Integrand size = 15, antiderivative size = 122 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^6} \, dx=-\frac {\left (a+b \sqrt [3]{x}\right )^{11}}{5 a x^5}+\frac {2 b \left (a+b \sqrt [3]{x}\right )^{11}}{35 a^2 x^{14/3}}-\frac {6 b^2 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^3 x^{13/3}}+\frac {b^3 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^4 x^4}-\frac {b^4 \left (a+b \sqrt [3]{x}\right )^{11}}{5005 a^5 x^{11/3}} \]
-1/5*(a+b*x^(1/3))^11/a/x^5+2/35*b*(a+b*x^(1/3))^11/a^2/x^(14/3)-6/455*b^2 *(a+b*x^(1/3))^11/a^3/x^(13/3)+1/455*b^3*(a+b*x^(1/3))^11/a^4/x^4-1/5005*b ^4*(a+b*x^(1/3))^11/a^5/x^(11/3)
Time = 0.08 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^6} \, dx=\frac {-1001 a^{10}-10725 a^9 b \sqrt [3]{x}-51975 a^8 b^2 x^{2/3}-150150 a^7 b^3 x-286650 a^6 b^4 x^{4/3}-378378 a^5 b^5 x^{5/3}-350350 a^4 b^6 x^2-225225 a^3 b^7 x^{7/3}-96525 a^2 b^8 x^{8/3}-25025 a b^9 x^3-3003 b^{10} x^{10/3}}{5005 x^5} \]
(-1001*a^10 - 10725*a^9*b*x^(1/3) - 51975*a^8*b^2*x^(2/3) - 150150*a^7*b^3 *x - 286650*a^6*b^4*x^(4/3) - 378378*a^5*b^5*x^(5/3) - 350350*a^4*b^6*x^2 - 225225*a^3*b^7*x^(7/3) - 96525*a^2*b^8*x^(8/3) - 25025*a*b^9*x^3 - 3003* b^10*x^(10/3))/(5005*x^5)
Time = 0.20 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {798, 55, 55, 55, 55, 48}
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.
\(\displaystyle \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^6} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 3 \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^{16/3}}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle 3 \left (-\frac {4 b \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^5}d\sqrt [3]{x}}{15 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{11}}{15 a x^5}\right )\) |
\(\Big \downarrow \) 55 |
\(\displaystyle 3 \left (-\frac {4 b \left (-\frac {3 b \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^{14/3}}d\sqrt [3]{x}}{14 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{11}}{14 a x^{14/3}}\right )}{15 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{11}}{15 a x^5}\right )\) |
\(\Big \downarrow \) 55 |
\(\displaystyle 3 \left (-\frac {4 b \left (-\frac {3 b \left (-\frac {2 b \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^{13/3}}d\sqrt [3]{x}}{13 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{11}}{13 a x^{13/3}}\right )}{14 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{11}}{14 a x^{14/3}}\right )}{15 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{11}}{15 a x^5}\right )\) |
\(\Big \downarrow \) 55 |
\(\displaystyle 3 \left (-\frac {4 b \left (-\frac {3 b \left (-\frac {2 b \left (-\frac {b \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^4}d\sqrt [3]{x}}{12 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{11}}{12 a x^4}\right )}{13 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{11}}{13 a x^{13/3}}\right )}{14 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{11}}{14 a x^{14/3}}\right )}{15 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{11}}{15 a x^5}\right )\) |
\(\Big \downarrow \) 48 |
\(\displaystyle 3 \left (-\frac {4 b \left (-\frac {3 b \left (-\frac {2 b \left (\frac {b \left (a+b \sqrt [3]{x}\right )^{11}}{132 a^2 x^{11/3}}-\frac {\left (a+b \sqrt [3]{x}\right )^{11}}{12 a x^4}\right )}{13 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{11}}{13 a x^{13/3}}\right )}{14 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{11}}{14 a x^{14/3}}\right )}{15 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{11}}{15 a x^5}\right )\) |
3*((-4*b*((-3*b*((-2*b*(-1/12*(a + b*x^(1/3))^11/(a*x^4) + (b*(a + b*x^(1/ 3))^11)/(132*a^2*x^(11/3))))/(13*a) - (a + b*x^(1/3))^11/(13*a*x^(13/3)))) /(14*a) - (a + b*x^(1/3))^11/(14*a*x^(14/3))))/(15*a) - (a + b*x^(1/3))^11 /(15*a*x^5))
3.24.34.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
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 = 3.61 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(-\frac {3 b^{10}}{5 x^{\frac {5}{3}}}-\frac {378 a^{5} b^{5}}{5 x^{\frac {10}{3}}}-\frac {45 a^{3} b^{7}}{x^{\frac {8}{3}}}-\frac {30 a^{7} b^{3}}{x^{4}}-\frac {5 a \,b^{9}}{x^{2}}-\frac {135 a^{8} b^{2}}{13 x^{\frac {13}{3}}}-\frac {135 a^{2} b^{8}}{7 x^{\frac {7}{3}}}-\frac {a^{10}}{5 x^{5}}-\frac {15 a^{9} b}{7 x^{\frac {14}{3}}}-\frac {70 a^{4} b^{6}}{x^{3}}-\frac {630 a^{6} b^{4}}{11 x^{\frac {11}{3}}}\) | \(113\) |
default | \(-\frac {3 b^{10}}{5 x^{\frac {5}{3}}}-\frac {378 a^{5} b^{5}}{5 x^{\frac {10}{3}}}-\frac {45 a^{3} b^{7}}{x^{\frac {8}{3}}}-\frac {30 a^{7} b^{3}}{x^{4}}-\frac {5 a \,b^{9}}{x^{2}}-\frac {135 a^{8} b^{2}}{13 x^{\frac {13}{3}}}-\frac {135 a^{2} b^{8}}{7 x^{\frac {7}{3}}}-\frac {a^{10}}{5 x^{5}}-\frac {15 a^{9} b}{7 x^{\frac {14}{3}}}-\frac {70 a^{4} b^{6}}{x^{3}}-\frac {630 a^{6} b^{4}}{11 x^{\frac {11}{3}}}\) | \(113\) |
trager | \(\frac {\left (-1+x \right ) \left (a^{9} x^{4}+150 a^{6} b^{3} x^{4}+350 a^{3} b^{6} x^{4}+25 b^{9} x^{4}+a^{9} x^{3}+150 a^{6} b^{3} x^{3}+350 a^{3} b^{6} x^{3}+25 b^{9} x^{3}+a^{9} x^{2}+150 a^{6} b^{3} x^{2}+350 a^{3} b^{6} x^{2}+a^{9} x +150 x \,a^{6} b^{3}+a^{9}\right ) a}{5 x^{5}}-\frac {3 \left (77 b^{9} x^{3}+5775 a^{3} b^{6} x^{2}+7350 x \,a^{6} b^{3}+275 a^{9}\right ) b}{385 x^{\frac {14}{3}}}-\frac {27 \left (325 b^{6} x^{2}+1274 a^{3} b^{3} x +175 a^{6}\right ) a^{2} b^{2}}{455 x^{\frac {13}{3}}}\) | \(206\) |
-3/5*b^10/x^(5/3)-378/5*a^5*b^5/x^(10/3)-45*a^3*b^7/x^(8/3)-30/x^4*a^7*b^3 -5/x^2*a*b^9-135/13*a^8*b^2/x^(13/3)-135/7*a^2*b^8/x^(7/3)-1/5*a^10/x^5-15 /7*a^9*b/x^(14/3)-70*a^4*b^6/x^3-630/11*a^6*b^4/x^(11/3)
Time = 0.28 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^6} \, dx=-\frac {25025 \, a b^{9} x^{3} + 350350 \, a^{4} b^{6} x^{2} + 150150 \, a^{7} b^{3} x + 1001 \, a^{10} + 297 \, {\left (325 \, a^{2} b^{8} x^{2} + 1274 \, a^{5} b^{5} x + 175 \, a^{8} b^{2}\right )} x^{\frac {2}{3}} + 39 \, {\left (77 \, b^{10} x^{3} + 5775 \, a^{3} b^{7} x^{2} + 7350 \, a^{6} b^{4} x + 275 \, a^{9} b\right )} x^{\frac {1}{3}}}{5005 \, x^{5}} \]
-1/5005*(25025*a*b^9*x^3 + 350350*a^4*b^6*x^2 + 150150*a^7*b^3*x + 1001*a^ 10 + 297*(325*a^2*b^8*x^2 + 1274*a^5*b^5*x + 175*a^8*b^2)*x^(2/3) + 39*(77 *b^10*x^3 + 5775*a^3*b^7*x^2 + 7350*a^6*b^4*x + 275*a^9*b)*x^(1/3))/x^5
Time = 0.68 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.17 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^6} \, dx=- \frac {a^{10}}{5 x^{5}} - \frac {15 a^{9} b}{7 x^{\frac {14}{3}}} - \frac {135 a^{8} b^{2}}{13 x^{\frac {13}{3}}} - \frac {30 a^{7} b^{3}}{x^{4}} - \frac {630 a^{6} b^{4}}{11 x^{\frac {11}{3}}} - \frac {378 a^{5} b^{5}}{5 x^{\frac {10}{3}}} - \frac {70 a^{4} b^{6}}{x^{3}} - \frac {45 a^{3} b^{7}}{x^{\frac {8}{3}}} - \frac {135 a^{2} b^{8}}{7 x^{\frac {7}{3}}} - \frac {5 a b^{9}}{x^{2}} - \frac {3 b^{10}}{5 x^{\frac {5}{3}}} \]
-a**10/(5*x**5) - 15*a**9*b/(7*x**(14/3)) - 135*a**8*b**2/(13*x**(13/3)) - 30*a**7*b**3/x**4 - 630*a**6*b**4/(11*x**(11/3)) - 378*a**5*b**5/(5*x**(1 0/3)) - 70*a**4*b**6/x**3 - 45*a**3*b**7/x**(8/3) - 135*a**2*b**8/(7*x**(7 /3)) - 5*a*b**9/x**2 - 3*b**10/(5*x**(5/3))
Time = 0.19 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^6} \, dx=-\frac {3003 \, b^{10} x^{\frac {10}{3}} + 25025 \, a b^{9} x^{3} + 96525 \, a^{2} b^{8} x^{\frac {8}{3}} + 225225 \, a^{3} b^{7} x^{\frac {7}{3}} + 350350 \, a^{4} b^{6} x^{2} + 378378 \, a^{5} b^{5} x^{\frac {5}{3}} + 286650 \, a^{6} b^{4} x^{\frac {4}{3}} + 150150 \, a^{7} b^{3} x + 51975 \, a^{8} b^{2} x^{\frac {2}{3}} + 10725 \, a^{9} b x^{\frac {1}{3}} + 1001 \, a^{10}}{5005 \, x^{5}} \]
-1/5005*(3003*b^10*x^(10/3) + 25025*a*b^9*x^3 + 96525*a^2*b^8*x^(8/3) + 22 5225*a^3*b^7*x^(7/3) + 350350*a^4*b^6*x^2 + 378378*a^5*b^5*x^(5/3) + 28665 0*a^6*b^4*x^(4/3) + 150150*a^7*b^3*x + 51975*a^8*b^2*x^(2/3) + 10725*a^9*b *x^(1/3) + 1001*a^10)/x^5
Time = 0.29 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^6} \, dx=-\frac {3003 \, b^{10} x^{\frac {10}{3}} + 25025 \, a b^{9} x^{3} + 96525 \, a^{2} b^{8} x^{\frac {8}{3}} + 225225 \, a^{3} b^{7} x^{\frac {7}{3}} + 350350 \, a^{4} b^{6} x^{2} + 378378 \, a^{5} b^{5} x^{\frac {5}{3}} + 286650 \, a^{6} b^{4} x^{\frac {4}{3}} + 150150 \, a^{7} b^{3} x + 51975 \, a^{8} b^{2} x^{\frac {2}{3}} + 10725 \, a^{9} b x^{\frac {1}{3}} + 1001 \, a^{10}}{5005 \, x^{5}} \]
-1/5005*(3003*b^10*x^(10/3) + 25025*a*b^9*x^3 + 96525*a^2*b^8*x^(8/3) + 22 5225*a^3*b^7*x^(7/3) + 350350*a^4*b^6*x^2 + 378378*a^5*b^5*x^(5/3) + 28665 0*a^6*b^4*x^(4/3) + 150150*a^7*b^3*x + 51975*a^8*b^2*x^(2/3) + 10725*a^9*b *x^(1/3) + 1001*a^10)/x^5
Time = 5.58 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^6} \, dx=-\frac {\frac {a^{10}}{5}+\frac {3\,b^{10}\,x^{10/3}}{5}+30\,a^7\,b^3\,x+5\,a\,b^9\,x^3+\frac {15\,a^9\,b\,x^{1/3}}{7}+70\,a^4\,b^6\,x^2+\frac {135\,a^8\,b^2\,x^{2/3}}{13}+\frac {630\,a^6\,b^4\,x^{4/3}}{11}+\frac {378\,a^5\,b^5\,x^{5/3}}{5}+45\,a^3\,b^7\,x^{7/3}+\frac {135\,a^2\,b^8\,x^{8/3}}{7}}{x^5} \]