Integrand size = 15, antiderivative size = 111 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^2} \, dx=\frac {2 a^7}{b^8 \left (a+b \sqrt {x}\right )}-\frac {12 a^5 \sqrt {x}}{b^7}+\frac {5 a^4 x}{b^6}-\frac {8 a^3 x^{3/2}}{3 b^5}+\frac {3 a^2 x^2}{2 b^4}-\frac {4 a x^{5/2}}{5 b^3}+\frac {x^3}{3 b^2}+\frac {14 a^6 \log \left (a+b \sqrt {x}\right )}{b^8} \] Output:
2*a^7/b^8/(a+b*x^(1/2))-12*a^5*x^(1/2)/b^7+5*a^4*x/b^6-8/3*a^3*x^(3/2)/b^5 +3/2*a^2*x^2/b^4-4/5*a*x^(5/2)/b^3+1/3*x^3/b^2+14*a^6*ln(a+b*x^(1/2))/b^8
Time = 0.06 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.07 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^2} \, dx=\frac {60 a^7-360 a^6 b \sqrt {x}-210 a^5 b^2 x+70 a^4 b^3 x^{3/2}-35 a^3 b^4 x^2+21 a^2 b^5 x^{5/2}-14 a b^6 x^3+10 b^7 x^{7/2}}{30 b^8 \left (a+b \sqrt {x}\right )}+\frac {14 a^6 \log \left (a+b \sqrt {x}\right )}{b^8} \] Input:
Integrate[x^3/(a + b*Sqrt[x])^2,x]
Output:
(60*a^7 - 360*a^6*b*Sqrt[x] - 210*a^5*b^2*x + 70*a^4*b^3*x^(3/2) - 35*a^3* b^4*x^2 + 21*a^2*b^5*x^(5/2) - 14*a*b^6*x^3 + 10*b^7*x^(7/2))/(30*b^8*(a + b*Sqrt[x])) + (14*a^6*Log[a + b*Sqrt[x]])/b^8
Time = 0.43 (sec) , antiderivative size = 114, 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.
\(\displaystyle \int \frac {x^3}{\left (a+b \sqrt {x}\right )^2} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 2 \int \frac {x^{7/2}}{\left (a+b \sqrt {x}\right )^2}d\sqrt {x}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle 2 \int \left (-\frac {a^7}{b^7 \left (a+b \sqrt {x}\right )^2}+\frac {7 a^6}{b^7 \left (a+b \sqrt {x}\right )}-\frac {6 a^5}{b^7}+\frac {5 \sqrt {x} a^4}{b^6}-\frac {4 x a^3}{b^5}+\frac {3 x^{3/2} a^2}{b^4}-\frac {2 x^2 a}{b^3}+\frac {x^{5/2}}{b^2}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (\frac {a^7}{b^8 \left (a+b \sqrt {x}\right )}+\frac {7 a^6 \log \left (a+b \sqrt {x}\right )}{b^8}-\frac {6 a^5 \sqrt {x}}{b^7}+\frac {5 a^4 x}{2 b^6}-\frac {4 a^3 x^{3/2}}{3 b^5}+\frac {3 a^2 x^2}{4 b^4}-\frac {2 a x^{5/2}}{5 b^3}+\frac {x^3}{6 b^2}\right )\) |
Input:
Int[x^3/(a + b*Sqrt[x])^2,x]
Output:
2*(a^7/(b^8*(a + b*Sqrt[x])) - (6*a^5*Sqrt[x])/b^7 + (5*a^4*x)/(2*b^6) - ( 4*a^3*x^(3/2))/(3*b^5) + (3*a^2*x^2)/(4*b^4) - (2*a*x^(5/2))/(5*b^3) + x^3 /(6*b^2) + (7*a^6*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 = 0.47 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(-\frac {2 \left (-\frac {x^{3} b^{5}}{6}+\frac {2 a \,x^{\frac {5}{2}} b^{4}}{5}-\frac {3 a^{2} b^{3} x^{2}}{4}+\frac {4 a^{3} x^{\frac {3}{2}} b^{2}}{3}-\frac {5 a^{4} b x}{2}+6 a^{5} \sqrt {x}\right )}{b^{7}}+\frac {2 a^{7}}{b^{8} \left (a +b \sqrt {x}\right )}+\frac {14 a^{6} \ln \left (a +b \sqrt {x}\right )}{b^{8}}\) | \(95\) |
default | \(-\frac {2 \left (-\frac {x^{3} b^{5}}{6}+\frac {2 a \,x^{\frac {5}{2}} b^{4}}{5}-\frac {3 a^{2} b^{3} x^{2}}{4}+\frac {4 a^{3} x^{\frac {3}{2}} b^{2}}{3}-\frac {5 a^{4} b x}{2}+6 a^{5} \sqrt {x}\right )}{b^{7}}+\frac {2 a^{7}}{b^{8} \left (a +b \sqrt {x}\right )}+\frac {14 a^{6} \ln \left (a +b \sqrt {x}\right )}{b^{8}}\) | \(95\) |
Input:
int(x^3/(a+b*x^(1/2))^2,x,method=_RETURNVERBOSE)
Output:
-2/b^7*(-1/6*x^3*b^5+2/5*a*x^(5/2)*b^4-3/4*a^2*b^3*x^2+4/3*a^3*x^(3/2)*b^2 -5/2*a^4*b*x+6*a^5*x^(1/2))+2*a^7/b^8/(a+b*x^(1/2))+14*a^6*ln(a+b*x^(1/2)) /b^8
Time = 0.08 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.15 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^2} \, dx=\frac {10 \, b^{8} x^{4} + 35 \, a^{2} b^{6} x^{3} + 105 \, a^{4} b^{4} x^{2} - 150 \, a^{6} b^{2} x - 60 \, a^{8} + 420 \, {\left (a^{6} b^{2} x - a^{8}\right )} \log \left (b \sqrt {x} + a\right ) - 4 \, {\left (6 \, a b^{7} x^{3} + 14 \, a^{3} b^{5} x^{2} + 70 \, a^{5} b^{3} x - 105 \, a^{7} b\right )} \sqrt {x}}{30 \, {\left (b^{10} x - a^{2} b^{8}\right )}} \] Input:
integrate(x^3/(a+b*x^(1/2))^2,x, algorithm="fricas")
Output:
1/30*(10*b^8*x^4 + 35*a^2*b^6*x^3 + 105*a^4*b^4*x^2 - 150*a^6*b^2*x - 60*a ^8 + 420*(a^6*b^2*x - a^8)*log(b*sqrt(x) + a) - 4*(6*a*b^7*x^3 + 14*a^3*b^ 5*x^2 + 70*a^5*b^3*x - 105*a^7*b)*sqrt(x))/(b^10*x - a^2*b^8)
Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (109) = 218\).
Time = 0.55 (sec) , antiderivative size = 272, normalized size of antiderivative = 2.45 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^2} \, dx=\begin {cases} \frac {420 a^{7} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{30 a b^{8} + 30 b^{9} \sqrt {x}} + \frac {420 a^{7}}{30 a b^{8} + 30 b^{9} \sqrt {x}} + \frac {420 a^{6} b \sqrt {x} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{30 a b^{8} + 30 b^{9} \sqrt {x}} - \frac {210 a^{5} b^{2} x}{30 a b^{8} + 30 b^{9} \sqrt {x}} + \frac {70 a^{4} b^{3} x^{\frac {3}{2}}}{30 a b^{8} + 30 b^{9} \sqrt {x}} - \frac {35 a^{3} b^{4} x^{2}}{30 a b^{8} + 30 b^{9} \sqrt {x}} + \frac {21 a^{2} b^{5} x^{\frac {5}{2}}}{30 a b^{8} + 30 b^{9} \sqrt {x}} - \frac {14 a b^{6} x^{3}}{30 a b^{8} + 30 b^{9} \sqrt {x}} + \frac {10 b^{7} x^{\frac {7}{2}}}{30 a b^{8} + 30 b^{9} \sqrt {x}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a^{2}} & \text {otherwise} \end {cases} \] Input:
integrate(x**3/(a+b*x**(1/2))**2,x)
Output:
Piecewise((420*a**7*log(a/b + sqrt(x))/(30*a*b**8 + 30*b**9*sqrt(x)) + 420 *a**7/(30*a*b**8 + 30*b**9*sqrt(x)) + 420*a**6*b*sqrt(x)*log(a/b + sqrt(x) )/(30*a*b**8 + 30*b**9*sqrt(x)) - 210*a**5*b**2*x/(30*a*b**8 + 30*b**9*sqr t(x)) + 70*a**4*b**3*x**(3/2)/(30*a*b**8 + 30*b**9*sqrt(x)) - 35*a**3*b**4 *x**2/(30*a*b**8 + 30*b**9*sqrt(x)) + 21*a**2*b**5*x**(5/2)/(30*a*b**8 + 3 0*b**9*sqrt(x)) - 14*a*b**6*x**3/(30*a*b**8 + 30*b**9*sqrt(x)) + 10*b**7*x **(7/2)/(30*a*b**8 + 30*b**9*sqrt(x)), Ne(b, 0)), (x**4/(4*a**2), True))
Time = 0.03 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.16 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^2} \, dx=\frac {14 \, a^{6} \log \left (b \sqrt {x} + a\right )}{b^{8}} + \frac {{\left (b \sqrt {x} + a\right )}^{6}}{3 \, b^{8}} - \frac {14 \, {\left (b \sqrt {x} + a\right )}^{5} a}{5 \, b^{8}} + \frac {21 \, {\left (b \sqrt {x} + a\right )}^{4} a^{2}}{2 \, b^{8}} - \frac {70 \, {\left (b \sqrt {x} + a\right )}^{3} a^{3}}{3 \, b^{8}} + \frac {35 \, {\left (b \sqrt {x} + a\right )}^{2} a^{4}}{b^{8}} - \frac {42 \, {\left (b \sqrt {x} + a\right )} a^{5}}{b^{8}} + \frac {2 \, a^{7}}{{\left (b \sqrt {x} + a\right )} b^{8}} \] Input:
integrate(x^3/(a+b*x^(1/2))^2,x, algorithm="maxima")
Output:
14*a^6*log(b*sqrt(x) + a)/b^8 + 1/3*(b*sqrt(x) + a)^6/b^8 - 14/5*(b*sqrt(x ) + a)^5*a/b^8 + 21/2*(b*sqrt(x) + a)^4*a^2/b^8 - 70/3*(b*sqrt(x) + a)^3*a ^3/b^8 + 35*(b*sqrt(x) + a)^2*a^4/b^8 - 42*(b*sqrt(x) + a)*a^5/b^8 + 2*a^7 /((b*sqrt(x) + a)*b^8)
Time = 0.12 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.90 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^2} \, dx=\frac {14 \, a^{6} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{b^{8}} + \frac {2 \, a^{7}}{{\left (b \sqrt {x} + a\right )} b^{8}} + \frac {10 \, b^{10} x^{3} - 24 \, a b^{9} x^{\frac {5}{2}} + 45 \, a^{2} b^{8} x^{2} - 80 \, a^{3} b^{7} x^{\frac {3}{2}} + 150 \, a^{4} b^{6} x - 360 \, a^{5} b^{5} \sqrt {x}}{30 \, b^{12}} \] Input:
integrate(x^3/(a+b*x^(1/2))^2,x, algorithm="giac")
Output:
14*a^6*log(abs(b*sqrt(x) + a))/b^8 + 2*a^7/((b*sqrt(x) + a)*b^8) + 1/30*(1 0*b^10*x^3 - 24*a*b^9*x^(5/2) + 45*a^2*b^8*x^2 - 80*a^3*b^7*x^(3/2) + 150* a^4*b^6*x - 360*a^5*b^5*sqrt(x))/b^12
Time = 0.05 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.89 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^2} \, dx=\frac {x^3}{3\,b^2}+\frac {2\,a^7}{b\,\left (a\,b^7+b^8\,\sqrt {x}\right )}-\frac {4\,a\,x^{5/2}}{5\,b^3}+\frac {5\,a^4\,x}{b^6}+\frac {14\,a^6\,\ln \left (a+b\,\sqrt {x}\right )}{b^8}+\frac {3\,a^2\,x^2}{2\,b^4}-\frac {8\,a^3\,x^{3/2}}{3\,b^5}-\frac {12\,a^5\,\sqrt {x}}{b^7} \] Input:
int(x^3/(a + b*x^(1/2))^2,x)
Output:
x^3/(3*b^2) + (2*a^7)/(b*(a*b^7 + b^8*x^(1/2))) - (4*a*x^(5/2))/(5*b^3) + (5*a^4*x)/b^6 + (14*a^6*log(a + b*x^(1/2)))/b^8 + (3*a^2*x^2)/(2*b^4) - (8 *a^3*x^(3/2))/(3*b^5) - (12*a^5*x^(1/2))/b^7
Time = 0.23 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.01 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^2} \, dx=\frac {420 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{6} b -420 \sqrt {x}\, a^{6} b +70 \sqrt {x}\, a^{4} b^{3} x +21 \sqrt {x}\, a^{2} b^{5} x^{2}+10 \sqrt {x}\, b^{7} x^{3}+420 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{7}-210 a^{5} b^{2} x -35 a^{3} b^{4} x^{2}-14 a \,b^{6} x^{3}}{30 b^{8} \left (\sqrt {x}\, b +a \right )} \] Input:
int(x^3/(a+b*x^(1/2))^2,x)
Output:
(420*sqrt(x)*log(sqrt(x)*b + a)*a**6*b - 420*sqrt(x)*a**6*b + 70*sqrt(x)*a **4*b**3*x + 21*sqrt(x)*a**2*b**5*x**2 + 10*sqrt(x)*b**7*x**3 + 420*log(sq rt(x)*b + a)*a**7 - 210*a**5*b**2*x - 35*a**3*b**4*x**2 - 14*a*b**6*x**3)/ (30*b**8*(sqrt(x)*b + a))