Integrand size = 15, antiderivative size = 155 \[ \int \frac {x^4}{\left (a+b \sqrt {x}\right )^5} \, dx=\frac {a^9}{2 b^{10} \left (a+b \sqrt {x}\right )^4}-\frac {6 a^8}{b^{10} \left (a+b \sqrt {x}\right )^3}+\frac {36 a^7}{b^{10} \left (a+b \sqrt {x}\right )^2}-\frac {168 a^6}{b^{10} \left (a+b \sqrt {x}\right )}+\frac {140 a^4 \sqrt {x}}{b^9}-\frac {35 a^3 x}{b^8}+\frac {10 a^2 x^{3/2}}{b^7}-\frac {5 a x^2}{2 b^6}+\frac {2 x^{5/2}}{5 b^5}-\frac {252 a^5 \log \left (a+b \sqrt {x}\right )}{b^{10}} \] Output:
1/2*a^9/b^10/(a+b*x^(1/2))^4-6*a^8/b^10/(a+b*x^(1/2))^3+36*a^7/b^10/(a+b*x ^(1/2))^2-168*a^6/b^10/(a+b*x^(1/2))+140*a^4*x^(1/2)/b^9-35*a^3*x/b^8+10*a ^2*x^(3/2)/b^7-5/2*a*x^2/b^6+2/5*x^(5/2)/b^5-252*a^5*ln(a+b*x^(1/2))/b^10
Time = 0.07 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.92 \[ \int \frac {x^4}{\left (a+b \sqrt {x}\right )^5} \, dx=\frac {-1375 a^9-2980 a^8 b \sqrt {x}+570 a^7 b^2 x+5420 a^6 b^3 x^{3/2}+3875 a^5 b^4 x^2+504 a^4 b^5 x^{5/2}-84 a^3 b^6 x^3+24 a^2 b^7 x^{7/2}-9 a b^8 x^4+4 b^9 x^{9/2}}{10 b^{10} \left (a+b \sqrt {x}\right )^4}-\frac {252 a^5 \log \left (a+b \sqrt {x}\right )}{b^{10}} \] Input:
Integrate[x^4/(a + b*Sqrt[x])^5,x]
Output:
(-1375*a^9 - 2980*a^8*b*Sqrt[x] + 570*a^7*b^2*x + 5420*a^6*b^3*x^(3/2) + 3 875*a^5*b^4*x^2 + 504*a^4*b^5*x^(5/2) - 84*a^3*b^6*x^3 + 24*a^2*b^7*x^(7/2 ) - 9*a*b^8*x^4 + 4*b^9*x^(9/2))/(10*b^10*(a + b*Sqrt[x])^4) - (252*a^5*Lo g[a + b*Sqrt[x]])/b^10
Time = 0.54 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.03, 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.
\(\displaystyle \int \frac {x^4}{\left (a+b \sqrt {x}\right )^5} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 2 \int \frac {x^{9/2}}{\left (a+b \sqrt {x}\right )^5}d\sqrt {x}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle 2 \int \left (-\frac {a^9}{b^9 \left (a+b \sqrt {x}\right )^5}+\frac {9 a^8}{b^9 \left (a+b \sqrt {x}\right )^4}-\frac {36 a^7}{b^9 \left (a+b \sqrt {x}\right )^3}+\frac {84 a^6}{b^9 \left (a+b \sqrt {x}\right )^2}-\frac {126 a^5}{b^9 \left (a+b \sqrt {x}\right )}+\frac {70 a^4}{b^9}-\frac {35 \sqrt {x} a^3}{b^8}+\frac {15 x a^2}{b^7}-\frac {5 x^{3/2} a}{b^6}+\frac {x^2}{b^5}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (\frac {a^9}{4 b^{10} \left (a+b \sqrt {x}\right )^4}-\frac {3 a^8}{b^{10} \left (a+b \sqrt {x}\right )^3}+\frac {18 a^7}{b^{10} \left (a+b \sqrt {x}\right )^2}-\frac {84 a^6}{b^{10} \left (a+b \sqrt {x}\right )}-\frac {126 a^5 \log \left (a+b \sqrt {x}\right )}{b^{10}}+\frac {70 a^4 \sqrt {x}}{b^9}-\frac {35 a^3 x}{2 b^8}+\frac {5 a^2 x^{3/2}}{b^7}-\frac {5 a x^2}{4 b^6}+\frac {x^{5/2}}{5 b^5}\right )\) |
Input:
Int[x^4/(a + b*Sqrt[x])^5,x]
Output:
2*(a^9/(4*b^10*(a + b*Sqrt[x])^4) - (3*a^8)/(b^10*(a + b*Sqrt[x])^3) + (18 *a^7)/(b^10*(a + b*Sqrt[x])^2) - (84*a^6)/(b^10*(a + b*Sqrt[x])) + (70*a^4 *Sqrt[x])/b^9 - (35*a^3*x)/(2*b^8) + (5*a^2*x^(3/2))/b^7 - (5*a*x^2)/(4*b^ 6) + x^(5/2)/(5*b^5) - (126*a^5*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 = 0.48 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.87
method | result | size |
derivativedivides | \(\frac {\frac {2 x^{\frac {5}{2}} b^{4}}{5}-\frac {5 a \,b^{3} x^{2}}{2}+10 x^{\frac {3}{2}} a^{2} b^{2}-35 a^{3} b x +140 \sqrt {x}\, a^{4}}{b^{9}}-\frac {6 a^{8}}{b^{10} \left (a +b \sqrt {x}\right )^{3}}+\frac {a^{9}}{2 b^{10} \left (a +b \sqrt {x}\right )^{4}}-\frac {252 a^{5} \ln \left (a +b \sqrt {x}\right )}{b^{10}}+\frac {36 a^{7}}{b^{10} \left (a +b \sqrt {x}\right )^{2}}-\frac {168 a^{6}}{b^{10} \left (a +b \sqrt {x}\right )}\) | \(135\) |
default | \(\frac {\frac {2 x^{\frac {5}{2}} b^{4}}{5}-\frac {5 a \,b^{3} x^{2}}{2}+10 x^{\frac {3}{2}} a^{2} b^{2}-35 a^{3} b x +140 \sqrt {x}\, a^{4}}{b^{9}}-\frac {6 a^{8}}{b^{10} \left (a +b \sqrt {x}\right )^{3}}+\frac {a^{9}}{2 b^{10} \left (a +b \sqrt {x}\right )^{4}}-\frac {252 a^{5} \ln \left (a +b \sqrt {x}\right )}{b^{10}}+\frac {36 a^{7}}{b^{10} \left (a +b \sqrt {x}\right )^{2}}-\frac {168 a^{6}}{b^{10} \left (a +b \sqrt {x}\right )}\) | \(135\) |
Input:
int(x^4/(a+b*x^(1/2))^5,x,method=_RETURNVERBOSE)
Output:
2/b^9*(1/5*x^(5/2)*b^4-5/4*a*b^3*x^2+5*x^(3/2)*a^2*b^2-35/2*a^3*b*x+70*x^( 1/2)*a^4)-6*a^8/b^10/(a+b*x^(1/2))^3+1/2*a^9/b^10/(a+b*x^(1/2))^4-252*a^5* ln(a+b*x^(1/2))/b^10+36*a^7/b^10/(a+b*x^(1/2))^2-168*a^6/b^10/(a+b*x^(1/2) )
Time = 0.10 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.58 \[ \int \frac {x^4}{\left (a+b \sqrt {x}\right )^5} \, dx=-\frac {25 \, a b^{12} x^{6} + 250 \, a^{3} b^{10} x^{5} - 1250 \, a^{5} b^{8} x^{4} - 40 \, a^{7} b^{6} x^{3} + 3840 \, a^{9} b^{4} x^{2} - 4240 \, a^{11} b^{2} x + 1375 \, a^{13} + 2520 \, {\left (a^{5} b^{8} x^{4} - 4 \, a^{7} b^{6} x^{3} + 6 \, a^{9} b^{4} x^{2} - 4 \, a^{11} b^{2} x + a^{13}\right )} \log \left (b \sqrt {x} + a\right ) - 4 \, {\left (b^{13} x^{6} + 21 \, a^{2} b^{11} x^{5} + 256 \, a^{4} b^{9} x^{4} - 1674 \, a^{6} b^{7} x^{3} + 3066 \, a^{8} b^{5} x^{2} - 2310 \, a^{10} b^{3} x + 630 \, a^{12} b\right )} \sqrt {x}}{10 \, {\left (b^{18} x^{4} - 4 \, a^{2} b^{16} x^{3} + 6 \, a^{4} b^{14} x^{2} - 4 \, a^{6} b^{12} x + a^{8} b^{10}\right )}} \] Input:
integrate(x^4/(a+b*x^(1/2))^5,x, algorithm="fricas")
Output:
-1/10*(25*a*b^12*x^6 + 250*a^3*b^10*x^5 - 1250*a^5*b^8*x^4 - 40*a^7*b^6*x^ 3 + 3840*a^9*b^4*x^2 - 4240*a^11*b^2*x + 1375*a^13 + 2520*(a^5*b^8*x^4 - 4 *a^7*b^6*x^3 + 6*a^9*b^4*x^2 - 4*a^11*b^2*x + a^13)*log(b*sqrt(x) + a) - 4 *(b^13*x^6 + 21*a^2*b^11*x^5 + 256*a^4*b^9*x^4 - 1674*a^6*b^7*x^3 + 3066*a ^8*b^5*x^2 - 2310*a^10*b^3*x + 630*a^12*b)*sqrt(x))/(b^18*x^4 - 4*a^2*b^16 *x^3 + 6*a^4*b^14*x^2 - 4*a^6*b^12*x + a^8*b^10)
Leaf count of result is larger than twice the leaf count of optimal. 966 vs. \(2 (151) = 302\).
Time = 1.51 (sec) , antiderivative size = 966, normalized size of antiderivative = 6.23 \[ \int \frac {x^4}{\left (a+b \sqrt {x}\right )^5} \, dx =\text {Too large to display} \] Input:
integrate(x**4/(a+b*x**(1/2))**5,x)
Output:
Piecewise((zoo*x**(5/2), Eq(a, 0) & Eq(b, 0)), (x**5/(5*a**5), Eq(b, 0)), (zoo*x**5, Eq(a, -b*sqrt(x))), (-2520*a**9*log(a/b + sqrt(x))/(10*a**4*b** 10 + 40*a**3*b**11*sqrt(x) + 60*a**2*b**12*x + 40*a*b**13*x**(3/2) + 10*b* *14*x**2) - 5250*a**9/(10*a**4*b**10 + 40*a**3*b**11*sqrt(x) + 60*a**2*b** 12*x + 40*a*b**13*x**(3/2) + 10*b**14*x**2) - 10080*a**8*b*sqrt(x)*log(a/b + sqrt(x))/(10*a**4*b**10 + 40*a**3*b**11*sqrt(x) + 60*a**2*b**12*x + 40* a*b**13*x**(3/2) + 10*b**14*x**2) - 18480*a**8*b*sqrt(x)/(10*a**4*b**10 + 40*a**3*b**11*sqrt(x) + 60*a**2*b**12*x + 40*a*b**13*x**(3/2) + 10*b**14*x **2) - 15120*a**7*b**2*x*log(a/b + sqrt(x))/(10*a**4*b**10 + 40*a**3*b**11 *sqrt(x) + 60*a**2*b**12*x + 40*a*b**13*x**(3/2) + 10*b**14*x**2) - 22680* a**7*b**2*x/(10*a**4*b**10 + 40*a**3*b**11*sqrt(x) + 60*a**2*b**12*x + 40* a*b**13*x**(3/2) + 10*b**14*x**2) - 10080*a**6*b**3*x**(3/2)*log(a/b + sqr t(x))/(10*a**4*b**10 + 40*a**3*b**11*sqrt(x) + 60*a**2*b**12*x + 40*a*b**1 3*x**(3/2) + 10*b**14*x**2) - 10080*a**6*b**3*x**(3/2)/(10*a**4*b**10 + 40 *a**3*b**11*sqrt(x) + 60*a**2*b**12*x + 40*a*b**13*x**(3/2) + 10*b**14*x** 2) - 2520*a**5*b**4*x**2*log(a/b + sqrt(x))/(10*a**4*b**10 + 40*a**3*b**11 *sqrt(x) + 60*a**2*b**12*x + 40*a*b**13*x**(3/2) + 10*b**14*x**2) + 504*a* *4*b**5*x**(5/2)/(10*a**4*b**10 + 40*a**3*b**11*sqrt(x) + 60*a**2*b**12*x + 40*a*b**13*x**(3/2) + 10*b**14*x**2) - 84*a**3*b**6*x**3/(10*a**4*b**10 + 40*a**3*b**11*sqrt(x) + 60*a**2*b**12*x + 40*a*b**13*x**(3/2) + 10*b*...
Time = 0.03 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.05 \[ \int \frac {x^4}{\left (a+b \sqrt {x}\right )^5} \, dx=-\frac {252 \, a^{5} \log \left (b \sqrt {x} + a\right )}{b^{10}} + \frac {2 \, {\left (b \sqrt {x} + a\right )}^{5}}{5 \, b^{10}} - \frac {9 \, {\left (b \sqrt {x} + a\right )}^{4} a}{2 \, b^{10}} + \frac {24 \, {\left (b \sqrt {x} + a\right )}^{3} a^{2}}{b^{10}} - \frac {84 \, {\left (b \sqrt {x} + a\right )}^{2} a^{3}}{b^{10}} + \frac {252 \, {\left (b \sqrt {x} + a\right )} a^{4}}{b^{10}} - \frac {168 \, a^{6}}{{\left (b \sqrt {x} + a\right )} b^{10}} + \frac {36 \, a^{7}}{{\left (b \sqrt {x} + a\right )}^{2} b^{10}} - \frac {6 \, a^{8}}{{\left (b \sqrt {x} + a\right )}^{3} b^{10}} + \frac {a^{9}}{2 \, {\left (b \sqrt {x} + a\right )}^{4} b^{10}} \] Input:
integrate(x^4/(a+b*x^(1/2))^5,x, algorithm="maxima")
Output:
-252*a^5*log(b*sqrt(x) + a)/b^10 + 2/5*(b*sqrt(x) + a)^5/b^10 - 9/2*(b*sqr t(x) + a)^4*a/b^10 + 24*(b*sqrt(x) + a)^3*a^2/b^10 - 84*(b*sqrt(x) + a)^2* a^3/b^10 + 252*(b*sqrt(x) + a)*a^4/b^10 - 168*a^6/((b*sqrt(x) + a)*b^10) + 36*a^7/((b*sqrt(x) + a)^2*b^10) - 6*a^8/((b*sqrt(x) + a)^3*b^10) + 1/2*a^ 9/((b*sqrt(x) + a)^4*b^10)
Time = 0.12 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.78 \[ \int \frac {x^4}{\left (a+b \sqrt {x}\right )^5} \, dx=-\frac {252 \, a^{5} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{b^{10}} - \frac {336 \, a^{6} b^{3} x^{\frac {3}{2}} + 936 \, a^{7} b^{2} x + 876 \, a^{8} b \sqrt {x} + 275 \, a^{9}}{2 \, {\left (b \sqrt {x} + a\right )}^{4} b^{10}} + \frac {4 \, b^{20} x^{\frac {5}{2}} - 25 \, a b^{19} x^{2} + 100 \, a^{2} b^{18} x^{\frac {3}{2}} - 350 \, a^{3} b^{17} x + 1400 \, a^{4} b^{16} \sqrt {x}}{10 \, b^{25}} \] Input:
integrate(x^4/(a+b*x^(1/2))^5,x, algorithm="giac")
Output:
-252*a^5*log(abs(b*sqrt(x) + a))/b^10 - 1/2*(336*a^6*b^3*x^(3/2) + 936*a^7 *b^2*x + 876*a^8*b*sqrt(x) + 275*a^9)/((b*sqrt(x) + a)^4*b^10) + 1/10*(4*b ^20*x^(5/2) - 25*a*b^19*x^2 + 100*a^2*b^18*x^(3/2) - 350*a^3*b^17*x + 1400 *a^4*b^16*sqrt(x))/b^25
Time = 0.33 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.95 \[ \int \frac {x^4}{\left (a+b \sqrt {x}\right )^5} \, dx=\frac {2\,x^{5/2}}{5\,b^5}-\frac {\frac {275\,a^9}{2\,b}+438\,a^8\,\sqrt {x}+168\,a^6\,b^2\,x^{3/2}+468\,a^7\,b\,x}{a^4\,b^9+b^{13}\,x^2+6\,a^2\,b^{11}\,x+4\,a\,b^{12}\,x^{3/2}+4\,a^3\,b^{10}\,\sqrt {x}}-\frac {5\,a\,x^2}{2\,b^6}-\frac {35\,a^3\,x}{b^8}-\frac {252\,a^5\,\ln \left (a+b\,\sqrt {x}\right )}{b^{10}}+\frac {10\,a^2\,x^{3/2}}{b^7}+\frac {140\,a^4\,\sqrt {x}}{b^9} \] Input:
int(x^4/(a + b*x^(1/2))^5,x)
Output:
(2*x^(5/2))/(5*b^5) - ((275*a^9)/(2*b) + 438*a^8*x^(1/2) + 168*a^6*b^2*x^( 3/2) + 468*a^7*b*x)/(a^4*b^9 + b^13*x^2 + 6*a^2*b^11*x + 4*a*b^12*x^(3/2) + 4*a^3*b^10*x^(1/2)) - (5*a*x^2)/(2*b^6) - (35*a^3*x)/b^8 - (252*a^5*log( a + b*x^(1/2)))/b^10 + (10*a^2*x^(3/2))/b^7 + (140*a^4*x^(1/2))/b^9
Time = 0.25 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.37 \[ \int \frac {x^4}{\left (a+b \sqrt {x}\right )^5} \, dx=\frac {-10080 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{8} b -10080 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{6} b^{3} x -8400 \sqrt {x}\, a^{8} b +504 \sqrt {x}\, a^{4} b^{5} x^{2}+24 \sqrt {x}\, a^{2} b^{7} x^{3}+4 \sqrt {x}\, b^{9} x^{4}-2520 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{9}-15120 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{7} b^{2} x -2520 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{5} b^{4} x^{2}-2730 a^{9}-7560 a^{7} b^{2} x +2520 a^{5} b^{4} x^{2}-84 a^{3} b^{6} x^{3}-9 a \,b^{8} x^{4}}{10 b^{10} \left (4 \sqrt {x}\, a^{3} b +4 \sqrt {x}\, a \,b^{3} x +a^{4}+6 a^{2} b^{2} x +b^{4} x^{2}\right )} \] Input:
int(x^4/(a+b*x^(1/2))^5,x)
Output:
( - 10080*sqrt(x)*log(sqrt(x)*b + a)*a**8*b - 10080*sqrt(x)*log(sqrt(x)*b + a)*a**6*b**3*x - 8400*sqrt(x)*a**8*b + 504*sqrt(x)*a**4*b**5*x**2 + 24*s qrt(x)*a**2*b**7*x**3 + 4*sqrt(x)*b**9*x**4 - 2520*log(sqrt(x)*b + a)*a**9 - 15120*log(sqrt(x)*b + a)*a**7*b**2*x - 2520*log(sqrt(x)*b + a)*a**5*b** 4*x**2 - 2730*a**9 - 7560*a**7*b**2*x + 2520*a**5*b**4*x**2 - 84*a**3*b**6 *x**3 - 9*a*b**8*x**4)/(10*b**10*(4*sqrt(x)*a**3*b + 4*sqrt(x)*a*b**3*x + a**4 + 6*a**2*b**2*x + b**4*x**2))