Integrand size = 15, antiderivative size = 157 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {2 a^7}{7 b^8 \left (a+b \sqrt {x}\right )^7}-\frac {7 a^6}{3 b^8 \left (a+b \sqrt {x}\right )^6}+\frac {42 a^5}{5 b^8 \left (a+b \sqrt {x}\right )^5}-\frac {35 a^4}{2 b^8 \left (a+b \sqrt {x}\right )^4}+\frac {70 a^3}{3 b^8 \left (a+b \sqrt {x}\right )^3}-\frac {21 a^2}{b^8 \left (a+b \sqrt {x}\right )^2}+\frac {14 a}{b^8 \left (a+b \sqrt {x}\right )}+\frac {2 \log \left (a+b \sqrt {x}\right )}{b^8} \] Output:
2/7*a^7/b^8/(a+b*x^(1/2))^7-7/3*a^6/b^8/(a+b*x^(1/2))^6+42/5*a^5/b^8/(a+b* x^(1/2))^5-35/2*a^4/b^8/(a+b*x^(1/2))^4+70/3*a^3/b^8/(a+b*x^(1/2))^3-21*a^ 2/b^8/(a+b*x^(1/2))^2+14*a/b^8/(a+b*x^(1/2))+2*ln(a+b*x^(1/2))/b^8
Time = 0.07 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.66 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {a \left (1089 a^6+7203 a^5 b \sqrt {x}+20139 a^4 b^2 x+30625 a^3 b^3 x^{3/2}+26950 a^2 b^4 x^2+13230 a b^5 x^{5/2}+2940 b^6 x^3\right )}{210 b^8 \left (a+b \sqrt {x}\right )^7}+\frac {2 \log \left (a+b \sqrt {x}\right )}{b^8} \] Input:
Integrate[x^3/(a + b*Sqrt[x])^8,x]
Output:
(a*(1089*a^6 + 7203*a^5*b*Sqrt[x] + 20139*a^4*b^2*x + 30625*a^3*b^3*x^(3/2 ) + 26950*a^2*b^4*x^2 + 13230*a*b^5*x^(5/2) + 2940*b^6*x^3))/(210*b^8*(a + b*Sqrt[x])^7) + (2*Log[a + b*Sqrt[x]])/b^8
Time = 0.50 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.02, 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.
\(\displaystyle \int \frac {x^3}{\left (a+b \sqrt {x}\right )^8} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 2 \int \frac {x^{7/2}}{\left (a+b \sqrt {x}\right )^8}d\sqrt {x}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle 2 \int \left (-\frac {a^7}{b^7 \left (a+b \sqrt {x}\right )^8}+\frac {7 a^6}{b^7 \left (a+b \sqrt {x}\right )^7}-\frac {21 a^5}{b^7 \left (a+b \sqrt {x}\right )^6}+\frac {35 a^4}{b^7 \left (a+b \sqrt {x}\right )^5}-\frac {35 a^3}{b^7 \left (a+b \sqrt {x}\right )^4}+\frac {21 a^2}{b^7 \left (a+b \sqrt {x}\right )^3}-\frac {7 a}{b^7 \left (a+b \sqrt {x}\right )^2}+\frac {1}{b^7 \left (a+b \sqrt {x}\right )}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (\frac {a^7}{7 b^8 \left (a+b \sqrt {x}\right )^7}-\frac {7 a^6}{6 b^8 \left (a+b \sqrt {x}\right )^6}+\frac {21 a^5}{5 b^8 \left (a+b \sqrt {x}\right )^5}-\frac {35 a^4}{4 b^8 \left (a+b \sqrt {x}\right )^4}+\frac {35 a^3}{3 b^8 \left (a+b \sqrt {x}\right )^3}-\frac {21 a^2}{2 b^8 \left (a+b \sqrt {x}\right )^2}+\frac {7 a}{b^8 \left (a+b \sqrt {x}\right )}+\frac {\log \left (a+b \sqrt {x}\right )}{b^8}\right )\) |
Input:
Int[x^3/(a + b*Sqrt[x])^8,x]
Output:
2*(a^7/(7*b^8*(a + b*Sqrt[x])^7) - (7*a^6)/(6*b^8*(a + b*Sqrt[x])^6) + (21 *a^5)/(5*b^8*(a + b*Sqrt[x])^5) - (35*a^4)/(4*b^8*(a + b*Sqrt[x])^4) + (35 *a^3)/(3*b^8*(a + b*Sqrt[x])^3) - (21*a^2)/(2*b^8*(a + b*Sqrt[x])^2) + (7* a)/(b^8*(a + b*Sqrt[x])) + Log[a + b*Sqrt[x]]/b^8)
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 = 0.45 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(\frac {2 a^{7}}{7 b^{8} \left (a +b \sqrt {x}\right )^{7}}-\frac {7 a^{6}}{3 b^{8} \left (a +b \sqrt {x}\right )^{6}}+\frac {42 a^{5}}{5 b^{8} \left (a +b \sqrt {x}\right )^{5}}-\frac {35 a^{4}}{2 b^{8} \left (a +b \sqrt {x}\right )^{4}}+\frac {70 a^{3}}{3 b^{8} \left (a +b \sqrt {x}\right )^{3}}-\frac {21 a^{2}}{b^{8} \left (a +b \sqrt {x}\right )^{2}}+\frac {14 a}{b^{8} \left (a +b \sqrt {x}\right )}+\frac {2 \ln \left (a +b \sqrt {x}\right )}{b^{8}}\) | \(132\) |
default | \(\frac {2 a^{7}}{7 b^{8} \left (a +b \sqrt {x}\right )^{7}}-\frac {7 a^{6}}{3 b^{8} \left (a +b \sqrt {x}\right )^{6}}+\frac {42 a^{5}}{5 b^{8} \left (a +b \sqrt {x}\right )^{5}}-\frac {35 a^{4}}{2 b^{8} \left (a +b \sqrt {x}\right )^{4}}+\frac {70 a^{3}}{3 b^{8} \left (a +b \sqrt {x}\right )^{3}}-\frac {21 a^{2}}{b^{8} \left (a +b \sqrt {x}\right )^{2}}+\frac {14 a}{b^{8} \left (a +b \sqrt {x}\right )}+\frac {2 \ln \left (a +b \sqrt {x}\right )}{b^{8}}\) | \(132\) |
Input:
int(x^3/(a+b*x^(1/2))^8,x,method=_RETURNVERBOSE)
Output:
2/7*a^7/b^8/(a+b*x^(1/2))^7-7/3*a^6/b^8/(a+b*x^(1/2))^6+42/5*a^5/b^8/(a+b* x^(1/2))^5-35/2*a^4/b^8/(a+b*x^(1/2))^4+70/3*a^3/b^8/(a+b*x^(1/2))^3-21*a^ 2/b^8/(a+b*x^(1/2))^2+14*a/b^8/(a+b*x^(1/2))+2*ln(a+b*x^(1/2))/b^8
Leaf count of result is larger than twice the leaf count of optimal. 315 vs. \(2 (131) = 262\).
Time = 0.12 (sec) , antiderivative size = 315, normalized size of antiderivative = 2.01 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^8} \, dx=-\frac {7350 \, a^{2} b^{12} x^{6} - 16905 \, a^{4} b^{10} x^{5} + 32585 \, a^{6} b^{8} x^{4} - 34370 \, a^{8} b^{6} x^{3} + 21504 \, a^{10} b^{4} x^{2} - 7413 \, a^{12} b^{2} x + 1089 \, a^{14} - 420 \, {\left (b^{14} x^{7} - 7 \, a^{2} b^{12} x^{6} + 21 \, a^{4} b^{10} x^{5} - 35 \, a^{6} b^{8} x^{4} + 35 \, a^{8} b^{6} x^{3} - 21 \, a^{10} b^{4} x^{2} + 7 \, a^{12} b^{2} x - a^{14}\right )} \log \left (b \sqrt {x} + a\right ) - 4 \, {\left (735 \, a b^{13} x^{6} - 980 \, a^{3} b^{11} x^{5} + 2891 \, a^{5} b^{9} x^{4} - 3072 \, a^{7} b^{7} x^{3} + 1981 \, a^{9} b^{5} x^{2} - 700 \, a^{11} b^{3} x + 105 \, a^{13} b\right )} \sqrt {x}}{210 \, {\left (b^{22} x^{7} - 7 \, a^{2} b^{20} x^{6} + 21 \, a^{4} b^{18} x^{5} - 35 \, a^{6} b^{16} x^{4} + 35 \, a^{8} b^{14} x^{3} - 21 \, a^{10} b^{12} x^{2} + 7 \, a^{12} b^{10} x - a^{14} b^{8}\right )}} \] Input:
integrate(x^3/(a+b*x^(1/2))^8,x, algorithm="fricas")
Output:
-1/210*(7350*a^2*b^12*x^6 - 16905*a^4*b^10*x^5 + 32585*a^6*b^8*x^4 - 34370 *a^8*b^6*x^3 + 21504*a^10*b^4*x^2 - 7413*a^12*b^2*x + 1089*a^14 - 420*(b^1 4*x^7 - 7*a^2*b^12*x^6 + 21*a^4*b^10*x^5 - 35*a^6*b^8*x^4 + 35*a^8*b^6*x^3 - 21*a^10*b^4*x^2 + 7*a^12*b^2*x - a^14)*log(b*sqrt(x) + a) - 4*(735*a*b^ 13*x^6 - 980*a^3*b^11*x^5 + 2891*a^5*b^9*x^4 - 3072*a^7*b^7*x^3 + 1981*a^9 *b^5*x^2 - 700*a^11*b^3*x + 105*a^13*b)*sqrt(x))/(b^22*x^7 - 7*a^2*b^20*x^ 6 + 21*a^4*b^18*x^5 - 35*a^6*b^16*x^4 + 35*a^8*b^14*x^3 - 21*a^10*b^12*x^2 + 7*a^12*b^10*x - a^14*b^8)
Leaf count of result is larger than twice the leaf count of optimal. 1629 vs. \(2 (150) = 300\).
Time = 2.18 (sec) , antiderivative size = 1629, normalized size of antiderivative = 10.38 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^8} \, dx=\text {Too large to display} \] Input:
integrate(x**3/(a+b*x**(1/2))**8,x)
Output:
Piecewise((420*a**7*log(a/b + sqrt(x))/(210*a**7*b**8 + 1470*a**6*b**9*sqr t(x) + 4410*a**5*b**10*x + 7350*a**4*b**11*x**(3/2) + 7350*a**3*b**12*x**2 + 4410*a**2*b**13*x**(5/2) + 1470*a*b**14*x**3 + 210*b**15*x**(7/2)) + 10 89*a**7/(210*a**7*b**8 + 1470*a**6*b**9*sqrt(x) + 4410*a**5*b**10*x + 7350 *a**4*b**11*x**(3/2) + 7350*a**3*b**12*x**2 + 4410*a**2*b**13*x**(5/2) + 1 470*a*b**14*x**3 + 210*b**15*x**(7/2)) + 2940*a**6*b*sqrt(x)*log(a/b + sqr t(x))/(210*a**7*b**8 + 1470*a**6*b**9*sqrt(x) + 4410*a**5*b**10*x + 7350*a **4*b**11*x**(3/2) + 7350*a**3*b**12*x**2 + 4410*a**2*b**13*x**(5/2) + 147 0*a*b**14*x**3 + 210*b**15*x**(7/2)) + 7203*a**6*b*sqrt(x)/(210*a**7*b**8 + 1470*a**6*b**9*sqrt(x) + 4410*a**5*b**10*x + 7350*a**4*b**11*x**(3/2) + 7350*a**3*b**12*x**2 + 4410*a**2*b**13*x**(5/2) + 1470*a*b**14*x**3 + 210* b**15*x**(7/2)) + 8820*a**5*b**2*x*log(a/b + sqrt(x))/(210*a**7*b**8 + 147 0*a**6*b**9*sqrt(x) + 4410*a**5*b**10*x + 7350*a**4*b**11*x**(3/2) + 7350* a**3*b**12*x**2 + 4410*a**2*b**13*x**(5/2) + 1470*a*b**14*x**3 + 210*b**15 *x**(7/2)) + 20139*a**5*b**2*x/(210*a**7*b**8 + 1470*a**6*b**9*sqrt(x) + 4 410*a**5*b**10*x + 7350*a**4*b**11*x**(3/2) + 7350*a**3*b**12*x**2 + 4410* a**2*b**13*x**(5/2) + 1470*a*b**14*x**3 + 210*b**15*x**(7/2)) + 14700*a**4 *b**3*x**(3/2)*log(a/b + sqrt(x))/(210*a**7*b**8 + 1470*a**6*b**9*sqrt(x) + 4410*a**5*b**10*x + 7350*a**4*b**11*x**(3/2) + 7350*a**3*b**12*x**2 + 44 10*a**2*b**13*x**(5/2) + 1470*a*b**14*x**3 + 210*b**15*x**(7/2)) + 3062...
Time = 0.03 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.83 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {2 \, \log \left (b \sqrt {x} + a\right )}{b^{8}} + \frac {14 \, a}{{\left (b \sqrt {x} + a\right )} b^{8}} - \frac {21 \, a^{2}}{{\left (b \sqrt {x} + a\right )}^{2} b^{8}} + \frac {70 \, a^{3}}{3 \, {\left (b \sqrt {x} + a\right )}^{3} b^{8}} - \frac {35 \, a^{4}}{2 \, {\left (b \sqrt {x} + a\right )}^{4} b^{8}} + \frac {42 \, a^{5}}{5 \, {\left (b \sqrt {x} + a\right )}^{5} b^{8}} - \frac {7 \, a^{6}}{3 \, {\left (b \sqrt {x} + a\right )}^{6} b^{8}} + \frac {2 \, a^{7}}{7 \, {\left (b \sqrt {x} + a\right )}^{7} b^{8}} \] Input:
integrate(x^3/(a+b*x^(1/2))^8,x, algorithm="maxima")
Output:
2*log(b*sqrt(x) + a)/b^8 + 14*a/((b*sqrt(x) + a)*b^8) - 21*a^2/((b*sqrt(x) + a)^2*b^8) + 70/3*a^3/((b*sqrt(x) + a)^3*b^8) - 35/2*a^4/((b*sqrt(x) + a )^4*b^8) + 42/5*a^5/((b*sqrt(x) + a)^5*b^8) - 7/3*a^6/((b*sqrt(x) + a)^6*b ^8) + 2/7*a^7/((b*sqrt(x) + a)^7*b^8)
Time = 0.12 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.61 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {2 \, \log \left ({\left | b \sqrt {x} + a \right |}\right )}{b^{8}} + \frac {2940 \, a b^{5} x^{3} + 13230 \, a^{2} b^{4} x^{\frac {5}{2}} + 26950 \, a^{3} b^{3} x^{2} + 30625 \, a^{4} b^{2} x^{\frac {3}{2}} + 20139 \, a^{5} b x + 7203 \, a^{6} \sqrt {x} + \frac {1089 \, a^{7}}{b}}{210 \, {\left (b \sqrt {x} + a\right )}^{7} b^{7}} \] Input:
integrate(x^3/(a+b*x^(1/2))^8,x, algorithm="giac")
Output:
2*log(abs(b*sqrt(x) + a))/b^8 + 1/210*(2940*a*b^5*x^3 + 13230*a^2*b^4*x^(5 /2) + 26950*a^3*b^3*x^2 + 30625*a^4*b^2*x^(3/2) + 20139*a^5*b*x + 7203*a^6 *sqrt(x) + 1089*a^7/b)/((b*sqrt(x) + a)^7*b^7)
Time = 0.44 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.01 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {\frac {363\,a^7}{70\,b^8}+\frac {14\,a\,x^3}{b^2}+\frac {959\,a^5\,x}{10\,b^6}+\frac {385\,a^3\,x^2}{3\,b^4}+\frac {63\,a^2\,x^{5/2}}{b^3}+\frac {875\,a^4\,x^{3/2}}{6\,b^5}+\frac {343\,a^6\,\sqrt {x}}{10\,b^7}}{a^7+b^7\,x^{7/2}+21\,a^5\,b^2\,x+7\,a\,b^6\,x^3+7\,a^6\,b\,\sqrt {x}+35\,a^3\,b^4\,x^2+35\,a^4\,b^3\,x^{3/2}+21\,a^2\,b^5\,x^{5/2}}+\frac {2\,\ln \left (a+b\,\sqrt {x}\right )}{b^8} \] Input:
int(x^3/(a + b*x^(1/2))^8,x)
Output:
((363*a^7)/(70*b^8) + (14*a*x^3)/b^2 + (959*a^5*x)/(10*b^6) + (385*a^3*x^2 )/(3*b^4) + (63*a^2*x^(5/2))/b^3 + (875*a^4*x^(3/2))/(6*b^5) + (343*a^6*x^ (1/2))/(10*b^7))/(a^7 + b^7*x^(7/2) + 21*a^5*b^2*x + 7*a*b^6*x^3 + 7*a^6*b *x^(1/2) + 35*a^3*b^4*x^2 + 35*a^4*b^3*x^(3/2) + 21*a^2*b^5*x^(5/2)) + (2* log(a + b*x^(1/2)))/b^8
Time = 0.20 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.79 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {2940 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{6} b +14700 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{4} b^{3} x +8820 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{2} b^{5} x^{2}+420 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, b +a \right ) b^{7} x^{3}+4263 \sqrt {x}\, a^{6} b +15925 \sqrt {x}\, a^{4} b^{3} x +4410 \sqrt {x}\, a^{2} b^{5} x^{2}-420 \sqrt {x}\, b^{7} x^{3}+420 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{7}+8820 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{5} b^{2} x +14700 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{3} b^{4} x^{2}+2940 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a \,b^{6} x^{3}+669 a^{7}+11319 a^{5} b^{2} x +12250 a^{3} b^{4} x^{2}}{210 b^{8} \left (7 \sqrt {x}\, a^{6} b +35 \sqrt {x}\, a^{4} b^{3} x +21 \sqrt {x}\, a^{2} b^{5} x^{2}+\sqrt {x}\, b^{7} x^{3}+a^{7}+21 a^{5} b^{2} x +35 a^{3} b^{4} x^{2}+7 a \,b^{6} x^{3}\right )} \] Input:
int(x^3/(a+b*x^(1/2))^8,x)
Output:
(2940*sqrt(x)*log(sqrt(x)*b + a)*a**6*b + 14700*sqrt(x)*log(sqrt(x)*b + a) *a**4*b**3*x + 8820*sqrt(x)*log(sqrt(x)*b + a)*a**2*b**5*x**2 + 420*sqrt(x )*log(sqrt(x)*b + a)*b**7*x**3 + 4263*sqrt(x)*a**6*b + 15925*sqrt(x)*a**4* b**3*x + 4410*sqrt(x)*a**2*b**5*x**2 - 420*sqrt(x)*b**7*x**3 + 420*log(sqr t(x)*b + a)*a**7 + 8820*log(sqrt(x)*b + a)*a**5*b**2*x + 14700*log(sqrt(x) *b + a)*a**3*b**4*x**2 + 2940*log(sqrt(x)*b + a)*a*b**6*x**3 + 669*a**7 + 11319*a**5*b**2*x + 12250*a**3*b**4*x**2)/(210*b**8*(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))