Integrand size = 15, antiderivative size = 123 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^2 x^4} \, dx=\frac {3 a^8}{b^9 \left (a+\frac {b}{\sqrt [3]{x}}\right )}-\frac {3}{7 b^2 x^{7/3}}+\frac {a}{b^3 x^2}-\frac {9 a^2}{5 b^4 x^{5/3}}+\frac {3 a^3}{b^5 x^{4/3}}-\frac {5 a^4}{b^6 x}+\frac {9 a^5}{b^7 x^{2/3}}-\frac {21 a^6}{b^8 \sqrt [3]{x}}+\frac {24 a^7 \log \left (a+\frac {b}{\sqrt [3]{x}}\right )}{b^9} \] Output:
3*a^8/b^9/(a+b/x^(1/3))-3/7/b^2/x^(7/3)+a/b^3/x^2-9/5*a^2/b^4/x^(5/3)+3*a^ 3/b^5/x^(4/3)-5*a^4/b^6/x+9*a^5/b^7/x^(2/3)-21*a^6/b^8/x^(1/3)+24*a^7*ln(a +b/x^(1/3))/b^9
Time = 0.19 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^2 x^4} \, dx=-\frac {\frac {b \left (15 b^7-20 a b^6 \sqrt [3]{x}+28 a^2 b^5 x^{2/3}-42 a^3 b^4 x+70 a^4 b^3 x^{4/3}-140 a^5 b^2 x^{5/3}+420 a^6 b x^2+840 a^7 x^{7/3}\right )}{\left (b+a \sqrt [3]{x}\right ) x^{7/3}}-840 a^7 \log \left (b+a \sqrt [3]{x}\right )+280 a^7 \log (x)}{35 b^9} \] Input:
Integrate[1/((a + b/x^(1/3))^2*x^4),x]
Output:
-1/35*((b*(15*b^7 - 20*a*b^6*x^(1/3) + 28*a^2*b^5*x^(2/3) - 42*a^3*b^4*x + 70*a^4*b^3*x^(4/3) - 140*a^5*b^2*x^(5/3) + 420*a^6*b*x^2 + 840*a^7*x^(7/3 )))/((b + a*x^(1/3))*x^(7/3)) - 840*a^7*Log[b + a*x^(1/3)] + 280*a^7*Log[x ])/b^9
Time = 0.47 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.16, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {795, 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.
| \(\displaystyle \int \frac {1}{x^4 \left (a+\frac {b}{\sqrt [3]{x}}\right )^2} \, dx\) |
| \(\Big \downarrow \) 795 |
| \(\displaystyle \int \frac {1}{x^{10/3} \left (a \sqrt [3]{x}+b\right )^2}dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle 3 \int \frac {1}{\left (\sqrt [3]{x} a+b\right )^2 x^{8/3}}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle 3 \int \left (\frac {8 a^8}{b^9 \left (\sqrt [3]{x} a+b\right )}+\frac {a^8}{b^8 \left (\sqrt [3]{x} a+b\right )^2}-\frac {8 a^7}{b^9 \sqrt [3]{x}}+\frac {7 a^6}{b^8 x^{2/3}}-\frac {6 a^5}{b^7 x}+\frac {5 a^4}{b^6 x^{4/3}}-\frac {4 a^3}{b^5 x^{5/3}}+\frac {3 a^2}{b^4 x^2}-\frac {2 a}{b^3 x^{7/3}}+\frac {1}{b^2 x^{8/3}}\right )d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \left (\frac {8 a^7 \log \left (a \sqrt [3]{x}+b\right )}{b^9}-\frac {8 a^7 \log \left (\sqrt [3]{x}\right )}{b^9}-\frac {a^7}{b^8 \left (a \sqrt [3]{x}+b\right )}-\frac {7 a^6}{b^8 \sqrt [3]{x}}+\frac {3 a^5}{b^7 x^{2/3}}-\frac {5 a^4}{3 b^6 x}+\frac {a^3}{b^5 x^{4/3}}-\frac {3 a^2}{5 b^4 x^{5/3}}+\frac {a}{3 b^3 x^2}-\frac {1}{7 b^2 x^{7/3}}\right )\) |
Input:
Int[1/((a + b/x^(1/3))^2*x^4),x]
Output:
3*(-(a^7/(b^8*(b + a*x^(1/3)))) - 1/(7*b^2*x^(7/3)) + a/(3*b^3*x^2) - (3*a ^2)/(5*b^4*x^(5/3)) + a^3/(b^5*x^(4/3)) - (5*a^4)/(3*b^6*x) + (3*a^5)/(b^7 *x^(2/3)) - (7*a^6)/(b^8*x^(1/3)) + (8*a^7*Log[b + a*x^(1/3)])/b^9 - (8*a^ 7*Log[x^(1/3)])/b^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] :> Int[x^(m + n*p)* (b + a/x^n)^p, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && NegQ[n]
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 = 116, normalized size of antiderivative = 0.94
| method | result | size |
| derivativedivides | \(-\frac {3}{7 b^{2} x^{\frac {7}{3}}}-\frac {9 a^{2}}{5 b^{4} x^{\frac {5}{3}}}-\frac {5 a^{4}}{b^{6} x}-\frac {8 a^{7} \ln \left (x \right )}{b^{9}}-\frac {21 a^{6}}{b^{8} x^{\frac {1}{3}}}+\frac {9 a^{5}}{b^{7} x^{\frac {2}{3}}}+\frac {3 a^{3}}{b^{5} x^{\frac {4}{3}}}+\frac {a}{b^{3} x^{2}}-\frac {3 a^{7}}{b^{8} \left (b +a \,x^{\frac {1}{3}}\right )}+\frac {24 a^{7} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{b^{9}}\) | \(116\) |
| default | \(-\frac {3}{7 b^{2} x^{\frac {7}{3}}}-\frac {9 a^{2}}{5 b^{4} x^{\frac {5}{3}}}-\frac {5 a^{4}}{b^{6} x}-\frac {8 a^{7} \ln \left (x \right )}{b^{9}}-\frac {21 a^{6}}{b^{8} x^{\frac {1}{3}}}+\frac {9 a^{5}}{b^{7} x^{\frac {2}{3}}}+\frac {3 a^{3}}{b^{5} x^{\frac {4}{3}}}+\frac {a}{b^{3} x^{2}}-\frac {3 a^{7}}{b^{8} \left (b +a \,x^{\frac {1}{3}}\right )}+\frac {24 a^{7} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{b^{9}}\) | \(116\) |
Input:
int(1/(a+b/x^(1/3))^2/x^4,x,method=_RETURNVERBOSE)
Output:
-3/7/b^2/x^(7/3)-9/5*a^2/b^4/x^(5/3)-5*a^4/b^6/x-8/b^9*a^7*ln(x)-21*a^6/b^ 8/x^(1/3)+9*a^5/b^7/x^(2/3)+3*a^3/b^5/x^(4/3)+a/b^3/x^2-3*a^7/b^8/(b+a*x^( 1/3))+24/b^9*a^7*ln(b+a*x^(1/3))
Time = 0.08 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.46 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^2 x^4} \, dx=-\frac {280 \, a^{7} b^{3} x^{3} + 140 \, a^{4} b^{6} x^{2} - 35 \, a b^{9} x - 840 \, {\left (a^{10} x^{4} + a^{7} b^{3} x^{3}\right )} \log \left (a x^{\frac {1}{3}} + b\right ) + 840 \, {\left (a^{10} x^{4} + a^{7} b^{3} x^{3}\right )} \log \left (x^{\frac {1}{3}}\right ) + 15 \, {\left (56 \, a^{9} b x^{3} + 42 \, a^{6} b^{4} x^{2} - 6 \, a^{3} b^{7} x + b^{10}\right )} x^{\frac {2}{3}} - 21 \, {\left (20 \, a^{8} b^{2} x^{3} + 12 \, a^{5} b^{5} x^{2} - 3 \, a^{2} b^{8} x\right )} x^{\frac {1}{3}}}{35 \, {\left (a^{3} b^{9} x^{4} + b^{12} x^{3}\right )}} \] Input:
integrate(1/(a+b/x^(1/3))^2/x^4,x, algorithm="fricas")
Output:
-1/35*(280*a^7*b^3*x^3 + 140*a^4*b^6*x^2 - 35*a*b^9*x - 840*(a^10*x^4 + a^ 7*b^3*x^3)*log(a*x^(1/3) + b) + 840*(a^10*x^4 + a^7*b^3*x^3)*log(x^(1/3)) + 15*(56*a^9*b*x^3 + 42*a^6*b^4*x^2 - 6*a^3*b^7*x + b^10)*x^(2/3) - 21*(20 *a^8*b^2*x^3 + 12*a^5*b^5*x^2 - 3*a^2*b^8*x)*x^(1/3))/(a^3*b^9*x^4 + b^12* x^3)
Leaf count of result is larger than twice the leaf count of optimal. 440 vs. \(2 (121) = 242\).
Time = 4.62 (sec) , antiderivative size = 440, normalized size of antiderivative = 3.58 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^2 x^4} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {7}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {3}{7 b^{2} x^{\frac {7}{3}}} & \text {for}\: a = 0 \\- \frac {1}{3 a^{2} x^{3}} & \text {for}\: b = 0 \\- \frac {280 a^{8} x^{4} \log {\left (x \right )}}{35 a b^{9} x^{4} + 35 b^{10} x^{\frac {11}{3}}} + \frac {840 a^{8} x^{4} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{35 a b^{9} x^{4} + 35 b^{10} x^{\frac {11}{3}}} - \frac {280 a^{7} b x^{\frac {11}{3}} \log {\left (x \right )}}{35 a b^{9} x^{4} + 35 b^{10} x^{\frac {11}{3}}} + \frac {840 a^{7} b x^{\frac {11}{3}} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{35 a b^{9} x^{4} + 35 b^{10} x^{\frac {11}{3}}} - \frac {840 a^{7} b x^{\frac {11}{3}}}{35 a b^{9} x^{4} + 35 b^{10} x^{\frac {11}{3}}} - \frac {420 a^{6} b^{2} x^{\frac {10}{3}}}{35 a b^{9} x^{4} + 35 b^{10} x^{\frac {11}{3}}} + \frac {140 a^{5} b^{3} x^{3}}{35 a b^{9} x^{4} + 35 b^{10} x^{\frac {11}{3}}} - \frac {70 a^{4} b^{4} x^{\frac {8}{3}}}{35 a b^{9} x^{4} + 35 b^{10} x^{\frac {11}{3}}} + \frac {42 a^{3} b^{5} x^{\frac {7}{3}}}{35 a b^{9} x^{4} + 35 b^{10} x^{\frac {11}{3}}} - \frac {28 a^{2} b^{6} x^{2}}{35 a b^{9} x^{4} + 35 b^{10} x^{\frac {11}{3}}} + \frac {20 a b^{7} x^{\frac {5}{3}}}{35 a b^{9} x^{4} + 35 b^{10} x^{\frac {11}{3}}} - \frac {15 b^{8} x^{\frac {4}{3}}}{35 a b^{9} x^{4} + 35 b^{10} x^{\frac {11}{3}}} & \text {otherwise} \end {cases} \] Input:
integrate(1/(a+b/x**(1/3))**2/x**4,x)
Output:
Piecewise((zoo/x**(7/3), Eq(a, 0) & Eq(b, 0)), (-3/(7*b**2*x**(7/3)), Eq(a , 0)), (-1/(3*a**2*x**3), Eq(b, 0)), (-280*a**8*x**4*log(x)/(35*a*b**9*x** 4 + 35*b**10*x**(11/3)) + 840*a**8*x**4*log(x**(1/3) + b/a)/(35*a*b**9*x** 4 + 35*b**10*x**(11/3)) - 280*a**7*b*x**(11/3)*log(x)/(35*a*b**9*x**4 + 35 *b**10*x**(11/3)) + 840*a**7*b*x**(11/3)*log(x**(1/3) + b/a)/(35*a*b**9*x* *4 + 35*b**10*x**(11/3)) - 840*a**7*b*x**(11/3)/(35*a*b**9*x**4 + 35*b**10 *x**(11/3)) - 420*a**6*b**2*x**(10/3)/(35*a*b**9*x**4 + 35*b**10*x**(11/3) ) + 140*a**5*b**3*x**3/(35*a*b**9*x**4 + 35*b**10*x**(11/3)) - 70*a**4*b** 4*x**(8/3)/(35*a*b**9*x**4 + 35*b**10*x**(11/3)) + 42*a**3*b**5*x**(7/3)/( 35*a*b**9*x**4 + 35*b**10*x**(11/3)) - 28*a**2*b**6*x**2/(35*a*b**9*x**4 + 35*b**10*x**(11/3)) + 20*a*b**7*x**(5/3)/(35*a*b**9*x**4 + 35*b**10*x**(1 1/3)) - 15*b**8*x**(4/3)/(35*a*b**9*x**4 + 35*b**10*x**(11/3)), True))
Time = 0.03 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.19 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^2 x^4} \, dx=\frac {24 \, a^{7} \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{b^{9}} - \frac {3 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{7}}{7 \, b^{9}} + \frac {4 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{6} a}{b^{9}} - \frac {84 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{5} a^{2}}{5 \, b^{9}} + \frac {42 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{4} a^{3}}{b^{9}} - \frac {70 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{3} a^{4}}{b^{9}} + \frac {84 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{2} a^{5}}{b^{9}} - \frac {84 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )} a^{6}}{b^{9}} + \frac {3 \, a^{8}}{{\left (a + \frac {b}{x^{\frac {1}{3}}}\right )} b^{9}} \] Input:
integrate(1/(a+b/x^(1/3))^2/x^4,x, algorithm="maxima")
Output:
24*a^7*log(a + b/x^(1/3))/b^9 - 3/7*(a + b/x^(1/3))^7/b^9 + 4*(a + b/x^(1/ 3))^6*a/b^9 - 84/5*(a + b/x^(1/3))^5*a^2/b^9 + 42*(a + b/x^(1/3))^4*a^3/b^ 9 - 70*(a + b/x^(1/3))^3*a^4/b^9 + 84*(a + b/x^(1/3))^2*a^5/b^9 - 84*(a + b/x^(1/3))*a^6/b^9 + 3*a^8/((a + b/x^(1/3))*b^9)
Time = 0.13 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^2 x^4} \, dx=\frac {24 \, a^{7} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{b^{9}} - \frac {8 \, a^{7} \log \left ({\left | x \right |}\right )}{b^{9}} - \frac {840 \, a^{7} b x^{\frac {7}{3}} + 420 \, a^{6} b^{2} x^{2} - 140 \, a^{5} b^{3} x^{\frac {5}{3}} + 70 \, a^{4} b^{4} x^{\frac {4}{3}} - 42 \, a^{3} b^{5} x + 28 \, a^{2} b^{6} x^{\frac {2}{3}} - 20 \, a b^{7} x^{\frac {1}{3}} + 15 \, b^{8}}{35 \, {\left (a x^{\frac {1}{3}} + b\right )} b^{9} x^{\frac {7}{3}}} \] Input:
integrate(1/(a+b/x^(1/3))^2/x^4,x, algorithm="giac")
Output:
24*a^7*log(abs(a*x^(1/3) + b))/b^9 - 8*a^7*log(abs(x))/b^9 - 1/35*(840*a^7 *b*x^(7/3) + 420*a^6*b^2*x^2 - 140*a^5*b^3*x^(5/3) + 70*a^4*b^4*x^(4/3) - 42*a^3*b^5*x + 28*a^2*b^6*x^(2/3) - 20*a*b^7*x^(1/3) + 15*b^8)/((a*x^(1/3) + b)*b^9*x^(7/3))
Time = 0.13 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^2 x^4} \, dx=\frac {48\,a^7\,\mathrm {atanh}\left (\frac {2\,a\,x^{1/3}}{b}+1\right )}{b^9}-\frac {\frac {3}{7\,b}-\frac {4\,a\,x^{1/3}}{7\,b^2}-\frac {6\,a^3\,x}{5\,b^4}+\frac {4\,a^2\,x^{2/3}}{5\,b^3}+\frac {12\,a^6\,x^2}{b^7}+\frac {2\,a^4\,x^{4/3}}{b^5}-\frac {4\,a^5\,x^{5/3}}{b^6}+\frac {24\,a^7\,x^{7/3}}{b^8}}{a\,x^{8/3}+b\,x^{7/3}} \] Input:
int(1/(x^4*(a + b/x^(1/3))^2),x)
Output:
(48*a^7*atanh((2*a*x^(1/3))/b + 1))/b^9 - (3/(7*b) - (4*a*x^(1/3))/(7*b^2) - (6*a^3*x)/(5*b^4) + (4*a^2*x^(2/3))/(5*b^3) + (12*a^6*x^2)/b^7 + (2*a^4 *x^(4/3))/b^5 - (4*a^5*x^(5/3))/b^6 + (24*a^7*x^(7/3))/b^8)/(a*x^(8/3) + b *x^(7/3))
Time = 0.22 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.23 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^2 x^4} \, dx=\frac {-840 x^{\frac {8}{3}} \mathrm {log}\left (x^{\frac {1}{3}}\right ) a^{8}+840 x^{\frac {8}{3}} \mathrm {log}\left (x^{\frac {1}{3}} a +b \right ) a^{8}+840 x^{\frac {8}{3}} a^{8}+140 x^{\frac {5}{3}} a^{5} b^{3}-28 x^{\frac {2}{3}} a^{2} b^{6}-840 x^{\frac {7}{3}} \mathrm {log}\left (x^{\frac {1}{3}}\right ) a^{7} b +840 x^{\frac {7}{3}} \mathrm {log}\left (x^{\frac {1}{3}} a +b \right ) a^{7} b -70 x^{\frac {4}{3}} a^{4} b^{4}+20 x^{\frac {1}{3}} a \,b^{7}-420 a^{6} b^{2} x^{2}+42 a^{3} b^{5} x -15 b^{8}}{35 x^{\frac {7}{3}} b^{9} \left (x^{\frac {1}{3}} a +b \right )} \] Input:
int(1/(a+b/x^(1/3))^2/x^4,x)
Output:
( - 840*x**(2/3)*log(x**(1/3))*a**8*x**2 + 840*x**(2/3)*log(x**(1/3)*a + b )*a**8*x**2 + 840*x**(2/3)*a**8*x**2 + 140*x**(2/3)*a**5*b**3*x - 28*x**(2 /3)*a**2*b**6 - 840*x**(1/3)*log(x**(1/3))*a**7*b*x**2 + 840*x**(1/3)*log( x**(1/3)*a + b)*a**7*b*x**2 - 70*x**(1/3)*a**4*b**4*x + 20*x**(1/3)*a*b**7 - 420*a**6*b**2*x**2 + 42*a**3*b**5*x - 15*b**8)/(35*x**(1/3)*b**9*x**2*( x**(1/3)*a + b))