Integrand size = 15, antiderivative size = 136 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^4} \, dx=\frac {3 a^8}{2 b^9 \left (a+\frac {b}{\sqrt [3]{x}}\right )^2}-\frac {24 a^7}{b^9 \left (a+\frac {b}{\sqrt [3]{x}}\right )}-\frac {1}{2 b^3 x^2}+\frac {9 a}{5 b^4 x^{5/3}}-\frac {9 a^2}{2 b^5 x^{4/3}}+\frac {10 a^3}{b^6 x}-\frac {45 a^4}{2 b^7 x^{2/3}}+\frac {63 a^5}{b^8 \sqrt [3]{x}}-\frac {84 a^6 \log \left (a+\frac {b}{\sqrt [3]{x}}\right )}{b^9} \] Output:
3/2*a^8/b^9/(a+b/x^(1/3))^2-24*a^7/b^9/(a+b/x^(1/3))-1/2/b^3/x^2+9/5*a/b^4 /x^(5/3)-9/2*a^2/b^5/x^(4/3)+10*a^3/b^6/x-45/2*a^4/b^7/x^(2/3)+63*a^5/b^8/ x^(1/3)-84*a^6*ln(a+b/x^(1/3))/b^9
Time = 0.13 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^4} \, dx=\frac {\frac {b \left (-5 b^7+8 a b^6 \sqrt [3]{x}-14 a^2 b^5 x^{2/3}+28 a^3 b^4 x-70 a^4 b^3 x^{4/3}+280 a^5 b^2 x^{5/3}+1260 a^6 b x^2+840 a^7 x^{7/3}\right )}{\left (b+a \sqrt [3]{x}\right )^2 x^2}-840 a^6 \log \left (b+a \sqrt [3]{x}\right )+280 a^6 \log (x)}{10 b^9} \] Input:
Integrate[1/((a + b/x^(1/3))^3*x^4),x]
Output:
((b*(-5*b^7 + 8*a*b^6*x^(1/3) - 14*a^2*b^5*x^(2/3) + 28*a^3*b^4*x - 70*a^4 *b^3*x^(4/3) + 280*a^5*b^2*x^(5/3) + 1260*a^6*b*x^2 + 840*a^7*x^(7/3)))/(( b + a*x^(1/3))^2*x^2) - 840*a^6*Log[b + a*x^(1/3)] + 280*a^6*Log[x])/(10*b ^9)
Time = 0.50 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.13, 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.
\(\displaystyle \int \frac {1}{x^4 \left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx\) |
\(\Big \downarrow \) 795 |
\(\displaystyle \int \frac {1}{x^3 \left (a \sqrt [3]{x}+b\right )^3}dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 3 \int \frac {1}{\left (\sqrt [3]{x} a+b\right )^3 x^{7/3}}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle 3 \int \left (-\frac {28 a^7}{b^9 \left (\sqrt [3]{x} a+b\right )}-\frac {7 a^7}{b^8 \left (\sqrt [3]{x} a+b\right )^2}-\frac {a^7}{b^7 \left (\sqrt [3]{x} a+b\right )^3}+\frac {28 a^6}{b^9 \sqrt [3]{x}}-\frac {21 a^5}{b^8 x^{2/3}}+\frac {15 a^4}{b^7 x}-\frac {10 a^3}{b^6 x^{4/3}}+\frac {6 a^2}{b^5 x^{5/3}}-\frac {3 a}{b^4 x^2}+\frac {1}{b^3 x^{7/3}}\right )d\sqrt [3]{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \left (-\frac {28 a^6 \log \left (a \sqrt [3]{x}+b\right )}{b^9}+\frac {28 a^6 \log \left (\sqrt [3]{x}\right )}{b^9}+\frac {7 a^6}{b^8 \left (a \sqrt [3]{x}+b\right )}+\frac {a^6}{2 b^7 \left (a \sqrt [3]{x}+b\right )^2}+\frac {21 a^5}{b^8 \sqrt [3]{x}}-\frac {15 a^4}{2 b^7 x^{2/3}}+\frac {10 a^3}{3 b^6 x}-\frac {3 a^2}{2 b^5 x^{4/3}}+\frac {3 a}{5 b^4 x^{5/3}}-\frac {1}{6 b^3 x^2}\right )\) |
Input:
Int[1/((a + b/x^(1/3))^3*x^4),x]
Output:
3*(a^6/(2*b^7*(b + a*x^(1/3))^2) + (7*a^6)/(b^8*(b + a*x^(1/3))) - 1/(6*b^ 3*x^2) + (3*a)/(5*b^4*x^(5/3)) - (3*a^2)/(2*b^5*x^(4/3)) + (10*a^3)/(3*b^6 *x) - (15*a^4)/(2*b^7*x^(2/3)) + (21*a^5)/(b^8*x^(1/3)) - (28*a^6*Log[b + a*x^(1/3)])/b^9 + (28*a^6*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 = 0.49 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.90
method | result | size |
derivativedivides | \(-\frac {1}{2 b^{3} x^{2}}+\frac {28 a^{6} \ln \left (x \right )}{b^{9}}+\frac {63 a^{5}}{b^{8} x^{\frac {1}{3}}}-\frac {45 a^{4}}{2 b^{7} x^{\frac {2}{3}}}+\frac {10 a^{3}}{b^{6} x}-\frac {9 a^{2}}{2 b^{5} x^{\frac {4}{3}}}+\frac {9 a}{5 b^{4} x^{\frac {5}{3}}}-\frac {84 a^{6} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{b^{9}}+\frac {21 a^{6}}{b^{8} \left (b +a \,x^{\frac {1}{3}}\right )}+\frac {3 a^{6}}{2 b^{7} \left (b +a \,x^{\frac {1}{3}}\right )^{2}}\) | \(123\) |
default | \(-\frac {1}{2 b^{3} x^{2}}+\frac {28 a^{6} \ln \left (x \right )}{b^{9}}+\frac {63 a^{5}}{b^{8} x^{\frac {1}{3}}}-\frac {45 a^{4}}{2 b^{7} x^{\frac {2}{3}}}+\frac {10 a^{3}}{b^{6} x}-\frac {9 a^{2}}{2 b^{5} x^{\frac {4}{3}}}+\frac {9 a}{5 b^{4} x^{\frac {5}{3}}}-\frac {84 a^{6} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{b^{9}}+\frac {21 a^{6}}{b^{8} \left (b +a \,x^{\frac {1}{3}}\right )}+\frac {3 a^{6}}{2 b^{7} \left (b +a \,x^{\frac {1}{3}}\right )^{2}}\) | \(123\) |
Input:
int(1/(a+b/x^(1/3))^3/x^4,x,method=_RETURNVERBOSE)
Output:
-1/2/b^3/x^2+28/b^9*a^6*ln(x)+63*a^5/b^8/x^(1/3)-45/2*a^4/b^7/x^(2/3)+10*a ^3/b^6/x-9/2*a^2/b^5/x^(4/3)+9/5*a/b^4/x^(5/3)-84/b^9*a^6*ln(b+a*x^(1/3))+ 21/b^8*a^6/(b+a*x^(1/3))+3/2*a^6/b^7/(b+a*x^(1/3))^2
Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (112) = 224\).
Time = 0.09 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.69 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^4} \, dx=\frac {280 \, a^{9} b^{3} x^{3} + 420 \, a^{6} b^{6} x^{2} + 90 \, a^{3} b^{9} x - 5 \, b^{12} - 840 \, {\left (a^{12} x^{4} + 2 \, a^{9} b^{3} x^{3} + a^{6} b^{6} x^{2}\right )} \log \left (a x^{\frac {1}{3}} + b\right ) + 840 \, {\left (a^{12} x^{4} + 2 \, a^{9} b^{3} x^{3} + a^{6} b^{6} x^{2}\right )} \log \left (x^{\frac {1}{3}}\right ) + 15 \, {\left (56 \, a^{11} b x^{3} + 98 \, a^{8} b^{4} x^{2} + 36 \, a^{5} b^{7} x - 3 \, a^{2} b^{10}\right )} x^{\frac {2}{3}} - 3 \, {\left (140 \, a^{10} b^{2} x^{3} + 224 \, a^{7} b^{5} x^{2} + 63 \, a^{4} b^{8} x - 6 \, a b^{11}\right )} x^{\frac {1}{3}}}{10 \, {\left (a^{6} b^{9} x^{4} + 2 \, a^{3} b^{12} x^{3} + b^{15} x^{2}\right )}} \] Input:
integrate(1/(a+b/x^(1/3))^3/x^4,x, algorithm="fricas")
Output:
1/10*(280*a^9*b^3*x^3 + 420*a^6*b^6*x^2 + 90*a^3*b^9*x - 5*b^12 - 840*(a^1 2*x^4 + 2*a^9*b^3*x^3 + a^6*b^6*x^2)*log(a*x^(1/3) + b) + 840*(a^12*x^4 + 2*a^9*b^3*x^3 + a^6*b^6*x^2)*log(x^(1/3)) + 15*(56*a^11*b*x^3 + 98*a^8*b^4 *x^2 + 36*a^5*b^7*x - 3*a^2*b^10)*x^(2/3) - 3*(140*a^10*b^2*x^3 + 224*a^7* b^5*x^2 + 63*a^4*b^8*x - 6*a*b^11)*x^(1/3))/(a^6*b^9*x^4 + 2*a^3*b^12*x^3 + b^15*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 707 vs. \(2 (133) = 266\).
Time = 7.71 (sec) , antiderivative size = 707, normalized size of antiderivative = 5.20 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^4} \, dx =\text {Too large to display} \] Input:
integrate(1/(a+b/x**(1/3))**3/x**4,x)
Output:
Piecewise((zoo/x**2, Eq(a, 0) & Eq(b, 0)), (-1/(2*b**3*x**2), Eq(a, 0)), ( -1/(3*a**3*x**3), Eq(b, 0)), (280*a**8*x**(13/3)*log(x)/(10*a**2*b**9*x**( 13/3) + 20*a*b**10*x**4 + 10*b**11*x**(11/3)) - 840*a**8*x**(13/3)*log(x** (1/3) + b/a)/(10*a**2*b**9*x**(13/3) + 20*a*b**10*x**4 + 10*b**11*x**(11/3 )) + 560*a**7*b*x**4*log(x)/(10*a**2*b**9*x**(13/3) + 20*a*b**10*x**4 + 10 *b**11*x**(11/3)) - 1680*a**7*b*x**4*log(x**(1/3) + b/a)/(10*a**2*b**9*x** (13/3) + 20*a*b**10*x**4 + 10*b**11*x**(11/3)) + 840*a**7*b*x**4/(10*a**2* b**9*x**(13/3) + 20*a*b**10*x**4 + 10*b**11*x**(11/3)) + 280*a**6*b**2*x** (11/3)*log(x)/(10*a**2*b**9*x**(13/3) + 20*a*b**10*x**4 + 10*b**11*x**(11/ 3)) - 840*a**6*b**2*x**(11/3)*log(x**(1/3) + b/a)/(10*a**2*b**9*x**(13/3) + 20*a*b**10*x**4 + 10*b**11*x**(11/3)) + 1260*a**6*b**2*x**(11/3)/(10*a** 2*b**9*x**(13/3) + 20*a*b**10*x**4 + 10*b**11*x**(11/3)) + 280*a**5*b**3*x **(10/3)/(10*a**2*b**9*x**(13/3) + 20*a*b**10*x**4 + 10*b**11*x**(11/3)) - 70*a**4*b**4*x**3/(10*a**2*b**9*x**(13/3) + 20*a*b**10*x**4 + 10*b**11*x* *(11/3)) + 28*a**3*b**5*x**(8/3)/(10*a**2*b**9*x**(13/3) + 20*a*b**10*x**4 + 10*b**11*x**(11/3)) - 14*a**2*b**6*x**(7/3)/(10*a**2*b**9*x**(13/3) + 2 0*a*b**10*x**4 + 10*b**11*x**(11/3)) + 8*a*b**7*x**2/(10*a**2*b**9*x**(13/ 3) + 20*a*b**10*x**4 + 10*b**11*x**(11/3)) - 5*b**8*x**(5/3)/(10*a**2*b**9 *x**(13/3) + 20*a*b**10*x**4 + 10*b**11*x**(11/3)), True))
Time = 0.03 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^4} \, dx=-\frac {84 \, a^{6} \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{b^{9}} - \frac {{\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{6}}{2 \, b^{9}} + \frac {24 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{5} a}{5 \, b^{9}} - \frac {21 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{4} a^{2}}{b^{9}} + \frac {56 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{3} a^{3}}{b^{9}} - \frac {105 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{2} a^{4}}{b^{9}} + \frac {168 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )} a^{5}}{b^{9}} - \frac {24 \, a^{7}}{{\left (a + \frac {b}{x^{\frac {1}{3}}}\right )} b^{9}} + \frac {3 \, a^{8}}{2 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{2} b^{9}} \] Input:
integrate(1/(a+b/x^(1/3))^3/x^4,x, algorithm="maxima")
Output:
-84*a^6*log(a + b/x^(1/3))/b^9 - 1/2*(a + b/x^(1/3))^6/b^9 + 24/5*(a + b/x ^(1/3))^5*a/b^9 - 21*(a + b/x^(1/3))^4*a^2/b^9 + 56*(a + b/x^(1/3))^3*a^3/ b^9 - 105*(a + b/x^(1/3))^2*a^4/b^9 + 168*(a + b/x^(1/3))*a^5/b^9 - 24*a^7 /((a + b/x^(1/3))*b^9) + 3/2*a^8/((a + b/x^(1/3))^2*b^9)
Time = 0.12 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^4} \, dx=-\frac {84 \, a^{6} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{b^{9}} + \frac {28 \, a^{6} \log \left ({\left | x \right |}\right )}{b^{9}} + \frac {840 \, a^{7} b x^{\frac {7}{3}} + 1260 \, a^{6} b^{2} x^{2} + 280 \, a^{5} b^{3} x^{\frac {5}{3}} - 70 \, a^{4} b^{4} x^{\frac {4}{3}} + 28 \, a^{3} b^{5} x - 14 \, a^{2} b^{6} x^{\frac {2}{3}} + 8 \, a b^{7} x^{\frac {1}{3}} - 5 \, b^{8}}{10 \, {\left (a x^{\frac {1}{3}} + b\right )}^{2} b^{9} x^{2}} \] Input:
integrate(1/(a+b/x^(1/3))^3/x^4,x, algorithm="giac")
Output:
-84*a^6*log(abs(a*x^(1/3) + b))/b^9 + 28*a^6*log(abs(x))/b^9 + 1/10*(840*a ^7*b*x^(7/3) + 1260*a^6*b^2*x^2 + 280*a^5*b^3*x^(5/3) - 70*a^4*b^4*x^(4/3) + 28*a^3*b^5*x - 14*a^2*b^6*x^(2/3) + 8*a*b^7*x^(1/3) - 5*b^8)/((a*x^(1/3 ) + b)^2*b^9*x^2)
Time = 0.43 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^4} \, dx=\frac {\frac {4\,a\,x^{1/3}}{5\,b^2}-\frac {1}{2\,b}+\frac {14\,a^3\,x}{5\,b^4}-\frac {7\,a^2\,x^{2/3}}{5\,b^3}+\frac {126\,a^6\,x^2}{b^7}-\frac {7\,a^4\,x^{4/3}}{b^5}+\frac {28\,a^5\,x^{5/3}}{b^6}+\frac {84\,a^7\,x^{7/3}}{b^8}}{a^2\,x^{8/3}+b^2\,x^2+2\,a\,b\,x^{7/3}}-\frac {168\,a^6\,\mathrm {atanh}\left (\frac {2\,a\,x^{1/3}}{b}+1\right )}{b^9} \] Input:
int(1/(x^4*(a + b/x^(1/3))^3),x)
Output:
((4*a*x^(1/3))/(5*b^2) - 1/(2*b) + (14*a^3*x)/(5*b^4) - (7*a^2*x^(2/3))/(5 *b^3) + (126*a^6*x^2)/b^7 - (7*a^4*x^(4/3))/b^5 + (28*a^5*x^(5/3))/b^6 + ( 84*a^7*x^(7/3))/b^8)/(a^2*x^(8/3) + b^2*x^2 + 2*a*b*x^(7/3)) - (168*a^6*at anh((2*a*x^(1/3))/b + 1))/b^9
Time = 0.21 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.44 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^4} \, dx=\frac {840 x^{\frac {8}{3}} \mathrm {log}\left (x^{\frac {1}{3}}\right ) a^{8}-840 x^{\frac {8}{3}} \mathrm {log}\left (x^{\frac {1}{3}} a +b \right ) a^{8}-420 x^{\frac {8}{3}} a^{8}+280 x^{\frac {5}{3}} a^{5} b^{3}-14 x^{\frac {2}{3}} a^{2} b^{6}+1680 x^{\frac {7}{3}} \mathrm {log}\left (x^{\frac {1}{3}}\right ) a^{7} b -1680 x^{\frac {7}{3}} \mathrm {log}\left (x^{\frac {1}{3}} a +b \right ) a^{7} b -70 x^{\frac {4}{3}} a^{4} b^{4}+8 x^{\frac {1}{3}} a \,b^{7}+840 \,\mathrm {log}\left (x^{\frac {1}{3}}\right ) a^{6} b^{2} x^{2}-840 \,\mathrm {log}\left (x^{\frac {1}{3}} a +b \right ) a^{6} b^{2} x^{2}+840 a^{6} b^{2} x^{2}+28 a^{3} b^{5} x -5 b^{8}}{10 b^{9} x^{2} \left (x^{\frac {2}{3}} a^{2}+2 x^{\frac {1}{3}} a b +b^{2}\right )} \] Input:
int(1/(a+b/x^(1/3))^3/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 - 420*x**(2/3)*a**8*x**2 + 280*x**(2/3)*a**5*b**3*x - 14*x**(2/3) *a**2*b**6 + 1680*x**(1/3)*log(x**(1/3))*a**7*b*x**2 - 1680*x**(1/3)*log(x **(1/3)*a + b)*a**7*b*x**2 - 70*x**(1/3)*a**4*b**4*x + 8*x**(1/3)*a*b**7 + 840*log(x**(1/3))*a**6*b**2*x**2 - 840*log(x**(1/3)*a + b)*a**6*b**2*x**2 + 840*a**6*b**2*x**2 + 28*a**3*b**5*x - 5*b**8)/(10*b**9*x**2*(x**(2/3)*a **2 + 2*x**(1/3)*a*b + b**2))