Integrand size = 15, antiderivative size = 131 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^4} \, dx=-\frac {a^{10}}{3 x^3}-\frac {15 a^9 b}{4 x^{8/3}}-\frac {135 a^8 b^2}{7 x^{7/3}}-\frac {60 a^7 b^3}{x^2}-\frac {126 a^6 b^4}{x^{5/3}}-\frac {189 a^5 b^5}{x^{4/3}}-\frac {210 a^4 b^6}{x}-\frac {180 a^3 b^7}{x^{2/3}}-\frac {135 a^2 b^8}{\sqrt [3]{x}}+3 b^{10} \sqrt [3]{x}+10 a b^9 \log (x) \]
-1/3*a^10/x^3-15/4*a^9*b/x^(8/3)-135/7*a^8*b^2/x^(7/3)-60*a^7*b^3/x^2-126* a^6*b^4/x^(5/3)-189*a^5*b^5/x^(4/3)-210*a^4*b^6/x-180*a^3*b^7/x^(2/3)-135* a^2*b^8/x^(1/3)+3*b^10*x^(1/3)+10*a*b^9*ln(x)
Time = 0.12 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^4} \, dx=\frac {-28 a^{10}-315 a^9 b \sqrt [3]{x}-1620 a^8 b^2 x^{2/3}-5040 a^7 b^3 x-10584 a^6 b^4 x^{4/3}-15876 a^5 b^5 x^{5/3}-17640 a^4 b^6 x^2-15120 a^3 b^7 x^{7/3}-11340 a^2 b^8 x^{8/3}+252 b^{10} x^{10/3}}{84 x^3}+30 a b^9 \log \left (\sqrt [3]{x}\right ) \]
(-28*a^10 - 315*a^9*b*x^(1/3) - 1620*a^8*b^2*x^(2/3) - 5040*a^7*b^3*x - 10 584*a^6*b^4*x^(4/3) - 15876*a^5*b^5*x^(5/3) - 17640*a^4*b^6*x^2 - 15120*a^ 3*b^7*x^(7/3) - 11340*a^2*b^8*x^(8/3) + 252*b^10*x^(10/3))/(84*x^3) + 30*a *b^9*Log[x^(1/3)]
Time = 0.26 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.04, 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.
\(\displaystyle \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^4} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 3 \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^{10/3}}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle 3 \int \left (\frac {a^{10}}{x^{10/3}}+\frac {10 b a^9}{x^3}+\frac {45 b^2 a^8}{x^{8/3}}+\frac {120 b^3 a^7}{x^{7/3}}+\frac {210 b^4 a^6}{x^2}+\frac {252 b^5 a^5}{x^{5/3}}+\frac {210 b^6 a^4}{x^{4/3}}+\frac {120 b^7 a^3}{x}+\frac {45 b^8 a^2}{x^{2/3}}+\frac {10 b^9 a}{\sqrt [3]{x}}+b^{10}\right )d\sqrt [3]{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \left (-\frac {a^{10}}{9 x^3}-\frac {5 a^9 b}{4 x^{8/3}}-\frac {45 a^8 b^2}{7 x^{7/3}}-\frac {20 a^7 b^3}{x^2}-\frac {42 a^6 b^4}{x^{5/3}}-\frac {63 a^5 b^5}{x^{4/3}}-\frac {70 a^4 b^6}{x}-\frac {60 a^3 b^7}{x^{2/3}}-\frac {45 a^2 b^8}{\sqrt [3]{x}}+10 a b^9 \log \left (\sqrt [3]{x}\right )+b^{10} \sqrt [3]{x}\right )\) |
3*(-1/9*a^10/x^3 - (5*a^9*b)/(4*x^(8/3)) - (45*a^8*b^2)/(7*x^(7/3)) - (20* a^7*b^3)/x^2 - (42*a^6*b^4)/x^(5/3) - (63*a^5*b^5)/x^(4/3) - (70*a^4*b^6)/ x - (60*a^3*b^7)/x^(2/3) - (45*a^2*b^8)/x^(1/3) + b^10*x^(1/3) + 10*a*b^9* Log[x^(1/3)])
3.24.32.3.1 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 = 3.68 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(-\frac {a^{10}}{3 x^{3}}-\frac {15 a^{9} b}{4 x^{\frac {8}{3}}}-\frac {135 a^{8} b^{2}}{7 x^{\frac {7}{3}}}-\frac {60 a^{7} b^{3}}{x^{2}}-\frac {126 a^{6} b^{4}}{x^{\frac {5}{3}}}-\frac {189 a^{5} b^{5}}{x^{\frac {4}{3}}}-\frac {210 a^{4} b^{6}}{x}-\frac {180 a^{3} b^{7}}{x^{\frac {2}{3}}}-\frac {135 a^{2} b^{8}}{x^{\frac {1}{3}}}+3 b^{10} x^{\frac {1}{3}}+10 a \,b^{9} \ln \left (x \right )\) | \(112\) |
default | \(-\frac {a^{10}}{3 x^{3}}-\frac {15 a^{9} b}{4 x^{\frac {8}{3}}}-\frac {135 a^{8} b^{2}}{7 x^{\frac {7}{3}}}-\frac {60 a^{7} b^{3}}{x^{2}}-\frac {126 a^{6} b^{4}}{x^{\frac {5}{3}}}-\frac {189 a^{5} b^{5}}{x^{\frac {4}{3}}}-\frac {210 a^{4} b^{6}}{x}-\frac {180 a^{3} b^{7}}{x^{\frac {2}{3}}}-\frac {135 a^{2} b^{8}}{x^{\frac {1}{3}}}+3 b^{10} x^{\frac {1}{3}}+10 a \,b^{9} \ln \left (x \right )\) | \(112\) |
trager | \(\frac {\left (-1+x \right ) \left (a^{6} x^{2}+180 a^{3} b^{3} x^{2}+630 b^{6} x^{2}+a^{6} x +180 a^{3} b^{3} x +a^{6}\right ) a^{4}}{3 x^{3}}-\frac {3 \left (-4 b^{9} x^{3}+240 a^{3} b^{6} x^{2}+168 x \,a^{6} b^{3}+5 a^{9}\right ) b}{4 x^{\frac {8}{3}}}-\frac {27 \left (35 b^{6} x^{2}+49 a^{3} b^{3} x +5 a^{6}\right ) a^{2} b^{2}}{7 x^{\frac {7}{3}}}+10 a \,b^{9} \ln \left (x \right )\) | \(139\) |
-1/3*a^10/x^3-15/4*a^9*b/x^(8/3)-135/7*a^8*b^2/x^(7/3)-60*a^7*b^3/x^2-126* a^6*b^4/x^(5/3)-189*a^5*b^5/x^(4/3)-210*a^4*b^6/x-180*a^3*b^7/x^(2/3)-135* a^2*b^8/x^(1/3)+3*b^10*x^(1/3)+10*a*b^9*ln(x)
Time = 0.28 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^4} \, dx=\frac {2520 \, a b^{9} x^{3} \log \left (x^{\frac {1}{3}}\right ) - 17640 \, a^{4} b^{6} x^{2} - 5040 \, a^{7} b^{3} x - 28 \, a^{10} - 324 \, {\left (35 \, a^{2} b^{8} x^{2} + 49 \, a^{5} b^{5} x + 5 \, a^{8} b^{2}\right )} x^{\frac {2}{3}} + 63 \, {\left (4 \, b^{10} x^{3} - 240 \, a^{3} b^{7} x^{2} - 168 \, a^{6} b^{4} x - 5 \, a^{9} b\right )} x^{\frac {1}{3}}}{84 \, x^{3}} \]
1/84*(2520*a*b^9*x^3*log(x^(1/3)) - 17640*a^4*b^6*x^2 - 5040*a^7*b^3*x - 2 8*a^10 - 324*(35*a^2*b^8*x^2 + 49*a^5*b^5*x + 5*a^8*b^2)*x^(2/3) + 63*(4*b ^10*x^3 - 240*a^3*b^7*x^2 - 168*a^6*b^4*x - 5*a^9*b)*x^(1/3))/x^3
Time = 0.40 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^4} \, dx=- \frac {a^{10}}{3 x^{3}} - \frac {15 a^{9} b}{4 x^{\frac {8}{3}}} - \frac {135 a^{8} b^{2}}{7 x^{\frac {7}{3}}} - \frac {60 a^{7} b^{3}}{x^{2}} - \frac {126 a^{6} b^{4}}{x^{\frac {5}{3}}} - \frac {189 a^{5} b^{5}}{x^{\frac {4}{3}}} - \frac {210 a^{4} b^{6}}{x} - \frac {180 a^{3} b^{7}}{x^{\frac {2}{3}}} - \frac {135 a^{2} b^{8}}{\sqrt [3]{x}} + 10 a b^{9} \log {\left (x \right )} + 3 b^{10} \sqrt [3]{x} \]
-a**10/(3*x**3) - 15*a**9*b/(4*x**(8/3)) - 135*a**8*b**2/(7*x**(7/3)) - 60 *a**7*b**3/x**2 - 126*a**6*b**4/x**(5/3) - 189*a**5*b**5/x**(4/3) - 210*a* *4*b**6/x - 180*a**3*b**7/x**(2/3) - 135*a**2*b**8/x**(1/3) + 10*a*b**9*lo g(x) + 3*b**10*x**(1/3)
Time = 0.19 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^4} \, dx=10 \, a b^{9} \log \left (x\right ) + 3 \, b^{10} x^{\frac {1}{3}} - \frac {11340 \, a^{2} b^{8} x^{\frac {8}{3}} + 15120 \, a^{3} b^{7} x^{\frac {7}{3}} + 17640 \, a^{4} b^{6} x^{2} + 15876 \, a^{5} b^{5} x^{\frac {5}{3}} + 10584 \, a^{6} b^{4} x^{\frac {4}{3}} + 5040 \, a^{7} b^{3} x + 1620 \, a^{8} b^{2} x^{\frac {2}{3}} + 315 \, a^{9} b x^{\frac {1}{3}} + 28 \, a^{10}}{84 \, x^{3}} \]
10*a*b^9*log(x) + 3*b^10*x^(1/3) - 1/84*(11340*a^2*b^8*x^(8/3) + 15120*a^3 *b^7*x^(7/3) + 17640*a^4*b^6*x^2 + 15876*a^5*b^5*x^(5/3) + 10584*a^6*b^4*x ^(4/3) + 5040*a^7*b^3*x + 1620*a^8*b^2*x^(2/3) + 315*a^9*b*x^(1/3) + 28*a^ 10)/x^3
Time = 0.30 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^4} \, dx=10 \, a b^{9} \log \left ({\left | x \right |}\right ) + 3 \, b^{10} x^{\frac {1}{3}} - \frac {11340 \, a^{2} b^{8} x^{\frac {8}{3}} + 15120 \, a^{3} b^{7} x^{\frac {7}{3}} + 17640 \, a^{4} b^{6} x^{2} + 15876 \, a^{5} b^{5} x^{\frac {5}{3}} + 10584 \, a^{6} b^{4} x^{\frac {4}{3}} + 5040 \, a^{7} b^{3} x + 1620 \, a^{8} b^{2} x^{\frac {2}{3}} + 315 \, a^{9} b x^{\frac {1}{3}} + 28 \, a^{10}}{84 \, x^{3}} \]
10*a*b^9*log(abs(x)) + 3*b^10*x^(1/3) - 1/84*(11340*a^2*b^8*x^(8/3) + 1512 0*a^3*b^7*x^(7/3) + 17640*a^4*b^6*x^2 + 15876*a^5*b^5*x^(5/3) + 10584*a^6* b^4*x^(4/3) + 5040*a^7*b^3*x + 1620*a^8*b^2*x^(2/3) + 315*a^9*b*x^(1/3) + 28*a^10)/x^3
Time = 0.08 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^4} \, dx=3\,b^{10}\,x^{1/3}-\frac {a^{10}}{3\,x^3}-\frac {15\,a^9\,b}{4\,x^{8/3}}-\frac {210\,a^4\,b^6}{x}-\frac {60\,a^7\,b^3}{x^2}-\frac {135\,a^2\,b^8}{x^{1/3}}-\frac {180\,a^3\,b^7}{x^{2/3}}-\frac {189\,a^5\,b^5}{x^{4/3}}-\frac {126\,a^6\,b^4}{x^{5/3}}-\frac {135\,a^8\,b^2}{7\,x^{7/3}}+30\,a\,b^9\,\ln \left (x^{1/3}\right ) \]