Integrand size = 15, antiderivative size = 85 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^2} \, dx=\frac {b^2}{a^3 \left (a+b \sqrt {x}\right )^2}+\frac {6 b^2}{a^4 \left (a+b \sqrt {x}\right )}-\frac {1}{a^3 x}+\frac {6 b}{a^4 \sqrt {x}}-\frac {12 b^2 \log \left (a+b \sqrt {x}\right )}{a^5}+\frac {6 b^2 \log (x)}{a^5} \] Output:
b^2/a^3/(a+b*x^(1/2))^2+6*b^2/a^4/(a+b*x^(1/2))-1/a^3/x+6*b/a^4/x^(1/2)-12 *b^2*ln(a+b*x^(1/2))/a^5+6*b^2*ln(x)/a^5
Time = 0.10 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^2} \, dx=\frac {\frac {a \left (-a^3+4 a^2 b \sqrt {x}+18 a b^2 x+12 b^3 x^{3/2}\right )}{\left (a+b \sqrt {x}\right )^2 x}-12 b^2 \log \left (a+b \sqrt {x}\right )+6 b^2 \log (x)}{a^5} \] Input:
Integrate[1/((a + b*Sqrt[x])^3*x^2),x]
Output:
((a*(-a^3 + 4*a^2*b*Sqrt[x] + 18*a*b^2*x + 12*b^3*x^(3/2)))/((a + b*Sqrt[x ])^2*x) - 12*b^2*Log[a + b*Sqrt[x]] + 6*b^2*Log[x])/a^5
Time = 0.39 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.13, 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.
\(\displaystyle \int \frac {1}{x^2 \left (a+b \sqrt {x}\right )^3} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 2 \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^{3/2}}d\sqrt {x}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle 2 \int \left (-\frac {6 b^3}{a^5 \left (a+b \sqrt {x}\right )}-\frac {3 b^3}{a^4 \left (a+b \sqrt {x}\right )^2}-\frac {b^3}{a^3 \left (a+b \sqrt {x}\right )^3}+\frac {6 b^2}{a^5 \sqrt {x}}-\frac {3 b}{a^4 x}+\frac {1}{a^3 x^{3/2}}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (-\frac {6 b^2 \log \left (a+b \sqrt {x}\right )}{a^5}+\frac {6 b^2 \log \left (\sqrt {x}\right )}{a^5}+\frac {3 b^2}{a^4 \left (a+b \sqrt {x}\right )}+\frac {3 b}{a^4 \sqrt {x}}+\frac {b^2}{2 a^3 \left (a+b \sqrt {x}\right )^2}-\frac {1}{2 a^3 x}\right )\) |
Input:
Int[1/((a + b*Sqrt[x])^3*x^2),x]
Output:
2*(b^2/(2*a^3*(a + b*Sqrt[x])^2) + (3*b^2)/(a^4*(a + b*Sqrt[x])) - 1/(2*a^ 3*x) + (3*b)/(a^4*Sqrt[x]) - (6*b^2*Log[a + b*Sqrt[x]])/a^5 + (6*b^2*Log[S qrt[x]])/a^5)
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 = 0.47 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.92
method | result | size |
derivativedivides | \(\frac {b^{2}}{a^{3} \left (a +b \sqrt {x}\right )^{2}}+\frac {6 b^{2}}{a^{4} \left (a +b \sqrt {x}\right )}-\frac {1}{a^{3} x}+\frac {6 b}{a^{4} \sqrt {x}}-\frac {12 b^{2} \ln \left (a +b \sqrt {x}\right )}{a^{5}}+\frac {6 b^{2} \ln \left (x \right )}{a^{5}}\) | \(78\) |
default | \(\frac {b^{2}}{a^{3} \left (a +b \sqrt {x}\right )^{2}}+\frac {6 b^{2}}{a^{4} \left (a +b \sqrt {x}\right )}-\frac {1}{a^{3} x}+\frac {6 b}{a^{4} \sqrt {x}}-\frac {12 b^{2} \ln \left (a +b \sqrt {x}\right )}{a^{5}}+\frac {6 b^{2} \ln \left (x \right )}{a^{5}}\) | \(78\) |
Input:
int(1/(a+b*x^(1/2))^3/x^2,x,method=_RETURNVERBOSE)
Output:
b^2/a^3/(a+b*x^(1/2))^2+6*b^2/a^4/(a+b*x^(1/2))-1/a^3/x+6*b/a^4/x^(1/2)-12 *b^2*ln(a+b*x^(1/2))/a^5+6*b^2*ln(x)/a^5
Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (77) = 154\).
Time = 0.09 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.82 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^2} \, dx=-\frac {6 \, a^{2} b^{4} x^{2} - 9 \, a^{4} b^{2} x + a^{6} + 12 \, {\left (b^{6} x^{3} - 2 \, a^{2} b^{4} x^{2} + a^{4} b^{2} x\right )} \log \left (b \sqrt {x} + a\right ) - 12 \, {\left (b^{6} x^{3} - 2 \, a^{2} b^{4} x^{2} + a^{4} b^{2} x\right )} \log \left (\sqrt {x}\right ) - 2 \, {\left (6 \, a b^{5} x^{2} - 10 \, a^{3} b^{3} x + 3 \, a^{5} b\right )} \sqrt {x}}{a^{5} b^{4} x^{3} - 2 \, a^{7} b^{2} x^{2} + a^{9} x} \] Input:
integrate(1/(a+b*x^(1/2))^3/x^2,x, algorithm="fricas")
Output:
-(6*a^2*b^4*x^2 - 9*a^4*b^2*x + a^6 + 12*(b^6*x^3 - 2*a^2*b^4*x^2 + a^4*b^ 2*x)*log(b*sqrt(x) + a) - 12*(b^6*x^3 - 2*a^2*b^4*x^2 + a^4*b^2*x)*log(sqr t(x)) - 2*(6*a*b^5*x^2 - 10*a^3*b^3*x + 3*a^5*b)*sqrt(x))/(a^5*b^4*x^3 - 2 *a^7*b^2*x^2 + a^9*x)
Leaf count of result is larger than twice the leaf count of optimal. 481 vs. \(2 (82) = 164\).
Time = 1.61 (sec) , antiderivative size = 481, normalized size of antiderivative = 5.66 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^2} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {1}{a^{3} x} & \text {for}\: b = 0 \\- \frac {2}{5 b^{3} x^{\frac {5}{2}}} & \text {for}\: a = 0 \\- \frac {a^{4} \sqrt {x}}{a^{7} x^{\frac {3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac {5}{2}}} + \frac {4 a^{3} b x}{a^{7} x^{\frac {3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac {5}{2}}} + \frac {6 a^{2} b^{2} x^{\frac {3}{2}} \log {\left (x \right )}}{a^{7} x^{\frac {3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac {5}{2}}} - \frac {12 a^{2} b^{2} x^{\frac {3}{2}} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a^{7} x^{\frac {3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac {5}{2}}} + \frac {18 a^{2} b^{2} x^{\frac {3}{2}}}{a^{7} x^{\frac {3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac {5}{2}}} + \frac {12 a b^{3} x^{2} \log {\left (x \right )}}{a^{7} x^{\frac {3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac {5}{2}}} - \frac {24 a b^{3} x^{2} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a^{7} x^{\frac {3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac {5}{2}}} + \frac {12 a b^{3} x^{2}}{a^{7} x^{\frac {3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac {5}{2}}} + \frac {6 b^{4} x^{\frac {5}{2}} \log {\left (x \right )}}{a^{7} x^{\frac {3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac {5}{2}}} - \frac {12 b^{4} x^{\frac {5}{2}} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a^{7} x^{\frac {3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac {5}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate(1/(a+b*x**(1/2))**3/x**2,x)
Output:
Piecewise((zoo/x**(5/2), Eq(a, 0) & Eq(b, 0)), (-1/(a**3*x), Eq(b, 0)), (- 2/(5*b**3*x**(5/2)), Eq(a, 0)), (-a**4*sqrt(x)/(a**7*x**(3/2) + 2*a**6*b*x **2 + a**5*b**2*x**(5/2)) + 4*a**3*b*x/(a**7*x**(3/2) + 2*a**6*b*x**2 + a* *5*b**2*x**(5/2)) + 6*a**2*b**2*x**(3/2)*log(x)/(a**7*x**(3/2) + 2*a**6*b* x**2 + a**5*b**2*x**(5/2)) - 12*a**2*b**2*x**(3/2)*log(a/b + sqrt(x))/(a** 7*x**(3/2) + 2*a**6*b*x**2 + a**5*b**2*x**(5/2)) + 18*a**2*b**2*x**(3/2)/( a**7*x**(3/2) + 2*a**6*b*x**2 + a**5*b**2*x**(5/2)) + 12*a*b**3*x**2*log(x )/(a**7*x**(3/2) + 2*a**6*b*x**2 + a**5*b**2*x**(5/2)) - 24*a*b**3*x**2*lo g(a/b + sqrt(x))/(a**7*x**(3/2) + 2*a**6*b*x**2 + a**5*b**2*x**(5/2)) + 12 *a*b**3*x**2/(a**7*x**(3/2) + 2*a**6*b*x**2 + a**5*b**2*x**(5/2)) + 6*b**4 *x**(5/2)*log(x)/(a**7*x**(3/2) + 2*a**6*b*x**2 + a**5*b**2*x**(5/2)) - 12 *b**4*x**(5/2)*log(a/b + sqrt(x))/(a**7*x**(3/2) + 2*a**6*b*x**2 + a**5*b* *2*x**(5/2)), True))
Time = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^2} \, dx=\frac {12 \, b^{3} x^{\frac {3}{2}} + 18 \, a b^{2} x + 4 \, a^{2} b \sqrt {x} - a^{3}}{a^{4} b^{2} x^{2} + 2 \, a^{5} b x^{\frac {3}{2}} + a^{6} x} - \frac {12 \, b^{2} \log \left (b \sqrt {x} + a\right )}{a^{5}} + \frac {6 \, b^{2} \log \left (x\right )}{a^{5}} \] Input:
integrate(1/(a+b*x^(1/2))^3/x^2,x, algorithm="maxima")
Output:
(12*b^3*x^(3/2) + 18*a*b^2*x + 4*a^2*b*sqrt(x) - a^3)/(a^4*b^2*x^2 + 2*a^5 *b*x^(3/2) + a^6*x) - 12*b^2*log(b*sqrt(x) + a)/a^5 + 6*b^2*log(x)/a^5
Time = 0.12 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^2} \, dx=-\frac {12 \, b^{2} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{a^{5}} + \frac {6 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{5}} + \frac {12 \, b^{3} x^{\frac {3}{2}} + 18 \, a b^{2} x + 4 \, a^{2} b \sqrt {x} - a^{3}}{{\left (b x + a \sqrt {x}\right )}^{2} a^{4}} \] Input:
integrate(1/(a+b*x^(1/2))^3/x^2,x, algorithm="giac")
Output:
-12*b^2*log(abs(b*sqrt(x) + a))/a^5 + 6*b^2*log(abs(x))/a^5 + (12*b^3*x^(3 /2) + 18*a*b^2*x + 4*a^2*b*sqrt(x) - a^3)/((b*x + a*sqrt(x))^2*a^4)
Time = 0.36 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^2} \, dx=\frac {\frac {4\,b\,\sqrt {x}}{a^2}-\frac {1}{a}+\frac {18\,b^2\,x}{a^3}+\frac {12\,b^3\,x^{3/2}}{a^4}}{a^2\,x+b^2\,x^2+2\,a\,b\,x^{3/2}}-\frac {24\,b^2\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a^5} \] Input:
int(1/(x^2*(a + b*x^(1/2))^3),x)
Output:
((4*b*x^(1/2))/a^2 - 1/a + (18*b^2*x)/a^3 + (12*b^3*x^(3/2))/a^4)/(a^2*x + b^2*x^2 + 2*a*b*x^(3/2)) - (24*b^2*atanh((2*b*x^(1/2))/a + 1))/a^5
Time = 0.21 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.61 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^2} \, dx=\frac {-24 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, b +a \right ) a \,b^{3} x +24 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\right ) a \,b^{3} x +4 \sqrt {x}\, a^{3} b -12 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{2} b^{2} x -12 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) b^{4} x^{2}+12 \,\mathrm {log}\left (\sqrt {x}\right ) a^{2} b^{2} x +12 \,\mathrm {log}\left (\sqrt {x}\right ) b^{4} x^{2}-a^{4}+12 a^{2} b^{2} x -6 b^{4} x^{2}}{a^{5} x \left (2 \sqrt {x}\, a b +a^{2}+b^{2} x \right )} \] Input:
int(1/(a+b*x^(1/2))^3/x^2,x)
Output:
( - 24*sqrt(x)*log(sqrt(x)*b + a)*a*b**3*x + 24*sqrt(x)*log(sqrt(x))*a*b** 3*x + 4*sqrt(x)*a**3*b - 12*log(sqrt(x)*b + a)*a**2*b**2*x - 12*log(sqrt(x )*b + a)*b**4*x**2 + 12*log(sqrt(x))*a**2*b**2*x + 12*log(sqrt(x))*b**4*x* *2 - a**4 + 12*a**2*b**2*x - 6*b**4*x**2)/(a**5*x*(2*sqrt(x)*a*b + a**2 + b**2*x))