Integrand size = 15, antiderivative size = 96 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^8} \, dx=-\frac {\left (a+b \sqrt {x}\right )^{11}}{7 a x^7}+\frac {3 b \left (a+b \sqrt {x}\right )^{11}}{91 a^2 x^{13/2}}-\frac {b^2 \left (a+b \sqrt {x}\right )^{11}}{182 a^3 x^6}+\frac {b^3 \left (a+b \sqrt {x}\right )^{11}}{2002 a^4 x^{11/2}} \]
-1/7*(a+b*x^(1/2))^11/a/x^7+3/91*b*(a+b*x^(1/2))^11/a^2/x^(13/2)-1/182*b^2 *(a+b*x^(1/2))^11/a^3/x^6+1/2002*b^3*(a+b*x^(1/2))^11/a^4/x^(11/2)
Time = 0.07 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.29 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^8} \, dx=\frac {-286 a^{10}-3080 a^9 b \sqrt {x}-15015 a^8 b^2 x-43680 a^7 b^3 x^{3/2}-84084 a^6 b^4 x^2-112112 a^5 b^5 x^{5/2}-105105 a^4 b^6 x^3-68640 a^3 b^7 x^{7/2}-30030 a^2 b^8 x^4-8008 a b^9 x^{9/2}-1001 b^{10} x^5}{2002 x^7} \]
(-286*a^10 - 3080*a^9*b*Sqrt[x] - 15015*a^8*b^2*x - 43680*a^7*b^3*x^(3/2) - 84084*a^6*b^4*x^2 - 112112*a^5*b^5*x^(5/2) - 105105*a^4*b^6*x^3 - 68640* a^3*b^7*x^(7/2) - 30030*a^2*b^8*x^4 - 8008*a*b^9*x^(9/2) - 1001*b^10*x^5)/ (2002*x^7)
Time = 0.20 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.15, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {798, 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.
\(\displaystyle \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^8} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 2 \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{15/2}}d\sqrt {x}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle 2 \left (-\frac {3 b \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^7}d\sqrt {x}}{14 a}-\frac {\left (a+b \sqrt {x}\right )^{11}}{14 a x^7}\right )\) |
\(\Big \downarrow \) 55 |
\(\displaystyle 2 \left (-\frac {3 b \left (-\frac {2 b \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{13/2}}d\sqrt {x}}{13 a}-\frac {\left (a+b \sqrt {x}\right )^{11}}{13 a x^{13/2}}\right )}{14 a}-\frac {\left (a+b \sqrt {x}\right )^{11}}{14 a x^7}\right )\) |
\(\Big \downarrow \) 55 |
\(\displaystyle 2 \left (-\frac {3 b \left (-\frac {2 b \left (-\frac {b \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^6}d\sqrt {x}}{12 a}-\frac {\left (a+b \sqrt {x}\right )^{11}}{12 a x^6}\right )}{13 a}-\frac {\left (a+b \sqrt {x}\right )^{11}}{13 a x^{13/2}}\right )}{14 a}-\frac {\left (a+b \sqrt {x}\right )^{11}}{14 a x^7}\right )\) |
\(\Big \downarrow \) 48 |
\(\displaystyle 2 \left (-\frac {3 b \left (-\frac {2 b \left (\frac {b \left (a+b \sqrt {x}\right )^{11}}{132 a^2 x^{11/2}}-\frac {\left (a+b \sqrt {x}\right )^{11}}{12 a x^6}\right )}{13 a}-\frac {\left (a+b \sqrt {x}\right )^{11}}{13 a x^{13/2}}\right )}{14 a}-\frac {\left (a+b \sqrt {x}\right )^{11}}{14 a x^7}\right )\) |
2*((-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^(13/2))))/(14*a) - (a + b*Sqrt[x])^11/(14*a*x^7))
3.22.65.3.1 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 = 3.51 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.18
method | result | size |
derivativedivides | \(-\frac {b^{10}}{2 x^{2}}-\frac {20 a^{9} b}{13 x^{\frac {13}{2}}}-\frac {a^{10}}{7 x^{7}}-\frac {105 a^{4} b^{6}}{2 x^{4}}-\frac {240 a^{3} b^{7}}{7 x^{\frac {7}{2}}}-\frac {240 a^{7} b^{3}}{11 x^{\frac {11}{2}}}-\frac {42 a^{6} b^{4}}{x^{5}}-\frac {4 a \,b^{9}}{x^{\frac {5}{2}}}-\frac {15 a^{8} b^{2}}{2 x^{6}}-\frac {15 a^{2} b^{8}}{x^{3}}-\frac {56 a^{5} b^{5}}{x^{\frac {9}{2}}}\) | \(113\) |
default | \(-\frac {b^{10}}{2 x^{2}}-\frac {20 a^{9} b}{13 x^{\frac {13}{2}}}-\frac {a^{10}}{7 x^{7}}-\frac {105 a^{4} b^{6}}{2 x^{4}}-\frac {240 a^{3} b^{7}}{7 x^{\frac {7}{2}}}-\frac {240 a^{7} b^{3}}{11 x^{\frac {11}{2}}}-\frac {42 a^{6} b^{4}}{x^{5}}-\frac {4 a \,b^{9}}{x^{\frac {5}{2}}}-\frac {15 a^{8} b^{2}}{2 x^{6}}-\frac {15 a^{2} b^{8}}{x^{3}}-\frac {56 a^{5} b^{5}}{x^{\frac {9}{2}}}\) | \(113\) |
trager | \(\frac {\left (-1+x \right ) \left (2 a^{10} x^{6}+105 a^{8} b^{2} x^{6}+588 a^{6} b^{4} x^{6}+735 a^{4} b^{6} x^{6}+210 a^{2} b^{8} x^{6}+7 b^{10} x^{6}+2 a^{10} x^{5}+105 a^{8} b^{2} x^{5}+588 a^{6} b^{4} x^{5}+735 a^{4} b^{6} x^{5}+210 a^{2} b^{8} x^{5}+7 b^{10} x^{5}+2 a^{10} x^{4}+105 a^{8} b^{2} x^{4}+588 a^{6} b^{4} x^{4}+735 x^{4} a^{4} b^{6}+210 a^{2} b^{8} x^{4}+2 a^{10} x^{3}+105 a^{8} b^{2} x^{3}+588 a^{6} b^{4} x^{3}+735 a^{4} b^{6} x^{3}+2 a^{10} x^{2}+105 a^{8} b^{2} x^{2}+588 x^{2} a^{6} b^{4}+2 a^{10} x +105 a^{8} b^{2} x +2 a^{10}\right )}{14 x^{7}}-\frac {4 \left (1001 b^{8} x^{4}+8580 a^{2} b^{6} x^{3}+14014 a^{4} b^{4} x^{2}+5460 a^{6} b^{2} x +385 a^{8}\right ) a b}{1001 x^{\frac {13}{2}}}\) | \(326\) |
-1/2*b^10/x^2-20/13*a^9*b/x^(13/2)-1/7*a^10/x^7-105/2*a^4*b^6/x^4-240/7*a^ 3*b^7/x^(7/2)-240/11*a^7*b^3/x^(11/2)-42*a^6*b^4/x^5-4*a*b^9/x^(5/2)-15/2* a^8*b^2/x^6-15*a^2*b^8/x^3-56*a^5*b^5/x^(9/2)
Time = 0.26 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.18 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^8} \, dx=-\frac {1001 \, b^{10} x^{5} + 30030 \, a^{2} b^{8} x^{4} + 105105 \, a^{4} b^{6} x^{3} + 84084 \, a^{6} b^{4} x^{2} + 15015 \, a^{8} b^{2} x + 286 \, a^{10} + 8 \, {\left (1001 \, a b^{9} x^{4} + 8580 \, a^{3} b^{7} x^{3} + 14014 \, a^{5} b^{5} x^{2} + 5460 \, a^{7} b^{3} x + 385 \, a^{9} b\right )} \sqrt {x}}{2002 \, x^{7}} \]
-1/2002*(1001*b^10*x^5 + 30030*a^2*b^8*x^4 + 105105*a^4*b^6*x^3 + 84084*a^ 6*b^4*x^2 + 15015*a^8*b^2*x + 286*a^10 + 8*(1001*a*b^9*x^4 + 8580*a^3*b^7* x^3 + 14014*a^5*b^5*x^2 + 5460*a^7*b^3*x + 385*a^9*b)*sqrt(x))/x^7
Time = 0.53 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.44 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^8} \, dx=- \frac {a^{10}}{7 x^{7}} - \frac {20 a^{9} b}{13 x^{\frac {13}{2}}} - \frac {15 a^{8} b^{2}}{2 x^{6}} - \frac {240 a^{7} b^{3}}{11 x^{\frac {11}{2}}} - \frac {42 a^{6} b^{4}}{x^{5}} - \frac {56 a^{5} b^{5}}{x^{\frac {9}{2}}} - \frac {105 a^{4} b^{6}}{2 x^{4}} - \frac {240 a^{3} b^{7}}{7 x^{\frac {7}{2}}} - \frac {15 a^{2} b^{8}}{x^{3}} - \frac {4 a b^{9}}{x^{\frac {5}{2}}} - \frac {b^{10}}{2 x^{2}} \]
-a**10/(7*x**7) - 20*a**9*b/(13*x**(13/2)) - 15*a**8*b**2/(2*x**6) - 240*a **7*b**3/(11*x**(11/2)) - 42*a**6*b**4/x**5 - 56*a**5*b**5/x**(9/2) - 105* a**4*b**6/(2*x**4) - 240*a**3*b**7/(7*x**(7/2)) - 15*a**2*b**8/x**3 - 4*a* b**9/x**(5/2) - b**10/(2*x**2)
Time = 0.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.17 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^8} \, dx=-\frac {1001 \, b^{10} x^{5} + 8008 \, a b^{9} x^{\frac {9}{2}} + 30030 \, a^{2} b^{8} x^{4} + 68640 \, a^{3} b^{7} x^{\frac {7}{2}} + 105105 \, a^{4} b^{6} x^{3} + 112112 \, a^{5} b^{5} x^{\frac {5}{2}} + 84084 \, a^{6} b^{4} x^{2} + 43680 \, a^{7} b^{3} x^{\frac {3}{2}} + 15015 \, a^{8} b^{2} x + 3080 \, a^{9} b \sqrt {x} + 286 \, a^{10}}{2002 \, x^{7}} \]
-1/2002*(1001*b^10*x^5 + 8008*a*b^9*x^(9/2) + 30030*a^2*b^8*x^4 + 68640*a^ 3*b^7*x^(7/2) + 105105*a^4*b^6*x^3 + 112112*a^5*b^5*x^(5/2) + 84084*a^6*b^ 4*x^2 + 43680*a^7*b^3*x^(3/2) + 15015*a^8*b^2*x + 3080*a^9*b*sqrt(x) + 286 *a^10)/x^7
Time = 0.28 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.17 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^8} \, dx=-\frac {1001 \, b^{10} x^{5} + 8008 \, a b^{9} x^{\frac {9}{2}} + 30030 \, a^{2} b^{8} x^{4} + 68640 \, a^{3} b^{7} x^{\frac {7}{2}} + 105105 \, a^{4} b^{6} x^{3} + 112112 \, a^{5} b^{5} x^{\frac {5}{2}} + 84084 \, a^{6} b^{4} x^{2} + 43680 \, a^{7} b^{3} x^{\frac {3}{2}} + 15015 \, a^{8} b^{2} x + 3080 \, a^{9} b \sqrt {x} + 286 \, a^{10}}{2002 \, x^{7}} \]
-1/2002*(1001*b^10*x^5 + 8008*a*b^9*x^(9/2) + 30030*a^2*b^8*x^4 + 68640*a^ 3*b^7*x^(7/2) + 105105*a^4*b^6*x^3 + 112112*a^5*b^5*x^(5/2) + 84084*a^6*b^ 4*x^2 + 43680*a^7*b^3*x^(3/2) + 15015*a^8*b^2*x + 3080*a^9*b*sqrt(x) + 286 *a^10)/x^7
Time = 5.82 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.17 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^8} \, dx=-\frac {\frac {a^{10}}{7}+\frac {b^{10}\,x^5}{2}+\frac {15\,a^8\,b^2\,x}{2}+\frac {20\,a^9\,b\,\sqrt {x}}{13}+4\,a\,b^9\,x^{9/2}+42\,a^6\,b^4\,x^2+\frac {105\,a^4\,b^6\,x^3}{2}+15\,a^2\,b^8\,x^4+\frac {240\,a^7\,b^3\,x^{3/2}}{11}+56\,a^5\,b^5\,x^{5/2}+\frac {240\,a^3\,b^7\,x^{7/2}}{7}}{x^7} \]