Integrand size = 24, antiderivative size = 56 \[ \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^4} \, dx=-\frac {3 a^2}{7 b^3 \left (a+b \sqrt [3]{x}\right )^7}+\frac {a}{b^3 \left (a+b \sqrt [3]{x}\right )^6}-\frac {3}{5 b^3 \left (a+b \sqrt [3]{x}\right )^5} \] Output:
-3/7*a^2/b^3/(a+b*x^(1/3))^7+a/b^3/(a+b*x^(1/3))^6-3/5/b^3/(a+b*x^(1/3))^5
Time = 0.05 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^4} \, dx=\frac {-a^2-7 a b \sqrt [3]{x}-21 b^2 x^{2/3}}{35 b^3 \left (a+b \sqrt [3]{x}\right )^7} \] Input:
Integrate[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^(-4),x]
Output:
(-a^2 - 7*a*b*x^(1/3) - 21*b^2*x^(2/3))/(35*b^3*(a + b*x^(1/3))^7)
Time = 0.21 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1379, 774, 27, 53, 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.
\(\displaystyle \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^4} \, dx\) |
\(\Big \downarrow \) 1379 |
\(\displaystyle b^8 \int \frac {1}{\left (\sqrt [3]{x} b^2+a b\right )^8}dx\) |
\(\Big \downarrow \) 774 |
\(\displaystyle 3 b^8 \int \frac {x^{2/3}}{b^8 \left (a+b \sqrt [3]{x}\right )^8}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 3 \int \frac {x^{2/3}}{\left (a+b \sqrt [3]{x}\right )^8}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle 3 \int \left (\frac {a^2}{b^2 \left (a+b \sqrt [3]{x}\right )^8}-\frac {2 a}{b^2 \left (a+b \sqrt [3]{x}\right )^7}+\frac {1}{b^2 \left (a+b \sqrt [3]{x}\right )^6}\right )d\sqrt [3]{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \left (-\frac {a^2}{7 b^3 \left (a+b \sqrt [3]{x}\right )^7}+\frac {a}{3 b^3 \left (a+b \sqrt [3]{x}\right )^6}-\frac {1}{5 b^3 \left (a+b \sqrt [3]{x}\right )^5}\right )\) |
Input:
Int[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^(-4),x]
Output:
3*(-1/7*a^2/(b^3*(a + b*x^(1/3))^7) + a/(3*b^3*(a + b*x^(1/3))^6) - 1/(5*b ^3*(a + b*x^(1/3))^5))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Simp[k Subst[Int[x^(k - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; Fre eQ[{a, b, p}, x] && FractionQ[n]
Int[((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/ c^p Int[(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n 2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p] && NeQ[p, 1]
Time = 0.39 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(-\frac {3 a^{2}}{7 b^{3} \left (a +b \,x^{\frac {1}{3}}\right )^{7}}+\frac {a}{b^{3} \left (a +b \,x^{\frac {1}{3}}\right )^{6}}-\frac {3}{5 b^{3} \left (a +b \,x^{\frac {1}{3}}\right )^{5}}\) | \(47\) |
trager | \(\frac {\left (x -1\right ) \left (a^{15} b^{18} x^{6}-28 a^{12} b^{21} x^{6}+1876 a^{9} b^{24} x^{6}-6020 a^{6} b^{27} x^{6}+2870 a^{3} b^{30} x^{6}-140 b^{33} x^{6}+7 a^{18} b^{15} x^{5}-195 a^{15} b^{18} x^{5}+13104 a^{12} b^{21} x^{5}-40264 a^{9} b^{24} x^{5}+14070 a^{6} b^{27} x^{5}+1890 a^{3} b^{30} x^{5}-140 b^{33} x^{5}+161 a^{21} b^{12} x^{4}+399 a^{18} b^{15} x^{4}+42141 a^{15} b^{18} x^{4}-108416 a^{12} b^{21} x^{4}+24906 a^{9} b^{24} x^{4}+14070 a^{6} b^{27} x^{4}+2870 a^{3} b^{30} x^{4}-2835 a^{24} b^{9} x^{3}-20909 a^{21} b^{12} x^{3}+5789 a^{18} b^{15} x^{3}-269009 a^{15} b^{18} x^{3}-108416 a^{12} b^{21} x^{3}-40264 a^{9} b^{24} x^{3}-6020 a^{6} b^{27} x^{3}+6055 a^{27} b^{6} x^{2}+38325 a^{24} b^{9} x^{2}+171171 a^{21} b^{12} x^{2}+5789 a^{18} b^{15} x^{2}+42141 a^{15} b^{18} x^{2}+13104 a^{12} b^{21} x^{2}+1876 a^{9} b^{24} x^{2}-1855 a^{30} b^{3} x -7665 a^{27} b^{6} x +38325 a^{24} b^{9} x -20909 a^{21} b^{12} x +399 a^{18} b^{15} x -195 a^{15} b^{18} x -28 a^{12} b^{21} x +35 a^{33}-1855 a^{30} b^{3}+6055 a^{27} b^{6}-2835 a^{24} b^{9}+161 a^{21} b^{12}+7 a^{18} b^{15}+a^{15} b^{18}\right ) a}{35 \left (b^{3} x +a^{3}\right )^{7} \left (a^{21}+7 a^{18} b^{3}+21 a^{15} b^{6}+35 a^{12} b^{9}+35 a^{9} b^{12}+21 a^{6} b^{15}+7 a^{3} b^{18}+b^{21}\right )}-\frac {3 b \,x^{\frac {4}{3}} \left (7 b^{12} x^{4}-476 a^{3} b^{9} x^{3}+1932 x^{2} b^{6} a^{6}-1160 a^{9} b^{3} x +70 a^{12}\right )}{35 \left (b^{3} x +a^{3}\right )^{7}}+\frac {27 b^{2} a^{2} x^{\frac {5}{3}} \left (-20 b^{9} x^{3}+168 a^{3} b^{6} x^{2}-189 a^{6} b^{3} x +28 a^{9}\right )}{35 \left (b^{3} x +a^{3}\right )^{7}}\) | \(677\) |
default | \(\text {Expression too large to display}\) | \(4530\) |
Input:
int(1/(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^4,x,method=_RETURNVERBOSE)
Output:
-3/7*a^2/b^3/(a+b*x^(1/3))^7+a/b^3/(a+b*x^(1/3))^6-3/5/b^3/(a+b*x^(1/3))^5
Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (46) = 92\).
Time = 0.18 (sec) , antiderivative size = 244, normalized size of antiderivative = 4.36 \[ \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^4} \, dx=\frac {140 \, a b^{15} x^{5} - 2870 \, a^{4} b^{12} x^{4} + 6020 \, a^{7} b^{9} x^{3} - 1876 \, a^{10} b^{6} x^{2} + 28 \, a^{13} b^{3} x - a^{16} - 27 \, {\left (20 \, a^{2} b^{14} x^{4} - 168 \, a^{5} b^{11} x^{3} + 189 \, a^{8} b^{8} x^{2} - 28 \, a^{11} b^{5} x\right )} x^{\frac {2}{3}} - 3 \, {\left (7 \, b^{16} x^{5} - 476 \, a^{3} b^{13} x^{4} + 1932 \, a^{6} b^{10} x^{3} - 1160 \, a^{9} b^{7} x^{2} + 70 \, a^{12} b^{4} x\right )} x^{\frac {1}{3}}}{35 \, {\left (b^{24} x^{7} + 7 \, a^{3} b^{21} x^{6} + 21 \, a^{6} b^{18} x^{5} + 35 \, a^{9} b^{15} x^{4} + 35 \, a^{12} b^{12} x^{3} + 21 \, a^{15} b^{9} x^{2} + 7 \, a^{18} b^{6} x + a^{21} b^{3}\right )}} \] Input:
integrate(1/(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^4,x, algorithm="fricas")
Output:
1/35*(140*a*b^15*x^5 - 2870*a^4*b^12*x^4 + 6020*a^7*b^9*x^3 - 1876*a^10*b^ 6*x^2 + 28*a^13*b^3*x - a^16 - 27*(20*a^2*b^14*x^4 - 168*a^5*b^11*x^3 + 18 9*a^8*b^8*x^2 - 28*a^11*b^5*x)*x^(2/3) - 3*(7*b^16*x^5 - 476*a^3*b^13*x^4 + 1932*a^6*b^10*x^3 - 1160*a^9*b^7*x^2 + 70*a^12*b^4*x)*x^(1/3))/(b^24*x^7 + 7*a^3*b^21*x^6 + 21*a^6*b^18*x^5 + 35*a^9*b^15*x^4 + 35*a^12*b^12*x^3 + 21*a^15*b^9*x^2 + 7*a^18*b^6*x + a^21*b^3)
Leaf count of result is larger than twice the leaf count of optimal. 309 vs. \(2 (51) = 102\).
Time = 1.56 (sec) , antiderivative size = 309, normalized size of antiderivative = 5.52 \[ \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^4} \, dx=\begin {cases} - \frac {a^{2}}{35 a^{7} b^{3} + 245 a^{6} b^{4} \sqrt [3]{x} + 735 a^{5} b^{5} x^{\frac {2}{3}} + 1225 a^{4} b^{6} x + 1225 a^{3} b^{7} x^{\frac {4}{3}} + 735 a^{2} b^{8} x^{\frac {5}{3}} + 245 a b^{9} x^{2} + 35 b^{10} x^{\frac {7}{3}}} - \frac {7 a b \sqrt [3]{x}}{35 a^{7} b^{3} + 245 a^{6} b^{4} \sqrt [3]{x} + 735 a^{5} b^{5} x^{\frac {2}{3}} + 1225 a^{4} b^{6} x + 1225 a^{3} b^{7} x^{\frac {4}{3}} + 735 a^{2} b^{8} x^{\frac {5}{3}} + 245 a b^{9} x^{2} + 35 b^{10} x^{\frac {7}{3}}} - \frac {21 b^{2} x^{\frac {2}{3}}}{35 a^{7} b^{3} + 245 a^{6} b^{4} \sqrt [3]{x} + 735 a^{5} b^{5} x^{\frac {2}{3}} + 1225 a^{4} b^{6} x + 1225 a^{3} b^{7} x^{\frac {4}{3}} + 735 a^{2} b^{8} x^{\frac {5}{3}} + 245 a b^{9} x^{2} + 35 b^{10} x^{\frac {7}{3}}} & \text {for}\: b \neq 0 \\\frac {x}{a^{8}} & \text {otherwise} \end {cases} \] Input:
integrate(1/(a**2+2*a*b*x**(1/3)+b**2*x**(2/3))**4,x)
Output:
Piecewise((-a**2/(35*a**7*b**3 + 245*a**6*b**4*x**(1/3) + 735*a**5*b**5*x* *(2/3) + 1225*a**4*b**6*x + 1225*a**3*b**7*x**(4/3) + 735*a**2*b**8*x**(5/ 3) + 245*a*b**9*x**2 + 35*b**10*x**(7/3)) - 7*a*b*x**(1/3)/(35*a**7*b**3 + 245*a**6*b**4*x**(1/3) + 735*a**5*b**5*x**(2/3) + 1225*a**4*b**6*x + 1225 *a**3*b**7*x**(4/3) + 735*a**2*b**8*x**(5/3) + 245*a*b**9*x**2 + 35*b**10* x**(7/3)) - 21*b**2*x**(2/3)/(35*a**7*b**3 + 245*a**6*b**4*x**(1/3) + 735* a**5*b**5*x**(2/3) + 1225*a**4*b**6*x + 1225*a**3*b**7*x**(4/3) + 735*a**2 *b**8*x**(5/3) + 245*a*b**9*x**2 + 35*b**10*x**(7/3)), Ne(b, 0)), (x/a**8, True))
Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (46) = 92\).
Time = 0.03 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.79 \[ \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^4} \, dx=-\frac {21 \, b^{2} x^{\frac {2}{3}} + 7 \, a b x^{\frac {1}{3}} + a^{2}}{35 \, {\left (b^{10} x^{\frac {7}{3}} + 7 \, a b^{9} x^{2} + 21 \, a^{2} b^{8} x^{\frac {5}{3}} + 35 \, a^{3} b^{7} x^{\frac {4}{3}} + 35 \, a^{4} b^{6} x + 21 \, a^{5} b^{5} x^{\frac {2}{3}} + 7 \, a^{6} b^{4} x^{\frac {1}{3}} + a^{7} b^{3}\right )}} \] Input:
integrate(1/(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^4,x, algorithm="maxima")
Output:
-1/35*(21*b^2*x^(2/3) + 7*a*b*x^(1/3) + a^2)/(b^10*x^(7/3) + 7*a*b^9*x^2 + 21*a^2*b^8*x^(5/3) + 35*a^3*b^7*x^(4/3) + 35*a^4*b^6*x + 21*a^5*b^5*x^(2/ 3) + 7*a^6*b^4*x^(1/3) + a^7*b^3)
Time = 0.12 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.59 \[ \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^4} \, dx=-\frac {21 \, b^{2} x^{\frac {2}{3}} + 7 \, a b x^{\frac {1}{3}} + a^{2}}{35 \, {\left (b x^{\frac {1}{3}} + a\right )}^{7} b^{3}} \] Input:
integrate(1/(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^4,x, algorithm="giac")
Output:
-1/35*(21*b^2*x^(2/3) + 7*a*b*x^(1/3) + a^2)/((b*x^(1/3) + a)^7*b^3)
Time = 0.12 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.80 \[ \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^4} \, dx=-\frac {\frac {a^2}{35\,b^3}+\frac {3\,x^{2/3}}{5\,b}+\frac {a\,x^{1/3}}{5\,b^2}}{a^7+b^7\,x^{7/3}+35\,a^4\,b^3\,x+7\,a\,b^6\,x^2+7\,a^6\,b\,x^{1/3}+21\,a^5\,b^2\,x^{2/3}+35\,a^3\,b^4\,x^{4/3}+21\,a^2\,b^5\,x^{5/3}} \] Input:
int(1/(a^2 + b^2*x^(2/3) + 2*a*b*x^(1/3))^4,x)
Output:
-(a^2/(35*b^3) + (3*x^(2/3))/(5*b) + (a*x^(1/3))/(5*b^2))/(a^7 + b^7*x^(7/ 3) + 35*a^4*b^3*x + 7*a*b^6*x^2 + 7*a^6*b*x^(1/3) + 21*a^5*b^2*x^(2/3) + 3 5*a^3*b^4*x^(4/3) + 21*a^2*b^5*x^(5/3))
Time = 0.16 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.77 \[ \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^4} \, dx=\frac {-21 x^{\frac {2}{3}} b^{2}-7 x^{\frac {1}{3}} a b -a^{2}}{35 b^{3} \left (21 x^{\frac {2}{3}} a^{5} b^{2}+21 x^{\frac {5}{3}} a^{2} b^{5}+7 x^{\frac {1}{3}} a^{6} b +35 x^{\frac {4}{3}} a^{3} b^{4}+x^{\frac {7}{3}} b^{7}+a^{7}+35 a^{4} b^{3} x +7 a \,b^{6} x^{2}\right )} \] Input:
int(1/(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^4,x)
Output:
( - 21*x**(2/3)*b**2 - 7*x**(1/3)*a*b - a**2)/(35*b**3*(21*x**(2/3)*a**5*b **2 + 21*x**(2/3)*a**2*b**5*x + 7*x**(1/3)*a**6*b + 35*x**(1/3)*a**3*b**4* x + x**(1/3)*b**7*x**2 + a**7 + 35*a**4*b**3*x + 7*a*b**6*x**2))