Integrand size = 15, antiderivative size = 139 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^4} \, dx=\frac {b^6}{a^7 \left (a+b \sqrt {x}\right )^2}+\frac {14 b^6}{a^8 \left (a+b \sqrt {x}\right )}-\frac {1}{3 a^3 x^3}+\frac {6 b}{5 a^4 x^{5/2}}-\frac {3 b^2}{a^5 x^2}+\frac {20 b^3}{3 a^6 x^{3/2}}-\frac {15 b^4}{a^7 x}+\frac {42 b^5}{a^8 \sqrt {x}}-\frac {56 b^6 \log \left (a+b \sqrt {x}\right )}{a^9}+\frac {28 b^6 \log (x)}{a^9} \] Output:
b^6/a^7/(a+b*x^(1/2))^2+14*b^6/a^8/(a+b*x^(1/2))-1/3/a^3/x^3+6/5*b/a^4/x^( 5/2)-3*b^2/a^5/x^2+20/3*b^3/a^6/x^(3/2)-15*b^4/a^7/x+42*b^5/a^8/x^(1/2)-56 *b^6*ln(a+b*x^(1/2))/a^9+28*b^6*ln(x)/a^9
Time = 0.13 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^4} \, dx=\frac {\frac {a \left (-5 a^7+8 a^6 b \sqrt {x}-14 a^5 b^2 x+28 a^4 b^3 x^{3/2}-70 a^3 b^4 x^2+280 a^2 b^5 x^{5/2}+1260 a b^6 x^3+840 b^7 x^{7/2}\right )}{\left (a+b \sqrt {x}\right )^2 x^3}-840 b^6 \log \left (a+b \sqrt {x}\right )+420 b^6 \log (x)}{15 a^9} \] Input:
Integrate[1/((a + b*Sqrt[x])^3*x^4),x]
Output:
((a*(-5*a^7 + 8*a^6*b*Sqrt[x] - 14*a^5*b^2*x + 28*a^4*b^3*x^(3/2) - 70*a^3 *b^4*x^2 + 280*a^2*b^5*x^(5/2) + 1260*a*b^6*x^3 + 840*b^7*x^(7/2)))/((a + b*Sqrt[x])^2*x^3) - 840*b^6*Log[a + b*Sqrt[x]] + 420*b^6*Log[x])/(15*a^9)
Time = 0.49 (sec) , antiderivative size = 152, 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.
\(\displaystyle \int \frac {1}{x^4 \left (a+b \sqrt {x}\right )^3} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 2 \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^{7/2}}d\sqrt {x}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle 2 \int \left (-\frac {28 b^7}{a^9 \left (a+b \sqrt {x}\right )}-\frac {7 b^7}{a^8 \left (a+b \sqrt {x}\right )^2}-\frac {b^7}{a^7 \left (a+b \sqrt {x}\right )^3}+\frac {28 b^6}{a^9 \sqrt {x}}-\frac {21 b^5}{a^8 x}+\frac {15 b^4}{a^7 x^{3/2}}-\frac {10 b^3}{a^6 x^2}+\frac {6 b^2}{a^5 x^{5/2}}-\frac {3 b}{a^4 x^3}+\frac {1}{a^3 x^{7/2}}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (-\frac {28 b^6 \log \left (a+b \sqrt {x}\right )}{a^9}+\frac {28 b^6 \log \left (\sqrt {x}\right )}{a^9}+\frac {7 b^6}{a^8 \left (a+b \sqrt {x}\right )}+\frac {21 b^5}{a^8 \sqrt {x}}+\frac {b^6}{2 a^7 \left (a+b \sqrt {x}\right )^2}-\frac {15 b^4}{2 a^7 x}+\frac {10 b^3}{3 a^6 x^{3/2}}-\frac {3 b^2}{2 a^5 x^2}+\frac {3 b}{5 a^4 x^{5/2}}-\frac {1}{6 a^3 x^3}\right )\) |
Input:
Int[1/((a + b*Sqrt[x])^3*x^4),x]
Output:
2*(b^6/(2*a^7*(a + b*Sqrt[x])^2) + (7*b^6)/(a^8*(a + b*Sqrt[x])) - 1/(6*a^ 3*x^3) + (3*b)/(5*a^4*x^(5/2)) - (3*b^2)/(2*a^5*x^2) + (10*b^3)/(3*a^6*x^( 3/2)) - (15*b^4)/(2*a^7*x) + (21*b^5)/(a^8*Sqrt[x]) - (28*b^6*Log[a + b*Sq rt[x]])/a^9 + (28*b^6*Log[Sqrt[x]])/a^9)
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 = 122, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {b^{6}}{a^{7} \left (a +b \sqrt {x}\right )^{2}}+\frac {14 b^{6}}{a^{8} \left (a +b \sqrt {x}\right )}-\frac {1}{3 a^{3} x^{3}}+\frac {6 b}{5 a^{4} x^{\frac {5}{2}}}-\frac {3 b^{2}}{a^{5} x^{2}}+\frac {20 b^{3}}{3 a^{6} x^{\frac {3}{2}}}-\frac {15 b^{4}}{a^{7} x}+\frac {42 b^{5}}{a^{8} \sqrt {x}}-\frac {56 b^{6} \ln \left (a +b \sqrt {x}\right )}{a^{9}}+\frac {28 b^{6} \ln \left (x \right )}{a^{9}}\) | \(122\) |
default | \(\frac {b^{6}}{a^{7} \left (a +b \sqrt {x}\right )^{2}}+\frac {14 b^{6}}{a^{8} \left (a +b \sqrt {x}\right )}-\frac {1}{3 a^{3} x^{3}}+\frac {6 b}{5 a^{4} x^{\frac {5}{2}}}-\frac {3 b^{2}}{a^{5} x^{2}}+\frac {20 b^{3}}{3 a^{6} x^{\frac {3}{2}}}-\frac {15 b^{4}}{a^{7} x}+\frac {42 b^{5}}{a^{8} \sqrt {x}}-\frac {56 b^{6} \ln \left (a +b \sqrt {x}\right )}{a^{9}}+\frac {28 b^{6} \ln \left (x \right )}{a^{9}}\) | \(122\) |
Input:
int(1/(a+b*x^(1/2))^3/x^4,x,method=_RETURNVERBOSE)
Output:
b^6/a^7/(a+b*x^(1/2))^2+14*b^6/a^8/(a+b*x^(1/2))-1/3/a^3/x^3+6/5*b/a^4/x^( 5/2)-3*b^2/a^5/x^2+20/3*b^3/a^6/x^(3/2)-15*b^4/a^7/x+42*b^5/a^8/x^(1/2)-56 *b^6*ln(a+b*x^(1/2))/a^9+28*b^6*ln(x)/a^9
Time = 0.09 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.49 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^4} \, dx=-\frac {420 \, a^{2} b^{8} x^{4} - 630 \, a^{4} b^{6} x^{3} + 140 \, a^{6} b^{4} x^{2} + 35 \, a^{8} b^{2} x + 5 \, a^{10} + 840 \, {\left (b^{10} x^{5} - 2 \, a^{2} b^{8} x^{4} + a^{4} b^{6} x^{3}\right )} \log \left (b \sqrt {x} + a\right ) - 840 \, {\left (b^{10} x^{5} - 2 \, a^{2} b^{8} x^{4} + a^{4} b^{6} x^{3}\right )} \log \left (\sqrt {x}\right ) - 2 \, {\left (420 \, a b^{9} x^{4} - 700 \, a^{3} b^{7} x^{3} + 224 \, a^{5} b^{5} x^{2} + 32 \, a^{7} b^{3} x + 9 \, a^{9} b\right )} \sqrt {x}}{15 \, {\left (a^{9} b^{4} x^{5} - 2 \, a^{11} b^{2} x^{4} + a^{13} x^{3}\right )}} \] Input:
integrate(1/(a+b*x^(1/2))^3/x^4,x, algorithm="fricas")
Output:
-1/15*(420*a^2*b^8*x^4 - 630*a^4*b^6*x^3 + 140*a^6*b^4*x^2 + 35*a^8*b^2*x + 5*a^10 + 840*(b^10*x^5 - 2*a^2*b^8*x^4 + a^4*b^6*x^3)*log(b*sqrt(x) + a) - 840*(b^10*x^5 - 2*a^2*b^8*x^4 + a^4*b^6*x^3)*log(sqrt(x)) - 2*(420*a*b^ 9*x^4 - 700*a^3*b^7*x^3 + 224*a^5*b^5*x^2 + 32*a^7*b^3*x + 9*a^9*b)*sqrt(x ))/(a^9*b^4*x^5 - 2*a^11*b^2*x^4 + a^13*x^3)
Leaf count of result is larger than twice the leaf count of optimal. 707 vs. \(2 (138) = 276\).
Time = 2.98 (sec) , antiderivative size = 707, normalized size of antiderivative = 5.09 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^4} \, dx =\text {Too large to display} \] Input:
integrate(1/(a+b*x**(1/2))**3/x**4,x)
Output:
Piecewise((zoo/x**(9/2), Eq(a, 0) & Eq(b, 0)), (-1/(3*a**3*x**3), Eq(b, 0) ), (-2/(9*b**3*x**(9/2)), Eq(a, 0)), (-5*a**8*sqrt(x)/(15*a**11*x**(7/2) + 30*a**10*b*x**4 + 15*a**9*b**2*x**(9/2)) + 8*a**7*b*x/(15*a**11*x**(7/2) + 30*a**10*b*x**4 + 15*a**9*b**2*x**(9/2)) - 14*a**6*b**2*x**(3/2)/(15*a** 11*x**(7/2) + 30*a**10*b*x**4 + 15*a**9*b**2*x**(9/2)) + 28*a**5*b**3*x**2 /(15*a**11*x**(7/2) + 30*a**10*b*x**4 + 15*a**9*b**2*x**(9/2)) - 70*a**4*b **4*x**(5/2)/(15*a**11*x**(7/2) + 30*a**10*b*x**4 + 15*a**9*b**2*x**(9/2)) + 280*a**3*b**5*x**3/(15*a**11*x**(7/2) + 30*a**10*b*x**4 + 15*a**9*b**2* x**(9/2)) + 420*a**2*b**6*x**(7/2)*log(x)/(15*a**11*x**(7/2) + 30*a**10*b* x**4 + 15*a**9*b**2*x**(9/2)) - 840*a**2*b**6*x**(7/2)*log(a/b + sqrt(x))/ (15*a**11*x**(7/2) + 30*a**10*b*x**4 + 15*a**9*b**2*x**(9/2)) + 1260*a**2* b**6*x**(7/2)/(15*a**11*x**(7/2) + 30*a**10*b*x**4 + 15*a**9*b**2*x**(9/2) ) + 840*a*b**7*x**4*log(x)/(15*a**11*x**(7/2) + 30*a**10*b*x**4 + 15*a**9* b**2*x**(9/2)) - 1680*a*b**7*x**4*log(a/b + sqrt(x))/(15*a**11*x**(7/2) + 30*a**10*b*x**4 + 15*a**9*b**2*x**(9/2)) + 840*a*b**7*x**4/(15*a**11*x**(7 /2) + 30*a**10*b*x**4 + 15*a**9*b**2*x**(9/2)) + 420*b**8*x**(9/2)*log(x)/ (15*a**11*x**(7/2) + 30*a**10*b*x**4 + 15*a**9*b**2*x**(9/2)) - 840*b**8*x **(9/2)*log(a/b + sqrt(x))/(15*a**11*x**(7/2) + 30*a**10*b*x**4 + 15*a**9* b**2*x**(9/2)), True))
Time = 0.03 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^4} \, dx=\frac {840 \, b^{7} x^{\frac {7}{2}} + 1260 \, a b^{6} x^{3} + 280 \, a^{2} b^{5} x^{\frac {5}{2}} - 70 \, a^{3} b^{4} x^{2} + 28 \, a^{4} b^{3} x^{\frac {3}{2}} - 14 \, a^{5} b^{2} x + 8 \, a^{6} b \sqrt {x} - 5 \, a^{7}}{15 \, {\left (a^{8} b^{2} x^{4} + 2 \, a^{9} b x^{\frac {7}{2}} + a^{10} x^{3}\right )}} - \frac {56 \, b^{6} \log \left (b \sqrt {x} + a\right )}{a^{9}} + \frac {28 \, b^{6} \log \left (x\right )}{a^{9}} \] Input:
integrate(1/(a+b*x^(1/2))^3/x^4,x, algorithm="maxima")
Output:
1/15*(840*b^7*x^(7/2) + 1260*a*b^6*x^3 + 280*a^2*b^5*x^(5/2) - 70*a^3*b^4* x^2 + 28*a^4*b^3*x^(3/2) - 14*a^5*b^2*x + 8*a^6*b*sqrt(x) - 5*a^7)/(a^8*b^ 2*x^4 + 2*a^9*b*x^(7/2) + a^10*x^3) - 56*b^6*log(b*sqrt(x) + a)/a^9 + 28*b ^6*log(x)/a^9
Time = 0.12 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^4} \, dx=-\frac {56 \, b^{6} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{a^{9}} + \frac {28 \, b^{6} \log \left ({\left | x \right |}\right )}{a^{9}} + \frac {840 \, a b^{7} x^{\frac {7}{2}} + 1260 \, a^{2} b^{6} x^{3} + 280 \, a^{3} b^{5} x^{\frac {5}{2}} - 70 \, a^{4} b^{4} x^{2} + 28 \, a^{5} b^{3} x^{\frac {3}{2}} - 14 \, a^{6} b^{2} x + 8 \, a^{7} b \sqrt {x} - 5 \, a^{8}}{15 \, {\left (b \sqrt {x} + a\right )}^{2} a^{9} x^{3}} \] Input:
integrate(1/(a+b*x^(1/2))^3/x^4,x, algorithm="giac")
Output:
-56*b^6*log(abs(b*sqrt(x) + a))/a^9 + 28*b^6*log(abs(x))/a^9 + 1/15*(840*a *b^7*x^(7/2) + 1260*a^2*b^6*x^3 + 280*a^3*b^5*x^(5/2) - 70*a^4*b^4*x^2 + 2 8*a^5*b^3*x^(3/2) - 14*a^6*b^2*x + 8*a^7*b*sqrt(x) - 5*a^8)/((b*sqrt(x) + a)^2*a^9*x^3)
Time = 0.42 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^4} \, dx=\frac {\frac {8\,b\,\sqrt {x}}{15\,a^2}-\frac {1}{3\,a}-\frac {14\,b^2\,x}{15\,a^3}-\frac {14\,b^4\,x^2}{3\,a^5}+\frac {28\,b^3\,x^{3/2}}{15\,a^4}+\frac {84\,b^6\,x^3}{a^7}+\frac {56\,b^5\,x^{5/2}}{3\,a^6}+\frac {56\,b^7\,x^{7/2}}{a^8}}{a^2\,x^3+b^2\,x^4+2\,a\,b\,x^{7/2}}-\frac {112\,b^6\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a^9} \] Input:
int(1/(x^4*(a + b*x^(1/2))^3),x)
Output:
((8*b*x^(1/2))/(15*a^2) - 1/(3*a) - (14*b^2*x)/(15*a^3) - (14*b^4*x^2)/(3* a^5) + (28*b^3*x^(3/2))/(15*a^4) + (84*b^6*x^3)/a^7 + (56*b^5*x^(5/2))/(3* a^6) + (56*b^7*x^(7/2))/a^8)/(a^2*x^3 + b^2*x^4 + 2*a*b*x^(7/2)) - (112*b^ 6*atanh((2*b*x^(1/2))/a + 1))/a^9
Time = 0.22 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.38 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^4} \, dx=\frac {-1680 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, b +a \right ) a \,b^{7} x^{3}+1680 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\right ) a \,b^{7} x^{3}+8 \sqrt {x}\, a^{7} b +28 \sqrt {x}\, a^{5} b^{3} x +280 \sqrt {x}\, a^{3} b^{5} x^{2}-840 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{2} b^{6} x^{3}-840 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) b^{8} x^{4}+840 \,\mathrm {log}\left (\sqrt {x}\right ) a^{2} b^{6} x^{3}+840 \,\mathrm {log}\left (\sqrt {x}\right ) b^{8} x^{4}-5 a^{8}-14 a^{6} b^{2} x -70 a^{4} b^{4} x^{2}+840 a^{2} b^{6} x^{3}-420 b^{8} x^{4}}{15 a^{9} x^{3} \left (2 \sqrt {x}\, a b +a^{2}+b^{2} x \right )} \] Input:
int(1/(a+b*x^(1/2))^3/x^4,x)
Output:
( - 1680*sqrt(x)*log(sqrt(x)*b + a)*a*b**7*x**3 + 1680*sqrt(x)*log(sqrt(x) )*a*b**7*x**3 + 8*sqrt(x)*a**7*b + 28*sqrt(x)*a**5*b**3*x + 280*sqrt(x)*a* *3*b**5*x**2 - 840*log(sqrt(x)*b + a)*a**2*b**6*x**3 - 840*log(sqrt(x)*b + a)*b**8*x**4 + 840*log(sqrt(x))*a**2*b**6*x**3 + 840*log(sqrt(x))*b**8*x* *4 - 5*a**8 - 14*a**6*b**2*x - 70*a**4*b**4*x**2 + 840*a**2*b**6*x**3 - 42 0*b**8*x**4)/(15*a**9*x**3*(2*sqrt(x)*a*b + a**2 + b**2*x))