Integrand size = 15, antiderivative size = 88 \[ \int \frac {x^2}{\left (a+b \sqrt {x}\right )^3} \, dx=\frac {a^5}{b^6 \left (a+b \sqrt {x}\right )^2}-\frac {10 a^4}{b^6 \left (a+b \sqrt {x}\right )}+\frac {12 a^2 \sqrt {x}}{b^5}-\frac {3 a x}{b^4}+\frac {2 x^{3/2}}{3 b^3}-\frac {20 a^3 \log \left (a+b \sqrt {x}\right )}{b^6} \] Output:
a^5/b^6/(a+b*x^(1/2))^2-10*a^4/b^6/(a+b*x^(1/2))+12*a^2*x^(1/2)/b^5-3*a*x/ b^4+2/3*x^(3/2)/b^3-20*a^3*ln(a+b*x^(1/2))/b^6
Time = 0.06 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.08 \[ \int \frac {x^2}{\left (a+b \sqrt {x}\right )^3} \, dx=\frac {-27 a^5+6 a^4 b \sqrt {x}+63 a^3 b^2 x+20 a^2 b^3 x^{3/2}-5 a b^4 x^2+2 b^5 x^{5/2}}{3 b^6 \left (a+b \sqrt {x}\right )^2}-\frac {20 a^3 \log \left (a+b \sqrt {x}\right )}{b^6} \] Input:
Integrate[x^2/(a + b*Sqrt[x])^3,x]
Output:
(-27*a^5 + 6*a^4*b*Sqrt[x] + 63*a^3*b^2*x + 20*a^2*b^3*x^(3/2) - 5*a*b^4*x ^2 + 2*b^5*x^(5/2))/(3*b^6*(a + b*Sqrt[x])^2) - (20*a^3*Log[a + b*Sqrt[x]] )/b^6
Time = 0.39 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.08, 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.
\(\displaystyle \int \frac {x^2}{\left (a+b \sqrt {x}\right )^3} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 2 \int \frac {x^{5/2}}{\left (a+b \sqrt {x}\right )^3}d\sqrt {x}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle 2 \int \left (-\frac {a^5}{b^5 \left (a+b \sqrt {x}\right )^3}+\frac {5 a^4}{b^5 \left (a+b \sqrt {x}\right )^2}-\frac {10 a^3}{b^5 \left (a+b \sqrt {x}\right )}+\frac {6 a^2}{b^5}-\frac {3 \sqrt {x} a}{b^4}+\frac {x}{b^3}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (\frac {a^5}{2 b^6 \left (a+b \sqrt {x}\right )^2}-\frac {5 a^4}{b^6 \left (a+b \sqrt {x}\right )}-\frac {10 a^3 \log \left (a+b \sqrt {x}\right )}{b^6}+\frac {6 a^2 \sqrt {x}}{b^5}-\frac {3 a x}{2 b^4}+\frac {x^{3/2}}{3 b^3}\right )\) |
Input:
Int[x^2/(a + b*Sqrt[x])^3,x]
Output:
2*(a^5/(2*b^6*(a + b*Sqrt[x])^2) - (5*a^4)/(b^6*(a + b*Sqrt[x])) + (6*a^2* Sqrt[x])/b^5 - (3*a*x)/(2*b^4) + x^(3/2)/(3*b^3) - (10*a^3*Log[a + b*Sqrt[ x]])/b^6)
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 = 78, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {\frac {2 b^{2} x^{\frac {3}{2}}}{3}-3 a b x +12 a^{2} \sqrt {x}}{b^{5}}-\frac {20 a^{3} \ln \left (a +b \sqrt {x}\right )}{b^{6}}+\frac {a^{5}}{b^{6} \left (a +b \sqrt {x}\right )^{2}}-\frac {10 a^{4}}{b^{6} \left (a +b \sqrt {x}\right )}\) | \(78\) |
default | \(\frac {\frac {2 b^{2} x^{\frac {3}{2}}}{3}-3 a b x +12 a^{2} \sqrt {x}}{b^{5}}-\frac {20 a^{3} \ln \left (a +b \sqrt {x}\right )}{b^{6}}+\frac {a^{5}}{b^{6} \left (a +b \sqrt {x}\right )^{2}}-\frac {10 a^{4}}{b^{6} \left (a +b \sqrt {x}\right )}\) | \(78\) |
Input:
int(x^2/(a+b*x^(1/2))^3,x,method=_RETURNVERBOSE)
Output:
2/b^5*(1/3*b^2*x^(3/2)-3/2*a*b*x+6*a^2*x^(1/2))-20*a^3*ln(a+b*x^(1/2))/b^6 +a^5/b^6/(a+b*x^(1/2))^2-10*a^4/b^6/(a+b*x^(1/2))
Time = 0.09 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.53 \[ \int \frac {x^2}{\left (a+b \sqrt {x}\right )^3} \, dx=-\frac {9 \, a b^{6} x^{3} - 18 \, a^{3} b^{4} x^{2} - 24 \, a^{5} b^{2} x + 27 \, a^{7} + 60 \, {\left (a^{3} b^{4} x^{2} - 2 \, a^{5} b^{2} x + a^{7}\right )} \log \left (b \sqrt {x} + a\right ) - 2 \, {\left (b^{7} x^{3} + 16 \, a^{2} b^{5} x^{2} - 50 \, a^{4} b^{3} x + 30 \, a^{6} b\right )} \sqrt {x}}{3 \, {\left (b^{10} x^{2} - 2 \, a^{2} b^{8} x + a^{4} b^{6}\right )}} \] Input:
integrate(x^2/(a+b*x^(1/2))^3,x, algorithm="fricas")
Output:
-1/3*(9*a*b^6*x^3 - 18*a^3*b^4*x^2 - 24*a^5*b^2*x + 27*a^7 + 60*(a^3*b^4*x ^2 - 2*a^5*b^2*x + a^7)*log(b*sqrt(x) + a) - 2*(b^7*x^3 + 16*a^2*b^5*x^2 - 50*a^4*b^3*x + 30*a^6*b)*sqrt(x))/(b^10*x^2 - 2*a^2*b^8*x + a^4*b^6)
Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (85) = 170\).
Time = 0.51 (sec) , antiderivative size = 333, normalized size of antiderivative = 3.78 \[ \int \frac {x^2}{\left (a+b \sqrt {x}\right )^3} \, dx=\begin {cases} - \frac {60 a^{5} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{3 a^{2} b^{6} + 6 a b^{7} \sqrt {x} + 3 b^{8} x} - \frac {90 a^{5}}{3 a^{2} b^{6} + 6 a b^{7} \sqrt {x} + 3 b^{8} x} - \frac {120 a^{4} b \sqrt {x} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{3 a^{2} b^{6} + 6 a b^{7} \sqrt {x} + 3 b^{8} x} - \frac {120 a^{4} b \sqrt {x}}{3 a^{2} b^{6} + 6 a b^{7} \sqrt {x} + 3 b^{8} x} - \frac {60 a^{3} b^{2} x \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{3 a^{2} b^{6} + 6 a b^{7} \sqrt {x} + 3 b^{8} x} + \frac {20 a^{2} b^{3} x^{\frac {3}{2}}}{3 a^{2} b^{6} + 6 a b^{7} \sqrt {x} + 3 b^{8} x} - \frac {5 a b^{4} x^{2}}{3 a^{2} b^{6} + 6 a b^{7} \sqrt {x} + 3 b^{8} x} + \frac {2 b^{5} x^{\frac {5}{2}}}{3 a^{2} b^{6} + 6 a b^{7} \sqrt {x} + 3 b^{8} x} & \text {for}\: b \neq 0 \\\frac {x^{3}}{3 a^{3}} & \text {otherwise} \end {cases} \] Input:
integrate(x**2/(a+b*x**(1/2))**3,x)
Output:
Piecewise((-60*a**5*log(a/b + sqrt(x))/(3*a**2*b**6 + 6*a*b**7*sqrt(x) + 3 *b**8*x) - 90*a**5/(3*a**2*b**6 + 6*a*b**7*sqrt(x) + 3*b**8*x) - 120*a**4* b*sqrt(x)*log(a/b + sqrt(x))/(3*a**2*b**6 + 6*a*b**7*sqrt(x) + 3*b**8*x) - 120*a**4*b*sqrt(x)/(3*a**2*b**6 + 6*a*b**7*sqrt(x) + 3*b**8*x) - 60*a**3* b**2*x*log(a/b + sqrt(x))/(3*a**2*b**6 + 6*a*b**7*sqrt(x) + 3*b**8*x) + 20 *a**2*b**3*x**(3/2)/(3*a**2*b**6 + 6*a*b**7*sqrt(x) + 3*b**8*x) - 5*a*b**4 *x**2/(3*a**2*b**6 + 6*a*b**7*sqrt(x) + 3*b**8*x) + 2*b**5*x**(5/2)/(3*a** 2*b**6 + 6*a*b**7*sqrt(x) + 3*b**8*x), Ne(b, 0)), (x**3/(3*a**3), True))
Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.07 \[ \int \frac {x^2}{\left (a+b \sqrt {x}\right )^3} \, dx=-\frac {20 \, a^{3} \log \left (b \sqrt {x} + a\right )}{b^{6}} + \frac {2 \, {\left (b \sqrt {x} + a\right )}^{3}}{3 \, b^{6}} - \frac {5 \, {\left (b \sqrt {x} + a\right )}^{2} a}{b^{6}} + \frac {20 \, {\left (b \sqrt {x} + a\right )} a^{2}}{b^{6}} - \frac {10 \, a^{4}}{{\left (b \sqrt {x} + a\right )} b^{6}} + \frac {a^{5}}{{\left (b \sqrt {x} + a\right )}^{2} b^{6}} \] Input:
integrate(x^2/(a+b*x^(1/2))^3,x, algorithm="maxima")
Output:
-20*a^3*log(b*sqrt(x) + a)/b^6 + 2/3*(b*sqrt(x) + a)^3/b^6 - 5*(b*sqrt(x) + a)^2*a/b^6 + 20*(b*sqrt(x) + a)*a^2/b^6 - 10*a^4/((b*sqrt(x) + a)*b^6) + a^5/((b*sqrt(x) + a)^2*b^6)
Time = 0.12 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.90 \[ \int \frac {x^2}{\left (a+b \sqrt {x}\right )^3} \, dx=-\frac {20 \, a^{3} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{b^{6}} - \frac {10 \, a^{4} b \sqrt {x} + 9 \, a^{5}}{{\left (b \sqrt {x} + a\right )}^{2} b^{6}} + \frac {2 \, b^{6} x^{\frac {3}{2}} - 9 \, a b^{5} x + 36 \, a^{2} b^{4} \sqrt {x}}{3 \, b^{9}} \] Input:
integrate(x^2/(a+b*x^(1/2))^3,x, algorithm="giac")
Output:
-20*a^3*log(abs(b*sqrt(x) + a))/b^6 - (10*a^4*b*sqrt(x) + 9*a^5)/((b*sqrt( x) + a)^2*b^6) + 1/3*(2*b^6*x^(3/2) - 9*a*b^5*x + 36*a^2*b^4*sqrt(x))/b^9
Time = 0.04 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.98 \[ \int \frac {x^2}{\left (a+b \sqrt {x}\right )^3} \, dx=\frac {2\,x^{3/2}}{3\,b^3}-\frac {\frac {9\,a^5}{b}+10\,a^4\,\sqrt {x}}{b^7\,x+a^2\,b^5+2\,a\,b^6\,\sqrt {x}}-\frac {20\,a^3\,\ln \left (a+b\,\sqrt {x}\right )}{b^6}+\frac {12\,a^2\,\sqrt {x}}{b^5}-\frac {3\,a\,x}{b^4} \] Input:
int(x^2/(a + b*x^(1/2))^3,x)
Output:
(2*x^(3/2))/(3*b^3) - ((9*a^5)/b + 10*a^4*x^(1/2))/(b^7*x + a^2*b^5 + 2*a* b^6*x^(1/2)) - (20*a^3*log(a + b*x^(1/2)))/b^6 + (12*a^2*x^(1/2))/b^5 - (3 *a*x)/b^4
Time = 0.20 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.25 \[ \int \frac {x^2}{\left (a+b \sqrt {x}\right )^3} \, dx=\frac {-120 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{4} b +20 \sqrt {x}\, a^{2} b^{3} x +2 \sqrt {x}\, b^{5} x^{2}-60 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{5}-60 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{3} b^{2} x -30 a^{5}+60 a^{3} b^{2} x -5 a \,b^{4} x^{2}}{3 b^{6} \left (2 \sqrt {x}\, a b +a^{2}+b^{2} x \right )} \] Input:
int(x^2/(a+b*x^(1/2))^3,x)
Output:
( - 120*sqrt(x)*log(sqrt(x)*b + a)*a**4*b + 20*sqrt(x)*a**2*b**3*x + 2*sqr t(x)*b**5*x**2 - 60*log(sqrt(x)*b + a)*a**5 - 60*log(sqrt(x)*b + a)*a**3*b **2*x - 30*a**5 + 60*a**3*b**2*x - 5*a*b**4*x**2)/(3*b**6*(2*sqrt(x)*a*b + a**2 + b**2*x))