Integrand size = 15, antiderivative size = 114 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^3} \, dx=\frac {a^7}{b^8 \left (a+b \sqrt {x}\right )^2}-\frac {14 a^6}{b^8 \left (a+b \sqrt {x}\right )}+\frac {30 a^4 \sqrt {x}}{b^7}-\frac {10 a^3 x}{b^6}+\frac {4 a^2 x^{3/2}}{b^5}-\frac {3 a x^2}{2 b^4}+\frac {2 x^{5/2}}{5 b^3}-\frac {42 a^5 \log \left (a+b \sqrt {x}\right )}{b^8} \] Output:
a^7/b^8/(a+b*x^(1/2))^2-14*a^6/b^8/(a+b*x^(1/2))+30*a^4*x^(1/2)/b^7-10*a^3 *x/b^6+4*a^2*x^(3/2)/b^5-3/2*a*x^2/b^4+2/5*x^(5/2)/b^3-42*a^5*ln(a+b*x^(1/ 2))/b^8
Time = 0.07 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.04 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^3} \, dx=\frac {-130 a^7+160 a^6 b \sqrt {x}+500 a^5 b^2 x+140 a^4 b^3 x^{3/2}-35 a^3 b^4 x^2+14 a^2 b^5 x^{5/2}-7 a b^6 x^3+4 b^7 x^{7/2}}{10 b^8 \left (a+b \sqrt {x}\right )^2}-\frac {42 a^5 \log \left (a+b \sqrt {x}\right )}{b^8} \] Input:
Integrate[x^3/(a + b*Sqrt[x])^3,x]
Output:
(-130*a^7 + 160*a^6*b*Sqrt[x] + 500*a^5*b^2*x + 140*a^4*b^3*x^(3/2) - 35*a ^3*b^4*x^2 + 14*a^2*b^5*x^(5/2) - 7*a*b^6*x^3 + 4*b^7*x^(7/2))/(10*b^8*(a + b*Sqrt[x])^2) - (42*a^5*Log[a + b*Sqrt[x]])/b^8
Time = 0.45 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.04, 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.
\(\displaystyle \int \frac {x^3}{\left (a+b \sqrt {x}\right )^3} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 2 \int \frac {x^{7/2}}{\left (a+b \sqrt {x}\right )^3}d\sqrt {x}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle 2 \int \left (-\frac {a^7}{b^7 \left (a+b \sqrt {x}\right )^3}+\frac {7 a^6}{b^7 \left (a+b \sqrt {x}\right )^2}-\frac {21 a^5}{b^7 \left (a+b \sqrt {x}\right )}+\frac {15 a^4}{b^7}-\frac {10 \sqrt {x} a^3}{b^6}+\frac {6 x a^2}{b^5}-\frac {3 x^{3/2} a}{b^4}+\frac {x^2}{b^3}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (\frac {a^7}{2 b^8 \left (a+b \sqrt {x}\right )^2}-\frac {7 a^6}{b^8 \left (a+b \sqrt {x}\right )}-\frac {21 a^5 \log \left (a+b \sqrt {x}\right )}{b^8}+\frac {15 a^4 \sqrt {x}}{b^7}-\frac {5 a^3 x}{b^6}+\frac {2 a^2 x^{3/2}}{b^5}-\frac {3 a x^2}{4 b^4}+\frac {x^{5/2}}{5 b^3}\right )\) |
Input:
Int[x^3/(a + b*Sqrt[x])^3,x]
Output:
2*(a^7/(2*b^8*(a + b*Sqrt[x])^2) - (7*a^6)/(b^8*(a + b*Sqrt[x])) + (15*a^4 *Sqrt[x])/b^7 - (5*a^3*x)/b^6 + (2*a^2*x^(3/2))/b^5 - (3*a*x^2)/(4*b^4) + x^(5/2)/(5*b^3) - (21*a^5*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 = 0.46 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {\frac {2 x^{\frac {5}{2}} b^{4}}{5}-\frac {3 a \,b^{3} x^{2}}{2}+4 x^{\frac {3}{2}} a^{2} b^{2}-10 a^{3} b x +30 \sqrt {x}\, a^{4}}{b^{7}}+\frac {a^{7}}{b^{8} \left (a +b \sqrt {x}\right )^{2}}-\frac {42 a^{5} \ln \left (a +b \sqrt {x}\right )}{b^{8}}-\frac {14 a^{6}}{b^{8} \left (a +b \sqrt {x}\right )}\) | \(100\) |
default | \(\frac {\frac {2 x^{\frac {5}{2}} b^{4}}{5}-\frac {3 a \,b^{3} x^{2}}{2}+4 x^{\frac {3}{2}} a^{2} b^{2}-10 a^{3} b x +30 \sqrt {x}\, a^{4}}{b^{7}}+\frac {a^{7}}{b^{8} \left (a +b \sqrt {x}\right )^{2}}-\frac {42 a^{5} \ln \left (a +b \sqrt {x}\right )}{b^{8}}-\frac {14 a^{6}}{b^{8} \left (a +b \sqrt {x}\right )}\) | \(100\) |
Input:
int(x^3/(a+b*x^(1/2))^3,x,method=_RETURNVERBOSE)
Output:
2/b^7*(1/5*x^(5/2)*b^4-3/4*a*b^3*x^2+2*x^(3/2)*a^2*b^2-5*a^3*b*x+15*x^(1/2 )*a^4)+a^7/b^8/(a+b*x^(1/2))^2-42*a^5*ln(a+b*x^(1/2))/b^8-14*a^6/b^8/(a+b* x^(1/2))
Time = 0.08 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.38 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^3} \, dx=-\frac {15 \, a b^{8} x^{4} + 70 \, a^{3} b^{6} x^{3} - 185 \, a^{5} b^{4} x^{2} - 50 \, a^{7} b^{2} x + 130 \, a^{9} + 420 \, {\left (a^{5} b^{4} x^{2} - 2 \, a^{7} b^{2} x + a^{9}\right )} \log \left (b \sqrt {x} + a\right ) - 4 \, {\left (b^{9} x^{4} + 8 \, a^{2} b^{7} x^{3} + 56 \, a^{4} b^{5} x^{2} - 175 \, a^{6} b^{3} x + 105 \, a^{8} b\right )} \sqrt {x}}{10 \, {\left (b^{12} x^{2} - 2 \, a^{2} b^{10} x + a^{4} b^{8}\right )}} \] Input:
integrate(x^3/(a+b*x^(1/2))^3,x, algorithm="fricas")
Output:
-1/10*(15*a*b^8*x^4 + 70*a^3*b^6*x^3 - 185*a^5*b^4*x^2 - 50*a^7*b^2*x + 13 0*a^9 + 420*(a^5*b^4*x^2 - 2*a^7*b^2*x + a^9)*log(b*sqrt(x) + a) - 4*(b^9* x^4 + 8*a^2*b^7*x^3 + 56*a^4*b^5*x^2 - 175*a^6*b^3*x + 105*a^8*b)*sqrt(x)) /(b^12*x^2 - 2*a^2*b^10*x + a^4*b^8)
Leaf count of result is larger than twice the leaf count of optimal. 413 vs. \(2 (112) = 224\).
Time = 0.64 (sec) , antiderivative size = 413, normalized size of antiderivative = 3.62 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^3} \, dx=\begin {cases} - \frac {420 a^{7} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt {x} + 10 b^{10} x} - \frac {630 a^{7}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt {x} + 10 b^{10} x} - \frac {840 a^{6} b \sqrt {x} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt {x} + 10 b^{10} x} - \frac {840 a^{6} b \sqrt {x}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt {x} + 10 b^{10} x} - \frac {420 a^{5} b^{2} x \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt {x} + 10 b^{10} x} + \frac {140 a^{4} b^{3} x^{\frac {3}{2}}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt {x} + 10 b^{10} x} - \frac {35 a^{3} b^{4} x^{2}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt {x} + 10 b^{10} x} + \frac {14 a^{2} b^{5} x^{\frac {5}{2}}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt {x} + 10 b^{10} x} - \frac {7 a b^{6} x^{3}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt {x} + 10 b^{10} x} + \frac {4 b^{7} x^{\frac {7}{2}}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt {x} + 10 b^{10} x} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a^{3}} & \text {otherwise} \end {cases} \] Input:
integrate(x**3/(a+b*x**(1/2))**3,x)
Output:
Piecewise((-420*a**7*log(a/b + sqrt(x))/(10*a**2*b**8 + 20*a*b**9*sqrt(x) + 10*b**10*x) - 630*a**7/(10*a**2*b**8 + 20*a*b**9*sqrt(x) + 10*b**10*x) - 840*a**6*b*sqrt(x)*log(a/b + sqrt(x))/(10*a**2*b**8 + 20*a*b**9*sqrt(x) + 10*b**10*x) - 840*a**6*b*sqrt(x)/(10*a**2*b**8 + 20*a*b**9*sqrt(x) + 10*b **10*x) - 420*a**5*b**2*x*log(a/b + sqrt(x))/(10*a**2*b**8 + 20*a*b**9*sqr t(x) + 10*b**10*x) + 140*a**4*b**3*x**(3/2)/(10*a**2*b**8 + 20*a*b**9*sqrt (x) + 10*b**10*x) - 35*a**3*b**4*x**2/(10*a**2*b**8 + 20*a*b**9*sqrt(x) + 10*b**10*x) + 14*a**2*b**5*x**(5/2)/(10*a**2*b**8 + 20*a*b**9*sqrt(x) + 10 *b**10*x) - 7*a*b**6*x**3/(10*a**2*b**8 + 20*a*b**9*sqrt(x) + 10*b**10*x) + 4*b**7*x**(7/2)/(10*a**2*b**8 + 20*a*b**9*sqrt(x) + 10*b**10*x), Ne(b, 0 )), (x**4/(4*a**3), True))
Time = 0.03 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.12 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^3} \, dx=-\frac {42 \, a^{5} \log \left (b \sqrt {x} + a\right )}{b^{8}} + \frac {2 \, {\left (b \sqrt {x} + a\right )}^{5}}{5 \, b^{8}} - \frac {7 \, {\left (b \sqrt {x} + a\right )}^{4} a}{2 \, b^{8}} + \frac {14 \, {\left (b \sqrt {x} + a\right )}^{3} a^{2}}{b^{8}} - \frac {35 \, {\left (b \sqrt {x} + a\right )}^{2} a^{3}}{b^{8}} + \frac {70 \, {\left (b \sqrt {x} + a\right )} a^{4}}{b^{8}} - \frac {14 \, a^{6}}{{\left (b \sqrt {x} + a\right )} b^{8}} + \frac {a^{7}}{{\left (b \sqrt {x} + a\right )}^{2} b^{8}} \] Input:
integrate(x^3/(a+b*x^(1/2))^3,x, algorithm="maxima")
Output:
-42*a^5*log(b*sqrt(x) + a)/b^8 + 2/5*(b*sqrt(x) + a)^5/b^8 - 7/2*(b*sqrt(x ) + a)^4*a/b^8 + 14*(b*sqrt(x) + a)^3*a^2/b^8 - 35*(b*sqrt(x) + a)^2*a^3/b ^8 + 70*(b*sqrt(x) + a)*a^4/b^8 - 14*a^6/((b*sqrt(x) + a)*b^8) + a^7/((b*s qrt(x) + a)^2*b^8)
Time = 0.12 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.89 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^3} \, dx=-\frac {42 \, a^{5} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{b^{8}} - \frac {14 \, a^{6} b \sqrt {x} + 13 \, a^{7}}{{\left (b \sqrt {x} + a\right )}^{2} b^{8}} + \frac {4 \, b^{12} x^{\frac {5}{2}} - 15 \, a b^{11} x^{2} + 40 \, a^{2} b^{10} x^{\frac {3}{2}} - 100 \, a^{3} b^{9} x + 300 \, a^{4} b^{8} \sqrt {x}}{10 \, b^{15}} \] Input:
integrate(x^3/(a+b*x^(1/2))^3,x, algorithm="giac")
Output:
-42*a^5*log(abs(b*sqrt(x) + a))/b^8 - (14*a^6*b*sqrt(x) + 13*a^7)/((b*sqrt (x) + a)^2*b^8) + 1/10*(4*b^12*x^(5/2) - 15*a*b^11*x^2 + 40*a^2*b^10*x^(3/ 2) - 100*a^3*b^9*x + 300*a^4*b^8*sqrt(x))/b^15
Time = 0.30 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.95 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^3} \, dx=\frac {2\,x^{5/2}}{5\,b^3}-\frac {\frac {13\,a^7}{b}+14\,a^6\,\sqrt {x}}{b^9\,x+a^2\,b^7+2\,a\,b^8\,\sqrt {x}}-\frac {3\,a\,x^2}{2\,b^4}-\frac {10\,a^3\,x}{b^6}-\frac {42\,a^5\,\ln \left (a+b\,\sqrt {x}\right )}{b^8}+\frac {4\,a^2\,x^{3/2}}{b^5}+\frac {30\,a^4\,\sqrt {x}}{b^7} \] Input:
int(x^3/(a + b*x^(1/2))^3,x)
Output:
(2*x^(5/2))/(5*b^3) - ((13*a^7)/b + 14*a^6*x^(1/2))/(b^9*x + a^2*b^7 + 2*a *b^8*x^(1/2)) - (3*a*x^2)/(2*b^4) - (10*a^3*x)/b^6 - (42*a^5*log(a + b*x^( 1/2)))/b^8 + (4*a^2*x^(3/2))/b^5 + (30*a^4*x^(1/2))/b^7
Time = 0.24 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.18 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^3} \, dx=\frac {-840 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{6} b +140 \sqrt {x}\, a^{4} b^{3} x +14 \sqrt {x}\, a^{2} b^{5} x^{2}+4 \sqrt {x}\, b^{7} x^{3}-420 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{7}-420 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{5} b^{2} x -210 a^{7}+420 a^{5} b^{2} x -35 a^{3} b^{4} x^{2}-7 a \,b^{6} x^{3}}{10 b^{8} \left (2 \sqrt {x}\, a b +a^{2}+b^{2} x \right )} \] Input:
int(x^3/(a+b*x^(1/2))^3,x)
Output:
( - 840*sqrt(x)*log(sqrt(x)*b + a)*a**6*b + 140*sqrt(x)*a**4*b**3*x + 14*s qrt(x)*a**2*b**5*x**2 + 4*sqrt(x)*b**7*x**3 - 420*log(sqrt(x)*b + a)*a**7 - 420*log(sqrt(x)*b + a)*a**5*b**2*x - 210*a**7 + 420*a**5*b**2*x - 35*a** 3*b**4*x**2 - 7*a*b**6*x**3)/(10*b**8*(2*sqrt(x)*a*b + a**2 + b**2*x))