Integrand size = 43, antiderivative size = 93 \[ \int \frac {\left (-\sqrt [4]{2}+\sqrt {3} x+\sqrt [4]{2} x^2\right )^2}{\left (\sqrt {3}+2 \sqrt [4]{2} x\right )^3} \, dx=-\frac {41+24 \sqrt {2}}{64\ 2^{3/4} \left (\sqrt {3}+2 \sqrt [4]{2} x\right )^2}+\frac {\left (\sqrt {3}+2 \sqrt [4]{2} x\right )^2}{64\ 2^{3/4}}-\frac {\left (3+4 \sqrt {2}\right ) \log \left (\sqrt {3}+2 \sqrt [4]{2} x\right )}{16\ 2^{3/4}} \] Output:
-1/128*(41+24*2^(1/2))*2^(1/4)/(3^(1/2)+2*2^(1/4)*x)^2+1/128*(3^(1/2)+2*2^ (1/4)*x)^2*2^(1/4)-1/32*(3+4*2^(1/2))*ln(3^(1/2)+2*2^(1/4)*x)*2^(1/4)
Time = 0.18 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.38 \[ \int \frac {\left (-\sqrt [4]{2}+\sqrt {3} x+\sqrt [4]{2} x^2\right )^2}{\left (\sqrt {3}+2 \sqrt [4]{2} x\right )^3} \, dx=\frac {1}{128} \left (\frac {8 \sqrt {3} \left (2833+1968 \sqrt {2}\right ) x}{3936+2833 \sqrt {2}}+4\ 2^{3/4} x^2-\frac {2^{3/4} \left (210617+148680 \sqrt {2}\right )}{\left (3936+2833 \sqrt {2}\right ) \left (\sqrt {3}+2 \sqrt [4]{2} x\right )^2}-\frac {4\ 2^{3/4} \left (24243+17236 \sqrt {2}\right ) \log \left (\sqrt {3}+2 \sqrt [4]{2} x\right )}{3936+2833 \sqrt {2}}\right ) \] Input:
Integrate[(-2^(1/4) + Sqrt[3]*x + 2^(1/4)*x^2)^2/(Sqrt[3] + 2*2^(1/4)*x)^3 ,x]
Output:
((8*Sqrt[3]*(2833 + 1968*Sqrt[2])*x)/(3936 + 2833*Sqrt[2]) + 4*2^(3/4)*x^2 - (2^(3/4)*(210617 + 148680*Sqrt[2]))/((3936 + 2833*Sqrt[2])*(Sqrt[3] + 2 *2^(1/4)*x)^2) - (4*2^(3/4)*(24243 + 17236*Sqrt[2])*Log[Sqrt[3] + 2*2^(1/4 )*x])/(3936 + 2833*Sqrt[2]))/128
Time = 0.54 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {1107, 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.
| \(\displaystyle \int \frac {\left (\sqrt [4]{2} x^2+\sqrt {3} x-\sqrt [4]{2}\right )^2}{\left (2 \sqrt [4]{2} x+\sqrt {3}\right )^3} \, dx\) |
| \(\Big \downarrow \) 1107 |
| \(\displaystyle \int \left (\frac {2 \sqrt [4]{2} x+\sqrt {3}}{16 \sqrt {2}}+\frac {-8-3 \sqrt {2}}{16 \left (2 \sqrt [4]{2} x+\sqrt {3}\right )}+\frac {48+41 \sqrt {2}}{32 \left (2 \sqrt [4]{2} x+\sqrt {3}\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (2 \sqrt [4]{2} x+\sqrt {3}\right )^2}{64\ 2^{3/4}}-\frac {41+24 \sqrt {2}}{64\ 2^{3/4} \left (2 \sqrt [4]{2} x+\sqrt {3}\right )^2}-\frac {\left (3+4 \sqrt {2}\right ) \log \left (2 \sqrt [4]{2} x+\sqrt {3}\right )}{16\ 2^{3/4}}\) |
Input:
Int[(-2^(1/4) + Sqrt[3]*x + 2^(1/4)*x^2)^2/(Sqrt[3] + 2*2^(1/4)*x)^3,x]
Output:
-1/64*(41 + 24*Sqrt[2])/(2^(3/4)*(Sqrt[3] + 2*2^(1/4)*x)^2) + (Sqrt[3] + 2 *2^(1/4)*x)^2/(64*2^(3/4)) - ((3 + 4*Sqrt[2])*Log[Sqrt[3] + 2*2^(1/4)*x])/ (16*2^(3/4))
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; F reeQ[{a, b, c, d, e, m}, x] && EqQ[2*c*d - b*e, 0] && IGtQ[p, 0] && !(EqQ[ m, 3] && NeQ[p, 1])
Time = 0.50 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.73
| method | result | size |
| default | \(\frac {2^{\frac {3}{4}} x^{2}}{32}+\frac {\sqrt {2}\, \sqrt {3}\, x}{32}-\frac {2^{\frac {1}{4}} \left (41 \sqrt {2}+48\right )}{64 \left (4 x +\sqrt {3}\, 2^{\frac {3}{4}}\right )^{2}}-\frac {2^{\frac {1}{4}} \left (3+4 \sqrt {2}\right ) \ln \left (4 x +\sqrt {3}\, 2^{\frac {3}{4}}\right )}{32}\) | \(68\) |
| risch | \(\frac {\sqrt {2}\, \sqrt {3}\, x}{32}+\frac {2^{\frac {3}{4}} x^{2}}{32}-\frac {41 \,2^{\frac {3}{4}}}{1024 \left (x +\frac {\sqrt {3}\, 2^{\frac {3}{4}}}{4}\right )^{2}}-\frac {3 \,2^{\frac {1}{4}}}{64 \left (x +\frac {\sqrt {3}\, 2^{\frac {3}{4}}}{4}\right )^{2}}-\frac {\ln \left (4 x +\sqrt {3}\, 2^{\frac {3}{4}}\right ) 2^{\frac {3}{4}}}{8}-\frac {3 \ln \left (4 x +\sqrt {3}\, 2^{\frac {3}{4}}\right ) 2^{\frac {1}{4}}}{32}\) | \(87\) |
| meijerg | \(\frac {3 \,2^{\frac {1}{4}} \left (-\frac {2 x 2^{\frac {1}{4}} \sqrt {3}\, \left (-\frac {40 \,2^{\frac {3}{4}} \sqrt {3}\, x^{3}}{9}+\frac {80 \sqrt {2}\, x^{2}}{3}+60 \,2^{\frac {1}{4}} \sqrt {3}\, x +60\right )}{15 \left (1+\frac {2 \,2^{\frac {1}{4}} \sqrt {3}\, x}{3}\right )^{2}}+12 \ln \left (1+\frac {2 \,2^{\frac {1}{4}} \sqrt {3}\, x}{3}\right )\right )}{128}+\frac {2^{\frac {1}{4}} \left (\sqrt {3}+\sqrt {3+4 \sqrt {2}}\right ) \sqrt {3}\, \left (\frac {x \left (\frac {16 \sqrt {2}\, x^{2}}{3}+12 \,2^{\frac {1}{4}} \sqrt {3}\, x +12\right ) 2^{\frac {1}{4}} \sqrt {3}}{3 \left (1+\frac {2 \,2^{\frac {1}{4}} \sqrt {3}\, x}{3}\right )^{2}}-6 \ln \left (1+\frac {2 \,2^{\frac {1}{4}} \sqrt {3}\, x}{3}\right )\right )}{64}+\frac {2^{\frac {1}{4}} \left (\sqrt {3}+\sqrt {3+4 \sqrt {2}}\right )^{2} \left (-\frac {2 \,2^{\frac {1}{4}} \sqrt {3}\, x \left (6 \,2^{\frac {1}{4}} \sqrt {3}\, x +6\right )}{9 \left (1+\frac {2 \,2^{\frac {1}{4}} \sqrt {3}\, x}{3}\right )^{2}}+2 \ln \left (1+\frac {2 \,2^{\frac {1}{4}} \sqrt {3}\, x}{3}\right )\right )}{128}+\frac {2^{\frac {1}{4}} \left (\sqrt {3}-\sqrt {3+4 \sqrt {2}}\right ) \sqrt {3}\, \left (\frac {x \left (\frac {16 \sqrt {2}\, x^{2}}{3}+12 \,2^{\frac {1}{4}} \sqrt {3}\, x +12\right ) 2^{\frac {1}{4}} \sqrt {3}}{3 \left (1+\frac {2 \,2^{\frac {1}{4}} \sqrt {3}\, x}{3}\right )^{2}}-6 \ln \left (1+\frac {2 \,2^{\frac {1}{4}} \sqrt {3}\, x}{3}\right )\right )}{64}+\frac {9 \,2^{\frac {1}{4}} \left (\frac {\sqrt {3}}{18}-\frac {\sqrt {3+4 \sqrt {2}}}{18}\right ) \left (\sqrt {3}+\sqrt {3+4 \sqrt {2}}\right ) \left (-\frac {2 \,2^{\frac {1}{4}} \sqrt {3}\, x \left (6 \,2^{\frac {1}{4}} \sqrt {3}\, x +6\right )}{9 \left (1+\frac {2 \,2^{\frac {1}{4}} \sqrt {3}\, x}{3}\right )^{2}}+2 \ln \left (1+\frac {2 \,2^{\frac {1}{4}} \sqrt {3}\, x}{3}\right )\right )}{16}+\frac {2^{\frac {3}{4}} \left (\sqrt {3}-\sqrt {3+4 \sqrt {2}}\right ) \left (\sqrt {3}+\sqrt {3+4 \sqrt {2}}\right )^{2} \sqrt {3}\, x^{2}}{144 \left (1+\frac {2 \,2^{\frac {1}{4}} \sqrt {3}\, x}{3}\right )^{2}}+\frac {2^{\frac {1}{4}} \left (\sqrt {3}-\sqrt {3+4 \sqrt {2}}\right )^{2} \left (-\frac {2 \,2^{\frac {1}{4}} \sqrt {3}\, x \left (6 \,2^{\frac {1}{4}} \sqrt {3}\, x +6\right )}{9 \left (1+\frac {2 \,2^{\frac {1}{4}} \sqrt {3}\, x}{3}\right )^{2}}+2 \ln \left (1+\frac {2 \,2^{\frac {1}{4}} \sqrt {3}\, x}{3}\right )\right )}{128}+\frac {2^{\frac {3}{4}} \left (\sqrt {3}-\sqrt {3+4 \sqrt {2}}\right )^{2} \left (\sqrt {3}+\sqrt {3+4 \sqrt {2}}\right ) \sqrt {3}\, x^{2}}{144 \left (1+\frac {2 \,2^{\frac {1}{4}} \sqrt {3}\, x}{3}\right )^{2}}+\frac {\sqrt {6}\, \left (\sqrt {3}-\sqrt {3+4 \sqrt {2}}\right )^{2} \left (\sqrt {3}+\sqrt {3+4 \sqrt {2}}\right )^{2} x \left (\frac {2 \,2^{\frac {1}{4}} \sqrt {3}\, x}{3}+2\right )}{576 \left (1+\frac {2 \,2^{\frac {1}{4}} \sqrt {3}\, x}{3}\right )^{2}}\) | \(621\) |
Input:
int((-2^(1/4)+3^(1/2)*x+2^(1/4)*x^2)^2/(3^(1/2)+2*2^(1/4)*x)^3,x,method=_R ETURNVERBOSE)
Output:
1/32*2^(3/4)*x^2+1/32*2^(1/2)*3^(1/2)*x-1/64*2^(1/4)*(41*2^(1/2)+48)/(4*x+ 3^(1/2)*2^(3/4))^2-1/32*2^(1/4)*(3+4*2^(1/2))*ln(4*x+3^(1/2)*2^(3/4))
Leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (65) = 130\).
Time = 0.07 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.69 \[ \int \frac {\left (-\sqrt [4]{2}+\sqrt {3} x+\sqrt [4]{2} x^2\right )^2}{\left (\sqrt {3}+2 \sqrt [4]{2} x\right )^3} \, dx=-\frac {4 \, {\left (4 \cdot 2^{\frac {3}{4}} {\left (1024 \, x^{8} - 576 \, x^{4} + 81\right )} + 3 \cdot 2^{\frac {1}{4}} {\left (1024 \, x^{8} - 576 \, x^{4} + 81\right )}\right )} \log \left (\sqrt {3} 2^{\frac {3}{4}} + 4 \, x\right ) - 8 \cdot 2^{\frac {3}{4}} {\left (512 \, x^{10} - 944 \, x^{6} - 864 \, x^{4} - 513 \, x^{2} - 81\right )} - 8 \, \sqrt {3} {\left (768 \, x^{5} + 984 \, x^{3} + \sqrt {2} {\left (512 \, x^{9} + 368 \, x^{5} + 576 \, x^{3} + 225 \, x\right )} + 216 \, x\right )} + 3 \cdot 2^{\frac {1}{4}} {\left (2048 \, x^{6} + 3936 \, x^{4} + 1728 \, x^{2} + 369\right )}}{128 \, {\left (1024 \, x^{8} - 576 \, x^{4} + 81\right )}} \] Input:
integrate((-2^(1/4)+3^(1/2)*x+x^2*2^(1/4))^2/(3^(1/2)+2*2^(1/4)*x)^3,x, al gorithm="fricas")
Output:
-1/128*(4*(4*2^(3/4)*(1024*x^8 - 576*x^4 + 81) + 3*2^(1/4)*(1024*x^8 - 576 *x^4 + 81))*log(sqrt(3)*2^(3/4) + 4*x) - 8*2^(3/4)*(512*x^10 - 944*x^6 - 8 64*x^4 - 513*x^2 - 81) - 8*sqrt(3)*(768*x^5 + 984*x^3 + sqrt(2)*(512*x^9 + 368*x^5 + 576*x^3 + 225*x) + 216*x) + 3*2^(1/4)*(2048*x^6 + 3936*x^4 + 17 28*x^2 + 369))/(1024*x^8 - 576*x^4 + 81)
Time = 0.46 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.99 \[ \int \frac {\left (-\sqrt [4]{2}+\sqrt {3} x+\sqrt [4]{2} x^2\right )^2}{\left (\sqrt {3}+2 \sqrt [4]{2} x\right )^3} \, dx=\frac {2^{\frac {3}{4}} x^{2}}{32} + \frac {\sqrt {6} x}{32} + \frac {\sqrt [4]{2} \left (- 4 \sqrt {2} - 3\right ) \log {\left (16 \cdot 2^{\frac {3}{4}} x + 8 \sqrt {6} \right )}}{32} + \frac {- 328 \cdot 2^{\frac {3}{4}} - 384 \cdot \sqrt [4]{2}}{8192 x^{2} + 4096 \cdot 2^{\frac {3}{4}} \sqrt {3} x + 3072 \sqrt {2}} \] Input:
integrate((-2**(1/4)+3**(1/2)*x+x**2*2**(1/4))**2/(3**(1/2)+2*2**(1/4)*x)* *3,x)
Output:
2**(3/4)*x**2/32 + sqrt(6)*x/32 + 2**(1/4)*(-4*sqrt(2) - 3)*log(16*2**(3/4 )*x + 8*sqrt(6))/32 + (-328*2**(3/4) - 384*2**(1/4))/(8192*x**2 + 4096*2** (3/4)*sqrt(3)*x + 3072*sqrt(2))
Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.78 \[ \int \frac {\left (-\sqrt [4]{2}+\sqrt {3} x+\sqrt [4]{2} x^2\right )^2}{\left (\sqrt {3}+2 \sqrt [4]{2} x\right )^3} \, dx=-\frac {1}{32} \cdot 2^{\frac {1}{4}} {\left (4 \, \sqrt {2} + 3\right )} \log \left (2 \cdot 2^{\frac {1}{4}} x + \sqrt {3}\right ) + \frac {1}{32} \, \sqrt {2} {\left (2^{\frac {1}{4}} x^{2} + \sqrt {3} x\right )} - \frac {24 \, \sqrt {2} + 41}{64 \, {\left (8 \cdot 2^{\frac {1}{4}} x^{2} + 8 \, \sqrt {3} x + 3 \cdot 2^{\frac {3}{4}}\right )}} \] Input:
integrate((-2^(1/4)+3^(1/2)*x+x^2*2^(1/4))^2/(3^(1/2)+2*2^(1/4)*x)^3,x, al gorithm="maxima")
Output:
-1/32*2^(1/4)*(4*sqrt(2) + 3)*log(2*2^(1/4)*x + sqrt(3)) + 1/32*sqrt(2)*(2 ^(1/4)*x^2 + sqrt(3)*x) - 1/64*(24*sqrt(2) + 41)/(8*2^(1/4)*x^2 + 8*sqrt(3 )*x + 3*2^(3/4))
Exception generated. \[ \int \frac {\left (-\sqrt [4]{2}+\sqrt {3} x+\sqrt [4]{2} x^2\right )^2}{\left (\sqrt {3}+2 \sqrt [4]{2} x\right )^3} \, dx=\text {Exception raised: NotImplementedError} \] Input:
integrate((-2^(1/4)+3^(1/2)*x+x^2*2^(1/4))^2/(3^(1/2)+2*2^(1/4)*x)^3,x, al gorithm="giac")
Output:
Exception raised: NotImplementedError >> unable to parse Giac output: (roo tof([[3107414276067094328724118783432456182672461636123978958372864,0,-362 51401980421570635609481139600442826134612068444647200915456,0,144735628146 4232130181365104903
Time = 11.72 (sec) , antiderivative size = 483, normalized size of antiderivative = 5.19 \[ \int \frac {\left (-\sqrt [4]{2}+\sqrt {3} x+\sqrt [4]{2} x^2\right )^2}{\left (\sqrt {3}+2 \sqrt [4]{2} x\right )^3} \, dx=x\,\left (\frac {\sqrt {2}\,\sqrt {3}}{8}-\frac {3\,\sqrt {6}}{32}\right )+\frac {2^{3/4}\,x^2}{32}+\left (\sum _{k=1}^3\ln \left (\frac {9\,2^{1/4}\,\sqrt {3}}{1024}+\frac {27\,2^{3/4}\,\sqrt {3}}{8192}+\frac {207\,2^{1/4}\,\sqrt {6}}{32768}+\frac {23\,2^{3/4}\,\sqrt {6}}{8192}+\mathrm {root}\left (127401984\,2^{3/4}\,\sqrt {3}\,\sqrt {6}\,z^3-764411904\,2^{1/4}\,z^3+31850496\,\sqrt {2}\,\sqrt {3}\,\sqrt {6}\,z^2+23887872\,\sqrt {3}\,\sqrt {6}\,z^2-71663616\,\sqrt {2}\,z^2-191102976\,z^2+5971968\,2^{1/4}\,\sqrt {3}\,\sqrt {6}\,z-17915904\,2^{3/4}\,z-127800\,\sqrt {2}\,\sqrt {3}\,\sqrt {6}+260352\,\sqrt {3}\,\sqrt {6}-186336\,\sqrt {2}+1609268,z,k\right )\,\left (\frac {27\,\sqrt {2}\,\sqrt {3}}{128}-\frac {9\,\sqrt {2}\,\sqrt {6}}{128}+x\,\left (\frac {9\,2^{1/4}}{32}+\frac {15\,2^{3/4}}{512}+\frac {9\,2^{1/4}\,\sqrt {3}\,\sqrt {6}}{128}\right )+\mathrm {root}\left (127401984\,2^{3/4}\,\sqrt {3}\,\sqrt {6}\,z^3-764411904\,2^{1/4}\,z^3+31850496\,\sqrt {2}\,\sqrt {3}\,\sqrt {6}\,z^2+23887872\,\sqrt {3}\,\sqrt {6}\,z^2-71663616\,\sqrt {2}\,z^2-191102976\,z^2+5971968\,2^{1/4}\,\sqrt {3}\,\sqrt {6}\,z-17915904\,2^{3/4}\,z-127800\,\sqrt {2}\,\sqrt {3}\,\sqrt {6}+260352\,\sqrt {3}\,\sqrt {6}-186336\,\sqrt {2}+1609268,z,k\right )\,\left (\frac {27\,2^{1/4}\,\sqrt {3}}{16}-\frac {27\,2^{3/4}\,\sqrt {6}}{32}\right )+\frac {9\,\sqrt {3}}{32}-\frac {93\,\sqrt {6}}{1024}\right )+x\,\left (\frac {3\,\sqrt {3}\,\sqrt {6}}{512}+\frac {23\,\sqrt {2}}{2048}-\frac {27\,\sqrt {2}\,\sqrt {3}\,\sqrt {6}}{4096}+\frac {639}{8192}\right )\right )\,\mathrm {root}\left (127401984\,2^{3/4}\,\sqrt {3}\,\sqrt {6}\,z^3-764411904\,2^{1/4}\,z^3+31850496\,\sqrt {2}\,\sqrt {3}\,\sqrt {6}\,z^2+23887872\,\sqrt {3}\,\sqrt {6}\,z^2-71663616\,\sqrt {2}\,z^2-191102976\,z^2+5971968\,2^{1/4}\,\sqrt {3}\,\sqrt {6}\,z-17915904\,2^{3/4}\,z-127800\,\sqrt {2}\,\sqrt {3}\,\sqrt {6}+260352\,\sqrt {3}\,\sqrt {6}-186336\,\sqrt {2}+1609268,z,k\right )\right ) \] Input:
int((3^(1/2)*x - 2^(1/4) + 2^(1/4)*x^2)^2/(2*2^(1/4)*x + 3^(1/2))^3,x)
Output:
x*((2^(1/2)*3^(1/2))/8 - (3*6^(1/2))/32) + (2^(3/4)*x^2)/32 + symsum(log(( 9*2^(1/4)*3^(1/2))/1024 + (27*2^(3/4)*3^(1/2))/8192 + (207*2^(1/4)*6^(1/2) )/32768 + (23*2^(3/4)*6^(1/2))/8192 + root(127401984*2^(3/4)*3^(1/2)*6^(1/ 2)*z^3 - 764411904*2^(1/4)*z^3 + 31850496*2^(1/2)*3^(1/2)*6^(1/2)*z^2 + 23 887872*3^(1/2)*6^(1/2)*z^2 - 71663616*2^(1/2)*z^2 - 191102976*z^2 + 597196 8*2^(1/4)*3^(1/2)*6^(1/2)*z - 17915904*2^(3/4)*z - 127800*2^(1/2)*3^(1/2)* 6^(1/2) + 260352*3^(1/2)*6^(1/2) - 186336*2^(1/2) + 1609268, z, k)*((27*2^ (1/2)*3^(1/2))/128 - (9*2^(1/2)*6^(1/2))/128 + x*((9*2^(1/4))/32 + (15*2^( 3/4))/512 + (9*2^(1/4)*3^(1/2)*6^(1/2))/128) + root(127401984*2^(3/4)*3^(1 /2)*6^(1/2)*z^3 - 764411904*2^(1/4)*z^3 + 31850496*2^(1/2)*3^(1/2)*6^(1/2) *z^2 + 23887872*3^(1/2)*6^(1/2)*z^2 - 71663616*2^(1/2)*z^2 - 191102976*z^2 + 5971968*2^(1/4)*3^(1/2)*6^(1/2)*z - 17915904*2^(3/4)*z - 127800*2^(1/2) *3^(1/2)*6^(1/2) + 260352*3^(1/2)*6^(1/2) - 186336*2^(1/2) + 1609268, z, k )*((27*2^(1/4)*3^(1/2))/16 - (27*2^(3/4)*6^(1/2))/32) + (9*3^(1/2))/32 - ( 93*6^(1/2))/1024) + x*((3*3^(1/2)*6^(1/2))/512 + (23*2^(1/2))/2048 - (27*2 ^(1/2)*3^(1/2)*6^(1/2))/4096 + 639/8192))*root(127401984*2^(3/4)*3^(1/2)*6 ^(1/2)*z^3 - 764411904*2^(1/4)*z^3 + 31850496*2^(1/2)*3^(1/2)*6^(1/2)*z^2 + 23887872*3^(1/2)*6^(1/2)*z^2 - 71663616*2^(1/2)*z^2 - 191102976*z^2 + 59 71968*2^(1/4)*3^(1/2)*6^(1/2)*z - 17915904*2^(3/4)*z - 127800*2^(1/2)*3^(1 /2)*6^(1/2) + 260352*3^(1/2)*6^(1/2) - 186336*2^(1/2) + 1609268, z, k),...
Time = 0.32 (sec) , antiderivative size = 1488, normalized size of antiderivative = 16.00 \[ \int \frac {\left (-\sqrt [4]{2}+\sqrt {3} x+\sqrt [4]{2} x^2\right )^2}{\left (\sqrt {3}+2 \sqrt [4]{2} x\right )^3} \, dx =\text {Too large to display} \] Input:
int((-2^(1/4)+3^(1/2)*x+x^2*2^(1/4))^2/(3^(1/2)+2*2^(1/4)*x)^3,x)
Output:
(4026531840*sqrt(6)*x**21 - 503316480*sqrt(6)*x**17 + 4529848320*sqrt(6)*x **15 + 283115520*sqrt(6)*x**13 - 3822059520*sqrt(6)*x**11 - 895795200*sqrt (6)*x**9 + 1074954240*sqrt(6)*x**7 + 355518720*sqrt(6)*x**5 - 100776960*sq rt(6)*x**3 - 39366000*sqrt(6)*x + 6039797760*sqrt(3)*x**17 + 7738490880*sq rt(3)*x**15 - 3397386240*sqrt(3)*x**13 - 6529351680*sqrt(3)*x**11 + 183638 0160*sqrt(3)*x**7 + 268738560*sqrt(3)*x**5 - 172160640*sqrt(3)*x**3 - 3779 1360*sqrt(3)*x - 2516582400*sqrt(2)*2**(3/4)*log(128*sqrt(2)*x**6 + 108*sq rt(2)*x**2 - 288*x**4 - 27)*x**20 + 3538944000*sqrt(2)*2**(3/4)*log(128*sq rt(2)*x**6 + 108*sqrt(2)*x**2 - 288*x**4 - 27)*x**16 - 1990656000*sqrt(2)* 2**(3/4)*log(128*sqrt(2)*x**6 + 108*sqrt(2)*x**2 - 288*x**4 - 27)*x**12 + 559872000*sqrt(2)*2**(3/4)*log(128*sqrt(2)*x**6 + 108*sqrt(2)*x**2 - 288*x **4 - 27)*x**8 - 78732000*sqrt(2)*2**(3/4)*log(128*sqrt(2)*x**6 + 108*sqrt (2)*x**2 - 288*x**4 - 27)*x**4 + 4428675*sqrt(2)*2**(3/4)*log(128*sqrt(2)* x**6 + 108*sqrt(2)*x**2 - 288*x**4 - 27) - 2063597568*sqrt(2)*2**(3/4)*x** 20 - 2548039680*sqrt(2)*2**(3/4)*x**14 + 1088225280*sqrt(2)*2**(3/4)*x**12 + 2149908480*sqrt(2)*2**(3/4)*x**10 - 459095040*sqrt(2)*2**(3/4)*x**8 - 6 04661760*sqrt(2)*2**(3/4)*x**6 + 64560240*sqrt(2)*2**(3/4)*x**4 + 56687040 *sqrt(2)*2**(3/4)*x**2 - 2421009*sqrt(2)*2**(3/4) + 4026531840*2**(3/4)*x* *22 - 2415919104*2**(3/4)*x**20 - 10821304320*2**(3/4)*x**18 + 7537950720* 2**(3/4)*x**14 - 2120048640*2**(3/4)*x**10 + 537477120*2**(3/4)*x**8 + ...