Integrand size = 15, antiderivative size = 146 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^9} \, dx=-\frac {\left (a+b \sqrt {x}\right )^{11}}{8 a x^8}+\frac {b \left (a+b \sqrt {x}\right )^{11}}{24 a^2 x^{15/2}}-\frac {b^2 \left (a+b \sqrt {x}\right )^{11}}{84 a^3 x^7}+\frac {b^3 \left (a+b \sqrt {x}\right )^{11}}{364 a^4 x^{13/2}}-\frac {b^4 \left (a+b \sqrt {x}\right )^{11}}{2184 a^5 x^6}+\frac {b^5 \left (a+b \sqrt {x}\right )^{11}}{24024 a^6 x^{11/2}} \] Output:
-1/8*(a+b*x^(1/2))^11/a/x^8+1/24*b*(a+b*x^(1/2))^11/a^2/x^(15/2)-1/84*b^2* (a+b*x^(1/2))^11/a^3/x^7+1/364*b^3*(a+b*x^(1/2))^11/a^4/x^(13/2)-1/2184*b^ 4*(a+b*x^(1/2))^11/a^5/x^6+1/24024*b^5*(a+b*x^(1/2))^11/a^6/x^(11/2)
Time = 0.06 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^9} \, dx=\frac {-3003 a^{10}-32032 a^9 b \sqrt {x}-154440 a^8 b^2 x-443520 a^7 b^3 x^{3/2}-840840 a^6 b^4 x^2-1100736 a^5 b^5 x^{5/2}-1009008 a^4 b^6 x^3-640640 a^3 b^7 x^{7/2}-270270 a^2 b^8 x^4-68640 a b^9 x^{9/2}-8008 b^{10} x^5}{24024 x^8} \] Input:
Integrate[(a + b*Sqrt[x])^10/x^9,x]
Output:
(-3003*a^10 - 32032*a^9*b*Sqrt[x] - 154440*a^8*b^2*x - 443520*a^7*b^3*x^(3 /2) - 840840*a^6*b^4*x^2 - 1100736*a^5*b^5*x^(5/2) - 1009008*a^4*b^6*x^3 - 640640*a^3*b^7*x^(7/2) - 270270*a^2*b^8*x^4 - 68640*a*b^9*x^(9/2) - 8008* b^10*x^5)/(24024*x^8)
Time = 0.36 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.18, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {798, 55, 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.
| \(\displaystyle \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^9} \, dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle 2 \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{17/2}}d\sqrt {x}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle 2 \left (-\frac {5 b \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^8}d\sqrt {x}}{16 a}-\frac {\left (a+b \sqrt {x}\right )^{11}}{16 a x^8}\right )\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle 2 \left (-\frac {5 b \left (-\frac {4 b \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{15/2}}d\sqrt {x}}{15 a}-\frac {\left (a+b \sqrt {x}\right )^{11}}{15 a x^{15/2}}\right )}{16 a}-\frac {\left (a+b \sqrt {x}\right )^{11}}{16 a x^8}\right )\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle 2 \left (-\frac {5 b \left (-\frac {4 b \left (-\frac {3 b \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^7}d\sqrt {x}}{14 a}-\frac {\left (a+b \sqrt {x}\right )^{11}}{14 a x^7}\right )}{15 a}-\frac {\left (a+b \sqrt {x}\right )^{11}}{15 a x^{15/2}}\right )}{16 a}-\frac {\left (a+b \sqrt {x}\right )^{11}}{16 a x^8}\right )\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle 2 \left (-\frac {5 b \left (-\frac {4 b \left (-\frac {3 b \left (-\frac {2 b \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{13/2}}d\sqrt {x}}{13 a}-\frac {\left (a+b \sqrt {x}\right )^{11}}{13 a x^{13/2}}\right )}{14 a}-\frac {\left (a+b \sqrt {x}\right )^{11}}{14 a x^7}\right )}{15 a}-\frac {\left (a+b \sqrt {x}\right )^{11}}{15 a x^{15/2}}\right )}{16 a}-\frac {\left (a+b \sqrt {x}\right )^{11}}{16 a x^8}\right )\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle 2 \left (-\frac {5 b \left (-\frac {4 b \left (-\frac {3 b \left (-\frac {2 b \left (-\frac {b \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^6}d\sqrt {x}}{12 a}-\frac {\left (a+b \sqrt {x}\right )^{11}}{12 a x^6}\right )}{13 a}-\frac {\left (a+b \sqrt {x}\right )^{11}}{13 a x^{13/2}}\right )}{14 a}-\frac {\left (a+b \sqrt {x}\right )^{11}}{14 a x^7}\right )}{15 a}-\frac {\left (a+b \sqrt {x}\right )^{11}}{15 a x^{15/2}}\right )}{16 a}-\frac {\left (a+b \sqrt {x}\right )^{11}}{16 a x^8}\right )\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle 2 \left (-\frac {5 b \left (-\frac {4 b \left (-\frac {3 b \left (-\frac {2 b \left (\frac {b \left (a+b \sqrt {x}\right )^{11}}{132 a^2 x^{11/2}}-\frac {\left (a+b \sqrt {x}\right )^{11}}{12 a x^6}\right )}{13 a}-\frac {\left (a+b \sqrt {x}\right )^{11}}{13 a x^{13/2}}\right )}{14 a}-\frac {\left (a+b \sqrt {x}\right )^{11}}{14 a x^7}\right )}{15 a}-\frac {\left (a+b \sqrt {x}\right )^{11}}{15 a x^{15/2}}\right )}{16 a}-\frac {\left (a+b \sqrt {x}\right )^{11}}{16 a x^8}\right )\) |
Input:
Int[(a + b*Sqrt[x])^10/x^9,x]
Output:
2*((-5*b*((-4*b*((-3*b*((-2*b*(-1/12*(a + b*Sqrt[x])^11/(a*x^6) + (b*(a + b*Sqrt[x])^11)/(132*a^2*x^(11/2))))/(13*a) - (a + b*Sqrt[x])^11/(13*a*x^(1 3/2))))/(14*a) - (a + b*Sqrt[x])^11/(14*a*x^7)))/(15*a) - (a + b*Sqrt[x])^ 11/(15*a*x^(15/2))))/(16*a) - (a + b*Sqrt[x])^11/(16*a*x^8))
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 = 3.88 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.77
| method | result | size |
| derivativedivides | \(-\frac {504 a^{5} b^{5}}{11 x^{\frac {11}{2}}}-\frac {45 a^{8} b^{2}}{7 x^{7}}-\frac {b^{10}}{3 x^{3}}-\frac {a^{10}}{8 x^{8}}-\frac {240 a^{7} b^{3}}{13 x^{\frac {13}{2}}}-\frac {4 a^{9} b}{3 x^{\frac {15}{2}}}-\frac {35 a^{6} b^{4}}{x^{6}}-\frac {42 a^{4} b^{6}}{x^{5}}-\frac {45 a^{2} b^{8}}{4 x^{4}}-\frac {80 a^{3} b^{7}}{3 x^{\frac {9}{2}}}-\frac {20 a \,b^{9}}{7 x^{\frac {7}{2}}}\) | \(113\) |
| default | \(-\frac {504 a^{5} b^{5}}{11 x^{\frac {11}{2}}}-\frac {45 a^{8} b^{2}}{7 x^{7}}-\frac {b^{10}}{3 x^{3}}-\frac {a^{10}}{8 x^{8}}-\frac {240 a^{7} b^{3}}{13 x^{\frac {13}{2}}}-\frac {4 a^{9} b}{3 x^{\frac {15}{2}}}-\frac {35 a^{6} b^{4}}{x^{6}}-\frac {42 a^{4} b^{6}}{x^{5}}-\frac {45 a^{2} b^{8}}{4 x^{4}}-\frac {80 a^{3} b^{7}}{3 x^{\frac {9}{2}}}-\frac {20 a \,b^{9}}{7 x^{\frac {7}{2}}}\) | \(113\) |
| orering | \(-\frac {\left (-85800 b^{18} x^{9}+559130 a^{2} b^{16} x^{8}-1867320 a^{4} b^{14} x^{7}+3854340 a^{6} b^{12} x^{6}-5284296 a^{8} b^{10} x^{5}+4930875 a^{10} b^{8} x^{4}-3110940 a^{12} b^{6} x^{3}+1274550 a^{14} b^{4} x^{2}-306900 a^{16} b^{2} x +33033 a^{18}\right ) \left (a +b \sqrt {x}\right )^{10}}{120120 x^{8} \left (-b^{2} x +a^{2}\right )^{9}}-\frac {\left (-5720 b^{18} x^{9}+32890 a^{2} b^{16} x^{8}-98280 a^{4} b^{14} x^{7}+183540 a^{6} b^{12} x^{6}-229752 a^{8} b^{10} x^{5}+197235 a^{10} b^{8} x^{4}-115220 a^{12} b^{6} x^{3}+43950 a^{14} b^{4} x^{2}-9900 a^{16} b^{2} x +1001 a^{18}\right ) x^{2} \left (\frac {5 \left (a +b \sqrt {x}\right )^{9} b}{x^{\frac {19}{2}}}-\frac {9 \left (a +b \sqrt {x}\right )^{10}}{x^{10}}\right )}{60060 \left (-b^{2} x +a^{2}\right )^{9}}\) | \(275\) |
| trager | \(\frac {\left (-1+x \right ) \left (21 a^{10} x^{7}+1080 a^{8} b^{2} x^{7}+5880 a^{6} b^{4} x^{7}+7056 a^{4} b^{6} x^{7}+1890 a^{2} b^{8} x^{7}+56 b^{10} x^{7}+21 a^{10} x^{6}+1080 a^{8} b^{2} x^{6}+5880 a^{6} b^{4} x^{6}+7056 a^{4} b^{6} x^{6}+1890 a^{2} b^{8} x^{6}+56 b^{10} x^{6}+21 a^{10} x^{5}+1080 a^{8} b^{2} x^{5}+5880 a^{6} b^{4} x^{5}+7056 a^{4} b^{6} x^{5}+1890 a^{2} b^{8} x^{5}+56 b^{10} x^{5}+21 a^{10} x^{4}+1080 a^{8} b^{2} x^{4}+5880 a^{6} b^{4} x^{4}+7056 a^{4} b^{6} x^{4}+1890 a^{2} b^{8} x^{4}+21 a^{10} x^{3}+1080 a^{8} b^{2} x^{3}+5880 a^{6} b^{4} x^{3}+7056 a^{4} b^{6} x^{3}+21 a^{10} x^{2}+1080 a^{8} b^{2} x^{2}+5880 a^{6} b^{4} x^{2}+21 a^{10} x +1080 a^{8} b^{2} x +21 a^{10}\right )}{168 x^{8}}-\frac {4 \left (2145 b^{8} x^{4}+20020 a^{2} b^{6} x^{3}+34398 a^{4} b^{4} x^{2}+13860 a^{6} b^{2} x +1001 a^{8}\right ) a b}{3003 x^{\frac {15}{2}}}\) | \(386\) |
Input:
int((a+b*x^(1/2))^10/x^9,x,method=_RETURNVERBOSE)
Output:
-504/11*a^5*b^5/x^(11/2)-45/7*a^8*b^2/x^7-1/3*b^10/x^3-1/8*a^10/x^8-240/13 *a^7*b^3/x^(13/2)-4/3*a^9*b/x^(15/2)-35*a^6*b^4/x^6-42*a^4*b^6/x^5-45/4*a^ 2*b^8/x^4-80/3*a^3*b^7/x^(9/2)-20/7*a*b^9/x^(7/2)
Time = 0.07 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^9} \, dx=-\frac {8008 \, b^{10} x^{5} + 270270 \, a^{2} b^{8} x^{4} + 1009008 \, a^{4} b^{6} x^{3} + 840840 \, a^{6} b^{4} x^{2} + 154440 \, a^{8} b^{2} x + 3003 \, a^{10} + 32 \, {\left (2145 \, a b^{9} x^{4} + 20020 \, a^{3} b^{7} x^{3} + 34398 \, a^{5} b^{5} x^{2} + 13860 \, a^{7} b^{3} x + 1001 \, a^{9} b\right )} \sqrt {x}}{24024 \, x^{8}} \] Input:
integrate((a+b*x^(1/2))^10/x^9,x, algorithm="fricas")
Output:
-1/24024*(8008*b^10*x^5 + 270270*a^2*b^8*x^4 + 1009008*a^4*b^6*x^3 + 84084 0*a^6*b^4*x^2 + 154440*a^8*b^2*x + 3003*a^10 + 32*(2145*a*b^9*x^4 + 20020* a^3*b^7*x^3 + 34398*a^5*b^5*x^2 + 13860*a^7*b^3*x + 1001*a^9*b)*sqrt(x))/x ^8
Time = 0.87 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^9} \, dx=- \frac {a^{10}}{8 x^{8}} - \frac {4 a^{9} b}{3 x^{\frac {15}{2}}} - \frac {45 a^{8} b^{2}}{7 x^{7}} - \frac {240 a^{7} b^{3}}{13 x^{\frac {13}{2}}} - \frac {35 a^{6} b^{4}}{x^{6}} - \frac {504 a^{5} b^{5}}{11 x^{\frac {11}{2}}} - \frac {42 a^{4} b^{6}}{x^{5}} - \frac {80 a^{3} b^{7}}{3 x^{\frac {9}{2}}} - \frac {45 a^{2} b^{8}}{4 x^{4}} - \frac {20 a b^{9}}{7 x^{\frac {7}{2}}} - \frac {b^{10}}{3 x^{3}} \] Input:
integrate((a+b*x**(1/2))**10/x**9,x)
Output:
-a**10/(8*x**8) - 4*a**9*b/(3*x**(15/2)) - 45*a**8*b**2/(7*x**7) - 240*a** 7*b**3/(13*x**(13/2)) - 35*a**6*b**4/x**6 - 504*a**5*b**5/(11*x**(11/2)) - 42*a**4*b**6/x**5 - 80*a**3*b**7/(3*x**(9/2)) - 45*a**2*b**8/(4*x**4) - 2 0*a*b**9/(7*x**(7/2)) - b**10/(3*x**3)
Time = 0.03 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^9} \, dx=-\frac {8008 \, b^{10} x^{5} + 68640 \, a b^{9} x^{\frac {9}{2}} + 270270 \, a^{2} b^{8} x^{4} + 640640 \, a^{3} b^{7} x^{\frac {7}{2}} + 1009008 \, a^{4} b^{6} x^{3} + 1100736 \, a^{5} b^{5} x^{\frac {5}{2}} + 840840 \, a^{6} b^{4} x^{2} + 443520 \, a^{7} b^{3} x^{\frac {3}{2}} + 154440 \, a^{8} b^{2} x + 32032 \, a^{9} b \sqrt {x} + 3003 \, a^{10}}{24024 \, x^{8}} \] Input:
integrate((a+b*x^(1/2))^10/x^9,x, algorithm="maxima")
Output:
-1/24024*(8008*b^10*x^5 + 68640*a*b^9*x^(9/2) + 270270*a^2*b^8*x^4 + 64064 0*a^3*b^7*x^(7/2) + 1009008*a^4*b^6*x^3 + 1100736*a^5*b^5*x^(5/2) + 840840 *a^6*b^4*x^2 + 443520*a^7*b^3*x^(3/2) + 154440*a^8*b^2*x + 32032*a^9*b*sqr t(x) + 3003*a^10)/x^8
Time = 0.13 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^9} \, dx=-\frac {8008 \, b^{10} x^{5} + 68640 \, a b^{9} x^{\frac {9}{2}} + 270270 \, a^{2} b^{8} x^{4} + 640640 \, a^{3} b^{7} x^{\frac {7}{2}} + 1009008 \, a^{4} b^{6} x^{3} + 1100736 \, a^{5} b^{5} x^{\frac {5}{2}} + 840840 \, a^{6} b^{4} x^{2} + 443520 \, a^{7} b^{3} x^{\frac {3}{2}} + 154440 \, a^{8} b^{2} x + 32032 \, a^{9} b \sqrt {x} + 3003 \, a^{10}}{24024 \, x^{8}} \] Input:
integrate((a+b*x^(1/2))^10/x^9,x, algorithm="giac")
Output:
-1/24024*(8008*b^10*x^5 + 68640*a*b^9*x^(9/2) + 270270*a^2*b^8*x^4 + 64064 0*a^3*b^7*x^(7/2) + 1009008*a^4*b^6*x^3 + 1100736*a^5*b^5*x^(5/2) + 840840 *a^6*b^4*x^2 + 443520*a^7*b^3*x^(3/2) + 154440*a^8*b^2*x + 32032*a^9*b*sqr t(x) + 3003*a^10)/x^8
Time = 0.35 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^9} \, dx=-\frac {\frac {a^{10}}{8}+\frac {b^{10}\,x^5}{3}+\frac {45\,a^8\,b^2\,x}{7}+\frac {4\,a^9\,b\,\sqrt {x}}{3}+\frac {20\,a\,b^9\,x^{9/2}}{7}+35\,a^6\,b^4\,x^2+42\,a^4\,b^6\,x^3+\frac {45\,a^2\,b^8\,x^4}{4}+\frac {240\,a^7\,b^3\,x^{3/2}}{13}+\frac {504\,a^5\,b^5\,x^{5/2}}{11}+\frac {80\,a^3\,b^7\,x^{7/2}}{3}}{x^8} \] Input:
int((a + b*x^(1/2))^10/x^9,x)
Output:
-(a^10/8 + (b^10*x^5)/3 + (45*a^8*b^2*x)/7 + (4*a^9*b*x^(1/2))/3 + (20*a*b ^9*x^(9/2))/7 + 35*a^6*b^4*x^2 + 42*a^4*b^6*x^3 + (45*a^2*b^8*x^4)/4 + (24 0*a^7*b^3*x^(3/2))/13 + (504*a^5*b^5*x^(5/2))/11 + (80*a^3*b^7*x^(7/2))/3) /x^8
Time = 0.21 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^9} \, dx=\frac {-3003 \sqrt {x}\, a^{10}-154440 \sqrt {x}\, a^{8} b^{2} x -840840 \sqrt {x}\, a^{6} b^{4} x^{2}-1009008 \sqrt {x}\, a^{4} b^{6} x^{3}-270270 \sqrt {x}\, a^{2} b^{8} x^{4}-8008 \sqrt {x}\, b^{10} x^{5}-32032 a^{9} b x -443520 a^{7} b^{3} x^{2}-1100736 a^{5} b^{5} x^{3}-640640 a^{3} b^{7} x^{4}-68640 a \,b^{9} x^{5}}{24024 \sqrt {x}\, x^{8}} \] Input:
int((a+b*x^(1/2))^10/x^9,x)
Output:
( - 3003*sqrt(x)*a**10 - 154440*sqrt(x)*a**8*b**2*x - 840840*sqrt(x)*a**6* b**4*x**2 - 1009008*sqrt(x)*a**4*b**6*x**3 - 270270*sqrt(x)*a**2*b**8*x**4 - 8008*sqrt(x)*b**10*x**5 - 32032*a**9*b*x - 443520*a**7*b**3*x**2 - 1100 736*a**5*b**5*x**3 - 640640*a**3*b**7*x**4 - 68640*a*b**9*x**5)/(24024*sqr t(x)*x**8)