Integrand size = 15, antiderivative size = 89 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^5 x} \, dx=\frac {1}{2 a \left (a+b \sqrt {x}\right )^4}+\frac {2}{3 a^2 \left (a+b \sqrt {x}\right )^3}+\frac {1}{a^3 \left (a+b \sqrt {x}\right )^2}+\frac {2}{a^4 \left (a+b \sqrt {x}\right )}-\frac {2 \log \left (a+b \sqrt {x}\right )}{a^5}+\frac {\log (x)}{a^5} \] Output:
1/2/a/(a+b*x^(1/2))^4+2/3/a^2/(a+b*x^(1/2))^3+1/a^3/(a+b*x^(1/2))^2+2/a^4/ (a+b*x^(1/2))-2*ln(a+b*x^(1/2))/a^5+ln(x)/a^5
Time = 0.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^5 x} \, dx=\frac {\frac {a \left (25 a^3+52 a^2 b \sqrt {x}+42 a b^2 x+12 b^3 x^{3/2}\right )}{\left (a+b \sqrt {x}\right )^4}-12 \log \left (a+b \sqrt {x}\right )+6 \log (x)}{6 a^5} \] Input:
Integrate[1/((a + b*Sqrt[x])^5*x),x]
Output:
((a*(25*a^3 + 52*a^2*b*Sqrt[x] + 42*a*b^2*x + 12*b^3*x^(3/2)))/(a + b*Sqrt [x])^4 - 12*Log[a + b*Sqrt[x]] + 6*Log[x])/(6*a^5)
Time = 0.38 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.09, 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.
\(\displaystyle \int \frac {1}{x \left (a+b \sqrt {x}\right )^5} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 2 \int \frac {1}{\left (a+b \sqrt {x}\right )^5 \sqrt {x}}d\sqrt {x}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle 2 \int \left (-\frac {b}{a^5 \left (a+b \sqrt {x}\right )}-\frac {b}{a^4 \left (a+b \sqrt {x}\right )^2}-\frac {b}{a^3 \left (a+b \sqrt {x}\right )^3}-\frac {b}{a^2 \left (a+b \sqrt {x}\right )^4}-\frac {b}{a \left (a+b \sqrt {x}\right )^5}+\frac {1}{a^5 \sqrt {x}}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (-\frac {\log \left (a+b \sqrt {x}\right )}{a^5}+\frac {\log \left (\sqrt {x}\right )}{a^5}+\frac {1}{a^4 \left (a+b \sqrt {x}\right )}+\frac {1}{2 a^3 \left (a+b \sqrt {x}\right )^2}+\frac {1}{3 a^2 \left (a+b \sqrt {x}\right )^3}+\frac {1}{4 a \left (a+b \sqrt {x}\right )^4}\right )\) |
Input:
Int[1/((a + b*Sqrt[x])^5*x),x]
Output:
2*(1/(4*a*(a + b*Sqrt[x])^4) + 1/(3*a^2*(a + b*Sqrt[x])^3) + 1/(2*a^3*(a + b*Sqrt[x])^2) + 1/(a^4*(a + b*Sqrt[x])) - Log[a + b*Sqrt[x]]/a^5 + Log[Sq rt[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 = 0.45 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(\frac {1}{2 a \left (a +b \sqrt {x}\right )^{4}}+\frac {2}{3 a^{2} \left (a +b \sqrt {x}\right )^{3}}+\frac {1}{a^{3} \left (a +b \sqrt {x}\right )^{2}}+\frac {2}{a^{4} \left (a +b \sqrt {x}\right )}-\frac {2 \ln \left (a +b \sqrt {x}\right )}{a^{5}}+\frac {\ln \left (x \right )}{a^{5}}\) | \(76\) |
default | \(\frac {1}{2 a \left (a +b \sqrt {x}\right )^{4}}+\frac {2}{3 a^{2} \left (a +b \sqrt {x}\right )^{3}}+\frac {1}{a^{3} \left (a +b \sqrt {x}\right )^{2}}+\frac {2}{a^{4} \left (a +b \sqrt {x}\right )}-\frac {2 \ln \left (a +b \sqrt {x}\right )}{a^{5}}+\frac {\ln \left (x \right )}{a^{5}}\) | \(76\) |
Input:
int(1/(a+b*x^(1/2))^5/x,x,method=_RETURNVERBOSE)
Output:
1/2/a/(a+b*x^(1/2))^4+2/3/a^2/(a+b*x^(1/2))^3+1/a^3/(a+b*x^(1/2))^2+2/a^4/ (a+b*x^(1/2))-2*ln(a+b*x^(1/2))/a^5+ln(x)/a^5
Leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (75) = 150\).
Time = 0.09 (sec) , antiderivative size = 227, normalized size of antiderivative = 2.55 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^5 x} \, dx=-\frac {6 \, a^{2} b^{6} x^{3} - 21 \, a^{4} b^{4} x^{2} + 16 \, a^{6} b^{2} x - 25 \, a^{8} + 12 \, {\left (b^{8} x^{4} - 4 \, a^{2} b^{6} x^{3} + 6 \, a^{4} b^{4} x^{2} - 4 \, a^{6} b^{2} x + a^{8}\right )} \log \left (b \sqrt {x} + a\right ) - 12 \, {\left (b^{8} x^{4} - 4 \, a^{2} b^{6} x^{3} + 6 \, a^{4} b^{4} x^{2} - 4 \, a^{6} b^{2} x + a^{8}\right )} \log \left (\sqrt {x}\right ) - 4 \, {\left (3 \, a b^{7} x^{3} - 11 \, a^{3} b^{5} x^{2} + 14 \, a^{5} b^{3} x - 12 \, a^{7} b\right )} \sqrt {x}}{6 \, {\left (a^{5} b^{8} x^{4} - 4 \, a^{7} b^{6} x^{3} + 6 \, a^{9} b^{4} x^{2} - 4 \, a^{11} b^{2} x + a^{13}\right )}} \] Input:
integrate(1/(a+b*x^(1/2))^5/x,x, algorithm="fricas")
Output:
-1/6*(6*a^2*b^6*x^3 - 21*a^4*b^4*x^2 + 16*a^6*b^2*x - 25*a^8 + 12*(b^8*x^4 - 4*a^2*b^6*x^3 + 6*a^4*b^4*x^2 - 4*a^6*b^2*x + a^8)*log(b*sqrt(x) + a) - 12*(b^8*x^4 - 4*a^2*b^6*x^3 + 6*a^4*b^4*x^2 - 4*a^6*b^2*x + a^8)*log(sqrt (x)) - 4*(3*a*b^7*x^3 - 11*a^3*b^5*x^2 + 14*a^5*b^3*x - 12*a^7*b)*sqrt(x)) /(a^5*b^8*x^4 - 4*a^7*b^6*x^3 + 6*a^9*b^4*x^2 - 4*a^11*b^2*x + a^13)
Leaf count of result is larger than twice the leaf count of optimal. 1059 vs. \(2 (82) = 164\).
Time = 1.61 (sec) , antiderivative size = 1059, normalized size of antiderivative = 11.90 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^5 x} \, dx =\text {Too large to display} \] Input:
integrate(1/(a+b*x**(1/2))**5/x,x)
Output:
Piecewise((zoo/x**(5/2), Eq(a, 0) & Eq(b, 0)), (log(x)/a**5, Eq(b, 0)), (- 2/(5*b**5*x**(5/2)), Eq(a, 0)), (zoo*log(x), Eq(a, -b*sqrt(x))), (6*a**4*s qrt(x)*log(x)/(6*a**9*sqrt(x) + 24*a**8*b*x + 36*a**7*b**2*x**(3/2) + 24*a **6*b**3*x**2 + 6*a**5*b**4*x**(5/2)) - 12*a**4*sqrt(x)*log(a/b + sqrt(x)) /(6*a**9*sqrt(x) + 24*a**8*b*x + 36*a**7*b**2*x**(3/2) + 24*a**6*b**3*x**2 + 6*a**5*b**4*x**(5/2)) + 25*a**4*sqrt(x)/(6*a**9*sqrt(x) + 24*a**8*b*x + 36*a**7*b**2*x**(3/2) + 24*a**6*b**3*x**2 + 6*a**5*b**4*x**(5/2)) + 24*a* *3*b*x*log(x)/(6*a**9*sqrt(x) + 24*a**8*b*x + 36*a**7*b**2*x**(3/2) + 24*a **6*b**3*x**2 + 6*a**5*b**4*x**(5/2)) - 48*a**3*b*x*log(a/b + sqrt(x))/(6* a**9*sqrt(x) + 24*a**8*b*x + 36*a**7*b**2*x**(3/2) + 24*a**6*b**3*x**2 + 6 *a**5*b**4*x**(5/2)) + 52*a**3*b*x/(6*a**9*sqrt(x) + 24*a**8*b*x + 36*a**7 *b**2*x**(3/2) + 24*a**6*b**3*x**2 + 6*a**5*b**4*x**(5/2)) + 36*a**2*b**2* x**(3/2)*log(x)/(6*a**9*sqrt(x) + 24*a**8*b*x + 36*a**7*b**2*x**(3/2) + 24 *a**6*b**3*x**2 + 6*a**5*b**4*x**(5/2)) - 72*a**2*b**2*x**(3/2)*log(a/b + sqrt(x))/(6*a**9*sqrt(x) + 24*a**8*b*x + 36*a**7*b**2*x**(3/2) + 24*a**6*b **3*x**2 + 6*a**5*b**4*x**(5/2)) + 42*a**2*b**2*x**(3/2)/(6*a**9*sqrt(x) + 24*a**8*b*x + 36*a**7*b**2*x**(3/2) + 24*a**6*b**3*x**2 + 6*a**5*b**4*x** (5/2)) + 24*a*b**3*x**2*log(x)/(6*a**9*sqrt(x) + 24*a**8*b*x + 36*a**7*b** 2*x**(3/2) + 24*a**6*b**3*x**2 + 6*a**5*b**4*x**(5/2)) - 48*a*b**3*x**2*lo g(a/b + sqrt(x))/(6*a**9*sqrt(x) + 24*a**8*b*x + 36*a**7*b**2*x**(3/2) ...
Time = 0.04 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^5 x} \, dx=\frac {12 \, b^{3} x^{\frac {3}{2}} + 42 \, a b^{2} x + 52 \, a^{2} b \sqrt {x} + 25 \, a^{3}}{6 \, {\left (a^{4} b^{4} x^{2} + 4 \, a^{5} b^{3} x^{\frac {3}{2}} + 6 \, a^{6} b^{2} x + 4 \, a^{7} b \sqrt {x} + a^{8}\right )}} - \frac {2 \, \log \left (b \sqrt {x} + a\right )}{a^{5}} + \frac {\log \left (x\right )}{a^{5}} \] Input:
integrate(1/(a+b*x^(1/2))^5/x,x, algorithm="maxima")
Output:
1/6*(12*b^3*x^(3/2) + 42*a*b^2*x + 52*a^2*b*sqrt(x) + 25*a^3)/(a^4*b^4*x^2 + 4*a^5*b^3*x^(3/2) + 6*a^6*b^2*x + 4*a^7*b*sqrt(x) + a^8) - 2*log(b*sqrt (x) + a)/a^5 + log(x)/a^5
Time = 0.13 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^5 x} \, dx=-\frac {2 \, \log \left ({\left | b \sqrt {x} + a \right |}\right )}{a^{5}} + \frac {\log \left ({\left | x \right |}\right )}{a^{5}} + \frac {12 \, a b^{3} x^{\frac {3}{2}} + 42 \, a^{2} b^{2} x + 52 \, a^{3} b \sqrt {x} + 25 \, a^{4}}{6 \, {\left (b \sqrt {x} + a\right )}^{4} a^{5}} \] Input:
integrate(1/(a+b*x^(1/2))^5/x,x, algorithm="giac")
Output:
-2*log(abs(b*sqrt(x) + a))/a^5 + log(abs(x))/a^5 + 1/6*(12*a*b^3*x^(3/2) + 42*a^2*b^2*x + 52*a^3*b*sqrt(x) + 25*a^4)/((b*sqrt(x) + a)^4*a^5)
Time = 0.09 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^5 x} \, dx=\frac {\frac {25}{6\,a}+\frac {26\,b\,\sqrt {x}}{3\,a^2}+\frac {7\,b^2\,x}{a^3}+\frac {2\,b^3\,x^{3/2}}{a^4}}{a^4+b^4\,x^2+6\,a^2\,b^2\,x+4\,a^3\,b\,\sqrt {x}+4\,a\,b^3\,x^{3/2}}-\frac {4\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a^5} \] Input:
int(1/(x*(a + b*x^(1/2))^5),x)
Output:
(25/(6*a) + (26*b*x^(1/2))/(3*a^2) + (7*b^2*x)/a^3 + (2*b^3*x^(3/2))/a^4)/ (a^4 + b^4*x^2 + 6*a^2*b^2*x + 4*a^3*b*x^(1/2) + 4*a*b^3*x^(3/2)) - (4*ata nh((2*b*x^(1/2))/a + 1))/a^5
Time = 0.21 (sec) , antiderivative size = 203, normalized size of antiderivative = 2.28 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^5 x} \, dx=\frac {-48 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{3} b -48 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, b +a \right ) a \,b^{3} x +48 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\right ) a^{3} b +48 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\right ) a \,b^{3} x +40 \sqrt {x}\, a^{3} b -12 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{4}-72 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{2} b^{2} x -12 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) b^{4} x^{2}+12 \,\mathrm {log}\left (\sqrt {x}\right ) a^{4}+72 \,\mathrm {log}\left (\sqrt {x}\right ) a^{2} b^{2} x +12 \,\mathrm {log}\left (\sqrt {x}\right ) b^{4} x^{2}+22 a^{4}+24 a^{2} b^{2} x -3 b^{4} x^{2}}{6 a^{5} \left (4 \sqrt {x}\, a^{3} b +4 \sqrt {x}\, a \,b^{3} x +a^{4}+6 a^{2} b^{2} x +b^{4} x^{2}\right )} \] Input:
int(1/(a+b*x^(1/2))^5/x,x)
Output:
( - 48*sqrt(x)*log(sqrt(x)*b + a)*a**3*b - 48*sqrt(x)*log(sqrt(x)*b + a)*a *b**3*x + 48*sqrt(x)*log(sqrt(x))*a**3*b + 48*sqrt(x)*log(sqrt(x))*a*b**3* x + 40*sqrt(x)*a**3*b - 12*log(sqrt(x)*b + a)*a**4 - 72*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**4 + 72 *log(sqrt(x))*a**2*b**2*x + 12*log(sqrt(x))*b**4*x**2 + 22*a**4 + 24*a**2* b**2*x - 3*b**4*x**2)/(6*a**5*(4*sqrt(x)*a**3*b + 4*sqrt(x)*a*b**3*x + a** 4 + 6*a**2*b**2*x + b**4*x**2))