Integrand size = 15, antiderivative size = 249 \[ \int \frac {x^{13/2}}{\left (a+c x^4\right )^2} \, dx=-\frac {x^{7/2}}{4 c \left (a+c x^4\right )}-\frac {7 \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} \sqrt [8]{-a} c^{15/8}}+\frac {7 \arctan \left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} \sqrt [8]{-a} c^{15/8}}+\frac {7 \arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt [8]{-a} c^{15/8}}-\frac {7 \text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt [8]{-a} c^{15/8}}-\frac {7 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}}{\sqrt [4]{-a}+\sqrt [4]{c} x}\right )}{16 \sqrt {2} \sqrt [8]{-a} c^{15/8}} \] Output:
-1/4*x^(7/2)/c/(c*x^4+a)+7/32*arctan(-1+2^(1/2)*c^(1/8)*x^(1/2)/(-a)^(1/8) )*2^(1/2)/(-a)^(1/8)/c^(15/8)+7/32*arctan(1+2^(1/2)*c^(1/8)*x^(1/2)/(-a)^( 1/8))*2^(1/2)/(-a)^(1/8)/c^(15/8)+7/16*arctan(c^(1/8)*x^(1/2)/(-a)^(1/8))/ (-a)^(1/8)/c^(15/8)-7/16*arctanh(c^(1/8)*x^(1/2)/(-a)^(1/8))/(-a)^(1/8)/c^ (15/8)-7/32*arctanh(2^(1/2)*(-a)^(1/8)*c^(1/8)*x^(1/2)/((-a)^(1/4)+c^(1/4) *x))*2^(1/2)/(-a)^(1/8)/c^(15/8)
Time = 1.20 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.11 \[ \int \frac {x^{13/2}}{\left (a+c x^4\right )^2} \, dx=\frac {-\frac {8 c^{7/8} x^{7/2}}{a+c x^4}-\frac {7 \sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )}{\sqrt [8]{a}}-\frac {7 \sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )}{\sqrt [8]{a}}-\frac {7 \sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )}{\sqrt [8]{a}}-\frac {7 \sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [8]{c} \sqrt {-\left (\left (-2+\sqrt {2}\right ) x\right )}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )}{\sqrt [8]{a}}}{32 c^{15/8}} \] Input:
Integrate[x^(13/2)/(a + c*x^4)^2,x]
Output:
((-8*c^(7/8)*x^(7/2))/(a + c*x^4) - (7*Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[1 - 1/Sqrt[2]]*(a^(1/4) - c^(1/4)*x))/(a^(1/8)*c^(1/8)*Sqrt[x])])/a^(1/8) - (7 *Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[1 + 1/Sqrt[2]]*(a^(1/4) - c^(1/4)*x))/(a^( 1/8)*c^(1/8)*Sqrt[x])])/a^(1/8) - (7*Sqrt[2 + Sqrt[2]]*ArcTanh[(Sqrt[2 + S qrt[2]]*a^(1/8)*c^(1/8)*Sqrt[x])/(a^(1/4) + c^(1/4)*x)])/a^(1/8) - (7*Sqrt [2 - Sqrt[2]]*ArcTanh[(a^(1/8)*c^(1/8)*Sqrt[-((-2 + Sqrt[2])*x)])/(a^(1/4) + c^(1/4)*x)])/a^(1/8))/(32*c^(15/8))
Time = 1.01 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.41, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.933, Rules used = {817, 851, 830, 826, 827, 218, 221, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \frac {x^{13/2}}{\left (a+c x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle \frac {7 \int \frac {x^{5/2}}{c x^4+a}dx}{8 c}-\frac {x^{7/2}}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 851 |
| \(\displaystyle \frac {7 \int \frac {x^3}{c x^4+a}d\sqrt {x}}{4 c}-\frac {x^{7/2}}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 830 |
| \(\displaystyle \frac {7 \left (\frac {\int \frac {x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt {c}}-\frac {\int \frac {x}{\sqrt {-a}-\sqrt {c} x^2}d\sqrt {x}}{2 \sqrt {c}}\right )}{4 c}-\frac {x^{7/2}}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {7 \left (\frac {\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\int \frac {x}{\sqrt {-a}-\sqrt {c} x^2}d\sqrt {x}}{2 \sqrt {c}}\right )}{4 c}-\frac {x^{7/2}}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {7 \left (\frac {\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x}d\sqrt {x}}{2 \sqrt [4]{c}}-\frac {\int \frac {1}{\sqrt [4]{c} x+\sqrt [4]{-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{4 c}-\frac {x^{7/2}}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {7 \left (\frac {\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x}d\sqrt {x}}{2 \sqrt [4]{c}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )}{4 c}-\frac {x^{7/2}}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {7 \left (\frac {\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )}{4 c}-\frac {x^{7/2}}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {7 \left (\frac {\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt [4]{c}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )}{4 c}-\frac {x^{7/2}}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {7 \left (\frac {\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )}{4 c}-\frac {x^{7/2}}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {7 \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )}{4 c}-\frac {x^{7/2}}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {7 \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{c} \left (x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{c} \sqrt {x}+\sqrt [8]{-a}\right )}{\sqrt [8]{c} \left (x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )}{4 c}-\frac {x^{7/2}}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {7 \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{c} \left (x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{c} \sqrt {x}+\sqrt [8]{-a}\right )}{\sqrt [8]{c} \left (x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )}{4 c}-\frac {x^{7/2}}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{c} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}+\sqrt [8]{-a}}{x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt [8]{-a} \sqrt [4]{c}}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )}{4 c}-\frac {x^{7/2}}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {7 \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )}{4 c}-\frac {x^{7/2}}{4 c \left (a+c x^4\right )}\) |
Input:
Int[x^(13/2)/(a + c*x^4)^2,x]
Output:
-1/4*x^(7/2)/(c*(a + c*x^4)) + (7*(-1/2*(-1/2*ArcTan[(c^(1/8)*Sqrt[x])/(-a )^(1/8)]/((-a)^(1/8)*c^(3/8)) + ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(2*( -a)^(1/8)*c^(3/8)))/Sqrt[c] + ((-(ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a )^(1/8)]/(Sqrt[2]*(-a)^(1/8)*c^(1/8))) + ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[ x])/(-a)^(1/8)]/(Sqrt[2]*(-a)^(1/8)*c^(1/8)))/(2*c^(1/4)) - (-1/2*Log[(-a) ^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x]/(Sqrt[2]*(-a)^(1/ 8)*c^(1/8)) + Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4 )*x]/(2*Sqrt[2]*(-a)^(1/8)*c^(1/8)))/(2*c^(1/4)))/(2*Sqrt[c])))/(4*c)
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_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^( n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[c^n *((m - n + 1)/(b*n*(p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x ] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && ! ILtQ[(m + n*(p + 1) + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]]))
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{r = Numerator[Rt [-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[x^(m - n/2)/( r + s*x^(n/2)), x], x] - Simp[s/(2*b) Int[x^(m - n/2)/(r - s*x^(n/2)), x] , x]] /; FreeQ[{a, b}, x] && IGtQ[n/4, 0] && IGtQ[m, 0] && LeQ[n/2, m] && L tQ[m, n] && !GtQ[a/b, 0]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.52 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.19
| method | result | size |
| derivativedivides | \(-\frac {x^{\frac {7}{2}}}{4 c \left (c \,x^{4}+a \right )}+\frac {7 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}}\right )}{32 c^{2}}\) | \(47\) |
| default | \(-\frac {x^{\frac {7}{2}}}{4 c \left (c \,x^{4}+a \right )}+\frac {7 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}}\right )}{32 c^{2}}\) | \(47\) |
Input:
int(x^(13/2)/(c*x^4+a)^2,x,method=_RETURNVERBOSE)
Output:
-1/4*x^(7/2)/c/(c*x^4+a)+7/32/c^2*sum(1/_R*ln(x^(1/2)-_R),_R=RootOf(_Z^8*c +a))
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.64 \[ \int \frac {x^{13/2}}{\left (a+c x^4\right )^2} \, dx=-\frac {16 \, x^{\frac {7}{2}} + 7 \, \sqrt {2} {\left (\left (i - 1\right ) \, c^{2} x^{4} + \left (i - 1\right ) \, a c\right )} \left (-\frac {1}{a c^{15}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a c^{13} \left (-\frac {1}{a c^{15}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) + 7 \, \sqrt {2} {\left (-\left (i + 1\right ) \, c^{2} x^{4} - \left (i + 1\right ) \, a c\right )} \left (-\frac {1}{a c^{15}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a c^{13} \left (-\frac {1}{a c^{15}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) + 7 \, \sqrt {2} {\left (\left (i + 1\right ) \, c^{2} x^{4} + \left (i + 1\right ) \, a c\right )} \left (-\frac {1}{a c^{15}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a c^{13} \left (-\frac {1}{a c^{15}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) + 7 \, \sqrt {2} {\left (-\left (i - 1\right ) \, c^{2} x^{4} - \left (i - 1\right ) \, a c\right )} \left (-\frac {1}{a c^{15}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a c^{13} \left (-\frac {1}{a c^{15}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) - 14 \, {\left (c^{2} x^{4} + a c\right )} \left (-\frac {1}{a c^{15}}\right )^{\frac {1}{8}} \log \left (a c^{13} \left (-\frac {1}{a c^{15}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) + 14 \, {\left (i \, c^{2} x^{4} + i \, a c\right )} \left (-\frac {1}{a c^{15}}\right )^{\frac {1}{8}} \log \left (i \, a c^{13} \left (-\frac {1}{a c^{15}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) + 14 \, {\left (-i \, c^{2} x^{4} - i \, a c\right )} \left (-\frac {1}{a c^{15}}\right )^{\frac {1}{8}} \log \left (-i \, a c^{13} \left (-\frac {1}{a c^{15}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) + 14 \, {\left (c^{2} x^{4} + a c\right )} \left (-\frac {1}{a c^{15}}\right )^{\frac {1}{8}} \log \left (-a c^{13} \left (-\frac {1}{a c^{15}}\right )^{\frac {7}{8}} + \sqrt {x}\right )}{64 \, {\left (c^{2} x^{4} + a c\right )}} \] Input:
integrate(x^(13/2)/(c*x^4+a)^2,x, algorithm="fricas")
Output:
-1/64*(16*x^(7/2) + 7*sqrt(2)*((I - 1)*c^2*x^4 + (I - 1)*a*c)*(-1/(a*c^15) )^(1/8)*log((1/2*I + 1/2)*sqrt(2)*a*c^13*(-1/(a*c^15))^(7/8) + sqrt(x)) + 7*sqrt(2)*(-(I + 1)*c^2*x^4 - (I + 1)*a*c)*(-1/(a*c^15))^(1/8)*log(-(1/2*I - 1/2)*sqrt(2)*a*c^13*(-1/(a*c^15))^(7/8) + sqrt(x)) + 7*sqrt(2)*((I + 1) *c^2*x^4 + (I + 1)*a*c)*(-1/(a*c^15))^(1/8)*log((1/2*I - 1/2)*sqrt(2)*a*c^ 13*(-1/(a*c^15))^(7/8) + sqrt(x)) + 7*sqrt(2)*(-(I - 1)*c^2*x^4 - (I - 1)* a*c)*(-1/(a*c^15))^(1/8)*log(-(1/2*I + 1/2)*sqrt(2)*a*c^13*(-1/(a*c^15))^( 7/8) + sqrt(x)) - 14*(c^2*x^4 + a*c)*(-1/(a*c^15))^(1/8)*log(a*c^13*(-1/(a *c^15))^(7/8) + sqrt(x)) + 14*(I*c^2*x^4 + I*a*c)*(-1/(a*c^15))^(1/8)*log( I*a*c^13*(-1/(a*c^15))^(7/8) + sqrt(x)) + 14*(-I*c^2*x^4 - I*a*c)*(-1/(a*c ^15))^(1/8)*log(-I*a*c^13*(-1/(a*c^15))^(7/8) + sqrt(x)) + 14*(c^2*x^4 + a *c)*(-1/(a*c^15))^(1/8)*log(-a*c^13*(-1/(a*c^15))^(7/8) + sqrt(x)))/(c^2*x ^4 + a*c)
Timed out. \[ \int \frac {x^{13/2}}{\left (a+c x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate(x**(13/2)/(c*x**4+a)**2,x)
Output:
Timed out
\[ \int \frac {x^{13/2}}{\left (a+c x^4\right )^2} \, dx=\int { \frac {x^{\frac {13}{2}}}{{\left (c x^{4} + a\right )}^{2}} \,d x } \] Input:
integrate(x^(13/2)/(c*x^4+a)^2,x, algorithm="maxima")
Output:
-1/4*x^(7/2)/(c^2*x^4 + a*c) + 7*integrate(1/8*x^(5/2)/(c^2*x^4 + a*c), x)
Leaf count of result is larger than twice the leaf count of optimal. 486 vs. \(2 (168) = 336\).
Time = 0.26 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.95 \[ \int \frac {x^{13/2}}{\left (a+c x^4\right )^2} \, dx=-\frac {x^{\frac {7}{2}}}{4 \, {\left (c x^{4} + a\right )} c} + \frac {7 \, \left (\frac {a}{c}\right )^{\frac {7}{8}} \arctan \left (\frac {\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + 2 \, \sqrt {x}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{16 \, a c \sqrt {-2 \, \sqrt {2} + 4}} + \frac {7 \, \left (\frac {a}{c}\right )^{\frac {7}{8}} \arctan \left (-\frac {\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} - 2 \, \sqrt {x}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{16 \, a c \sqrt {-2 \, \sqrt {2} + 4}} + \frac {7 \, \left (\frac {a}{c}\right )^{\frac {7}{8}} \arctan \left (\frac {\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + 2 \, \sqrt {x}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{16 \, a c \sqrt {2 \, \sqrt {2} + 4}} + \frac {7 \, \left (\frac {a}{c}\right )^{\frac {7}{8}} \arctan \left (-\frac {\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} - 2 \, \sqrt {x}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{16 \, a c \sqrt {2 \, \sqrt {2} + 4}} - \frac {7 \, \left (\frac {a}{c}\right )^{\frac {7}{8}} \log \left (\sqrt {x} \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{32 \, a c \sqrt {-2 \, \sqrt {2} + 4}} + \frac {7 \, \left (\frac {a}{c}\right )^{\frac {7}{8}} \log \left (-\sqrt {x} \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{32 \, a c \sqrt {-2 \, \sqrt {2} + 4}} - \frac {7 \, \left (\frac {a}{c}\right )^{\frac {7}{8}} \log \left (\sqrt {x} \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{32 \, a c \sqrt {2 \, \sqrt {2} + 4}} + \frac {7 \, \left (\frac {a}{c}\right )^{\frac {7}{8}} \log \left (-\sqrt {x} \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{32 \, a c \sqrt {2 \, \sqrt {2} + 4}} \] Input:
integrate(x^(13/2)/(c*x^4+a)^2,x, algorithm="giac")
Output:
-1/4*x^(7/2)/((c*x^4 + a)*c) + 7/16*(a/c)^(7/8)*arctan((sqrt(-sqrt(2) + 2) *(a/c)^(1/8) + 2*sqrt(x))/(sqrt(sqrt(2) + 2)*(a/c)^(1/8)))/(a*c*sqrt(-2*sq rt(2) + 4)) + 7/16*(a/c)^(7/8)*arctan(-(sqrt(-sqrt(2) + 2)*(a/c)^(1/8) - 2 *sqrt(x))/(sqrt(sqrt(2) + 2)*(a/c)^(1/8)))/(a*c*sqrt(-2*sqrt(2) + 4)) + 7/ 16*(a/c)^(7/8)*arctan((sqrt(sqrt(2) + 2)*(a/c)^(1/8) + 2*sqrt(x))/(sqrt(-s qrt(2) + 2)*(a/c)^(1/8)))/(a*c*sqrt(2*sqrt(2) + 4)) + 7/16*(a/c)^(7/8)*arc tan(-(sqrt(sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(x))/(sqrt(-sqrt(2) + 2)*(a/c) ^(1/8)))/(a*c*sqrt(2*sqrt(2) + 4)) - 7/32*(a/c)^(7/8)*log(sqrt(x)*sqrt(sqr t(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a*c*sqrt(-2*sqrt(2) + 4)) + 7/32 *(a/c)^(7/8)*log(-sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4)) /(a*c*sqrt(-2*sqrt(2) + 4)) - 7/32*(a/c)^(7/8)*log(sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a*c*sqrt(2*sqrt(2) + 4)) + 7/32*(a/c)^ (7/8)*log(-sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a*c* sqrt(2*sqrt(2) + 4))
Time = 0.26 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.54 \[ \int \frac {x^{13/2}}{\left (a+c x^4\right )^2} \, dx=\frac {7\,\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}}{{\left (-a\right )}^{1/8}}\right )}{16\,{\left (-a\right )}^{1/8}\,c^{15/8}}-\frac {x^{7/2}}{4\,c\,\left (c\,x^4+a\right )}+\frac {\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,7{}\mathrm {i}}{16\,{\left (-a\right )}^{1/8}\,c^{15/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (\frac {7}{32}-\frac {7}{32}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}\,c^{15/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (\frac {7}{32}+\frac {7}{32}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}\,c^{15/8}} \] Input:
int(x^(13/2)/(a + c*x^4)^2,x)
Output:
(7*atan((c^(1/8)*x^(1/2))/(-a)^(1/8)))/(16*(-a)^(1/8)*c^(15/8)) - x^(7/2)/ (4*c*(a + c*x^4)) + (atan((c^(1/8)*x^(1/2)*1i)/(-a)^(1/8))*7i)/(16*(-a)^(1 /8)*c^(15/8)) + (2^(1/2)*atan((2^(1/2)*c^(1/8)*x^(1/2)*(1/2 - 1i/2))/(-a)^ (1/8))*(7/32 - 7i/32))/((-a)^(1/8)*c^(15/8)) + (2^(1/2)*atan((2^(1/2)*c^(1 /8)*x^(1/2)*(1/2 + 1i/2))/(-a)^(1/8))*(7/32 + 7i/32))/((-a)^(1/8)*c^(15/8) )
Time = 0.27 (sec) , antiderivative size = 775, normalized size of antiderivative = 3.11 \[ \int \frac {x^{13/2}}{\left (a+c x^4\right )^2} \, dx =\text {Too large to display} \] Input:
int(x^(13/2)/(c*x^4+a)^2,x)
Output:
( - 14*c**(1/8)*a**(7/8)*sqrt(sqrt(2) + 2)*atan((c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2) - 2*sqrt(x)*c**(1/4))/(c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2))) *a - 14*c**(1/8)*a**(7/8)*sqrt(sqrt(2) + 2)*atan((c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2) - 2*sqrt(x)*c**(1/4))/(c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2)) )*c*x**4 + 14*c**(1/8)*a**(7/8)*sqrt(sqrt(2) + 2)*atan((c**(1/8)*a**(1/8)* sqrt( - sqrt(2) + 2) + 2*sqrt(x)*c**(1/4))/(c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2)))*a + 14*c**(1/8)*a**(7/8)*sqrt(sqrt(2) + 2)*atan((c**(1/8)*a**(1/8) *sqrt( - sqrt(2) + 2) + 2*sqrt(x)*c**(1/4))/(c**(1/8)*a**(1/8)*sqrt(sqrt(2 ) + 2)))*c*x**4 - 14*c**(1/8)*a**(7/8)*sqrt( - sqrt(2) + 2)*atan((c**(1/8) *a**(1/8)*sqrt(sqrt(2) + 2) - 2*sqrt(x)*c**(1/4))/(c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2)))*a - 14*c**(1/8)*a**(7/8)*sqrt( - sqrt(2) + 2)*atan((c**( 1/8)*a**(1/8)*sqrt(sqrt(2) + 2) - 2*sqrt(x)*c**(1/4))/(c**(1/8)*a**(1/8)*s qrt( - sqrt(2) + 2)))*c*x**4 + 14*c**(1/8)*a**(7/8)*sqrt( - sqrt(2) + 2)*a tan((c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2) + 2*sqrt(x)*c**(1/4))/(c**(1/8)*a **(1/8)*sqrt( - sqrt(2) + 2)))*a + 14*c**(1/8)*a**(7/8)*sqrt( - sqrt(2) + 2)*atan((c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2) + 2*sqrt(x)*c**(1/4))/(c**(1/ 8)*a**(1/8)*sqrt( - sqrt(2) + 2)))*c*x**4 + 7*c**(1/8)*a**(7/8)*sqrt( - sq rt(2) + 2)*log( - sqrt(x)*c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2) + a**(1/4 ) + c**(1/4)*x)*a + 7*c**(1/8)*a**(7/8)*sqrt( - sqrt(2) + 2)*log( - sqrt(x )*c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2) + a**(1/4) + c**(1/4)*x)*c*x**...