Integrand size = 15, antiderivative size = 131 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^5} \, dx=\frac {a^7}{2 b^8 \left (a+b \sqrt {x}\right )^4}-\frac {14 a^6}{3 b^8 \left (a+b \sqrt {x}\right )^3}+\frac {21 a^5}{b^8 \left (a+b \sqrt {x}\right )^2}-\frac {70 a^4}{b^8 \left (a+b \sqrt {x}\right )}+\frac {30 a^2 \sqrt {x}}{b^7}-\frac {5 a x}{b^6}+\frac {2 x^{3/2}}{3 b^5}-\frac {70 a^3 \log \left (a+b \sqrt {x}\right )}{b^8} \] Output:
1/2*a^7/b^8/(a+b*x^(1/2))^4-14/3*a^6/b^8/(a+b*x^(1/2))^3+21*a^5/b^8/(a+b*x ^(1/2))^2-70*a^4/b^8/(a+b*x^(1/2))+30*a^2*x^(1/2)/b^7-5*a*x/b^6+2/3*x^(3/2 )/b^5-70*a^3*ln(a+b*x^(1/2))/b^8
Time = 0.07 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.91 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^5} \, dx=\frac {-319 a^7-856 a^6 b \sqrt {x}-444 a^5 b^2 x+544 a^4 b^3 x^{3/2}+556 a^3 b^4 x^2+84 a^2 b^5 x^{5/2}-14 a b^6 x^3+4 b^7 x^{7/2}}{6 b^8 \left (a+b \sqrt {x}\right )^4}-\frac {70 a^3 \log \left (a+b \sqrt {x}\right )}{b^8} \] Input:
Integrate[x^3/(a + b*Sqrt[x])^5,x]
Output:
(-319*a^7 - 856*a^6*b*Sqrt[x] - 444*a^5*b^2*x + 544*a^4*b^3*x^(3/2) + 556* a^3*b^4*x^2 + 84*a^2*b^5*x^(5/2) - 14*a*b^6*x^3 + 4*b^7*x^(7/2))/(6*b^8*(a + b*Sqrt[x])^4) - (70*a^3*Log[a + b*Sqrt[x]])/b^8
Time = 0.50 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.05, 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.
\(\displaystyle \int \frac {x^3}{\left (a+b \sqrt {x}\right )^5} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 2 \int \frac {x^{7/2}}{\left (a+b \sqrt {x}\right )^5}d\sqrt {x}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle 2 \int \left (-\frac {a^7}{b^7 \left (a+b \sqrt {x}\right )^5}+\frac {7 a^6}{b^7 \left (a+b \sqrt {x}\right )^4}-\frac {21 a^5}{b^7 \left (a+b \sqrt {x}\right )^3}+\frac {35 a^4}{b^7 \left (a+b \sqrt {x}\right )^2}-\frac {35 a^3}{b^7 \left (a+b \sqrt {x}\right )}+\frac {15 a^2}{b^7}-\frac {5 \sqrt {x} a}{b^6}+\frac {x}{b^5}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (\frac {a^7}{4 b^8 \left (a+b \sqrt {x}\right )^4}-\frac {7 a^6}{3 b^8 \left (a+b \sqrt {x}\right )^3}+\frac {21 a^5}{2 b^8 \left (a+b \sqrt {x}\right )^2}-\frac {35 a^4}{b^8 \left (a+b \sqrt {x}\right )}-\frac {35 a^3 \log \left (a+b \sqrt {x}\right )}{b^8}+\frac {15 a^2 \sqrt {x}}{b^7}-\frac {5 a x}{2 b^6}+\frac {x^{3/2}}{3 b^5}\right )\) |
Input:
Int[x^3/(a + b*Sqrt[x])^5,x]
Output:
2*(a^7/(4*b^8*(a + b*Sqrt[x])^4) - (7*a^6)/(3*b^8*(a + b*Sqrt[x])^3) + (21 *a^5)/(2*b^8*(a + b*Sqrt[x])^2) - (35*a^4)/(b^8*(a + b*Sqrt[x])) + (15*a^2 *Sqrt[x])/b^7 - (5*a*x)/(2*b^6) + x^(3/2)/(3*b^5) - (35*a^3*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 = 0.50 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(\frac {\frac {2 b^{2} x^{\frac {3}{2}}}{3}-5 a b x +30 a^{2} \sqrt {x}}{b^{7}}-\frac {14 a^{6}}{3 b^{8} \left (a +b \sqrt {x}\right )^{3}}+\frac {21 a^{5}}{b^{8} \left (a +b \sqrt {x}\right )^{2}}-\frac {70 a^{3} \ln \left (a +b \sqrt {x}\right )}{b^{8}}-\frac {70 a^{4}}{b^{8} \left (a +b \sqrt {x}\right )}+\frac {a^{7}}{2 b^{8} \left (a +b \sqrt {x}\right )^{4}}\) | \(113\) |
default | \(\frac {\frac {2 b^{2} x^{\frac {3}{2}}}{3}-5 a b x +30 a^{2} \sqrt {x}}{b^{7}}-\frac {14 a^{6}}{3 b^{8} \left (a +b \sqrt {x}\right )^{3}}+\frac {21 a^{5}}{b^{8} \left (a +b \sqrt {x}\right )^{2}}-\frac {70 a^{3} \ln \left (a +b \sqrt {x}\right )}{b^{8}}-\frac {70 a^{4}}{b^{8} \left (a +b \sqrt {x}\right )}+\frac {a^{7}}{2 b^{8} \left (a +b \sqrt {x}\right )^{4}}\) | \(113\) |
Input:
int(x^3/(a+b*x^(1/2))^5,x,method=_RETURNVERBOSE)
Output:
2/b^7*(1/3*b^2*x^(3/2)-5/2*a*b*x+15*a^2*x^(1/2))-14/3*a^6/b^8/(a+b*x^(1/2) )^3+21*a^5/b^8/(a+b*x^(1/2))^2-70*a^3*ln(a+b*x^(1/2))/b^8-70*a^4/b^8/(a+b* x^(1/2))+1/2*a^7/b^8/(a+b*x^(1/2))^4
Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (111) = 222\).
Time = 0.10 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.70 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^5} \, dx=-\frac {30 \, a b^{10} x^{5} - 120 \, a^{3} b^{8} x^{4} - 366 \, a^{5} b^{6} x^{3} + 1179 \, a^{7} b^{4} x^{2} - 1066 \, a^{9} b^{2} x + 319 \, a^{11} + 420 \, {\left (a^{3} b^{8} x^{4} - 4 \, a^{5} b^{6} x^{3} + 6 \, a^{7} b^{4} x^{2} - 4 \, a^{9} b^{2} x + a^{11}\right )} \log \left (b \sqrt {x} + a\right ) - 4 \, {\left (b^{11} x^{5} + 41 \, a^{2} b^{9} x^{4} - 279 \, a^{4} b^{7} x^{3} + 511 \, a^{6} b^{5} x^{2} - 385 \, a^{8} b^{3} x + 105 \, a^{10} b\right )} \sqrt {x}}{6 \, {\left (b^{16} x^{4} - 4 \, a^{2} b^{14} x^{3} + 6 \, a^{4} b^{12} x^{2} - 4 \, a^{6} b^{10} x + a^{8} b^{8}\right )}} \] Input:
integrate(x^3/(a+b*x^(1/2))^5,x, algorithm="fricas")
Output:
-1/6*(30*a*b^10*x^5 - 120*a^3*b^8*x^4 - 366*a^5*b^6*x^3 + 1179*a^7*b^4*x^2 - 1066*a^9*b^2*x + 319*a^11 + 420*(a^3*b^8*x^4 - 4*a^5*b^6*x^3 + 6*a^7*b^ 4*x^2 - 4*a^9*b^2*x + a^11)*log(b*sqrt(x) + a) - 4*(b^11*x^5 + 41*a^2*b^9* x^4 - 279*a^4*b^7*x^3 + 511*a^6*b^5*x^2 - 385*a^8*b^3*x + 105*a^10*b)*sqrt (x))/(b^16*x^4 - 4*a^2*b^14*x^3 + 6*a^4*b^12*x^2 - 4*a^6*b^10*x + a^8*b^8)
Leaf count of result is larger than twice the leaf count of optimal. 835 vs. \(2 (126) = 252\).
Time = 1.11 (sec) , antiderivative size = 835, normalized size of antiderivative = 6.37 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^5} \, dx =\text {Too large to display} \] Input:
integrate(x**3/(a+b*x**(1/2))**5,x)
Output:
Piecewise((zoo*x**(3/2), Eq(a, 0) & Eq(b, 0)), (x**4/(4*a**5), Eq(b, 0)), (zoo*x**4, Eq(a, -b*sqrt(x))), (-420*a**7*log(a/b + sqrt(x))/(6*a**4*b**8 + 24*a**3*b**9*sqrt(x) + 36*a**2*b**10*x + 24*a*b**11*x**(3/2) + 6*b**12*x **2) - 875*a**7/(6*a**4*b**8 + 24*a**3*b**9*sqrt(x) + 36*a**2*b**10*x + 24 *a*b**11*x**(3/2) + 6*b**12*x**2) - 1680*a**6*b*sqrt(x)*log(a/b + sqrt(x)) /(6*a**4*b**8 + 24*a**3*b**9*sqrt(x) + 36*a**2*b**10*x + 24*a*b**11*x**(3/ 2) + 6*b**12*x**2) - 3080*a**6*b*sqrt(x)/(6*a**4*b**8 + 24*a**3*b**9*sqrt( x) + 36*a**2*b**10*x + 24*a*b**11*x**(3/2) + 6*b**12*x**2) - 2520*a**5*b** 2*x*log(a/b + sqrt(x))/(6*a**4*b**8 + 24*a**3*b**9*sqrt(x) + 36*a**2*b**10 *x + 24*a*b**11*x**(3/2) + 6*b**12*x**2) - 3780*a**5*b**2*x/(6*a**4*b**8 + 24*a**3*b**9*sqrt(x) + 36*a**2*b**10*x + 24*a*b**11*x**(3/2) + 6*b**12*x* *2) - 1680*a**4*b**3*x**(3/2)*log(a/b + sqrt(x))/(6*a**4*b**8 + 24*a**3*b* *9*sqrt(x) + 36*a**2*b**10*x + 24*a*b**11*x**(3/2) + 6*b**12*x**2) - 1680* a**4*b**3*x**(3/2)/(6*a**4*b**8 + 24*a**3*b**9*sqrt(x) + 36*a**2*b**10*x + 24*a*b**11*x**(3/2) + 6*b**12*x**2) - 420*a**3*b**4*x**2*log(a/b + sqrt(x ))/(6*a**4*b**8 + 24*a**3*b**9*sqrt(x) + 36*a**2*b**10*x + 24*a*b**11*x**( 3/2) + 6*b**12*x**2) + 84*a**2*b**5*x**(5/2)/(6*a**4*b**8 + 24*a**3*b**9*s qrt(x) + 36*a**2*b**10*x + 24*a*b**11*x**(3/2) + 6*b**12*x**2) - 14*a*b**6 *x**3/(6*a**4*b**8 + 24*a**3*b**9*sqrt(x) + 36*a**2*b**10*x + 24*a*b**11*x **(3/2) + 6*b**12*x**2) + 4*b**7*x**(7/2)/(6*a**4*b**8 + 24*a**3*b**9*s...
Time = 0.04 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.98 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^5} \, dx=-\frac {70 \, a^{3} \log \left (b \sqrt {x} + a\right )}{b^{8}} + \frac {2 \, {\left (b \sqrt {x} + a\right )}^{3}}{3 \, b^{8}} - \frac {7 \, {\left (b \sqrt {x} + a\right )}^{2} a}{b^{8}} + \frac {42 \, {\left (b \sqrt {x} + a\right )} a^{2}}{b^{8}} - \frac {70 \, a^{4}}{{\left (b \sqrt {x} + a\right )} b^{8}} + \frac {21 \, a^{5}}{{\left (b \sqrt {x} + a\right )}^{2} b^{8}} - \frac {14 \, a^{6}}{3 \, {\left (b \sqrt {x} + a\right )}^{3} b^{8}} + \frac {a^{7}}{2 \, {\left (b \sqrt {x} + a\right )}^{4} b^{8}} \] Input:
integrate(x^3/(a+b*x^(1/2))^5,x, algorithm="maxima")
Output:
-70*a^3*log(b*sqrt(x) + a)/b^8 + 2/3*(b*sqrt(x) + a)^3/b^8 - 7*(b*sqrt(x) + a)^2*a/b^8 + 42*(b*sqrt(x) + a)*a^2/b^8 - 70*a^4/((b*sqrt(x) + a)*b^8) + 21*a^5/((b*sqrt(x) + a)^2*b^8) - 14/3*a^6/((b*sqrt(x) + a)^3*b^8) + 1/2*a ^7/((b*sqrt(x) + a)^4*b^8)
Time = 0.12 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.76 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^5} \, dx=-\frac {70 \, a^{3} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{b^{8}} - \frac {420 \, a^{4} b^{3} x^{\frac {3}{2}} + 1134 \, a^{5} b^{2} x + 1036 \, a^{6} b \sqrt {x} + 319 \, a^{7}}{6 \, {\left (b \sqrt {x} + a\right )}^{4} b^{8}} + \frac {2 \, b^{10} x^{\frac {3}{2}} - 15 \, a b^{9} x + 90 \, a^{2} b^{8} \sqrt {x}}{3 \, b^{15}} \] Input:
integrate(x^3/(a+b*x^(1/2))^5,x, algorithm="giac")
Output:
-70*a^3*log(abs(b*sqrt(x) + a))/b^8 - 1/6*(420*a^4*b^3*x^(3/2) + 1134*a^5* b^2*x + 1036*a^6*b*sqrt(x) + 319*a^7)/((b*sqrt(x) + a)^4*b^8) + 1/3*(2*b^1 0*x^(3/2) - 15*a*b^9*x + 90*a^2*b^8*sqrt(x))/b^15
Time = 0.32 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.96 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^5} \, dx=\frac {2\,x^{3/2}}{3\,b^5}-\frac {\frac {319\,a^7}{6\,b}+\frac {518\,a^6\,\sqrt {x}}{3}+70\,a^4\,b^2\,x^{3/2}+189\,a^5\,b\,x}{a^4\,b^7+b^{11}\,x^2+6\,a^2\,b^9\,x+4\,a\,b^{10}\,x^{3/2}+4\,a^3\,b^8\,\sqrt {x}}-\frac {70\,a^3\,\ln \left (a+b\,\sqrt {x}\right )}{b^8}+\frac {30\,a^2\,\sqrt {x}}{b^7}-\frac {5\,a\,x}{b^6} \] Input:
int(x^3/(a + b*x^(1/2))^5,x)
Output:
(2*x^(3/2))/(3*b^5) - ((319*a^7)/(6*b) + (518*a^6*x^(1/2))/3 + 70*a^4*b^2* x^(3/2) + 189*a^5*b*x)/(a^4*b^7 + b^11*x^2 + 6*a^2*b^9*x + 4*a*b^10*x^(3/2 ) + 4*a^3*b^8*x^(1/2)) - (70*a^3*log(a + b*x^(1/2)))/b^8 + (30*a^2*x^(1/2) )/b^7 - (5*a*x)/b^6
Time = 0.22 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.44 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^5} \, dx=\frac {-1680 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{6} b -1680 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{4} b^{3} x -1400 \sqrt {x}\, a^{6} b +84 \sqrt {x}\, a^{2} b^{5} x^{2}+4 \sqrt {x}\, b^{7} x^{3}-420 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{7}-2520 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{5} b^{2} x -420 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{3} b^{4} x^{2}-455 a^{7}-1260 a^{5} b^{2} x +420 a^{3} b^{4} x^{2}-14 a \,b^{6} x^{3}}{6 b^{8} \left (4 \sqrt {x}\, a^{3} b +4 \sqrt {x}\, a \,b^{3} x +a^{4}+6 a^{2} b^{2} x +b^{4} x^{2}\right )} \] Input:
int(x^3/(a+b*x^(1/2))^5,x)
Output:
( - 1680*sqrt(x)*log(sqrt(x)*b + a)*a**6*b - 1680*sqrt(x)*log(sqrt(x)*b + a)*a**4*b**3*x - 1400*sqrt(x)*a**6*b + 84*sqrt(x)*a**2*b**5*x**2 + 4*sqrt( x)*b**7*x**3 - 420*log(sqrt(x)*b + a)*a**7 - 2520*log(sqrt(x)*b + a)*a**5* b**2*x - 420*log(sqrt(x)*b + a)*a**3*b**4*x**2 - 455*a**7 - 1260*a**5*b**2 *x + 420*a**3*b**4*x**2 - 14*a*b**6*x**3)/(6*b**8*(4*sqrt(x)*a**3*b + 4*sq rt(x)*a*b**3*x + a**4 + 6*a**2*b**2*x + b**4*x**2))