Integrand size = 15, antiderivative size = 172 \[ \int \frac {x^4}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {2 a^9}{7 b^{10} \left (a+b \sqrt {x}\right )^7}-\frac {3 a^8}{b^{10} \left (a+b \sqrt {x}\right )^6}+\frac {72 a^7}{5 b^{10} \left (a+b \sqrt {x}\right )^5}-\frac {42 a^6}{b^{10} \left (a+b \sqrt {x}\right )^4}+\frac {84 a^5}{b^{10} \left (a+b \sqrt {x}\right )^3}-\frac {126 a^4}{b^{10} \left (a+b \sqrt {x}\right )^2}+\frac {168 a^3}{b^{10} \left (a+b \sqrt {x}\right )}-\frac {16 a \sqrt {x}}{b^9}+\frac {x}{b^8}+\frac {72 a^2 \log \left (a+b \sqrt {x}\right )}{b^{10}} \] Output:
2/7*a^9/b^10/(a+b*x^(1/2))^7-3*a^8/b^10/(a+b*x^(1/2))^6+72/5*a^7/b^10/(a+b *x^(1/2))^5-42*a^6/b^10/(a+b*x^(1/2))^4+84*a^5/b^10/(a+b*x^(1/2))^3-126*a^ 4/b^10/(a+b*x^(1/2))^2+168*a^3/b^10/(a+b*x^(1/2))-16*a*x^(1/2)/b^9+x/b^8+7 2*a^2*ln(a+b*x^(1/2))/b^10
Time = 0.08 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.83 \[ \int \frac {x^4}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {3349 a^9+20923 a^8 b \sqrt {x}+53949 a^7 b^2 x+72275 a^6 b^3 x^{3/2}+50225 a^5 b^4 x^2+12495 a^4 b^5 x^{5/2}-4655 a^3 b^6 x^3-3185 a^2 b^7 x^{7/2}-315 a b^8 x^4+35 b^9 x^{9/2}}{35 b^{10} \left (a+b \sqrt {x}\right )^7}+\frac {72 a^2 \log \left (a+b \sqrt {x}\right )}{b^{10}} \] Input:
Integrate[x^4/(a + b*Sqrt[x])^8,x]
Output:
(3349*a^9 + 20923*a^8*b*Sqrt[x] + 53949*a^7*b^2*x + 72275*a^6*b^3*x^(3/2) + 50225*a^5*b^4*x^2 + 12495*a^4*b^5*x^(5/2) - 4655*a^3*b^6*x^3 - 3185*a^2* b^7*x^(7/2) - 315*a*b^8*x^4 + 35*b^9*x^(9/2))/(35*b^10*(a + b*Sqrt[x])^7) + (72*a^2*Log[a + b*Sqrt[x]])/b^10
Time = 0.57 (sec) , antiderivative size = 179, 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.
| \(\displaystyle \int \frac {x^4}{\left (a+b \sqrt {x}\right )^8} \, dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle 2 \int \frac {x^{9/2}}{\left (a+b \sqrt {x}\right )^8}d\sqrt {x}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle 2 \int \left (-\frac {a^9}{b^9 \left (a+b \sqrt {x}\right )^8}+\frac {9 a^8}{b^9 \left (a+b \sqrt {x}\right )^7}-\frac {36 a^7}{b^9 \left (a+b \sqrt {x}\right )^6}+\frac {84 a^6}{b^9 \left (a+b \sqrt {x}\right )^5}-\frac {126 a^5}{b^9 \left (a+b \sqrt {x}\right )^4}+\frac {126 a^4}{b^9 \left (a+b \sqrt {x}\right )^3}-\frac {84 a^3}{b^9 \left (a+b \sqrt {x}\right )^2}+\frac {36 a^2}{b^9 \left (a+b \sqrt {x}\right )}-\frac {8 a}{b^9}+\frac {\sqrt {x}}{b^8}\right )d\sqrt {x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {a^9}{7 b^{10} \left (a+b \sqrt {x}\right )^7}-\frac {3 a^8}{2 b^{10} \left (a+b \sqrt {x}\right )^6}+\frac {36 a^7}{5 b^{10} \left (a+b \sqrt {x}\right )^5}-\frac {21 a^6}{b^{10} \left (a+b \sqrt {x}\right )^4}+\frac {42 a^5}{b^{10} \left (a+b \sqrt {x}\right )^3}-\frac {63 a^4}{b^{10} \left (a+b \sqrt {x}\right )^2}+\frac {84 a^3}{b^{10} \left (a+b \sqrt {x}\right )}+\frac {36 a^2 \log \left (a+b \sqrt {x}\right )}{b^{10}}-\frac {8 a \sqrt {x}}{b^9}+\frac {x}{2 b^8}\right )\) |
Input:
Int[x^4/(a + b*Sqrt[x])^8,x]
Output:
2*(a^9/(7*b^10*(a + b*Sqrt[x])^7) - (3*a^8)/(2*b^10*(a + b*Sqrt[x])^6) + ( 36*a^7)/(5*b^10*(a + b*Sqrt[x])^5) - (21*a^6)/(b^10*(a + b*Sqrt[x])^4) + ( 42*a^5)/(b^10*(a + b*Sqrt[x])^3) - (63*a^4)/(b^10*(a + b*Sqrt[x])^2) + (84 *a^3)/(b^10*(a + b*Sqrt[x])) - (8*a*Sqrt[x])/b^9 + x/(2*b^8) + (36*a^2*Log [a + b*Sqrt[x]])/b^10)
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 = 0.46 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.89
| method | result | size |
| derivativedivides | \(-\frac {2 \left (-\frac {b x}{2}+8 a \sqrt {x}\right )}{b^{9}}+\frac {72 a^{7}}{5 b^{10} \left (a +b \sqrt {x}\right )^{5}}-\frac {126 a^{4}}{b^{10} \left (a +b \sqrt {x}\right )^{2}}-\frac {3 a^{8}}{b^{10} \left (a +b \sqrt {x}\right )^{6}}-\frac {42 a^{6}}{b^{10} \left (a +b \sqrt {x}\right )^{4}}+\frac {84 a^{5}}{b^{10} \left (a +b \sqrt {x}\right )^{3}}+\frac {2 a^{9}}{7 b^{10} \left (a +b \sqrt {x}\right )^{7}}+\frac {168 a^{3}}{b^{10} \left (a +b \sqrt {x}\right )}+\frac {72 a^{2} \ln \left (a +b \sqrt {x}\right )}{b^{10}}\) | \(153\) |
| default | \(-\frac {2 \left (-\frac {b x}{2}+8 a \sqrt {x}\right )}{b^{9}}+\frac {72 a^{7}}{5 b^{10} \left (a +b \sqrt {x}\right )^{5}}-\frac {126 a^{4}}{b^{10} \left (a +b \sqrt {x}\right )^{2}}-\frac {3 a^{8}}{b^{10} \left (a +b \sqrt {x}\right )^{6}}-\frac {42 a^{6}}{b^{10} \left (a +b \sqrt {x}\right )^{4}}+\frac {84 a^{5}}{b^{10} \left (a +b \sqrt {x}\right )^{3}}+\frac {2 a^{9}}{7 b^{10} \left (a +b \sqrt {x}\right )^{7}}+\frac {168 a^{3}}{b^{10} \left (a +b \sqrt {x}\right )}+\frac {72 a^{2} \ln \left (a +b \sqrt {x}\right )}{b^{10}}\) | \(153\) |
Input:
int(x^4/(a+b*x^(1/2))^8,x,method=_RETURNVERBOSE)
Output:
-2/b^9*(-1/2*b*x+8*a*x^(1/2))+72/5*a^7/b^10/(a+b*x^(1/2))^5-126*a^4/b^10/( a+b*x^(1/2))^2-3*a^8/b^10/(a+b*x^(1/2))^6-42*a^6/b^10/(a+b*x^(1/2))^4+84*a ^5/b^10/(a+b*x^(1/2))^3+2/7*a^9/b^10/(a+b*x^(1/2))^7+168*a^3/b^10/(a+b*x^( 1/2))+72*a^2*ln(a+b*x^(1/2))/b^10
Leaf count of result is larger than twice the leaf count of optimal. 348 vs. \(2 (150) = 300\).
Time = 0.13 (sec) , antiderivative size = 348, normalized size of antiderivative = 2.02 \[ \int \frac {x^4}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {35 \, b^{16} x^{8} - 245 \, a^{2} b^{14} x^{7} - 9555 \, a^{4} b^{12} x^{6} + 41405 \, a^{6} b^{10} x^{5} - 83720 \, a^{8} b^{8} x^{4} + 94745 \, a^{10} b^{6} x^{3} - 62139 \, a^{12} b^{4} x^{2} + 22183 \, a^{14} b^{2} x - 3349 \, a^{16} + 2520 \, {\left (a^{2} b^{14} x^{7} - 7 \, a^{4} b^{12} x^{6} + 21 \, a^{6} b^{10} x^{5} - 35 \, a^{8} b^{8} x^{4} + 35 \, a^{10} b^{6} x^{3} - 21 \, a^{12} b^{4} x^{2} + 7 \, a^{14} b^{2} x - a^{16}\right )} \log \left (b \sqrt {x} + a\right ) - 8 \, {\left (70 \, a b^{15} x^{7} - 1225 \, a^{3} b^{13} x^{6} + 4410 \, a^{5} b^{11} x^{5} - 8393 \, a^{7} b^{9} x^{4} + 9216 \, a^{9} b^{7} x^{3} - 5943 \, a^{11} b^{5} x^{2} + 2100 \, a^{13} b^{3} x - 315 \, a^{15} b\right )} \sqrt {x}}{35 \, {\left (b^{24} x^{7} - 7 \, a^{2} b^{22} x^{6} + 21 \, a^{4} b^{20} x^{5} - 35 \, a^{6} b^{18} x^{4} + 35 \, a^{8} b^{16} x^{3} - 21 \, a^{10} b^{14} x^{2} + 7 \, a^{12} b^{12} x - a^{14} b^{10}\right )}} \] Input:
integrate(x^4/(a+b*x^(1/2))^8,x, algorithm="fricas")
Output:
1/35*(35*b^16*x^8 - 245*a^2*b^14*x^7 - 9555*a^4*b^12*x^6 + 41405*a^6*b^10* x^5 - 83720*a^8*b^8*x^4 + 94745*a^10*b^6*x^3 - 62139*a^12*b^4*x^2 + 22183* a^14*b^2*x - 3349*a^16 + 2520*(a^2*b^14*x^7 - 7*a^4*b^12*x^6 + 21*a^6*b^10 *x^5 - 35*a^8*b^8*x^4 + 35*a^10*b^6*x^3 - 21*a^12*b^4*x^2 + 7*a^14*b^2*x - a^16)*log(b*sqrt(x) + a) - 8*(70*a*b^15*x^7 - 1225*a^3*b^13*x^6 + 4410*a^ 5*b^11*x^5 - 8393*a^7*b^9*x^4 + 9216*a^9*b^7*x^3 - 5943*a^11*b^5*x^2 + 210 0*a^13*b^3*x - 315*a^15*b)*sqrt(x))/(b^24*x^7 - 7*a^2*b^22*x^6 + 21*a^4*b^ 20*x^5 - 35*a^6*b^18*x^4 + 35*a^8*b^16*x^3 - 21*a^10*b^14*x^2 + 7*a^12*b^1 2*x - a^14*b^10)
Leaf count of result is larger than twice the leaf count of optimal. 1839 vs. \(2 (167) = 334\).
Time = 2.65 (sec) , antiderivative size = 1839, normalized size of antiderivative = 10.69 \[ \int \frac {x^4}{\left (a+b \sqrt {x}\right )^8} \, dx=\text {Too large to display} \] Input:
integrate(x**4/(a+b*x**(1/2))**8,x)
Output:
Piecewise((2520*a**9*log(a/b + sqrt(x))/(35*a**7*b**10 + 245*a**6*b**11*sq rt(x) + 735*a**5*b**12*x + 1225*a**4*b**13*x**(3/2) + 1225*a**3*b**14*x**2 + 735*a**2*b**15*x**(5/2) + 245*a*b**16*x**3 + 35*b**17*x**(7/2)) + 6534* a**9/(35*a**7*b**10 + 245*a**6*b**11*sqrt(x) + 735*a**5*b**12*x + 1225*a** 4*b**13*x**(3/2) + 1225*a**3*b**14*x**2 + 735*a**2*b**15*x**(5/2) + 245*a* b**16*x**3 + 35*b**17*x**(7/2)) + 17640*a**8*b*sqrt(x)*log(a/b + sqrt(x))/ (35*a**7*b**10 + 245*a**6*b**11*sqrt(x) + 735*a**5*b**12*x + 1225*a**4*b** 13*x**(3/2) + 1225*a**3*b**14*x**2 + 735*a**2*b**15*x**(5/2) + 245*a*b**16 *x**3 + 35*b**17*x**(7/2)) + 43218*a**8*b*sqrt(x)/(35*a**7*b**10 + 245*a** 6*b**11*sqrt(x) + 735*a**5*b**12*x + 1225*a**4*b**13*x**(3/2) + 1225*a**3* b**14*x**2 + 735*a**2*b**15*x**(5/2) + 245*a*b**16*x**3 + 35*b**17*x**(7/2 )) + 52920*a**7*b**2*x*log(a/b + sqrt(x))/(35*a**7*b**10 + 245*a**6*b**11* sqrt(x) + 735*a**5*b**12*x + 1225*a**4*b**13*x**(3/2) + 1225*a**3*b**14*x* *2 + 735*a**2*b**15*x**(5/2) + 245*a*b**16*x**3 + 35*b**17*x**(7/2)) + 120 834*a**7*b**2*x/(35*a**7*b**10 + 245*a**6*b**11*sqrt(x) + 735*a**5*b**12*x + 1225*a**4*b**13*x**(3/2) + 1225*a**3*b**14*x**2 + 735*a**2*b**15*x**(5/ 2) + 245*a*b**16*x**3 + 35*b**17*x**(7/2)) + 88200*a**6*b**3*x**(3/2)*log( a/b + sqrt(x))/(35*a**7*b**10 + 245*a**6*b**11*sqrt(x) + 735*a**5*b**12*x + 1225*a**4*b**13*x**(3/2) + 1225*a**3*b**14*x**2 + 735*a**2*b**15*x**(5/2 ) + 245*a*b**16*x**3 + 35*b**17*x**(7/2)) + 183750*a**6*b**3*x**(3/2)/(...
Time = 0.03 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.94 \[ \int \frac {x^4}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {72 \, a^{2} \log \left (b \sqrt {x} + a\right )}{b^{10}} + \frac {{\left (b \sqrt {x} + a\right )}^{2}}{b^{10}} - \frac {18 \, {\left (b \sqrt {x} + a\right )} a}{b^{10}} + \frac {168 \, a^{3}}{{\left (b \sqrt {x} + a\right )} b^{10}} - \frac {126 \, a^{4}}{{\left (b \sqrt {x} + a\right )}^{2} b^{10}} + \frac {84 \, a^{5}}{{\left (b \sqrt {x} + a\right )}^{3} b^{10}} - \frac {42 \, a^{6}}{{\left (b \sqrt {x} + a\right )}^{4} b^{10}} + \frac {72 \, a^{7}}{5 \, {\left (b \sqrt {x} + a\right )}^{5} b^{10}} - \frac {3 \, a^{8}}{{\left (b \sqrt {x} + a\right )}^{6} b^{10}} + \frac {2 \, a^{9}}{7 \, {\left (b \sqrt {x} + a\right )}^{7} b^{10}} \] Input:
integrate(x^4/(a+b*x^(1/2))^8,x, algorithm="maxima")
Output:
72*a^2*log(b*sqrt(x) + a)/b^10 + (b*sqrt(x) + a)^2/b^10 - 18*(b*sqrt(x) + a)*a/b^10 + 168*a^3/((b*sqrt(x) + a)*b^10) - 126*a^4/((b*sqrt(x) + a)^2*b^ 10) + 84*a^5/((b*sqrt(x) + a)^3*b^10) - 42*a^6/((b*sqrt(x) + a)^4*b^10) + 72/5*a^7/((b*sqrt(x) + a)^5*b^10) - 3*a^8/((b*sqrt(x) + a)^6*b^10) + 2/7*a ^9/((b*sqrt(x) + a)^7*b^10)
Time = 0.12 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.69 \[ \int \frac {x^4}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {72 \, a^{2} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{b^{10}} + \frac {b^{8} x - 16 \, a b^{7} \sqrt {x}}{b^{16}} + \frac {5880 \, a^{3} b^{6} x^{3} + 30870 \, a^{4} b^{5} x^{\frac {5}{2}} + 69090 \, a^{5} b^{4} x^{2} + 83790 \, a^{6} b^{3} x^{\frac {3}{2}} + 57834 \, a^{7} b^{2} x + 21483 \, a^{8} b \sqrt {x} + 3349 \, a^{9}}{35 \, {\left (b \sqrt {x} + a\right )}^{7} b^{10}} \] Input:
integrate(x^4/(a+b*x^(1/2))^8,x, algorithm="giac")
Output:
72*a^2*log(abs(b*sqrt(x) + a))/b^10 + (b^8*x - 16*a*b^7*sqrt(x))/b^16 + 1/ 35*(5880*a^3*b^6*x^3 + 30870*a^4*b^5*x^(5/2) + 69090*a^5*b^4*x^2 + 83790*a ^6*b^3*x^(3/2) + 57834*a^7*b^2*x + 21483*a^8*b*sqrt(x) + 3349*a^9)/((b*sqr t(x) + a)^7*b^10)
Time = 0.10 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.04 \[ \int \frac {x^4}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {x}{b^8}+\frac {\frac {3349\,a^9}{35\,b}+\frac {3069\,a^8\,\sqrt {x}}{5}+1974\,a^5\,b^3\,x^2+168\,a^3\,b^5\,x^3+2394\,a^6\,b^2\,x^{3/2}+882\,a^4\,b^4\,x^{5/2}+\frac {8262\,a^7\,b\,x}{5}}{a^7\,b^9+b^{16}\,x^{7/2}+21\,a^5\,b^{11}\,x+7\,a\,b^{15}\,x^3+35\,a^3\,b^{13}\,x^2+7\,a^6\,b^{10}\,\sqrt {x}+35\,a^4\,b^{12}\,x^{3/2}+21\,a^2\,b^{14}\,x^{5/2}}-\frac {16\,a\,\sqrt {x}}{b^9}+\frac {72\,a^2\,\ln \left (a+b\,\sqrt {x}\right )}{b^{10}} \] Input:
int(x^4/(a + b*x^(1/2))^8,x)
Output:
x/b^8 + ((3349*a^9)/(35*b) + (3069*a^8*x^(1/2))/5 + 1974*a^5*b^3*x^2 + 168 *a^3*b^5*x^3 + 2394*a^6*b^2*x^(3/2) + 882*a^4*b^4*x^(5/2) + (8262*a^7*b*x) /5)/(a^7*b^9 + b^16*x^(7/2) + 21*a^5*b^11*x + 7*a*b^15*x^3 + 35*a^3*b^13*x ^2 + 7*a^6*b^10*x^(1/2) + 35*a^4*b^12*x^(3/2) + 21*a^2*b^14*x^(5/2)) - (16 *a*x^(1/2))/b^9 + (72*a^2*log(a + b*x^(1/2)))/b^10
Time = 0.27 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.79 \[ \int \frac {x^4}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {17640 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{8} b +88200 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{6} b^{3} x +52920 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{4} b^{5} x^{2}+2520 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{2} b^{7} x^{3}+25578 \sqrt {x}\, a^{8} b +95550 \sqrt {x}\, a^{6} b^{3} x +26460 \sqrt {x}\, a^{4} b^{5} x^{2}-2520 \sqrt {x}\, a^{2} b^{7} x^{3}+35 \sqrt {x}\, b^{9} x^{4}+2520 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{9}+52920 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{7} b^{2} x +88200 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{5} b^{4} x^{2}+17640 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{3} b^{6} x^{3}+4014 a^{9}+67914 a^{7} b^{2} x +73500 a^{5} b^{4} x^{2}-315 a \,b^{8} x^{4}}{35 b^{10} \left (7 \sqrt {x}\, a^{6} b +35 \sqrt {x}\, a^{4} b^{3} x +21 \sqrt {x}\, a^{2} b^{5} x^{2}+\sqrt {x}\, b^{7} x^{3}+a^{7}+21 a^{5} b^{2} x +35 a^{3} b^{4} x^{2}+7 a \,b^{6} x^{3}\right )} \] Input:
int(x^4/(a+b*x^(1/2))^8,x)
Output:
(17640*sqrt(x)*log(sqrt(x)*b + a)*a**8*b + 88200*sqrt(x)*log(sqrt(x)*b + a )*a**6*b**3*x + 52920*sqrt(x)*log(sqrt(x)*b + a)*a**4*b**5*x**2 + 2520*sqr t(x)*log(sqrt(x)*b + a)*a**2*b**7*x**3 + 25578*sqrt(x)*a**8*b + 95550*sqrt (x)*a**6*b**3*x + 26460*sqrt(x)*a**4*b**5*x**2 - 2520*sqrt(x)*a**2*b**7*x* *3 + 35*sqrt(x)*b**9*x**4 + 2520*log(sqrt(x)*b + a)*a**9 + 52920*log(sqrt( x)*b + a)*a**7*b**2*x + 88200*log(sqrt(x)*b + a)*a**5*b**4*x**2 + 17640*lo g(sqrt(x)*b + a)*a**3*b**6*x**3 + 4014*a**9 + 67914*a**7*b**2*x + 73500*a* *5*b**4*x**2 - 315*a*b**8*x**4)/(35*b**10*(7*sqrt(x)*a**6*b + 35*sqrt(x)*a **4*b**3*x + 21*sqrt(x)*a**2*b**5*x**2 + sqrt(x)*b**7*x**3 + a**7 + 21*a** 5*b**2*x + 35*a**3*b**4*x**2 + 7*a*b**6*x**3))