a^4/b^12 - 1386/5*(b*x^(1/3) + a) ^5*a^5/b^12 + 693/2*(b*x^(1/3) + a)^4*a^6/b^12 - 330*(b*x^(1/3) + a)^3*a^7 /b^12 + 495/2*(b*x^(1/3) + a)^2*a^8/b^12 - 165*(b*x^(1/3) + a)*a^9/b^12 + 3*a^11/((b*x^(1/3) + a)*b^12)
Time = 0.27 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.84 \[ \int \frac {x^3}{\left (a+b \sqrt [3]{x}\right )^2} \, dx=\frac {33 \, a^{10} \log \left ({\left | b x^{\frac {1}{3}} + a \right |}\right )}{b^{12}} + \frac {3 \, a^{11}}{{\left (b x^{\frac {1}{3}} + a\right )} b^{12}} + \frac {252 \, b^{18} x^{\frac {10}{3}} - 560 \, a b^{17} x^{3} + 945 \, a^{2} b^{16} x^{\frac {8}{3}} - 1440 \, a^{3} b^{15} x^{\frac {7}{3}} + 2100 \, a^{4} b^{14} x^{2} - 3024 \, a^{5} b^{13} x^{\frac {5}{3}} + 4410 \, a^{6} b^{12} x^{\frac {4}{3}} - 6720 \, a^{7} b^{11} x + 11340 \, a^{8} b^{10} x^{\frac {2}{3}} - 25200 \, a^{9} b^{9} x^{\frac {1}{3}}}{840 \, b^{20}} \]
33*a^10*log(abs(b*x^(1/3) + a))/b^12 + 3*a^11/((b*x^(1/3) + a)*b^12) + 1/8 40*(252*b^18*x^(10/3) - 560*a*b^17*x^3 + 945*a^2*b^16*x^(8/3) - 1440*a^3*b ^15*x^(7/3) + 2100*a^4*b^14*x^2 - 3024*a^5*b^13*x^(5/3) + 4410*a^6*b^12*x^ (4/3) - 6720*a^7*b^11*x + 11340*a^8*b^10*x^(2/3) - 25200*a^9*b^9*x^(1/3))/ b^20
Time = 0.09 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.84 \[ \int \frac {x^3}{\left (a+b \sqrt [3]{x}\right )^2} \, dx=\frac {3\,x^{10/3}}{10\,b^2}+\frac {3\,a^{11}}{b\,\left (a\,b^{11}+b^{12}\,x^{1/3}\right )}-\frac {2\,a\,x^3}{3\,b^3}-\frac {8\,a^7\,x}{b^9}+\frac {33\,a^{10}\,\ln \left (a+b\,x^{1/3}\right )}{b^{12}}+\frac {5\,a^4\,x^2}{2\,b^6}+\frac {9\,a^2\,x^{8/3}}{8\,b^4}-\frac {12\,a^3\,x^{7/3}}{7\,b^5}-\frac {18\,a^5\,x^{5/3}}{5\,b^7}+\frac {21\,a^6\,x^{4/3}}{4\,b^8}+\frac {27\,a^8\,x^{2/3}}{2\,b^{10}}-\frac {30\,a^9\,x^{1/3}}{b^{11}} \]
(3*x^(10/3))/(10*b^2) + (3*a^11)/(b*(a*b^11 + b^12*x^(1/3))) - (2*a*x^3)/( 3*b^3) - (8*a^7*x)/b^9 + (33*a^10*log(a + b*x^(1/3)))/b^12 + (5*a^4*x^2)/( 2*b^6) + (9*a^2*x^(8/3))/(8*b^4) - (12*a^3*x^(7/3))/(7*b^5) - (18*a^5*x^(5 /3))/(5*b^7) + (21*a^6*x^(4/3))/(4*b^8) + (27*a^8*x^(2/3))/(2*b^10) - (30* a^9*x^(1/3))/b^11