Integrand size = 15, antiderivative size = 95 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^3} \, dx=\frac {2 b^4}{a^5 \left (a+b \sqrt {x}\right )}-\frac {1}{2 a^2 x^2}+\frac {4 b}{3 a^3 x^{3/2}}-\frac {3 b^2}{a^4 x}+\frac {8 b^3}{a^5 \sqrt {x}}-\frac {10 b^4 \log \left (a+b \sqrt {x}\right )}{a^6}+\frac {5 b^4 \log (x)}{a^6} \] Output:
2*b^4/a^5/(a+b*x^(1/2))-1/2/a^2/x^2+4/3*b/a^3/x^(3/2)-3*b^2/a^4/x+8*b^3/a^ 5/x^(1/2)-10*b^4*ln(a+b*x^(1/2))/a^6+5*b^4*ln(x)/a^6
Time = 0.09 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^3} \, dx=\frac {\frac {a \left (-3 a^4+5 a^3 b \sqrt {x}-10 a^2 b^2 x+30 a b^3 x^{3/2}+60 b^4 x^2\right )}{\left (a+b \sqrt {x}\right ) x^2}-60 b^4 \log \left (a+b \sqrt {x}\right )+30 b^4 \log (x)}{6 a^6} \] Input:
Integrate[1/((a + b*Sqrt[x])^2*x^3),x]
Output:
((a*(-3*a^4 + 5*a^3*b*Sqrt[x] - 10*a^2*b^2*x + 30*a*b^3*x^(3/2) + 60*b^4*x ^2))/((a + b*Sqrt[x])*x^2) - 60*b^4*Log[a + b*Sqrt[x]] + 30*b^4*Log[x])/(6 *a^6)
Time = 0.40 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {798, 54, 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.
\(\displaystyle \int \frac {1}{x^3 \left (a+b \sqrt {x}\right )^2} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 2 \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^{5/2}}d\sqrt {x}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle 2 \int \left (-\frac {5 b^5}{a^6 \left (a+b \sqrt {x}\right )}-\frac {b^5}{a^5 \left (a+b \sqrt {x}\right )^2}+\frac {5 b^4}{a^6 \sqrt {x}}-\frac {4 b^3}{a^5 x}+\frac {3 b^2}{a^4 x^{3/2}}-\frac {2 b}{a^3 x^2}+\frac {1}{a^2 x^{5/2}}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (-\frac {5 b^4 \log \left (a+b \sqrt {x}\right )}{a^6}+\frac {5 b^4 \log \left (\sqrt {x}\right )}{a^6}+\frac {b^4}{a^5 \left (a+b \sqrt {x}\right )}+\frac {4 b^3}{a^5 \sqrt {x}}-\frac {3 b^2}{2 a^4 x}+\frac {2 b}{3 a^3 x^{3/2}}-\frac {1}{4 a^2 x^2}\right )\) |
Input:
Int[1/((a + b*Sqrt[x])^2*x^3),x]
Output:
2*(b^4/(a^5*(a + b*Sqrt[x])) - 1/(4*a^2*x^2) + (2*b)/(3*a^3*x^(3/2)) - (3* b^2)/(2*a^4*x) + (4*b^3)/(a^5*Sqrt[x]) - (5*b^4*Log[a + b*Sqrt[x]])/a^6 + (5*b^4*Log[Sqrt[x]])/a^6)
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[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 = 84, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {2 b^{4}}{a^{5} \left (a +b \sqrt {x}\right )}-\frac {1}{2 a^{2} x^{2}}+\frac {4 b}{3 a^{3} x^{\frac {3}{2}}}-\frac {3 b^{2}}{a^{4} x}+\frac {8 b^{3}}{a^{5} \sqrt {x}}-\frac {10 b^{4} \ln \left (a +b \sqrt {x}\right )}{a^{6}}+\frac {5 b^{4} \ln \left (x \right )}{a^{6}}\) | \(84\) |
default | \(\frac {2 b^{4}}{a^{5} \left (a +b \sqrt {x}\right )}-\frac {1}{2 a^{2} x^{2}}+\frac {4 b}{3 a^{3} x^{\frac {3}{2}}}-\frac {3 b^{2}}{a^{4} x}+\frac {8 b^{3}}{a^{5} \sqrt {x}}-\frac {10 b^{4} \ln \left (a +b \sqrt {x}\right )}{a^{6}}+\frac {5 b^{4} \ln \left (x \right )}{a^{6}}\) | \(84\) |
Input:
int(1/(a+b*x^(1/2))^2/x^3,x,method=_RETURNVERBOSE)
Output:
2*b^4/a^5/(a+b*x^(1/2))-1/2/a^2/x^2+4/3*b/a^3/x^(3/2)-3*b^2/a^4/x+8*b^3/a^ 5/x^(1/2)-10*b^4*ln(a+b*x^(1/2))/a^6+5*b^4*ln(x)/a^6
Time = 0.09 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.40 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^3} \, dx=-\frac {30 \, a^{2} b^{4} x^{2} - 15 \, a^{4} b^{2} x - 3 \, a^{6} + 60 \, {\left (b^{6} x^{3} - a^{2} b^{4} x^{2}\right )} \log \left (b \sqrt {x} + a\right ) - 60 \, {\left (b^{6} x^{3} - a^{2} b^{4} x^{2}\right )} \log \left (\sqrt {x}\right ) - 4 \, {\left (15 \, a b^{5} x^{2} - 10 \, a^{3} b^{3} x - 2 \, a^{5} b\right )} \sqrt {x}}{6 \, {\left (a^{6} b^{2} x^{3} - a^{8} x^{2}\right )}} \] Input:
integrate(1/(a+b*x^(1/2))^2/x^3,x, algorithm="fricas")
Output:
-1/6*(30*a^2*b^4*x^2 - 15*a^4*b^2*x - 3*a^6 + 60*(b^6*x^3 - a^2*b^4*x^2)*l og(b*sqrt(x) + a) - 60*(b^6*x^3 - a^2*b^4*x^2)*log(sqrt(x)) - 4*(15*a*b^5* x^2 - 10*a^3*b^3*x - 2*a^5*b)*sqrt(x))/(a^6*b^2*x^3 - a^8*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (94) = 188\).
Time = 0.88 (sec) , antiderivative size = 333, normalized size of antiderivative = 3.51 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^3} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{3}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {1}{2 a^{2} x^{2}} & \text {for}\: b = 0 \\- \frac {1}{3 b^{2} x^{3}} & \text {for}\: a = 0 \\- \frac {3 a^{5} \sqrt {x}}{6 a^{7} x^{\frac {5}{2}} + 6 a^{6} b x^{3}} + \frac {5 a^{4} b x}{6 a^{7} x^{\frac {5}{2}} + 6 a^{6} b x^{3}} - \frac {10 a^{3} b^{2} x^{\frac {3}{2}}}{6 a^{7} x^{\frac {5}{2}} + 6 a^{6} b x^{3}} + \frac {30 a^{2} b^{3} x^{2}}{6 a^{7} x^{\frac {5}{2}} + 6 a^{6} b x^{3}} + \frac {30 a b^{4} x^{\frac {5}{2}} \log {\left (x \right )}}{6 a^{7} x^{\frac {5}{2}} + 6 a^{6} b x^{3}} - \frac {60 a b^{4} x^{\frac {5}{2}} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{6 a^{7} x^{\frac {5}{2}} + 6 a^{6} b x^{3}} + \frac {60 a b^{4} x^{\frac {5}{2}}}{6 a^{7} x^{\frac {5}{2}} + 6 a^{6} b x^{3}} + \frac {30 b^{5} x^{3} \log {\left (x \right )}}{6 a^{7} x^{\frac {5}{2}} + 6 a^{6} b x^{3}} - \frac {60 b^{5} x^{3} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{6 a^{7} x^{\frac {5}{2}} + 6 a^{6} b x^{3}} & \text {otherwise} \end {cases} \] Input:
integrate(1/(a+b*x**(1/2))**2/x**3,x)
Output:
Piecewise((zoo/x**3, Eq(a, 0) & Eq(b, 0)), (-1/(2*a**2*x**2), Eq(b, 0)), ( -1/(3*b**2*x**3), Eq(a, 0)), (-3*a**5*sqrt(x)/(6*a**7*x**(5/2) + 6*a**6*b* x**3) + 5*a**4*b*x/(6*a**7*x**(5/2) + 6*a**6*b*x**3) - 10*a**3*b**2*x**(3/ 2)/(6*a**7*x**(5/2) + 6*a**6*b*x**3) + 30*a**2*b**3*x**2/(6*a**7*x**(5/2) + 6*a**6*b*x**3) + 30*a*b**4*x**(5/2)*log(x)/(6*a**7*x**(5/2) + 6*a**6*b*x **3) - 60*a*b**4*x**(5/2)*log(a/b + sqrt(x))/(6*a**7*x**(5/2) + 6*a**6*b*x **3) + 60*a*b**4*x**(5/2)/(6*a**7*x**(5/2) + 6*a**6*b*x**3) + 30*b**5*x**3 *log(x)/(6*a**7*x**(5/2) + 6*a**6*b*x**3) - 60*b**5*x**3*log(a/b + sqrt(x) )/(6*a**7*x**(5/2) + 6*a**6*b*x**3), True))
Time = 0.03 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^3} \, dx=\frac {60 \, b^{4} x^{2} + 30 \, a b^{3} x^{\frac {3}{2}} - 10 \, a^{2} b^{2} x + 5 \, a^{3} b \sqrt {x} - 3 \, a^{4}}{6 \, {\left (a^{5} b x^{\frac {5}{2}} + a^{6} x^{2}\right )}} - \frac {10 \, b^{4} \log \left (b \sqrt {x} + a\right )}{a^{6}} + \frac {5 \, b^{4} \log \left (x\right )}{a^{6}} \] Input:
integrate(1/(a+b*x^(1/2))^2/x^3,x, algorithm="maxima")
Output:
1/6*(60*b^4*x^2 + 30*a*b^3*x^(3/2) - 10*a^2*b^2*x + 5*a^3*b*sqrt(x) - 3*a^ 4)/(a^5*b*x^(5/2) + a^6*x^2) - 10*b^4*log(b*sqrt(x) + a)/a^6 + 5*b^4*log(x )/a^6
Time = 0.12 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^3} \, dx=-\frac {10 \, b^{4} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{a^{6}} + \frac {5 \, b^{4} \log \left ({\left | x \right |}\right )}{a^{6}} + \frac {60 \, a b^{4} x^{2} + 30 \, a^{2} b^{3} x^{\frac {3}{2}} - 10 \, a^{3} b^{2} x + 5 \, a^{4} b \sqrt {x} - 3 \, a^{5}}{6 \, {\left (b \sqrt {x} + a\right )} a^{6} x^{2}} \] Input:
integrate(1/(a+b*x^(1/2))^2/x^3,x, algorithm="giac")
Output:
-10*b^4*log(abs(b*sqrt(x) + a))/a^6 + 5*b^4*log(abs(x))/a^6 + 1/6*(60*a*b^ 4*x^2 + 30*a^2*b^3*x^(3/2) - 10*a^3*b^2*x + 5*a^4*b*sqrt(x) - 3*a^5)/((b*s qrt(x) + a)*a^6*x^2)
Time = 0.34 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^3} \, dx=\frac {\frac {5\,b\,\sqrt {x}}{6\,a^2}-\frac {1}{2\,a}-\frac {5\,b^2\,x}{3\,a^3}+\frac {10\,b^4\,x^2}{a^5}+\frac {5\,b^3\,x^{3/2}}{a^4}}{a\,x^2+b\,x^{5/2}}-\frac {20\,b^4\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a^6} \] Input:
int(1/(x^3*(a + b*x^(1/2))^2),x)
Output:
((5*b*x^(1/2))/(6*a^2) - 1/(2*a) - (5*b^2*x)/(3*a^3) + (10*b^4*x^2)/a^5 + (5*b^3*x^(3/2))/a^4)/(a*x^2 + b*x^(5/2)) - (20*b^4*atanh((2*b*x^(1/2))/a + 1))/a^6
Time = 0.20 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.24 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^3} \, dx=\frac {-60 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, b +a \right ) b^{5} x^{2}+60 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\right ) b^{5} x^{2}+5 \sqrt {x}\, a^{4} b +30 \sqrt {x}\, a^{2} b^{3} x -60 \sqrt {x}\, b^{5} x^{2}-60 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a \,b^{4} x^{2}+60 \,\mathrm {log}\left (\sqrt {x}\right ) a \,b^{4} x^{2}-3 a^{5}-10 a^{3} b^{2} x}{6 a^{6} x^{2} \left (\sqrt {x}\, b +a \right )} \] Input:
int(1/(a+b*x^(1/2))^2/x^3,x)
Output:
( - 60*sqrt(x)*log(sqrt(x)*b + a)*b**5*x**2 + 60*sqrt(x)*log(sqrt(x))*b**5 *x**2 + 5*sqrt(x)*a**4*b + 30*sqrt(x)*a**2*b**3*x - 60*sqrt(x)*b**5*x**2 - 60*log(sqrt(x)*b + a)*a*b**4*x**2 + 60*log(sqrt(x))*a*b**4*x**2 - 3*a**5 - 10*a**3*b**2*x)/(6*a**6*x**2*(sqrt(x)*b + a))