Integrand size = 15, antiderivative size = 72 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^7} \, dx=-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{6 a x^6}+\frac {b \left (a+b \sqrt [3]{x}\right )^{16}}{51 a^2 x^{17/3}}-\frac {b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{816 a^3 x^{16/3}} \]
-1/6*(a+b*x^(1/3))^16/a/x^6+1/51*b*(a+b*x^(1/3))^16/a^2/x^(17/3)-1/816*b^2 *(a+b*x^(1/3))^16/a^3/x^(16/3)
Leaf count is larger than twice the leaf count of optimal. \(189\) vs. \(2(72)=144\).
Time = 0.10 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.62 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^7} \, dx=\frac {-136 a^{15}-2160 a^{14} b \sqrt [3]{x}-16065 a^{13} b^2 x^{2/3}-74256 a^{12} b^3 x-238680 a^{11} b^4 x^{4/3}-565488 a^{10} b^5 x^{5/3}-1021020 a^9 b^6 x^2-1432080 a^8 b^7 x^{7/3}-1575288 a^7 b^8 x^{8/3}-1361360 a^6 b^9 x^3-918918 a^5 b^{10} x^{10/3}-477360 a^4 b^{11} x^{11/3}-185640 a^3 b^{12} x^4-51408 a^2 b^{13} x^{13/3}-9180 a b^{14} x^{14/3}-816 b^{15} x^5}{816 x^6} \]
(-136*a^15 - 2160*a^14*b*x^(1/3) - 16065*a^13*b^2*x^(2/3) - 74256*a^12*b^3 *x - 238680*a^11*b^4*x^(4/3) - 565488*a^10*b^5*x^(5/3) - 1021020*a^9*b^6*x ^2 - 1432080*a^8*b^7*x^(7/3) - 1575288*a^7*b^8*x^(8/3) - 1361360*a^6*b^9*x ^3 - 918918*a^5*b^10*x^(10/3) - 477360*a^4*b^11*x^(11/3) - 185640*a^3*b^12 *x^4 - 51408*a^2*b^13*x^(13/3) - 9180*a*b^14*x^(14/3) - 816*b^15*x^5)/(816 *x^6)
Time = 0.18 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.11, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {798, 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.
\(\displaystyle \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^7} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 3 \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^{19/3}}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle 3 \left (-\frac {b \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^6}d\sqrt [3]{x}}{9 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{18 a x^6}\right )\) |
\(\Big \downarrow \) 55 |
\(\displaystyle 3 \left (-\frac {b \left (-\frac {b \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^{17/3}}d\sqrt [3]{x}}{17 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{17 a x^{17/3}}\right )}{9 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{18 a x^6}\right )\) |
\(\Big \downarrow \) 48 |
\(\displaystyle 3 \left (-\frac {b \left (\frac {b \left (a+b \sqrt [3]{x}\right )^{16}}{272 a^2 x^{16/3}}-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{17 a x^{17/3}}\right )}{9 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{18 a x^6}\right )\) |
3*(-1/9*(b*(-1/17*(a + b*x^(1/3))^16/(a*x^(17/3)) + (b*(a + b*x^(1/3))^16) /(272*a^2*x^(16/3))))/a - (a + b*x^(1/3))^16/(18*a*x^6))
3.24.50.3.1 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]]
Leaf count of result is larger than twice the leaf count of optimal. \(167\) vs. \(2(56)=112\).
Time = 3.64 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.33
method | result | size |
derivativedivides | \(-\frac {5005 a^{6} b^{9}}{3 x^{3}}-\frac {585 a^{11} b^{4}}{2 x^{\frac {14}{3}}}-\frac {b^{15}}{x}-\frac {45 a \,b^{14}}{4 x^{\frac {4}{3}}}-\frac {693 a^{10} b^{5}}{x^{\frac {13}{3}}}-\frac {1755 a^{8} b^{7}}{x^{\frac {11}{3}}}-\frac {315 a^{13} b^{2}}{16 x^{\frac {16}{3}}}-\frac {45 a^{14} b}{17 x^{\frac {17}{3}}}-\frac {455 a^{3} b^{12}}{2 x^{2}}-\frac {9009 a^{5} b^{10}}{8 x^{\frac {8}{3}}}-\frac {a^{15}}{6 x^{6}}-\frac {585 a^{4} b^{11}}{x^{\frac {7}{3}}}-\frac {5005 a^{9} b^{6}}{4 x^{4}}-\frac {63 a^{2} b^{13}}{x^{\frac {5}{3}}}-\frac {3861 a^{7} b^{8}}{2 x^{\frac {10}{3}}}-\frac {91 a^{12} b^{3}}{x^{5}}\) | \(168\) |
default | \(-\frac {5005 a^{6} b^{9}}{3 x^{3}}-\frac {585 a^{11} b^{4}}{2 x^{\frac {14}{3}}}-\frac {b^{15}}{x}-\frac {45 a \,b^{14}}{4 x^{\frac {4}{3}}}-\frac {693 a^{10} b^{5}}{x^{\frac {13}{3}}}-\frac {1755 a^{8} b^{7}}{x^{\frac {11}{3}}}-\frac {315 a^{13} b^{2}}{16 x^{\frac {16}{3}}}-\frac {45 a^{14} b}{17 x^{\frac {17}{3}}}-\frac {455 a^{3} b^{12}}{2 x^{2}}-\frac {9009 a^{5} b^{10}}{8 x^{\frac {8}{3}}}-\frac {a^{15}}{6 x^{6}}-\frac {585 a^{4} b^{11}}{x^{\frac {7}{3}}}-\frac {5005 a^{9} b^{6}}{4 x^{4}}-\frac {63 a^{2} b^{13}}{x^{\frac {5}{3}}}-\frac {3861 a^{7} b^{8}}{2 x^{\frac {10}{3}}}-\frac {91 a^{12} b^{3}}{x^{5}}\) | \(168\) |
trager | \(\frac {\left (-1+x \right ) \left (2 a^{15} x^{5}+1092 a^{12} b^{3} x^{5}+15015 a^{9} b^{6} x^{5}+20020 a^{6} b^{9} x^{5}+2730 a^{3} b^{12} x^{5}+12 b^{15} x^{5}+2 x^{4} a^{15}+1092 a^{12} b^{3} x^{4}+15015 a^{9} b^{6} x^{4}+20020 a^{6} b^{9} x^{4}+2730 a^{3} b^{12} x^{4}+2 x^{3} a^{15}+1092 a^{12} b^{3} x^{3}+15015 a^{9} b^{6} x^{3}+20020 a^{6} b^{9} x^{3}+2 x^{2} a^{15}+1092 a^{12} b^{3} x^{2}+15015 a^{9} b^{6} x^{2}+2 x \,a^{15}+1092 a^{12} b^{3} x +2 a^{15}\right )}{12 x^{6}}-\frac {9 \left (952 b^{12} x^{4}+17017 a^{3} b^{9} x^{3}+26520 a^{6} b^{6} x^{2}+4420 a^{9} b^{3} x +40 a^{12}\right ) a^{2} b}{136 x^{\frac {17}{3}}}-\frac {9 \left (20 b^{12} x^{4}+1040 a^{3} b^{9} x^{3}+3432 a^{6} b^{6} x^{2}+1232 a^{9} b^{3} x +35 a^{12}\right ) a \,b^{2}}{16 x^{\frac {16}{3}}}\) | \(322\) |
-5005/3/x^3*a^6*b^9-585/2*a^11*b^4/x^(14/3)-1/x*b^15-45/4*a*b^14/x^(4/3)-6 93*a^10*b^5/x^(13/3)-1755*a^8*b^7/x^(11/3)-315/16*a^13*b^2/x^(16/3)-45/17* a^14*b/x^(17/3)-455/2*a^3*b^12/x^2-9009/8*a^5*b^10/x^(8/3)-1/6*a^15/x^6-58 5*a^4*b^11/x^(7/3)-5005/4*a^9*b^6/x^4-63*a^2*b^13/x^(5/3)-3861/2*a^7*b^8/x ^(10/3)-91/x^5*a^12*b^3
Leaf count of result is larger than twice the leaf count of optimal. 169 vs. \(2 (56) = 112\).
Time = 0.28 (sec) , antiderivative size = 169, normalized size of antiderivative = 2.35 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^7} \, dx=-\frac {816 \, b^{15} x^{5} + 185640 \, a^{3} b^{12} x^{4} + 1361360 \, a^{6} b^{9} x^{3} + 1021020 \, a^{9} b^{6} x^{2} + 74256 \, a^{12} b^{3} x + 136 \, a^{15} + 459 \, {\left (20 \, a b^{14} x^{4} + 1040 \, a^{4} b^{11} x^{3} + 3432 \, a^{7} b^{8} x^{2} + 1232 \, a^{10} b^{5} x + 35 \, a^{13} b^{2}\right )} x^{\frac {2}{3}} + 54 \, {\left (952 \, a^{2} b^{13} x^{4} + 17017 \, a^{5} b^{10} x^{3} + 26520 \, a^{8} b^{7} x^{2} + 4420 \, a^{11} b^{4} x + 40 \, a^{14} b\right )} x^{\frac {1}{3}}}{816 \, x^{6}} \]
-1/816*(816*b^15*x^5 + 185640*a^3*b^12*x^4 + 1361360*a^6*b^9*x^3 + 1021020 *a^9*b^6*x^2 + 74256*a^12*b^3*x + 136*a^15 + 459*(20*a*b^14*x^4 + 1040*a^4 *b^11*x^3 + 3432*a^7*b^8*x^2 + 1232*a^10*b^5*x + 35*a^13*b^2)*x^(2/3) + 54 *(952*a^2*b^13*x^4 + 17017*a^5*b^10*x^3 + 26520*a^8*b^7*x^2 + 4420*a^11*b^ 4*x + 40*a^14*b)*x^(1/3))/x^6
Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (61) = 122\).
Time = 0.88 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.90 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^7} \, dx=- \frac {a^{15}}{6 x^{6}} - \frac {45 a^{14} b}{17 x^{\frac {17}{3}}} - \frac {315 a^{13} b^{2}}{16 x^{\frac {16}{3}}} - \frac {91 a^{12} b^{3}}{x^{5}} - \frac {585 a^{11} b^{4}}{2 x^{\frac {14}{3}}} - \frac {693 a^{10} b^{5}}{x^{\frac {13}{3}}} - \frac {5005 a^{9} b^{6}}{4 x^{4}} - \frac {1755 a^{8} b^{7}}{x^{\frac {11}{3}}} - \frac {3861 a^{7} b^{8}}{2 x^{\frac {10}{3}}} - \frac {5005 a^{6} b^{9}}{3 x^{3}} - \frac {9009 a^{5} b^{10}}{8 x^{\frac {8}{3}}} - \frac {585 a^{4} b^{11}}{x^{\frac {7}{3}}} - \frac {455 a^{3} b^{12}}{2 x^{2}} - \frac {63 a^{2} b^{13}}{x^{\frac {5}{3}}} - \frac {45 a b^{14}}{4 x^{\frac {4}{3}}} - \frac {b^{15}}{x} \]
-a**15/(6*x**6) - 45*a**14*b/(17*x**(17/3)) - 315*a**13*b**2/(16*x**(16/3) ) - 91*a**12*b**3/x**5 - 585*a**11*b**4/(2*x**(14/3)) - 693*a**10*b**5/x** (13/3) - 5005*a**9*b**6/(4*x**4) - 1755*a**8*b**7/x**(11/3) - 3861*a**7*b* *8/(2*x**(10/3)) - 5005*a**6*b**9/(3*x**3) - 9009*a**5*b**10/(8*x**(8/3)) - 585*a**4*b**11/x**(7/3) - 455*a**3*b**12/(2*x**2) - 63*a**2*b**13/x**(5/ 3) - 45*a*b**14/(4*x**(4/3)) - b**15/x
Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (56) = 112\).
Time = 0.20 (sec) , antiderivative size = 167, normalized size of antiderivative = 2.32 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^7} \, dx=-\frac {816 \, b^{15} x^{5} + 9180 \, a b^{14} x^{\frac {14}{3}} + 51408 \, a^{2} b^{13} x^{\frac {13}{3}} + 185640 \, a^{3} b^{12} x^{4} + 477360 \, a^{4} b^{11} x^{\frac {11}{3}} + 918918 \, a^{5} b^{10} x^{\frac {10}{3}} + 1361360 \, a^{6} b^{9} x^{3} + 1575288 \, a^{7} b^{8} x^{\frac {8}{3}} + 1432080 \, a^{8} b^{7} x^{\frac {7}{3}} + 1021020 \, a^{9} b^{6} x^{2} + 565488 \, a^{10} b^{5} x^{\frac {5}{3}} + 238680 \, a^{11} b^{4} x^{\frac {4}{3}} + 74256 \, a^{12} b^{3} x + 16065 \, a^{13} b^{2} x^{\frac {2}{3}} + 2160 \, a^{14} b x^{\frac {1}{3}} + 136 \, a^{15}}{816 \, x^{6}} \]
-1/816*(816*b^15*x^5 + 9180*a*b^14*x^(14/3) + 51408*a^2*b^13*x^(13/3) + 18 5640*a^3*b^12*x^4 + 477360*a^4*b^11*x^(11/3) + 918918*a^5*b^10*x^(10/3) + 1361360*a^6*b^9*x^3 + 1575288*a^7*b^8*x^(8/3) + 1432080*a^8*b^7*x^(7/3) + 1021020*a^9*b^6*x^2 + 565488*a^10*b^5*x^(5/3) + 238680*a^11*b^4*x^(4/3) + 74256*a^12*b^3*x + 16065*a^13*b^2*x^(2/3) + 2160*a^14*b*x^(1/3) + 136*a^15 )/x^6
Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (56) = 112\).
Time = 0.28 (sec) , antiderivative size = 167, normalized size of antiderivative = 2.32 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^7} \, dx=-\frac {816 \, b^{15} x^{5} + 9180 \, a b^{14} x^{\frac {14}{3}} + 51408 \, a^{2} b^{13} x^{\frac {13}{3}} + 185640 \, a^{3} b^{12} x^{4} + 477360 \, a^{4} b^{11} x^{\frac {11}{3}} + 918918 \, a^{5} b^{10} x^{\frac {10}{3}} + 1361360 \, a^{6} b^{9} x^{3} + 1575288 \, a^{7} b^{8} x^{\frac {8}{3}} + 1432080 \, a^{8} b^{7} x^{\frac {7}{3}} + 1021020 \, a^{9} b^{6} x^{2} + 565488 \, a^{10} b^{5} x^{\frac {5}{3}} + 238680 \, a^{11} b^{4} x^{\frac {4}{3}} + 74256 \, a^{12} b^{3} x + 16065 \, a^{13} b^{2} x^{\frac {2}{3}} + 2160 \, a^{14} b x^{\frac {1}{3}} + 136 \, a^{15}}{816 \, x^{6}} \]
-1/816*(816*b^15*x^5 + 9180*a*b^14*x^(14/3) + 51408*a^2*b^13*x^(13/3) + 18 5640*a^3*b^12*x^4 + 477360*a^4*b^11*x^(11/3) + 918918*a^5*b^10*x^(10/3) + 1361360*a^6*b^9*x^3 + 1575288*a^7*b^8*x^(8/3) + 1432080*a^8*b^7*x^(7/3) + 1021020*a^9*b^6*x^2 + 565488*a^10*b^5*x^(5/3) + 238680*a^11*b^4*x^(4/3) + 74256*a^12*b^3*x + 16065*a^13*b^2*x^(2/3) + 2160*a^14*b*x^(1/3) + 136*a^15 )/x^6
Time = 6.04 (sec) , antiderivative size = 166, normalized size of antiderivative = 2.31 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^7} \, dx=-\frac {\frac {a^{15}}{6}+b^{15}\,x^5+91\,a^{12}\,b^3\,x+\frac {45\,a^{14}\,b\,x^{1/3}}{17}+\frac {45\,a\,b^{14}\,x^{14/3}}{4}+\frac {5005\,a^9\,b^6\,x^2}{4}+\frac {5005\,a^6\,b^9\,x^3}{3}+\frac {455\,a^3\,b^{12}\,x^4}{2}+\frac {315\,a^{13}\,b^2\,x^{2/3}}{16}+\frac {585\,a^{11}\,b^4\,x^{4/3}}{2}+693\,a^{10}\,b^5\,x^{5/3}+1755\,a^8\,b^7\,x^{7/3}+\frac {3861\,a^7\,b^8\,x^{8/3}}{2}+\frac {9009\,a^5\,b^{10}\,x^{10/3}}{8}+585\,a^4\,b^{11}\,x^{11/3}+63\,a^2\,b^{13}\,x^{13/3}}{x^6} \]
-(a^15/6 + b^15*x^5 + 91*a^12*b^3*x + (45*a^14*b*x^(1/3))/17 + (45*a*b^14* x^(14/3))/4 + (5005*a^9*b^6*x^2)/4 + (5005*a^6*b^9*x^3)/3 + (455*a^3*b^12* x^4)/2 + (315*a^13*b^2*x^(2/3))/16 + (585*a^11*b^4*x^(4/3))/2 + 693*a^10*b ^5*x^(5/3) + 1755*a^8*b^7*x^(7/3) + (3861*a^7*b^8*x^(8/3))/2 + (9009*a^5*b ^10*x^(10/3))/8 + 585*a^4*b^11*x^(11/3) + 63*a^2*b^13*x^(13/3))/x^6