Integrand size = 15, antiderivative size = 123 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^4} \, dx=\frac {2 b^6}{a^7 \left (a+b \sqrt {x}\right )}-\frac {1}{3 a^2 x^3}+\frac {4 b}{5 a^3 x^{5/2}}-\frac {3 b^2}{2 a^4 x^2}+\frac {8 b^3}{3 a^5 x^{3/2}}-\frac {5 b^4}{a^6 x}+\frac {12 b^5}{a^7 \sqrt {x}}-\frac {14 b^6 \log \left (a+b \sqrt {x}\right )}{a^8}+\frac {7 b^6 \log (x)}{a^8} \] Output:
2*b^6/a^7/(a+b*x^(1/2))-1/3/a^2/x^3+4/5*b/a^3/x^(5/2)-3/2*b^2/a^4/x^2+8/3* b^3/a^5/x^(3/2)-5*b^4/a^6/x+12*b^5/a^7/x^(1/2)-14*b^6*ln(a+b*x^(1/2))/a^8+ 7*b^6*ln(x)/a^8
Time = 0.11 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^4} \, dx=\frac {\frac {a \left (-10 a^6+14 a^5 b \sqrt {x}-21 a^4 b^2 x+35 a^3 b^3 x^{3/2}-70 a^2 b^4 x^2+210 a b^5 x^{5/2}+420 b^6 x^3\right )}{\left (a+b \sqrt {x}\right ) x^3}-420 b^6 \log \left (a+b \sqrt {x}\right )+210 b^6 \log (x)}{30 a^8} \] Input:
Integrate[1/((a + b*Sqrt[x])^2*x^4),x]
Output:
((a*(-10*a^6 + 14*a^5*b*Sqrt[x] - 21*a^4*b^2*x + 35*a^3*b^3*x^(3/2) - 70*a ^2*b^4*x^2 + 210*a*b^5*x^(5/2) + 420*b^6*x^3))/((a + b*Sqrt[x])*x^3) - 420 *b^6*Log[a + b*Sqrt[x]] + 210*b^6*Log[x])/(30*a^8)
Time = 0.45 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.06, 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.
\(\displaystyle \int \frac {1}{x^4 \left (a+b \sqrt {x}\right )^2} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 2 \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^{7/2}}d\sqrt {x}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle 2 \int \left (-\frac {7 b^7}{a^8 \left (a+b \sqrt {x}\right )}-\frac {b^7}{a^7 \left (a+b \sqrt {x}\right )^2}+\frac {7 b^6}{a^8 \sqrt {x}}-\frac {6 b^5}{a^7 x}+\frac {5 b^4}{a^6 x^{3/2}}-\frac {4 b^3}{a^5 x^2}+\frac {3 b^2}{a^4 x^{5/2}}-\frac {2 b}{a^3 x^3}+\frac {1}{a^2 x^{7/2}}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (-\frac {7 b^6 \log \left (a+b \sqrt {x}\right )}{a^8}+\frac {7 b^6 \log \left (\sqrt {x}\right )}{a^8}+\frac {b^6}{a^7 \left (a+b \sqrt {x}\right )}+\frac {6 b^5}{a^7 \sqrt {x}}-\frac {5 b^4}{2 a^6 x}+\frac {4 b^3}{3 a^5 x^{3/2}}-\frac {3 b^2}{4 a^4 x^2}+\frac {2 b}{5 a^3 x^{5/2}}-\frac {1}{6 a^2 x^3}\right )\) |
Input:
Int[1/((a + b*Sqrt[x])^2*x^4),x]
Output:
2*(b^6/(a^7*(a + b*Sqrt[x])) - 1/(6*a^2*x^3) + (2*b)/(5*a^3*x^(5/2)) - (3* b^2)/(4*a^4*x^2) + (4*b^3)/(3*a^5*x^(3/2)) - (5*b^4)/(2*a^6*x) + (6*b^5)/( a^7*Sqrt[x]) - (7*b^6*Log[a + b*Sqrt[x]])/a^8 + (7*b^6*Log[Sqrt[x]])/a^8)
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 = 106, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(\frac {2 b^{6}}{a^{7} \left (a +b \sqrt {x}\right )}-\frac {1}{3 a^{2} x^{3}}+\frac {4 b}{5 a^{3} x^{\frac {5}{2}}}-\frac {3 b^{2}}{2 a^{4} x^{2}}+\frac {8 b^{3}}{3 a^{5} x^{\frac {3}{2}}}-\frac {5 b^{4}}{a^{6} x}+\frac {12 b^{5}}{a^{7} \sqrt {x}}-\frac {14 b^{6} \ln \left (a +b \sqrt {x}\right )}{a^{8}}+\frac {7 b^{6} \ln \left (x \right )}{a^{8}}\) | \(106\) |
default | \(\frac {2 b^{6}}{a^{7} \left (a +b \sqrt {x}\right )}-\frac {1}{3 a^{2} x^{3}}+\frac {4 b}{5 a^{3} x^{\frac {5}{2}}}-\frac {3 b^{2}}{2 a^{4} x^{2}}+\frac {8 b^{3}}{3 a^{5} x^{\frac {3}{2}}}-\frac {5 b^{4}}{a^{6} x}+\frac {12 b^{5}}{a^{7} \sqrt {x}}-\frac {14 b^{6} \ln \left (a +b \sqrt {x}\right )}{a^{8}}+\frac {7 b^{6} \ln \left (x \right )}{a^{8}}\) | \(106\) |
Input:
int(1/(a+b*x^(1/2))^2/x^4,x,method=_RETURNVERBOSE)
Output:
2*b^6/a^7/(a+b*x^(1/2))-1/3/a^2/x^3+4/5*b/a^3/x^(5/2)-3/2*b^2/a^4/x^2+8/3* b^3/a^5/x^(3/2)-5*b^4/a^6/x+12*b^5/a^7/x^(1/2)-14*b^6*ln(a+b*x^(1/2))/a^8+ 7*b^6*ln(x)/a^8
Time = 0.08 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.26 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^4} \, dx=-\frac {210 \, a^{2} b^{6} x^{3} - 105 \, a^{4} b^{4} x^{2} - 35 \, a^{6} b^{2} x - 10 \, a^{8} + 420 \, {\left (b^{8} x^{4} - a^{2} b^{6} x^{3}\right )} \log \left (b \sqrt {x} + a\right ) - 420 \, {\left (b^{8} x^{4} - a^{2} b^{6} x^{3}\right )} \log \left (\sqrt {x}\right ) - 4 \, {\left (105 \, a b^{7} x^{3} - 70 \, a^{3} b^{5} x^{2} - 14 \, a^{5} b^{3} x - 6 \, a^{7} b\right )} \sqrt {x}}{30 \, {\left (a^{8} b^{2} x^{4} - a^{10} x^{3}\right )}} \] Input:
integrate(1/(a+b*x^(1/2))^2/x^4,x, algorithm="fricas")
Output:
-1/30*(210*a^2*b^6*x^3 - 105*a^4*b^4*x^2 - 35*a^6*b^2*x - 10*a^8 + 420*(b^ 8*x^4 - a^2*b^6*x^3)*log(b*sqrt(x) + a) - 420*(b^8*x^4 - a^2*b^6*x^3)*log( sqrt(x)) - 4*(105*a*b^7*x^3 - 70*a^3*b^5*x^2 - 14*a^5*b^3*x - 6*a^7*b)*sqr t(x))/(a^8*b^2*x^4 - a^10*x^3)
Leaf count of result is larger than twice the leaf count of optimal. 400 vs. \(2 (122) = 244\).
Time = 1.26 (sec) , antiderivative size = 400, normalized size of antiderivative = 3.25 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^4} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{4}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {1}{3 a^{2} x^{3}} & \text {for}\: b = 0 \\- \frac {1}{4 b^{2} x^{4}} & \text {for}\: a = 0 \\- \frac {10 a^{7} \sqrt {x}}{30 a^{9} x^{\frac {7}{2}} + 30 a^{8} b x^{4}} + \frac {14 a^{6} b x}{30 a^{9} x^{\frac {7}{2}} + 30 a^{8} b x^{4}} - \frac {21 a^{5} b^{2} x^{\frac {3}{2}}}{30 a^{9} x^{\frac {7}{2}} + 30 a^{8} b x^{4}} + \frac {35 a^{4} b^{3} x^{2}}{30 a^{9} x^{\frac {7}{2}} + 30 a^{8} b x^{4}} - \frac {70 a^{3} b^{4} x^{\frac {5}{2}}}{30 a^{9} x^{\frac {7}{2}} + 30 a^{8} b x^{4}} + \frac {210 a^{2} b^{5} x^{3}}{30 a^{9} x^{\frac {7}{2}} + 30 a^{8} b x^{4}} + \frac {210 a b^{6} x^{\frac {7}{2}} \log {\left (x \right )}}{30 a^{9} x^{\frac {7}{2}} + 30 a^{8} b x^{4}} - \frac {420 a b^{6} x^{\frac {7}{2}} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{30 a^{9} x^{\frac {7}{2}} + 30 a^{8} b x^{4}} + \frac {420 a b^{6} x^{\frac {7}{2}}}{30 a^{9} x^{\frac {7}{2}} + 30 a^{8} b x^{4}} + \frac {210 b^{7} x^{4} \log {\left (x \right )}}{30 a^{9} x^{\frac {7}{2}} + 30 a^{8} b x^{4}} - \frac {420 b^{7} x^{4} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{30 a^{9} x^{\frac {7}{2}} + 30 a^{8} b x^{4}} & \text {otherwise} \end {cases} \] Input:
integrate(1/(a+b*x**(1/2))**2/x**4,x)
Output:
Piecewise((zoo/x**4, Eq(a, 0) & Eq(b, 0)), (-1/(3*a**2*x**3), Eq(b, 0)), ( -1/(4*b**2*x**4), Eq(a, 0)), (-10*a**7*sqrt(x)/(30*a**9*x**(7/2) + 30*a**8 *b*x**4) + 14*a**6*b*x/(30*a**9*x**(7/2) + 30*a**8*b*x**4) - 21*a**5*b**2* x**(3/2)/(30*a**9*x**(7/2) + 30*a**8*b*x**4) + 35*a**4*b**3*x**2/(30*a**9* x**(7/2) + 30*a**8*b*x**4) - 70*a**3*b**4*x**(5/2)/(30*a**9*x**(7/2) + 30* a**8*b*x**4) + 210*a**2*b**5*x**3/(30*a**9*x**(7/2) + 30*a**8*b*x**4) + 21 0*a*b**6*x**(7/2)*log(x)/(30*a**9*x**(7/2) + 30*a**8*b*x**4) - 420*a*b**6* x**(7/2)*log(a/b + sqrt(x))/(30*a**9*x**(7/2) + 30*a**8*b*x**4) + 420*a*b* *6*x**(7/2)/(30*a**9*x**(7/2) + 30*a**8*b*x**4) + 210*b**7*x**4*log(x)/(30 *a**9*x**(7/2) + 30*a**8*b*x**4) - 420*b**7*x**4*log(a/b + sqrt(x))/(30*a* *9*x**(7/2) + 30*a**8*b*x**4), True))
Time = 0.03 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^4} \, dx=\frac {420 \, b^{6} x^{3} + 210 \, a b^{5} x^{\frac {5}{2}} - 70 \, a^{2} b^{4} x^{2} + 35 \, a^{3} b^{3} x^{\frac {3}{2}} - 21 \, a^{4} b^{2} x + 14 \, a^{5} b \sqrt {x} - 10 \, a^{6}}{30 \, {\left (a^{7} b x^{\frac {7}{2}} + a^{8} x^{3}\right )}} - \frac {14 \, b^{6} \log \left (b \sqrt {x} + a\right )}{a^{8}} + \frac {7 \, b^{6} \log \left (x\right )}{a^{8}} \] Input:
integrate(1/(a+b*x^(1/2))^2/x^4,x, algorithm="maxima")
Output:
1/30*(420*b^6*x^3 + 210*a*b^5*x^(5/2) - 70*a^2*b^4*x^2 + 35*a^3*b^3*x^(3/2 ) - 21*a^4*b^2*x + 14*a^5*b*sqrt(x) - 10*a^6)/(a^7*b*x^(7/2) + a^8*x^3) - 14*b^6*log(b*sqrt(x) + a)/a^8 + 7*b^6*log(x)/a^8
Time = 0.11 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^4} \, dx=-\frac {14 \, b^{6} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{a^{8}} + \frac {7 \, b^{6} \log \left ({\left | x \right |}\right )}{a^{8}} + \frac {420 \, a b^{6} x^{3} + 210 \, a^{2} b^{5} x^{\frac {5}{2}} - 70 \, a^{3} b^{4} x^{2} + 35 \, a^{4} b^{3} x^{\frac {3}{2}} - 21 \, a^{5} b^{2} x + 14 \, a^{6} b \sqrt {x} - 10 \, a^{7}}{30 \, {\left (b \sqrt {x} + a\right )} a^{8} x^{3}} \] Input:
integrate(1/(a+b*x^(1/2))^2/x^4,x, algorithm="giac")
Output:
-14*b^6*log(abs(b*sqrt(x) + a))/a^8 + 7*b^6*log(abs(x))/a^8 + 1/30*(420*a* b^6*x^3 + 210*a^2*b^5*x^(5/2) - 70*a^3*b^4*x^2 + 35*a^4*b^3*x^(3/2) - 21*a ^5*b^2*x + 14*a^6*b*sqrt(x) - 10*a^7)/((b*sqrt(x) + a)*a^8*x^3)
Time = 0.09 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^4} \, dx=\frac {\frac {7\,b\,\sqrt {x}}{15\,a^2}-\frac {1}{3\,a}-\frac {7\,b^2\,x}{10\,a^3}-\frac {7\,b^4\,x^2}{3\,a^5}+\frac {7\,b^3\,x^{3/2}}{6\,a^4}+\frac {14\,b^6\,x^3}{a^7}+\frac {7\,b^5\,x^{5/2}}{a^6}}{a\,x^3+b\,x^{7/2}}-\frac {28\,b^6\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a^8} \] Input:
int(1/(x^4*(a + b*x^(1/2))^2),x)
Output:
((7*b*x^(1/2))/(15*a^2) - 1/(3*a) - (7*b^2*x)/(10*a^3) - (7*b^4*x^2)/(3*a^ 5) + (7*b^3*x^(3/2))/(6*a^4) + (14*b^6*x^3)/a^7 + (7*b^5*x^(5/2))/a^6)/(a* x^3 + b*x^(7/2)) - (28*b^6*atanh((2*b*x^(1/2))/a + 1))/a^8
Time = 0.21 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.15 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^4} \, dx=\frac {-420 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, b +a \right ) b^{7} x^{3}+420 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\right ) b^{7} x^{3}+14 \sqrt {x}\, a^{6} b +35 \sqrt {x}\, a^{4} b^{3} x +210 \sqrt {x}\, a^{2} b^{5} x^{2}-420 \sqrt {x}\, b^{7} x^{3}-420 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a \,b^{6} x^{3}+420 \,\mathrm {log}\left (\sqrt {x}\right ) a \,b^{6} x^{3}-10 a^{7}-21 a^{5} b^{2} x -70 a^{3} b^{4} x^{2}}{30 a^{8} x^{3} \left (\sqrt {x}\, b +a \right )} \] Input:
int(1/(a+b*x^(1/2))^2/x^4,x)
Output:
( - 420*sqrt(x)*log(sqrt(x)*b + a)*b**7*x**3 + 420*sqrt(x)*log(sqrt(x))*b* *7*x**3 + 14*sqrt(x)*a**6*b + 35*sqrt(x)*a**4*b**3*x + 210*sqrt(x)*a**2*b* *5*x**2 - 420*sqrt(x)*b**7*x**3 - 420*log(sqrt(x)*b + a)*a*b**6*x**3 + 420 *log(sqrt(x))*a*b**6*x**3 - 10*a**7 - 21*a**5*b**2*x - 70*a**3*b**4*x**2)/ (30*a**8*x**3*(sqrt(x)*b + a))