Integrand size = 15, antiderivative size = 126 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^5 x^2} \, dx=\frac {b^2}{2 a^3 \left (a+b \sqrt {x}\right )^4}+\frac {2 b^2}{a^4 \left (a+b \sqrt {x}\right )^3}+\frac {6 b^2}{a^5 \left (a+b \sqrt {x}\right )^2}+\frac {20 b^2}{a^6 \left (a+b \sqrt {x}\right )}-\frac {1}{a^5 x}+\frac {10 b}{a^6 \sqrt {x}}-\frac {30 b^2 \log \left (a+b \sqrt {x}\right )}{a^7}+\frac {15 b^2 \log (x)}{a^7} \] Output:
1/2*b^2/a^3/(a+b*x^(1/2))^4+2*b^2/a^4/(a+b*x^(1/2))^3+6*b^2/a^5/(a+b*x^(1/ 2))^2+20*b^2/a^6/(a+b*x^(1/2))-1/a^5/x+10*b/a^6/x^(1/2)-30*b^2*ln(a+b*x^(1 /2))/a^7+15*b^2*ln(x)/a^7
Time = 0.13 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^5 x^2} \, dx=\frac {\frac {a \left (-2 a^5+12 a^4 b \sqrt {x}+125 a^3 b^2 x+260 a^2 b^3 x^{3/2}+210 a b^4 x^2+60 b^5 x^{5/2}\right )}{\left (a+b \sqrt {x}\right )^4 x}-60 b^2 \log \left (a+b \sqrt {x}\right )+30 b^2 \log (x)}{2 a^7} \] Input:
Integrate[1/((a + b*Sqrt[x])^5*x^2),x]
Output:
((a*(-2*a^5 + 12*a^4*b*Sqrt[x] + 125*a^3*b^2*x + 260*a^2*b^3*x^(3/2) + 210 *a*b^4*x^2 + 60*b^5*x^(5/2)))/((a + b*Sqrt[x])^4*x) - 60*b^2*Log[a + b*Sqr t[x]] + 30*b^2*Log[x])/(2*a^7)
Time = 0.47 (sec) , antiderivative size = 133, 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.
\(\displaystyle \int \frac {1}{x^2 \left (a+b \sqrt {x}\right )^5} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 2 \int \frac {1}{\left (a+b \sqrt {x}\right )^5 x^{3/2}}d\sqrt {x}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle 2 \int \left (-\frac {15 b^3}{a^7 \left (a+b \sqrt {x}\right )}-\frac {10 b^3}{a^6 \left (a+b \sqrt {x}\right )^2}-\frac {6 b^3}{a^5 \left (a+b \sqrt {x}\right )^3}-\frac {3 b^3}{a^4 \left (a+b \sqrt {x}\right )^4}-\frac {b^3}{a^3 \left (a+b \sqrt {x}\right )^5}+\frac {15 b^2}{a^7 \sqrt {x}}-\frac {5 b}{a^6 x}+\frac {1}{a^5 x^{3/2}}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (-\frac {15 b^2 \log \left (a+b \sqrt {x}\right )}{a^7}+\frac {15 b^2 \log \left (\sqrt {x}\right )}{a^7}+\frac {10 b^2}{a^6 \left (a+b \sqrt {x}\right )}+\frac {5 b}{a^6 \sqrt {x}}+\frac {3 b^2}{a^5 \left (a+b \sqrt {x}\right )^2}-\frac {1}{2 a^5 x}+\frac {b^2}{a^4 \left (a+b \sqrt {x}\right )^3}+\frac {b^2}{4 a^3 \left (a+b \sqrt {x}\right )^4}\right )\) |
Input:
Int[1/((a + b*Sqrt[x])^5*x^2),x]
Output:
2*(b^2/(4*a^3*(a + b*Sqrt[x])^4) + b^2/(a^4*(a + b*Sqrt[x])^3) + (3*b^2)/( a^5*(a + b*Sqrt[x])^2) + (10*b^2)/(a^6*(a + b*Sqrt[x])) - 1/(2*a^5*x) + (5 *b)/(a^6*Sqrt[x]) - (15*b^2*Log[a + b*Sqrt[x]])/a^7 + (15*b^2*Log[Sqrt[x]] )/a^7)
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 = 113, normalized size of antiderivative = 0.90
method | result | size |
derivativedivides | \(\frac {b^{2}}{2 a^{3} \left (a +b \sqrt {x}\right )^{4}}+\frac {2 b^{2}}{a^{4} \left (a +b \sqrt {x}\right )^{3}}+\frac {6 b^{2}}{a^{5} \left (a +b \sqrt {x}\right )^{2}}+\frac {20 b^{2}}{a^{6} \left (a +b \sqrt {x}\right )}-\frac {1}{a^{5} x}+\frac {10 b}{a^{6} \sqrt {x}}-\frac {30 b^{2} \ln \left (a +b \sqrt {x}\right )}{a^{7}}+\frac {15 b^{2} \ln \left (x \right )}{a^{7}}\) | \(113\) |
default | \(\frac {b^{2}}{2 a^{3} \left (a +b \sqrt {x}\right )^{4}}+\frac {2 b^{2}}{a^{4} \left (a +b \sqrt {x}\right )^{3}}+\frac {6 b^{2}}{a^{5} \left (a +b \sqrt {x}\right )^{2}}+\frac {20 b^{2}}{a^{6} \left (a +b \sqrt {x}\right )}-\frac {1}{a^{5} x}+\frac {10 b}{a^{6} \sqrt {x}}-\frac {30 b^{2} \ln \left (a +b \sqrt {x}\right )}{a^{7}}+\frac {15 b^{2} \ln \left (x \right )}{a^{7}}\) | \(113\) |
Input:
int(1/(a+b*x^(1/2))^5/x^2,x,method=_RETURNVERBOSE)
Output:
1/2*b^2/a^3/(a+b*x^(1/2))^4+2*b^2/a^4/(a+b*x^(1/2))^3+6*b^2/a^5/(a+b*x^(1/ 2))^2+20*b^2/a^6/(a+b*x^(1/2))-1/a^5/x+10*b/a^6/x^(1/2)-30*b^2*ln(a+b*x^(1 /2))/a^7+15*b^2*ln(x)/a^7
Leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (112) = 224\).
Time = 0.09 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.12 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^5 x^2} \, dx=-\frac {30 \, a^{2} b^{8} x^{4} - 105 \, a^{4} b^{6} x^{3} + 130 \, a^{6} b^{4} x^{2} - 65 \, a^{8} b^{2} x + 2 \, a^{10} + 60 \, {\left (b^{10} x^{5} - 4 \, a^{2} b^{8} x^{4} + 6 \, a^{4} b^{6} x^{3} - 4 \, a^{6} b^{4} x^{2} + a^{8} b^{2} x\right )} \log \left (b \sqrt {x} + a\right ) - 60 \, {\left (b^{10} x^{5} - 4 \, a^{2} b^{8} x^{4} + 6 \, a^{4} b^{6} x^{3} - 4 \, a^{6} b^{4} x^{2} + a^{8} b^{2} x\right )} \log \left (\sqrt {x}\right ) - 4 \, {\left (15 \, a b^{9} x^{4} - 55 \, a^{3} b^{7} x^{3} + 73 \, a^{5} b^{5} x^{2} - 40 \, a^{7} b^{3} x + 5 \, a^{9} b\right )} \sqrt {x}}{2 \, {\left (a^{7} b^{8} x^{5} - 4 \, a^{9} b^{6} x^{4} + 6 \, a^{11} b^{4} x^{3} - 4 \, a^{13} b^{2} x^{2} + a^{15} x\right )}} \] Input:
integrate(1/(a+b*x^(1/2))^5/x^2,x, algorithm="fricas")
Output:
-1/2*(30*a^2*b^8*x^4 - 105*a^4*b^6*x^3 + 130*a^6*b^4*x^2 - 65*a^8*b^2*x + 2*a^10 + 60*(b^10*x^5 - 4*a^2*b^8*x^4 + 6*a^4*b^6*x^3 - 4*a^6*b^4*x^2 + a^ 8*b^2*x)*log(b*sqrt(x) + a) - 60*(b^10*x^5 - 4*a^2*b^8*x^4 + 6*a^4*b^6*x^3 - 4*a^6*b^4*x^2 + a^8*b^2*x)*log(sqrt(x)) - 4*(15*a*b^9*x^4 - 55*a^3*b^7* x^3 + 73*a^5*b^5*x^2 - 40*a^7*b^3*x + 5*a^9*b)*sqrt(x))/(a^7*b^8*x^5 - 4*a ^9*b^6*x^4 + 6*a^11*b^4*x^3 - 4*a^13*b^2*x^2 + a^15*x)
Leaf count of result is larger than twice the leaf count of optimal. 1241 vs. \(2 (121) = 242\).
Time = 2.70 (sec) , antiderivative size = 1241, normalized size of antiderivative = 9.85 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^5 x^2} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+b*x**(1/2))**5/x**2,x)
Output:
Piecewise((zoo/x**(7/2), Eq(a, 0) & Eq(b, 0)), (-1/(a**5*x), Eq(b, 0)), (- 2/(7*b**5*x**(7/2)), Eq(a, 0)), (zoo/x, Eq(a, -b*sqrt(x))), (-2*a**6*sqrt( x)/(2*a**11*x**(3/2) + 8*a**10*b*x**2 + 12*a**9*b**2*x**(5/2) + 8*a**8*b** 3*x**3 + 2*a**7*b**4*x**(7/2)) + 12*a**5*b*x/(2*a**11*x**(3/2) + 8*a**10*b *x**2 + 12*a**9*b**2*x**(5/2) + 8*a**8*b**3*x**3 + 2*a**7*b**4*x**(7/2)) + 30*a**4*b**2*x**(3/2)*log(x)/(2*a**11*x**(3/2) + 8*a**10*b*x**2 + 12*a**9 *b**2*x**(5/2) + 8*a**8*b**3*x**3 + 2*a**7*b**4*x**(7/2)) - 60*a**4*b**2*x **(3/2)*log(a/b + sqrt(x))/(2*a**11*x**(3/2) + 8*a**10*b*x**2 + 12*a**9*b* *2*x**(5/2) + 8*a**8*b**3*x**3 + 2*a**7*b**4*x**(7/2)) + 125*a**4*b**2*x** (3/2)/(2*a**11*x**(3/2) + 8*a**10*b*x**2 + 12*a**9*b**2*x**(5/2) + 8*a**8* b**3*x**3 + 2*a**7*b**4*x**(7/2)) + 120*a**3*b**3*x**2*log(x)/(2*a**11*x** (3/2) + 8*a**10*b*x**2 + 12*a**9*b**2*x**(5/2) + 8*a**8*b**3*x**3 + 2*a**7 *b**4*x**(7/2)) - 240*a**3*b**3*x**2*log(a/b + sqrt(x))/(2*a**11*x**(3/2) + 8*a**10*b*x**2 + 12*a**9*b**2*x**(5/2) + 8*a**8*b**3*x**3 + 2*a**7*b**4* x**(7/2)) + 260*a**3*b**3*x**2/(2*a**11*x**(3/2) + 8*a**10*b*x**2 + 12*a** 9*b**2*x**(5/2) + 8*a**8*b**3*x**3 + 2*a**7*b**4*x**(7/2)) + 180*a**2*b**4 *x**(5/2)*log(x)/(2*a**11*x**(3/2) + 8*a**10*b*x**2 + 12*a**9*b**2*x**(5/2 ) + 8*a**8*b**3*x**3 + 2*a**7*b**4*x**(7/2)) - 360*a**2*b**4*x**(5/2)*log( a/b + sqrt(x))/(2*a**11*x**(3/2) + 8*a**10*b*x**2 + 12*a**9*b**2*x**(5/2) + 8*a**8*b**3*x**3 + 2*a**7*b**4*x**(7/2)) + 210*a**2*b**4*x**(5/2)/(2*...
Time = 0.05 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.03 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^5 x^2} \, dx=\frac {60 \, b^{5} x^{\frac {5}{2}} + 210 \, a b^{4} x^{2} + 260 \, a^{2} b^{3} x^{\frac {3}{2}} + 125 \, a^{3} b^{2} x + 12 \, a^{4} b \sqrt {x} - 2 \, a^{5}}{2 \, {\left (a^{6} b^{4} x^{3} + 4 \, a^{7} b^{3} x^{\frac {5}{2}} + 6 \, a^{8} b^{2} x^{2} + 4 \, a^{9} b x^{\frac {3}{2}} + a^{10} x\right )}} - \frac {30 \, b^{2} \log \left (b \sqrt {x} + a\right )}{a^{7}} + \frac {15 \, b^{2} \log \left (x\right )}{a^{7}} \] Input:
integrate(1/(a+b*x^(1/2))^5/x^2,x, algorithm="maxima")
Output:
1/2*(60*b^5*x^(5/2) + 210*a*b^4*x^2 + 260*a^2*b^3*x^(3/2) + 125*a^3*b^2*x + 12*a^4*b*sqrt(x) - 2*a^5)/(a^6*b^4*x^3 + 4*a^7*b^3*x^(5/2) + 6*a^8*b^2*x ^2 + 4*a^9*b*x^(3/2) + a^10*x) - 30*b^2*log(b*sqrt(x) + a)/a^7 + 15*b^2*lo g(x)/a^7
Time = 0.13 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^5 x^2} \, dx=-\frac {30 \, b^{2} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{a^{7}} + \frac {15 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{7}} + \frac {60 \, a b^{5} x^{\frac {5}{2}} + 210 \, a^{2} b^{4} x^{2} + 260 \, a^{3} b^{3} x^{\frac {3}{2}} + 125 \, a^{4} b^{2} x + 12 \, a^{5} b \sqrt {x} - 2 \, a^{6}}{2 \, {\left (b \sqrt {x} + a\right )}^{4} a^{7} x} \] Input:
integrate(1/(a+b*x^(1/2))^5/x^2,x, algorithm="giac")
Output:
-30*b^2*log(abs(b*sqrt(x) + a))/a^7 + 15*b^2*log(abs(x))/a^7 + 1/2*(60*a*b ^5*x^(5/2) + 210*a^2*b^4*x^2 + 260*a^3*b^3*x^(3/2) + 125*a^4*b^2*x + 12*a^ 5*b*sqrt(x) - 2*a^6)/((b*sqrt(x) + a)^4*a^7*x)
Time = 0.15 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^5 x^2} \, dx=\frac {\frac {6\,b\,\sqrt {x}}{a^2}-\frac {1}{a}+\frac {125\,b^2\,x}{2\,a^3}+\frac {105\,b^4\,x^2}{a^5}+\frac {130\,b^3\,x^{3/2}}{a^4}+\frac {30\,b^5\,x^{5/2}}{a^6}}{a^4\,x+b^4\,x^3+4\,a^3\,b\,x^{3/2}+4\,a\,b^3\,x^{5/2}+6\,a^2\,b^2\,x^2}-\frac {60\,b^2\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a^7} \] Input:
int(1/(x^2*(a + b*x^(1/2))^5),x)
Output:
((6*b*x^(1/2))/a^2 - 1/a + (125*b^2*x)/(2*a^3) + (105*b^4*x^2)/a^5 + (130* b^3*x^(3/2))/a^4 + (30*b^5*x^(5/2))/a^6)/(a^4*x + b^4*x^3 + 4*a^3*b*x^(3/2 ) + 4*a*b^3*x^(5/2) + 6*a^2*b^2*x^2) - (60*b^2*atanh((2*b*x^(1/2))/a + 1)) /a^7
Time = 0.23 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.98 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^5 x^2} \, dx=\frac {-240 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{3} b^{3} x -240 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, b +a \right ) a \,b^{5} x^{2}+240 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\right ) a^{3} b^{3} x +240 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\right ) a \,b^{5} x^{2}+12 \sqrt {x}\, a^{5} b +200 \sqrt {x}\, a^{3} b^{3} x -60 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{4} b^{2} x -360 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{2} b^{4} x^{2}-60 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) b^{6} x^{3}+60 \,\mathrm {log}\left (\sqrt {x}\right ) a^{4} b^{2} x +360 \,\mathrm {log}\left (\sqrt {x}\right ) a^{2} b^{4} x^{2}+60 \,\mathrm {log}\left (\sqrt {x}\right ) b^{6} x^{3}-2 a^{6}+110 a^{4} b^{2} x +120 a^{2} b^{4} x^{2}-15 b^{6} x^{3}}{2 a^{7} x \left (4 \sqrt {x}\, a^{3} b +4 \sqrt {x}\, a \,b^{3} x +a^{4}+6 a^{2} b^{2} x +b^{4} x^{2}\right )} \] Input:
int(1/(a+b*x^(1/2))^5/x^2,x)
Output:
( - 240*sqrt(x)*log(sqrt(x)*b + a)*a**3*b**3*x - 240*sqrt(x)*log(sqrt(x)*b + a)*a*b**5*x**2 + 240*sqrt(x)*log(sqrt(x))*a**3*b**3*x + 240*sqrt(x)*log (sqrt(x))*a*b**5*x**2 + 12*sqrt(x)*a**5*b + 200*sqrt(x)*a**3*b**3*x - 60*l og(sqrt(x)*b + a)*a**4*b**2*x - 360*log(sqrt(x)*b + a)*a**2*b**4*x**2 - 60 *log(sqrt(x)*b + a)*b**6*x**3 + 60*log(sqrt(x))*a**4*b**2*x + 360*log(sqrt (x))*a**2*b**4*x**2 + 60*log(sqrt(x))*b**6*x**3 - 2*a**6 + 110*a**4*b**2*x + 120*a**2*b**4*x**2 - 15*b**6*x**3)/(2*a**7*x*(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))