Integrand size = 11, antiderivative size = 94 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt {x}}\right )^4} \, dx=\frac {2 b^5}{3 a^6 \left (b+a \sqrt {x}\right )^3}-\frac {5 b^4}{a^6 \left (b+a \sqrt {x}\right )^2}+\frac {20 b^3}{a^6 \left (b+a \sqrt {x}\right )}-\frac {8 b \sqrt {x}}{a^5}+\frac {x}{a^4}+\frac {20 b^2 \log \left (b+a \sqrt {x}\right )}{a^6} \] Output:
2/3*b^5/a^6/(b+a*x^(1/2))^3-5*b^4/a^6/(b+a*x^(1/2))^2+20*b^3/a^6/(b+a*x^(1 /2))-8*b*x^(1/2)/a^5+x/a^4+20*b^2*ln(b+a*x^(1/2))/a^6
Time = 0.09 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.01 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt {x}}\right )^4} \, dx=\frac {47 b^5+81 a b^4 \sqrt {x}-9 a^2 b^3 x-63 a^3 b^2 x^{3/2}-15 a^4 b x^2+3 a^5 x^{5/2}}{3 a^6 \left (b+a \sqrt {x}\right )^3}+\frac {20 b^2 \log \left (b+a \sqrt {x}\right )}{a^6} \] Input:
Integrate[(a + b/Sqrt[x])^(-4),x]
Output:
(47*b^5 + 81*a*b^4*Sqrt[x] - 9*a^2*b^3*x - 63*a^3*b^2*x^(3/2) - 15*a^4*b*x ^2 + 3*a^5*x^(5/2))/(3*a^6*(b + a*Sqrt[x])^3) + (20*b^2*Log[b + a*Sqrt[x]] )/a^6
Time = 0.37 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {774, 795, 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.
\(\displaystyle \int \frac {1}{\left (a+\frac {b}{\sqrt {x}}\right )^4} \, dx\) |
\(\Big \downarrow \) 774 |
\(\displaystyle 2 \int \frac {\sqrt {x}}{\left (a+\frac {b}{\sqrt {x}}\right )^4}d\sqrt {x}\) |
\(\Big \downarrow \) 795 |
\(\displaystyle 2 \int \frac {x^{5/2}}{\left (\sqrt {x} a+b\right )^4}d\sqrt {x}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle 2 \int \left (-\frac {b^5}{a^5 \left (\sqrt {x} a+b\right )^4}+\frac {5 b^4}{a^5 \left (\sqrt {x} a+b\right )^3}-\frac {10 b^3}{a^5 \left (\sqrt {x} a+b\right )^2}+\frac {10 b^2}{a^5 \left (\sqrt {x} a+b\right )}-\frac {4 b}{a^5}+\frac {\sqrt {x}}{a^4}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (\frac {b^5}{3 a^6 \left (a \sqrt {x}+b\right )^3}-\frac {5 b^4}{2 a^6 \left (a \sqrt {x}+b\right )^2}+\frac {10 b^3}{a^6 \left (a \sqrt {x}+b\right )}+\frac {10 b^2 \log \left (a \sqrt {x}+b\right )}{a^6}-\frac {4 b \sqrt {x}}{a^5}+\frac {x}{2 a^4}\right )\) |
Input:
Int[(a + b/Sqrt[x])^(-4),x]
Output:
2*(b^5/(3*a^6*(b + a*Sqrt[x])^3) - (5*b^4)/(2*a^6*(b + a*Sqrt[x])^2) + (10 *b^3)/(a^6*(b + a*Sqrt[x])) - (4*b*Sqrt[x])/a^5 + x/(2*a^4) + (10*b^2*Log[ b + a*Sqrt[x]])/a^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[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Simp[k Subst[Int[x^(k - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; Fre eQ[{a, b, p}, x] && FractionQ[n]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)* (b + a/x^n)^p, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && NegQ[n]
Time = 0.37 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.90
method | result | size |
derivativedivides | \(\frac {a x -8 b \sqrt {x}}{a^{5}}+\frac {2 b^{5}}{3 a^{6} \left (b +a \sqrt {x}\right )^{3}}+\frac {20 b^{2} \ln \left (b +a \sqrt {x}\right )}{a^{6}}-\frac {5 b^{4}}{a^{6} \left (b +a \sqrt {x}\right )^{2}}+\frac {20 b^{3}}{a^{6} \left (b +a \sqrt {x}\right )}\) | \(85\) |
default | \(\frac {a x -8 b \sqrt {x}}{a^{5}}+\frac {2 b^{5}}{3 a^{6} \left (b +a \sqrt {x}\right )^{3}}+\frac {20 b^{2} \ln \left (b +a \sqrt {x}\right )}{a^{6}}-\frac {5 b^{4}}{a^{6} \left (b +a \sqrt {x}\right )^{2}}+\frac {20 b^{3}}{a^{6} \left (b +a \sqrt {x}\right )}\) | \(85\) |
Input:
int(1/(a+b/x^(1/2))^4,x,method=_RETURNVERBOSE)
Output:
2/a^5*(1/2*a*x-4*b*x^(1/2))+2/3*b^5/a^6/(b+a*x^(1/2))^3+20*b^2*ln(b+a*x^(1 /2))/a^6-5*b^4/a^6/(b+a*x^(1/2))^2+20*b^3/a^6/(b+a*x^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (82) = 164\).
Time = 0.08 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.83 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt {x}}\right )^4} \, dx=\frac {3 \, a^{8} x^{4} - 9 \, a^{6} b^{2} x^{3} - 66 \, a^{4} b^{4} x^{2} + 111 \, a^{2} b^{6} x - 47 \, b^{8} + 60 \, {\left (a^{6} b^{2} x^{3} - 3 \, a^{4} b^{4} x^{2} + 3 \, a^{2} b^{6} x - b^{8}\right )} \log \left (a \sqrt {x} + b\right ) - 4 \, {\left (6 \, a^{7} b x^{3} - 33 \, a^{5} b^{3} x^{2} + 40 \, a^{3} b^{5} x - 15 \, a b^{7}\right )} \sqrt {x}}{3 \, {\left (a^{12} x^{3} - 3 \, a^{10} b^{2} x^{2} + 3 \, a^{8} b^{4} x - a^{6} b^{6}\right )}} \] Input:
integrate(1/(a+b/x^(1/2))^4,x, algorithm="fricas")
Output:
1/3*(3*a^8*x^4 - 9*a^6*b^2*x^3 - 66*a^4*b^4*x^2 + 111*a^2*b^6*x - 47*b^8 + 60*(a^6*b^2*x^3 - 3*a^4*b^4*x^2 + 3*a^2*b^6*x - b^8)*log(a*sqrt(x) + b) - 4*(6*a^7*b*x^3 - 33*a^5*b^3*x^2 + 40*a^3*b^5*x - 15*a*b^7)*sqrt(x))/(a^12 *x^3 - 3*a^10*b^2*x^2 + 3*a^8*b^4*x - a^6*b^6)
Leaf count of result is larger than twice the leaf count of optimal. 503 vs. \(2 (90) = 180\).
Time = 0.52 (sec) , antiderivative size = 503, normalized size of antiderivative = 5.35 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt {x}}\right )^4} \, dx=\begin {cases} \frac {3 a^{5} x^{\frac {5}{2}}}{3 a^{9} x^{\frac {3}{2}} + 9 a^{8} b x + 9 a^{7} b^{2} \sqrt {x} + 3 a^{6} b^{3}} - \frac {15 a^{4} b x^{2}}{3 a^{9} x^{\frac {3}{2}} + 9 a^{8} b x + 9 a^{7} b^{2} \sqrt {x} + 3 a^{6} b^{3}} + \frac {60 a^{3} b^{2} x^{\frac {3}{2}} \log {\left (\sqrt {x} + \frac {b}{a} \right )}}{3 a^{9} x^{\frac {3}{2}} + 9 a^{8} b x + 9 a^{7} b^{2} \sqrt {x} + 3 a^{6} b^{3}} + \frac {180 a^{2} b^{3} x \log {\left (\sqrt {x} + \frac {b}{a} \right )}}{3 a^{9} x^{\frac {3}{2}} + 9 a^{8} b x + 9 a^{7} b^{2} \sqrt {x} + 3 a^{6} b^{3}} + \frac {180 a^{2} b^{3} x}{3 a^{9} x^{\frac {3}{2}} + 9 a^{8} b x + 9 a^{7} b^{2} \sqrt {x} + 3 a^{6} b^{3}} + \frac {180 a b^{4} \sqrt {x} \log {\left (\sqrt {x} + \frac {b}{a} \right )}}{3 a^{9} x^{\frac {3}{2}} + 9 a^{8} b x + 9 a^{7} b^{2} \sqrt {x} + 3 a^{6} b^{3}} + \frac {270 a b^{4} \sqrt {x}}{3 a^{9} x^{\frac {3}{2}} + 9 a^{8} b x + 9 a^{7} b^{2} \sqrt {x} + 3 a^{6} b^{3}} + \frac {60 b^{5} \log {\left (\sqrt {x} + \frac {b}{a} \right )}}{3 a^{9} x^{\frac {3}{2}} + 9 a^{8} b x + 9 a^{7} b^{2} \sqrt {x} + 3 a^{6} b^{3}} + \frac {110 b^{5}}{3 a^{9} x^{\frac {3}{2}} + 9 a^{8} b x + 9 a^{7} b^{2} \sqrt {x} + 3 a^{6} b^{3}} & \text {for}\: a \neq 0 \\\frac {x^{3}}{3 b^{4}} & \text {otherwise} \end {cases} \] Input:
integrate(1/(a+b/x**(1/2))**4,x)
Output:
Piecewise((3*a**5*x**(5/2)/(3*a**9*x**(3/2) + 9*a**8*b*x + 9*a**7*b**2*sqr t(x) + 3*a**6*b**3) - 15*a**4*b*x**2/(3*a**9*x**(3/2) + 9*a**8*b*x + 9*a** 7*b**2*sqrt(x) + 3*a**6*b**3) + 60*a**3*b**2*x**(3/2)*log(sqrt(x) + b/a)/( 3*a**9*x**(3/2) + 9*a**8*b*x + 9*a**7*b**2*sqrt(x) + 3*a**6*b**3) + 180*a* *2*b**3*x*log(sqrt(x) + b/a)/(3*a**9*x**(3/2) + 9*a**8*b*x + 9*a**7*b**2*s qrt(x) + 3*a**6*b**3) + 180*a**2*b**3*x/(3*a**9*x**(3/2) + 9*a**8*b*x + 9* a**7*b**2*sqrt(x) + 3*a**6*b**3) + 180*a*b**4*sqrt(x)*log(sqrt(x) + b/a)/( 3*a**9*x**(3/2) + 9*a**8*b*x + 9*a**7*b**2*sqrt(x) + 3*a**6*b**3) + 270*a* b**4*sqrt(x)/(3*a**9*x**(3/2) + 9*a**8*b*x + 9*a**7*b**2*sqrt(x) + 3*a**6* b**3) + 60*b**5*log(sqrt(x) + b/a)/(3*a**9*x**(3/2) + 9*a**8*b*x + 9*a**7* b**2*sqrt(x) + 3*a**6*b**3) + 110*b**5/(3*a**9*x**(3/2) + 9*a**8*b*x + 9*a **7*b**2*sqrt(x) + 3*a**6*b**3), Ne(a, 0)), (x**3/(3*b**4), True))
Time = 0.03 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.19 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt {x}}\right )^4} \, dx=\frac {3 \, a^{4} - \frac {15 \, a^{3} b}{\sqrt {x}} - \frac {110 \, a^{2} b^{2}}{x} - \frac {150 \, a b^{3}}{x^{\frac {3}{2}}} - \frac {60 \, b^{4}}{x^{2}}}{3 \, {\left (\frac {a^{8}}{x} + \frac {3 \, a^{7} b}{x^{\frac {3}{2}}} + \frac {3 \, a^{6} b^{2}}{x^{2}} + \frac {a^{5} b^{3}}{x^{\frac {5}{2}}}\right )}} + \frac {20 \, b^{2} \log \left (a + \frac {b}{\sqrt {x}}\right )}{a^{6}} + \frac {10 \, b^{2} \log \left (x\right )}{a^{6}} \] Input:
integrate(1/(a+b/x^(1/2))^4,x, algorithm="maxima")
Output:
1/3*(3*a^4 - 15*a^3*b/sqrt(x) - 110*a^2*b^2/x - 150*a*b^3/x^(3/2) - 60*b^4 /x^2)/(a^8/x + 3*a^7*b/x^(3/2) + 3*a^6*b^2/x^2 + a^5*b^3/x^(5/2)) + 20*b^2 *log(a + b/sqrt(x))/a^6 + 10*b^2*log(x)/a^6
Time = 0.12 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt {x}}\right )^4} \, dx=\frac {20 \, b^{2} \log \left ({\left | a \sqrt {x} + b \right |}\right )}{a^{6}} + \frac {a^{4} x - 8 \, a^{3} b \sqrt {x}}{a^{8}} + \frac {60 \, a^{2} b^{3} x + 105 \, a b^{4} \sqrt {x} + 47 \, b^{5}}{3 \, {\left (a \sqrt {x} + b\right )}^{3} a^{6}} \] Input:
integrate(1/(a+b/x^(1/2))^4,x, algorithm="giac")
Output:
20*b^2*log(abs(a*sqrt(x) + b))/a^6 + (a^4*x - 8*a^3*b*sqrt(x))/a^8 + 1/3*( 60*a^2*b^3*x + 105*a*b^4*sqrt(x) + 47*b^5)/((a*sqrt(x) + b)^3*a^6)
Time = 0.12 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.97 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt {x}}\right )^4} \, dx=\frac {x}{a^4}+\frac {\frac {47\,b^5}{3\,a}+35\,b^4\,\sqrt {x}+20\,a\,b^3\,x}{a^5\,b^3+a^8\,x^{3/2}+3\,a^6\,b^2\,\sqrt {x}+3\,a^7\,b\,x}-\frac {8\,b\,\sqrt {x}}{a^5}+\frac {20\,b^2\,\ln \left (b+a\,\sqrt {x}\right )}{a^6} \] Input:
int(1/(a + b/x^(1/2))^4,x)
Output:
x/a^4 + ((47*b^5)/(3*a) + 35*b^4*x^(1/2) + 20*a*b^3*x)/(a^5*b^3 + a^8*x^(3 /2) + 3*a^6*b^2*x^(1/2) + 3*a^7*b*x) - (8*b*x^(1/2))/a^5 + (20*b^2*log(b + a*x^(1/2)))/a^6
Time = 0.22 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.47 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt {x}}\right )^4} \, dx=\frac {60 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, a +b \right ) a^{3} b^{2} x +180 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, a +b \right ) a \,b^{4}+3 \sqrt {x}\, a^{5} x^{2}-60 \sqrt {x}\, a^{3} b^{2} x +90 \sqrt {x}\, a \,b^{4}+180 \,\mathrm {log}\left (\sqrt {x}\, a +b \right ) a^{2} b^{3} x +60 \,\mathrm {log}\left (\sqrt {x}\, a +b \right ) b^{5}-15 a^{4} b \,x^{2}+50 b^{5}}{3 a^{6} \left (\sqrt {x}\, a^{3} x +3 \sqrt {x}\, a \,b^{2}+3 a^{2} b x +b^{3}\right )} \] Input:
int(1/(a+b/x^(1/2))^4,x)
Output:
(60*sqrt(x)*log(sqrt(x)*a + b)*a**3*b**2*x + 180*sqrt(x)*log(sqrt(x)*a + b )*a*b**4 + 3*sqrt(x)*a**5*x**2 - 60*sqrt(x)*a**3*b**2*x + 90*sqrt(x)*a*b** 4 + 180*log(sqrt(x)*a + b)*a**2*b**3*x + 60*log(sqrt(x)*a + b)*b**5 - 15*a **4*b*x**2 + 50*b**5)/(3*a**6*(sqrt(x)*a**3*x + 3*sqrt(x)*a*b**2 + 3*a**2* b*x + b**3))