Integrand size = 15, antiderivative size = 146 \[ \int \frac {3+2 x}{\left (729-64 x^6\right )^2} \, dx=\frac {1}{708588 (3-2 x)}+\frac {x}{236196 \left (9-6 x+4 x^2\right )}-\frac {3+x}{708588 \left (9+6 x+4 x^2\right )}-\frac {\arctan \left (\frac {3-4 x}{3 \sqrt {3}}\right )}{157464 \sqrt {3}}+\frac {\arctan \left (\frac {3+4 x}{3 \sqrt {3}}\right )}{1417176 \sqrt {3}}-\frac {\log (3-2 x)}{472392}+\frac {\log (3+2 x)}{4251528}-\frac {\log \left (9-6 x+4 x^2\right )}{8503056}+\frac {\log \left (9+6 x+4 x^2\right )}{944784} \] Output:
1/(2125764-1417176*x)+x/(944784*x^2-1417176*x+2125764)-(3+x)/(2834352*x^2+ 4251528*x+6377292)-1/472392*arctan(1/9*(3-4*x)*3^(1/2))*3^(1/2)+1/4251528* arctan(1/9*(3+4*x)*3^(1/2))*3^(1/2)-1/472392*ln(3-2*x)+1/4251528*ln(3+2*x) -1/8503056*ln(4*x^2-6*x+9)+1/944784*ln(4*x^2+6*x+9)
Time = 0.05 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.83 \[ \int \frac {3+2 x}{\left (729-64 x^6\right )^2} \, dx=\frac {\frac {1944 x}{243-162 x+108 x^2-72 x^3+48 x^4-32 x^5}+18 \sqrt {3} \arctan \left (\frac {-3+4 x}{3 \sqrt {3}}\right )+2 \sqrt {3} \arctan \left (\frac {3+4 x}{3 \sqrt {3}}\right )-18 \log (3-2 x)+2 \log (3+2 x)-\log \left (9-6 x+4 x^2\right )+9 \log \left (9+6 x+4 x^2\right )}{8503056} \] Input:
Integrate[(3 + 2*x)/(729 - 64*x^6)^2,x]
Output:
((1944*x)/(243 - 162*x + 108*x^2 - 72*x^3 + 48*x^4 - 32*x^5) + 18*Sqrt[3]* ArcTan[(-3 + 4*x)/(3*Sqrt[3])] + 2*Sqrt[3]*ArcTan[(3 + 4*x)/(3*Sqrt[3])] - 18*Log[3 - 2*x] + 2*Log[3 + 2*x] - Log[9 - 6*x + 4*x^2] + 9*Log[9 + 6*x + 4*x^2])/8503056
Time = 0.63 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2019, 2462, 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.
| \(\displaystyle \int \frac {2 x+3}{\left (729-64 x^6\right )^2} \, dx\) |
| \(\Big \downarrow \) 2019 |
| \(\displaystyle \int \frac {1}{(2 x+3) \left (-32 x^5+48 x^4-72 x^3+108 x^2-162 x+243\right )^2}dx\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {33-2 x}{2125764 \left (4 x^2-6 x+9\right )}+\frac {6 x+7}{708588 \left (4 x^2+6 x+9\right )}+\frac {3-x}{39366 \left (4 x^2-6 x+9\right )^2}+\frac {x}{39366 \left (4 x^2+6 x+9\right )^2}-\frac {1}{236196 (2 x-3)}+\frac {1}{2125764 (2 x+3)}+\frac {1}{354294 (2 x-3)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\arctan \left (\frac {3-4 x}{3 \sqrt {3}}\right )}{157464 \sqrt {3}}+\frac {\arctan \left (\frac {4 x+3}{3 \sqrt {3}}\right )}{1417176 \sqrt {3}}+\frac {x}{236196 \left (4 x^2-6 x+9\right )}-\frac {x+3}{708588 \left (4 x^2+6 x+9\right )}-\frac {\log \left (4 x^2-6 x+9\right )}{8503056}+\frac {\log \left (4 x^2+6 x+9\right )}{944784}+\frac {1}{708588 (3-2 x)}-\frac {\log (3-2 x)}{472392}+\frac {\log (2 x+3)}{4251528}\) |
Input:
Int[(3 + 2*x)/(729 - 64*x^6)^2,x]
Output:
1/(708588*(3 - 2*x)) + x/(236196*(9 - 6*x + 4*x^2)) - (3 + x)/(708588*(9 + 6*x + 4*x^2)) - ArcTan[(3 - 4*x)/(3*Sqrt[3])]/(157464*Sqrt[3]) + ArcTan[( 3 + 4*x)/(3*Sqrt[3])]/(1417176*Sqrt[3]) - Log[3 - 2*x]/472392 + Log[3 + 2* x]/4251528 - Log[9 - 6*x + 4*x^2]/8503056 + Log[9 + 6*x + 4*x^2]/944784
Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px , Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Time = 0.13 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.71
| method | result | size |
| risch | \(-\frac {x}{139968 \left (x^{5}-\frac {3}{2} x^{4}+\frac {9}{4} x^{3}-\frac {27}{8} x^{2}+\frac {81}{16} x -\frac {243}{32}\right )}-\frac {\ln \left (16 x^{2}-24 x +36\right )}{8503056}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (4 x -3\right ) \sqrt {3}}{9}\right )}{472392}-\frac {\ln \left (2 x -3\right )}{472392}+\frac {\ln \left (2 x +3\right )}{4251528}+\frac {\ln \left (4 x^{2}+6 x +9\right )}{944784}+\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (2 x +\frac {3}{2}\right ) \sqrt {3}}{9}\right )}{4251528}\) | \(104\) |
| default | \(\frac {x}{944784 x^{2}-1417176 x +2125764}-\frac {\ln \left (4 x^{2}-6 x +9\right )}{8503056}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x -6\right ) \sqrt {3}}{18}\right )}{472392}-\frac {1}{708588 \left (2 x -3\right )}-\frac {\ln \left (2 x -3\right )}{472392}+\frac {-\frac {x}{4}-\frac {3}{4}}{708588 x^{2}+1062882 x +1594323}+\frac {\ln \left (4 x^{2}+6 x +9\right )}{944784}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x +6\right ) \sqrt {3}}{18}\right )}{4251528}+\frac {\ln \left (2 x +3\right )}{4251528}\) | \(115\) |
| meijerg | \(-\frac {\left (-1\right )^{\frac {5}{6}} \left (\frac {4 x \left (-1\right )^{\frac {1}{6}}}{6-\frac {128 x^{6}}{243}}-\frac {5 x \left (-1\right )^{\frac {1}{6}} \left (\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )-\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )+\frac {\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3-\left (x^{6}\right )^{\frac {1}{6}}}\right )-\frac {\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3+\left (x^{6}\right )^{\frac {1}{6}}}\right )\right )}{6 \left (x^{6}\right )^{\frac {1}{6}}}\right )}{708588}-\frac {\left (-1\right )^{\frac {2}{3}} \left (\frac {4 x^{2} \left (-1\right )^{\frac {1}{3}}}{3 \left (3-\frac {64 x^{6}}{243}\right )}-\frac {2 x^{2} \left (-1\right )^{\frac {1}{3}} \left (\ln \left (1-\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )-\frac {\ln \left (1+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}+\frac {16 \left (x^{6}\right )^{\frac {2}{3}}}{81}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (x^{6}\right )^{\frac {1}{3}}}{9 \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}\right )\right )}{3 \left (x^{6}\right )^{\frac {1}{3}}}\right )}{708588}\) | \(242\) |
Input:
int((2*x+3)/(-64*x^6+729)^2,x,method=_RETURNVERBOSE)
Output:
-1/139968*x/(x^5-3/2*x^4+9/4*x^3-27/8*x^2+81/16*x-243/32)-1/8503056*ln(16* x^2-24*x+36)+1/472392*3^(1/2)*arctan(1/9*(4*x-3)*3^(1/2))-1/472392*ln(2*x- 3)+1/4251528*ln(2*x+3)+1/944784*ln(4*x^2+6*x+9)+1/4251528*3^(1/2)*arctan(2 /9*(2*x+3/2)*3^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (116) = 232\).
Time = 0.10 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.76 \[ \int \frac {3+2 x}{\left (729-64 x^6\right )^2} \, dx=\frac {2 \, \sqrt {3} {\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + 18 \, \sqrt {3} {\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) + 9 \, {\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )} \log \left (4 \, x^{2} + 6 \, x + 9\right ) - {\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )} \log \left (4 \, x^{2} - 6 \, x + 9\right ) + 2 \, {\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )} \log \left (2 \, x + 3\right ) - 18 \, {\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )} \log \left (2 \, x - 3\right ) - 1944 \, x}{8503056 \, {\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )}} \] Input:
integrate((3+2*x)/(-64*x^6+729)^2,x, algorithm="fricas")
Output:
1/8503056*(2*sqrt(3)*(32*x^5 - 48*x^4 + 72*x^3 - 108*x^2 + 162*x - 243)*ar ctan(1/9*sqrt(3)*(4*x + 3)) + 18*sqrt(3)*(32*x^5 - 48*x^4 + 72*x^3 - 108*x ^2 + 162*x - 243)*arctan(1/9*sqrt(3)*(4*x - 3)) + 9*(32*x^5 - 48*x^4 + 72* x^3 - 108*x^2 + 162*x - 243)*log(4*x^2 + 6*x + 9) - (32*x^5 - 48*x^4 + 72* x^3 - 108*x^2 + 162*x - 243)*log(4*x^2 - 6*x + 9) + 2*(32*x^5 - 48*x^4 + 7 2*x^3 - 108*x^2 + 162*x - 243)*log(2*x + 3) - 18*(32*x^5 - 48*x^4 + 72*x^3 - 108*x^2 + 162*x - 243)*log(2*x - 3) - 1944*x)/(32*x^5 - 48*x^4 + 72*x^3 - 108*x^2 + 162*x - 243)
Time = 0.36 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.85 \[ \int \frac {3+2 x}{\left (729-64 x^6\right )^2} \, dx=- \frac {x}{139968 x^{5} - 209952 x^{4} + 314928 x^{3} - 472392 x^{2} + 708588 x - 1062882} - \frac {\log {\left (x - \frac {3}{2} \right )}}{472392} + \frac {\log {\left (x + \frac {3}{2} \right )}}{4251528} - \frac {\log {\left (x^{2} - \frac {3 x}{2} + \frac {9}{4} \right )}}{8503056} + \frac {\log {\left (x^{2} + \frac {3 x}{2} + \frac {9}{4} \right )}}{944784} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {4 \sqrt {3} x}{9} - \frac {\sqrt {3}}{3} \right )}}{472392} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {4 \sqrt {3} x}{9} + \frac {\sqrt {3}}{3} \right )}}{4251528} \] Input:
integrate((3+2*x)/(-64*x**6+729)**2,x)
Output:
-x/(139968*x**5 - 209952*x**4 + 314928*x**3 - 472392*x**2 + 708588*x - 106 2882) - log(x - 3/2)/472392 + log(x + 3/2)/4251528 - log(x**2 - 3*x/2 + 9/ 4)/8503056 + log(x**2 + 3*x/2 + 9/4)/944784 + sqrt(3)*atan(4*sqrt(3)*x/9 - sqrt(3)/3)/472392 + sqrt(3)*atan(4*sqrt(3)*x/9 + sqrt(3)/3)/4251528
Time = 0.11 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.72 \[ \int \frac {3+2 x}{\left (729-64 x^6\right )^2} \, dx=\frac {1}{4251528} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + \frac {1}{472392} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) - \frac {x}{4374 \, {\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )}} + \frac {1}{944784} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac {1}{8503056} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac {1}{4251528} \, \log \left (2 \, x + 3\right ) - \frac {1}{472392} \, \log \left (2 \, x - 3\right ) \] Input:
integrate((3+2*x)/(-64*x^6+729)^2,x, algorithm="maxima")
Output:
1/4251528*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 1/472392*sqrt(3)*arctan( 1/9*sqrt(3)*(4*x - 3)) - 1/4374*x/(32*x^5 - 48*x^4 + 72*x^3 - 108*x^2 + 16 2*x - 243) + 1/944784*log(4*x^2 + 6*x + 9) - 1/8503056*log(4*x^2 - 6*x + 9 ) + 1/4251528*log(2*x + 3) - 1/472392*log(2*x - 3)
Time = 0.12 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.76 \[ \int \frac {3+2 x}{\left (729-64 x^6\right )^2} \, dx=\frac {1}{4251528} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + \frac {1}{472392} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) - \frac {x}{4374 \, {\left (4 \, x^{2} + 6 \, x + 9\right )} {\left (4 \, x^{2} - 6 \, x + 9\right )} {\left (2 \, x - 3\right )}} + \frac {1}{944784} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac {1}{8503056} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac {1}{4251528} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - \frac {1}{472392} \, \log \left ({\left | 2 \, x - 3 \right |}\right ) \] Input:
integrate((3+2*x)/(-64*x^6+729)^2,x, algorithm="giac")
Output:
1/4251528*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 1/472392*sqrt(3)*arctan( 1/9*sqrt(3)*(4*x - 3)) - 1/4374*x/((4*x^2 + 6*x + 9)*(4*x^2 - 6*x + 9)*(2* x - 3)) + 1/944784*log(4*x^2 + 6*x + 9) - 1/8503056*log(4*x^2 - 6*x + 9) + 1/4251528*log(abs(2*x + 3)) - 1/472392*log(abs(2*x - 3))
Time = 5.86 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.83 \[ \int \frac {3+2 x}{\left (729-64 x^6\right )^2} \, dx=\frac {\ln \left (x+\frac {3}{2}\right )}{4251528}-\frac {\ln \left (x-\frac {3}{2}\right )}{472392}-\ln \left (x-\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {1}{8503056}+\frac {\sqrt {3}\,1{}\mathrm {i}}{944784}\right )-\ln \left (x+\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {1}{944784}+\frac {\sqrt {3}\,1{}\mathrm {i}}{8503056}\right )+\ln \left (x-\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {1}{8503056}+\frac {\sqrt {3}\,1{}\mathrm {i}}{944784}\right )+\ln \left (x+\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {1}{944784}+\frac {\sqrt {3}\,1{}\mathrm {i}}{8503056}\right )-\frac {x}{139968\,\left (x^5-\frac {3\,x^4}{2}+\frac {9\,x^3}{4}-\frac {27\,x^2}{8}+\frac {81\,x}{16}-\frac {243}{32}\right )} \] Input:
int((2*x + 3)/(64*x^6 - 729)^2,x)
Output:
log(x + 3/2)/4251528 - log(x - 3/2)/472392 - log(x - (3^(1/2)*3i)/4 - 3/4) *((3^(1/2)*1i)/944784 + 1/8503056) - log(x - (3^(1/2)*3i)/4 + 3/4)*((3^(1/ 2)*1i)/8503056 - 1/944784) + log(x + (3^(1/2)*3i)/4 - 3/4)*((3^(1/2)*1i)/9 44784 - 1/8503056) + log(x + (3^(1/2)*3i)/4 + 3/4)*((3^(1/2)*1i)/8503056 + 1/944784) - x/(139968*((81*x)/16 - (27*x^2)/8 + (9*x^3)/4 - (3*x^4)/2 + x ^5 - 243/32))
Time = 0.15 (sec) , antiderivative size = 554, normalized size of antiderivative = 3.79 \[ \int \frac {3+2 x}{\left (729-64 x^6\right )^2} \, dx =\text {Too large to display} \] Input:
int((3+2*x)/(-64*x^6+729)^2,x)
Output:
(576*sqrt(3)*atan((4*x - 3)/(3*sqrt(3)))*x**5 - 864*sqrt(3)*atan((4*x - 3) /(3*sqrt(3)))*x**4 + 1296*sqrt(3)*atan((4*x - 3)/(3*sqrt(3)))*x**3 - 1944* sqrt(3)*atan((4*x - 3)/(3*sqrt(3)))*x**2 + 2916*sqrt(3)*atan((4*x - 3)/(3* sqrt(3)))*x - 4374*sqrt(3)*atan((4*x - 3)/(3*sqrt(3))) + 64*sqrt(3)*atan(( 4*x + 3)/(3*sqrt(3)))*x**5 - 96*sqrt(3)*atan((4*x + 3)/(3*sqrt(3)))*x**4 + 144*sqrt(3)*atan((4*x + 3)/(3*sqrt(3)))*x**3 - 216*sqrt(3)*atan((4*x + 3) /(3*sqrt(3)))*x**2 + 324*sqrt(3)*atan((4*x + 3)/(3*sqrt(3)))*x - 486*sqrt( 3)*atan((4*x + 3)/(3*sqrt(3))) - 32*log(4*x**2 - 6*x + 9)*x**5 + 48*log(4* x**2 - 6*x + 9)*x**4 - 72*log(4*x**2 - 6*x + 9)*x**3 + 108*log(4*x**2 - 6* x + 9)*x**2 - 162*log(4*x**2 - 6*x + 9)*x + 243*log(4*x**2 - 6*x + 9) + 28 8*log(4*x**2 + 6*x + 9)*x**5 - 432*log(4*x**2 + 6*x + 9)*x**4 + 648*log(4* x**2 + 6*x + 9)*x**3 - 972*log(4*x**2 + 6*x + 9)*x**2 + 1458*log(4*x**2 + 6*x + 9)*x - 2187*log(4*x**2 + 6*x + 9) - 576*log(2*x - 3)*x**5 + 864*log( 2*x - 3)*x**4 - 1296*log(2*x - 3)*x**3 + 1944*log(2*x - 3)*x**2 - 2916*log (2*x - 3)*x + 4374*log(2*x - 3) + 64*log(2*x + 3)*x**5 - 96*log(2*x + 3)*x **4 + 144*log(2*x + 3)*x**3 - 216*log(2*x + 3)*x**2 + 324*log(2*x + 3)*x - 486*log(2*x + 3) - 1944*x)/(8503056*(32*x**5 - 48*x**4 + 72*x**3 - 108*x* *2 + 162*x - 243))