Integrand size = 15, antiderivative size = 144 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^7} \, dx=-\frac {a^{10}}{6 x^6}-\frac {30 a^9 b}{17 x^{17/3}}-\frac {135 a^8 b^2}{16 x^{16/3}}-\frac {24 a^7 b^3}{x^5}-\frac {45 a^6 b^4}{x^{14/3}}-\frac {756 a^5 b^5}{13 x^{13/3}}-\frac {105 a^4 b^6}{2 x^4}-\frac {360 a^3 b^7}{11 x^{11/3}}-\frac {27 a^2 b^8}{2 x^{10/3}}-\frac {10 a b^9}{3 x^3}-\frac {3 b^{10}}{8 x^{8/3}} \]
-1/6*a^10/x^6-30/17*a^9*b/x^(17/3)-135/16*a^8*b^2/x^(16/3)-24*a^7*b^3/x^5- 45*a^6*b^4/x^(14/3)-756/13*a^5*b^5/x^(13/3)-105/2*a^4*b^6/x^4-360/11*a^3*b ^7/x^(11/3)-27/2*a^2*b^8/x^(10/3)-10/3*a*b^9/x^3-3/8*b^10/x^(8/3)
Time = 0.09 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^7} \, dx=\frac {-19448 a^{10}-205920 a^9 b \sqrt [3]{x}-984555 a^8 b^2 x^{2/3}-2800512 a^7 b^3 x-5250960 a^6 b^4 x^{4/3}-6785856 a^5 b^5 x^{5/3}-6126120 a^4 b^6 x^2-3818880 a^3 b^7 x^{7/3}-1575288 a^2 b^8 x^{8/3}-388960 a b^9 x^3-43758 b^{10} x^{10/3}}{116688 x^6} \]
(-19448*a^10 - 205920*a^9*b*x^(1/3) - 984555*a^8*b^2*x^(2/3) - 2800512*a^7 *b^3*x - 5250960*a^6*b^4*x^(4/3) - 6785856*a^5*b^5*x^(5/3) - 6126120*a^4*b ^6*x^2 - 3818880*a^3*b^7*x^(7/3) - 1575288*a^2*b^8*x^(8/3) - 388960*a*b^9* x^3 - 43758*b^10*x^(10/3))/(116688*x^6)
Time = 0.27 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {798, 53, 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.
\(\displaystyle \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^7} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 3 \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^{19/3}}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle 3 \int \left (\frac {a^{10}}{x^{19/3}}+\frac {10 b a^9}{x^6}+\frac {45 b^2 a^8}{x^{17/3}}+\frac {120 b^3 a^7}{x^{16/3}}+\frac {210 b^4 a^6}{x^5}+\frac {252 b^5 a^5}{x^{14/3}}+\frac {210 b^6 a^4}{x^{13/3}}+\frac {120 b^7 a^3}{x^4}+\frac {45 b^8 a^2}{x^{11/3}}+\frac {10 b^9 a}{x^{10/3}}+\frac {b^{10}}{x^3}\right )d\sqrt [3]{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \left (-\frac {a^{10}}{18 x^6}-\frac {10 a^9 b}{17 x^{17/3}}-\frac {45 a^8 b^2}{16 x^{16/3}}-\frac {8 a^7 b^3}{x^5}-\frac {15 a^6 b^4}{x^{14/3}}-\frac {252 a^5 b^5}{13 x^{13/3}}-\frac {35 a^4 b^6}{2 x^4}-\frac {120 a^3 b^7}{11 x^{11/3}}-\frac {9 a^2 b^8}{2 x^{10/3}}-\frac {10 a b^9}{9 x^3}-\frac {b^{10}}{8 x^{8/3}}\right )\) |
3*(-1/18*a^10/x^6 - (10*a^9*b)/(17*x^(17/3)) - (45*a^8*b^2)/(16*x^(16/3)) - (8*a^7*b^3)/x^5 - (15*a^6*b^4)/x^(14/3) - (252*a^5*b^5)/(13*x^(13/3)) - (35*a^4*b^6)/(2*x^4) - (120*a^3*b^7)/(11*x^(11/3)) - (9*a^2*b^8)/(2*x^(10/ 3)) - (10*a*b^9)/(9*x^3) - b^10/(8*x^(8/3)))
3.24.35.3.1 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, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[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 = 3.70 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(-\frac {a^{10}}{6 x^{6}}-\frac {30 a^{9} b}{17 x^{\frac {17}{3}}}-\frac {135 a^{8} b^{2}}{16 x^{\frac {16}{3}}}-\frac {24 a^{7} b^{3}}{x^{5}}-\frac {45 a^{6} b^{4}}{x^{\frac {14}{3}}}-\frac {756 a^{5} b^{5}}{13 x^{\frac {13}{3}}}-\frac {105 a^{4} b^{6}}{2 x^{4}}-\frac {360 a^{3} b^{7}}{11 x^{\frac {11}{3}}}-\frac {27 a^{2} b^{8}}{2 x^{\frac {10}{3}}}-\frac {10 a \,b^{9}}{3 x^{3}}-\frac {3 b^{10}}{8 x^{\frac {8}{3}}}\) | \(113\) |
default | \(-\frac {a^{10}}{6 x^{6}}-\frac {30 a^{9} b}{17 x^{\frac {17}{3}}}-\frac {135 a^{8} b^{2}}{16 x^{\frac {16}{3}}}-\frac {24 a^{7} b^{3}}{x^{5}}-\frac {45 a^{6} b^{4}}{x^{\frac {14}{3}}}-\frac {756 a^{5} b^{5}}{13 x^{\frac {13}{3}}}-\frac {105 a^{4} b^{6}}{2 x^{4}}-\frac {360 a^{3} b^{7}}{11 x^{\frac {11}{3}}}-\frac {27 a^{2} b^{8}}{2 x^{\frac {10}{3}}}-\frac {10 a \,b^{9}}{3 x^{3}}-\frac {3 b^{10}}{8 x^{\frac {8}{3}}}\) | \(113\) |
trager | \(\frac {\left (-1+x \right ) \left (a^{9} x^{5}+144 a^{6} b^{3} x^{5}+315 a^{3} b^{6} x^{5}+20 b^{9} x^{5}+a^{9} x^{4}+144 a^{6} b^{3} x^{4}+315 a^{3} b^{6} x^{4}+20 b^{9} x^{4}+a^{9} x^{3}+144 a^{6} b^{3} x^{3}+315 a^{3} b^{6} x^{3}+20 b^{9} x^{3}+a^{9} x^{2}+144 a^{6} b^{3} x^{2}+315 a^{3} b^{6} x^{2}+a^{9} x +144 x \,a^{6} b^{3}+a^{9}\right ) a}{6 x^{6}}-\frac {3 \left (187 b^{9} x^{3}+16320 a^{3} b^{6} x^{2}+22440 x \,a^{6} b^{3}+880 a^{9}\right ) b}{1496 x^{\frac {17}{3}}}-\frac {27 \left (104 b^{6} x^{2}+448 a^{3} b^{3} x +65 a^{6}\right ) a^{2} b^{2}}{208 x^{\frac {16}{3}}}\) | \(243\) |
-1/6*a^10/x^6-30/17*a^9*b/x^(17/3)-135/16*a^8*b^2/x^(16/3)-24*a^7*b^3/x^5- 45*a^6*b^4/x^(14/3)-756/13*a^5*b^5/x^(13/3)-105/2*a^4*b^6/x^4-360/11*a^3*b ^7/x^(11/3)-27/2*a^2*b^8/x^(10/3)-10/3*a*b^9/x^3-3/8*b^10/x^(8/3)
Time = 0.26 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^7} \, dx=-\frac {388960 \, a b^{9} x^{3} + 6126120 \, a^{4} b^{6} x^{2} + 2800512 \, a^{7} b^{3} x + 19448 \, a^{10} + 15147 \, {\left (104 \, a^{2} b^{8} x^{2} + 448 \, a^{5} b^{5} x + 65 \, a^{8} b^{2}\right )} x^{\frac {2}{3}} + 234 \, {\left (187 \, b^{10} x^{3} + 16320 \, a^{3} b^{7} x^{2} + 22440 \, a^{6} b^{4} x + 880 \, a^{9} b\right )} x^{\frac {1}{3}}}{116688 \, x^{6}} \]
-1/116688*(388960*a*b^9*x^3 + 6126120*a^4*b^6*x^2 + 2800512*a^7*b^3*x + 19 448*a^10 + 15147*(104*a^2*b^8*x^2 + 448*a^5*b^5*x + 65*a^8*b^2)*x^(2/3) + 234*(187*b^10*x^3 + 16320*a^3*b^7*x^2 + 22440*a^6*b^4*x + 880*a^9*b)*x^(1/ 3))/x^6
Time = 0.86 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^7} \, dx=- \frac {a^{10}}{6 x^{6}} - \frac {30 a^{9} b}{17 x^{\frac {17}{3}}} - \frac {135 a^{8} b^{2}}{16 x^{\frac {16}{3}}} - \frac {24 a^{7} b^{3}}{x^{5}} - \frac {45 a^{6} b^{4}}{x^{\frac {14}{3}}} - \frac {756 a^{5} b^{5}}{13 x^{\frac {13}{3}}} - \frac {105 a^{4} b^{6}}{2 x^{4}} - \frac {360 a^{3} b^{7}}{11 x^{\frac {11}{3}}} - \frac {27 a^{2} b^{8}}{2 x^{\frac {10}{3}}} - \frac {10 a b^{9}}{3 x^{3}} - \frac {3 b^{10}}{8 x^{\frac {8}{3}}} \]
-a**10/(6*x**6) - 30*a**9*b/(17*x**(17/3)) - 135*a**8*b**2/(16*x**(16/3)) - 24*a**7*b**3/x**5 - 45*a**6*b**4/x**(14/3) - 756*a**5*b**5/(13*x**(13/3) ) - 105*a**4*b**6/(2*x**4) - 360*a**3*b**7/(11*x**(11/3)) - 27*a**2*b**8/( 2*x**(10/3)) - 10*a*b**9/(3*x**3) - 3*b**10/(8*x**(8/3))
Time = 0.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^7} \, dx=-\frac {43758 \, b^{10} x^{\frac {10}{3}} + 388960 \, a b^{9} x^{3} + 1575288 \, a^{2} b^{8} x^{\frac {8}{3}} + 3818880 \, a^{3} b^{7} x^{\frac {7}{3}} + 6126120 \, a^{4} b^{6} x^{2} + 6785856 \, a^{5} b^{5} x^{\frac {5}{3}} + 5250960 \, a^{6} b^{4} x^{\frac {4}{3}} + 2800512 \, a^{7} b^{3} x + 984555 \, a^{8} b^{2} x^{\frac {2}{3}} + 205920 \, a^{9} b x^{\frac {1}{3}} + 19448 \, a^{10}}{116688 \, x^{6}} \]
-1/116688*(43758*b^10*x^(10/3) + 388960*a*b^9*x^3 + 1575288*a^2*b^8*x^(8/3 ) + 3818880*a^3*b^7*x^(7/3) + 6126120*a^4*b^6*x^2 + 6785856*a^5*b^5*x^(5/3 ) + 5250960*a^6*b^4*x^(4/3) + 2800512*a^7*b^3*x + 984555*a^8*b^2*x^(2/3) + 205920*a^9*b*x^(1/3) + 19448*a^10)/x^6
Time = 0.28 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^7} \, dx=-\frac {43758 \, b^{10} x^{\frac {10}{3}} + 388960 \, a b^{9} x^{3} + 1575288 \, a^{2} b^{8} x^{\frac {8}{3}} + 3818880 \, a^{3} b^{7} x^{\frac {7}{3}} + 6126120 \, a^{4} b^{6} x^{2} + 6785856 \, a^{5} b^{5} x^{\frac {5}{3}} + 5250960 \, a^{6} b^{4} x^{\frac {4}{3}} + 2800512 \, a^{7} b^{3} x + 984555 \, a^{8} b^{2} x^{\frac {2}{3}} + 205920 \, a^{9} b x^{\frac {1}{3}} + 19448 \, a^{10}}{116688 \, x^{6}} \]
-1/116688*(43758*b^10*x^(10/3) + 388960*a*b^9*x^3 + 1575288*a^2*b^8*x^(8/3 ) + 3818880*a^3*b^7*x^(7/3) + 6126120*a^4*b^6*x^2 + 6785856*a^5*b^5*x^(5/3 ) + 5250960*a^6*b^4*x^(4/3) + 2800512*a^7*b^3*x + 984555*a^8*b^2*x^(2/3) + 205920*a^9*b*x^(1/3) + 19448*a^10)/x^6
Time = 5.69 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^7} \, dx=-\frac {\frac {a^{10}}{6}+\frac {3\,b^{10}\,x^{10/3}}{8}+24\,a^7\,b^3\,x+\frac {10\,a\,b^9\,x^3}{3}+\frac {30\,a^9\,b\,x^{1/3}}{17}+\frac {105\,a^4\,b^6\,x^2}{2}+\frac {135\,a^8\,b^2\,x^{2/3}}{16}+45\,a^6\,b^4\,x^{4/3}+\frac {756\,a^5\,b^5\,x^{5/3}}{13}+\frac {360\,a^3\,b^7\,x^{7/3}}{11}+\frac {27\,a^2\,b^8\,x^{8/3}}{2}}{x^6} \]