Integrand size = 24, antiderivative size = 345 \[ \int \frac {x^4}{\left (-c+a x^4\right )^2 \left (d+b x^4\right )} \, dx=\frac {x}{4 (b c+a d) \left (c-a x^4\right )}+\frac {(3 b c-a d) \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{c}}\right )}{8 \sqrt [4]{a} c^{3/4} (b c+a d)^2}+\frac {b^{3/4} \sqrt [4]{d} \arctan \left (\frac {\sqrt [4]{d}-\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{d}}\right )}{2 \sqrt {2} (b c+a d)^2}-\frac {b^{3/4} \sqrt [4]{d} \arctan \left (\frac {\sqrt [4]{d}+\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{d}}\right )}{2 \sqrt {2} (b c+a d)^2}+\frac {(3 b c-a d) \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{c}}\right )}{8 \sqrt [4]{a} c^{3/4} (b c+a d)^2}+\frac {b^{3/4} \sqrt [4]{d} \log \left (\sqrt {d}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{d} x+\sqrt {b} x^2\right )}{4 \sqrt {2} (b c+a d)^2}-\frac {b^{3/4} \sqrt [4]{d} \log \left (\sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{d} x+\sqrt {b} x^2\right )}{4 \sqrt {2} (b c+a d)^2} \] Output:
1/4*x/(a*d+b*c)/(-a*x^4+c)+1/8*(-a*d+3*b*c)*arctan(a^(1/4)*x/c^(1/4))/a^(1 /4)/c^(3/4)/(a*d+b*c)^2+1/4*b^(3/4)*d^(1/4)*arctan((d^(1/4)-2^(1/2)*b^(1/4 )*x)/d^(1/4))*2^(1/2)/(a*d+b*c)^2-1/4*b^(3/4)*d^(1/4)*arctan((d^(1/4)+2^(1 /2)*b^(1/4)*x)/d^(1/4))*2^(1/2)/(a*d+b*c)^2+1/8*(-a*d+3*b*c)*arctanh(a^(1/ 4)*x/c^(1/4))/a^(1/4)/c^(3/4)/(a*d+b*c)^2+1/8*b^(3/4)*d^(1/4)*ln(d^(1/2)-2 ^(1/2)*b^(1/4)*d^(1/4)*x+b^(1/2)*x^2)*2^(1/2)/(a*d+b*c)^2-1/8*b^(3/4)*d^(1 /4)*ln(d^(1/2)+2^(1/2)*b^(1/4)*d^(1/4)*x+b^(1/2)*x^2)*2^(1/2)/(a*d+b*c)^2
Time = 0.27 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.90 \[ \int \frac {x^4}{\left (-c+a x^4\right )^2 \left (d+b x^4\right )} \, dx=\frac {\frac {4 (b c+a d) x}{c-a x^4}-\frac {2 (-3 b c+a d) \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{c}}\right )}{\sqrt [4]{a} c^{3/4}}+4 \sqrt {2} b^{3/4} \sqrt [4]{d} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{d}}\right )-4 \sqrt {2} b^{3/4} \sqrt [4]{d} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{d}}\right )+\frac {(-3 b c+a d) \log \left (\sqrt [4]{c}-\sqrt [4]{a} x\right )}{\sqrt [4]{a} c^{3/4}}+\frac {(3 b c-a d) \log \left (\sqrt [4]{c}+\sqrt [4]{a} x\right )}{\sqrt [4]{a} c^{3/4}}+2 \sqrt {2} b^{3/4} \sqrt [4]{d} \log \left (\sqrt {d}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{d} x+\sqrt {b} x^2\right )-2 \sqrt {2} b^{3/4} \sqrt [4]{d} \log \left (\sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{d} x+\sqrt {b} x^2\right )}{16 (b c+a d)^2} \] Input:
Integrate[x^4/((-c + a*x^4)^2*(d + b*x^4)),x]
Output:
((4*(b*c + a*d)*x)/(c - a*x^4) - (2*(-3*b*c + a*d)*ArcTan[(a^(1/4)*x)/c^(1 /4)])/(a^(1/4)*c^(3/4)) + 4*Sqrt[2]*b^(3/4)*d^(1/4)*ArcTan[1 - (Sqrt[2]*b^ (1/4)*x)/d^(1/4)] - 4*Sqrt[2]*b^(3/4)*d^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)* x)/d^(1/4)] + ((-3*b*c + a*d)*Log[c^(1/4) - a^(1/4)*x])/(a^(1/4)*c^(3/4)) + ((3*b*c - a*d)*Log[c^(1/4) + a^(1/4)*x])/(a^(1/4)*c^(3/4)) + 2*Sqrt[2]*b ^(3/4)*d^(1/4)*Log[Sqrt[d] - Sqrt[2]*b^(1/4)*d^(1/4)*x + Sqrt[b]*x^2] - 2* Sqrt[2]*b^(3/4)*d^(1/4)*Log[Sqrt[d] + Sqrt[2]*b^(1/4)*d^(1/4)*x + Sqrt[b]* x^2])/(16*(b*c + a*d)^2)
Time = 1.04 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.95, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {971, 25, 1020, 755, 756, 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.
| \(\displaystyle \int \frac {x^4}{\left (a x^4-c\right )^2 \left (b x^4+d\right )} \, dx\) |
| \(\Big \downarrow \) 971 |
| \(\displaystyle \frac {\int -\frac {d-3 b x^4}{\left (c-a x^4\right ) \left (b x^4+d\right )}dx}{4 (a d+b c)}+\frac {x}{4 \left (c-a x^4\right ) (a d+b c)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x}{4 \left (c-a x^4\right ) (a d+b c)}-\frac {\int \frac {d-3 b x^4}{\left (c-a x^4\right ) \left (b x^4+d\right )}dx}{4 (a d+b c)}\) |
| \(\Big \downarrow \) 1020 |
| \(\displaystyle \frac {x}{4 \left (c-a x^4\right ) (a d+b c)}-\frac {\frac {4 b d \int \frac {1}{b x^4+d}dx}{a d+b c}-\frac {(3 b c-a d) \int \frac {1}{c-a x^4}dx}{a d+b c}}{4 (a d+b c)}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {x}{4 \left (c-a x^4\right ) (a d+b c)}-\frac {\frac {4 b d \left (\frac {\int \frac {\sqrt {d}-\sqrt {b} x^2}{b x^4+d}dx}{2 \sqrt {d}}+\frac {\int \frac {\sqrt {b} x^2+\sqrt {d}}{b x^4+d}dx}{2 \sqrt {d}}\right )}{a d+b c}-\frac {(3 b c-a d) \int \frac {1}{c-a x^4}dx}{a d+b c}}{4 (a d+b c)}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {x}{4 \left (c-a x^4\right ) (a d+b c)}-\frac {\frac {4 b d \left (\frac {\int \frac {\sqrt {d}-\sqrt {b} x^2}{b x^4+d}dx}{2 \sqrt {d}}+\frac {\int \frac {\sqrt {b} x^2+\sqrt {d}}{b x^4+d}dx}{2 \sqrt {d}}\right )}{a d+b c}-\frac {(3 b c-a d) \left (\frac {\int \frac {1}{\sqrt {c}-\sqrt {a} x^2}dx}{2 \sqrt {c}}+\frac {\int \frac {1}{\sqrt {a} x^2+\sqrt {c}}dx}{2 \sqrt {c}}\right )}{a d+b c}}{4 (a d+b c)}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {x}{4 \left (c-a x^4\right ) (a d+b c)}-\frac {\frac {4 b d \left (\frac {\int \frac {\sqrt {d}-\sqrt {b} x^2}{b x^4+d}dx}{2 \sqrt {d}}+\frac {\int \frac {\sqrt {b} x^2+\sqrt {d}}{b x^4+d}dx}{2 \sqrt {d}}\right )}{a d+b c}-\frac {(3 b c-a d) \left (\frac {\int \frac {1}{\sqrt {c}-\sqrt {a} x^2}dx}{2 \sqrt {c}}+\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{c}}\right )}{2 \sqrt [4]{a} c^{3/4}}\right )}{a d+b c}}{4 (a d+b c)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {x}{4 \left (c-a x^4\right ) (a d+b c)}-\frac {\frac {4 b d \left (\frac {\int \frac {\sqrt {d}-\sqrt {b} x^2}{b x^4+d}dx}{2 \sqrt {d}}+\frac {\int \frac {\sqrt {b} x^2+\sqrt {d}}{b x^4+d}dx}{2 \sqrt {d}}\right )}{a d+b c}-\frac {(3 b c-a d) \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{c}}\right )}{2 \sqrt [4]{a} c^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{c}}\right )}{2 \sqrt [4]{a} c^{3/4}}\right )}{a d+b c}}{4 (a d+b c)}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {x}{4 \left (c-a x^4\right ) (a d+b c)}-\frac {\frac {4 b d \left (\frac {\frac {\int \frac {1}{x^2-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{b}}+\frac {\sqrt {d}}{\sqrt {b}}}dx}{2 \sqrt {b}}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{b}}+\frac {\sqrt {d}}{\sqrt {b}}}dx}{2 \sqrt {b}}}{2 \sqrt {d}}+\frac {\int \frac {\sqrt {d}-\sqrt {b} x^2}{b x^4+d}dx}{2 \sqrt {d}}\right )}{a d+b c}-\frac {(3 b c-a d) \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{c}}\right )}{2 \sqrt [4]{a} c^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{c}}\right )}{2 \sqrt [4]{a} c^{3/4}}\right )}{a d+b c}}{4 (a d+b c)}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {x}{4 \left (c-a x^4\right ) (a d+b c)}-\frac {\frac {4 b d \left (\frac {\int \frac {\sqrt {d}-\sqrt {b} x^2}{b x^4+d}dx}{2 \sqrt {d}}+\frac {\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{d}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{d}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{d}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{d}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{d}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{d}}}{2 \sqrt {d}}\right )}{a d+b c}-\frac {(3 b c-a d) \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{c}}\right )}{2 \sqrt [4]{a} c^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{c}}\right )}{2 \sqrt [4]{a} c^{3/4}}\right )}{a d+b c}}{4 (a d+b c)}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {x}{4 \left (c-a x^4\right ) (a d+b c)}-\frac {\frac {4 b d \left (\frac {\int \frac {\sqrt {d}-\sqrt {b} x^2}{b x^4+d}dx}{2 \sqrt {d}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{d}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{d}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{d}}}{2 \sqrt {d}}\right )}{a d+b c}-\frac {(3 b c-a d) \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{c}}\right )}{2 \sqrt [4]{a} c^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{c}}\right )}{2 \sqrt [4]{a} c^{3/4}}\right )}{a d+b c}}{4 (a d+b c)}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {x}{4 \left (c-a x^4\right ) (a d+b c)}-\frac {\frac {4 b d \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{d}-2 \sqrt [4]{b} x}{\sqrt [4]{b} \left (x^2-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{b}}+\frac {\sqrt {d}}{\sqrt {b}}\right )}dx}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} x+\sqrt [4]{d}\right )}{\sqrt [4]{b} \left (x^2+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{b}}+\frac {\sqrt {d}}{\sqrt {b}}\right )}dx}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{d}}}{2 \sqrt {d}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{d}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{d}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{d}}}{2 \sqrt {d}}\right )}{a d+b c}-\frac {(3 b c-a d) \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{c}}\right )}{2 \sqrt [4]{a} c^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{c}}\right )}{2 \sqrt [4]{a} c^{3/4}}\right )}{a d+b c}}{4 (a d+b c)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x}{4 \left (c-a x^4\right ) (a d+b c)}-\frac {\frac {4 b d \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{d}-2 \sqrt [4]{b} x}{\sqrt [4]{b} \left (x^2-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{b}}+\frac {\sqrt {d}}{\sqrt {b}}\right )}dx}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} x+\sqrt [4]{d}\right )}{\sqrt [4]{b} \left (x^2+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{b}}+\frac {\sqrt {d}}{\sqrt {b}}\right )}dx}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{d}}}{2 \sqrt {d}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{d}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{d}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{d}}}{2 \sqrt {d}}\right )}{a d+b c}-\frac {(3 b c-a d) \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{c}}\right )}{2 \sqrt [4]{a} c^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{c}}\right )}{2 \sqrt [4]{a} c^{3/4}}\right )}{a d+b c}}{4 (a d+b c)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x}{4 \left (c-a x^4\right ) (a d+b c)}-\frac {\frac {4 b d \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{d}-2 \sqrt [4]{b} x}{x^2-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{b}}+\frac {\sqrt {d}}{\sqrt {b}}}dx}{2 \sqrt {2} \sqrt {b} \sqrt [4]{d}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} x+\sqrt [4]{d}}{x^2+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{b}}+\frac {\sqrt {d}}{\sqrt {b}}}dx}{2 \sqrt {b} \sqrt [4]{d}}}{2 \sqrt {d}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{d}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{d}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{d}}}{2 \sqrt {d}}\right )}{a d+b c}-\frac {(3 b c-a d) \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{c}}\right )}{2 \sqrt [4]{a} c^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{c}}\right )}{2 \sqrt [4]{a} c^{3/4}}\right )}{a d+b c}}{4 (a d+b c)}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {x}{4 \left (c-a x^4\right ) (a d+b c)}-\frac {\frac {4 b d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{d}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{d}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{d}}}{2 \sqrt {d}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{d} x+\sqrt {b} x^2+\sqrt {d}\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{d} x+\sqrt {b} x^2+\sqrt {d}\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{d}}}{2 \sqrt {d}}\right )}{a d+b c}-\frac {(3 b c-a d) \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{c}}\right )}{2 \sqrt [4]{a} c^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{c}}\right )}{2 \sqrt [4]{a} c^{3/4}}\right )}{a d+b c}}{4 (a d+b c)}\) |
Input:
Int[x^4/((-c + a*x^4)^2*(d + b*x^4)),x]
Output:
x/(4*(b*c + a*d)*(c - a*x^4)) - (-(((3*b*c - a*d)*(ArcTan[(a^(1/4)*x)/c^(1 /4)]/(2*a^(1/4)*c^(3/4)) + ArcTanh[(a^(1/4)*x)/c^(1/4)]/(2*a^(1/4)*c^(3/4) )))/(b*c + a*d)) + (4*b*d*((-(ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/d^(1/4)]/(Sqr t[2]*b^(1/4)*d^(1/4))) + ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/d^(1/4)]/(Sqrt[2]* b^(1/4)*d^(1/4)))/(2*Sqrt[d]) + (-1/2*Log[Sqrt[d] - Sqrt[2]*b^(1/4)*d^(1/4 )*x + Sqrt[b]*x^2]/(Sqrt[2]*b^(1/4)*d^(1/4)) + Log[Sqrt[d] + Sqrt[2]*b^(1/ 4)*d^(1/4)*x + Sqrt[b]*x^2]/(2*Sqrt[2]*b^(1/4)*d^(1/4)))/(2*Sqrt[d])))/(b* c + a*d))/(4*(b*c + a*d))
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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) 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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[e^(n - 1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)* ((c + d*x^n)^(q + 1)/(n*(b*c - a*d)*(p + 1))), x] - Simp[e^n/(n*(b*c - a*d) *(p + 1)) Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e , q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^( n_))), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^n), x], x ] - Simp[(d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^n), x], x] /; FreeQ[{a, b , c, d, e, f, n}, 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]
Time = 0.14 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.58
| method | result | size |
| default | \(\frac {\frac {\left (-\frac {a d}{4}-\frac {b c}{4}\right ) x}{a \,x^{4}-c}-\frac {\left (a d -3 b c \right ) \left (\frac {c}{a}\right )^{\frac {1}{4}} \left (\ln \left (\frac {x +\left (\frac {c}{a}\right )^{\frac {1}{4}}}{x -\left (\frac {c}{a}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {x}{\left (\frac {c}{a}\right )^{\frac {1}{4}}}\right )\right )}{16 c}}{\left (a d +b c \right )^{2}}-\frac {b \left (\frac {d}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {d}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {d}{b}}}{x^{2}-\left (\frac {d}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {d}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {d}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {d}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (a d +b c \right )^{2}}\) | \(200\) |
| risch | \(\text {Expression too large to display}\) | \(1272\) |
Input:
int(x^4/(a*x^4-c)^2/(b*x^4+d),x,method=_RETURNVERBOSE)
Output:
1/(a*d+b*c)^2*((-1/4*a*d-1/4*b*c)*x/(a*x^4-c)-1/16*(a*d-3*b*c)*(c/a)^(1/4) /c*(ln((x+(c/a)^(1/4))/(x-(c/a)^(1/4)))+2*arctan(x/(c/a)^(1/4))))-1/8*b/(a *d+b*c)^2*(d/b)^(1/4)*2^(1/2)*(ln((x^2+(d/b)^(1/4)*x*2^(1/2)+(d/b)^(1/2))/ (x^2-(d/b)^(1/4)*x*2^(1/2)+(d/b)^(1/2)))+2*arctan(2^(1/2)/(d/b)^(1/4)*x+1) +2*arctan(2^(1/2)/(d/b)^(1/4)*x-1))
Result contains complex when optimal does not.
Time = 1.94 (sec) , antiderivative size = 2801, normalized size of antiderivative = 8.12 \[ \int \frac {x^4}{\left (-c+a x^4\right )^2 \left (d+b x^4\right )} \, dx=\text {Too large to display} \] Input:
integrate(x^4/(a*x^4-c)^2/(b*x^4+d),x, algorithm="fricas")
Output:
-1/16*(((a*b*c + a^2*d)*x^4 - b*c^2 - a*c*d)*((81*b^4*c^4 - 108*a*b^3*c^3* d + 54*a^2*b^2*c^2*d^2 - 12*a^3*b*c*d^3 + a^4*d^4)/(a*b^8*c^11 + 8*a^2*b^7 *c^10*d + 28*a^3*b^6*c^9*d^2 + 56*a^4*b^5*c^8*d^3 + 70*a^5*b^4*c^7*d^4 + 5 6*a^6*b^3*c^6*d^5 + 28*a^7*b^2*c^5*d^6 + 8*a^8*b*c^4*d^7 + a^9*c^3*d^8))^( 1/4)*log(-(3*b*c - a*d)*x + (b^2*c^3 + 2*a*b*c^2*d + a^2*c*d^2)*((81*b^4*c ^4 - 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 - 12*a^3*b*c*d^3 + a^4*d^4)/(a*b ^8*c^11 + 8*a^2*b^7*c^10*d + 28*a^3*b^6*c^9*d^2 + 56*a^4*b^5*c^8*d^3 + 70* a^5*b^4*c^7*d^4 + 56*a^6*b^3*c^6*d^5 + 28*a^7*b^2*c^5*d^6 + 8*a^8*b*c^4*d^ 7 + a^9*c^3*d^8))^(1/4)) - ((a*b*c + a^2*d)*x^4 - b*c^2 - a*c*d)*((81*b^4* c^4 - 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 - 12*a^3*b*c*d^3 + a^4*d^4)/(a* b^8*c^11 + 8*a^2*b^7*c^10*d + 28*a^3*b^6*c^9*d^2 + 56*a^4*b^5*c^8*d^3 + 70 *a^5*b^4*c^7*d^4 + 56*a^6*b^3*c^6*d^5 + 28*a^7*b^2*c^5*d^6 + 8*a^8*b*c^4*d ^7 + a^9*c^3*d^8))^(1/4)*log(-(3*b*c - a*d)*x - (b^2*c^3 + 2*a*b*c^2*d + a ^2*c*d^2)*((81*b^4*c^4 - 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 - 12*a^3*b*c *d^3 + a^4*d^4)/(a*b^8*c^11 + 8*a^2*b^7*c^10*d + 28*a^3*b^6*c^9*d^2 + 56*a ^4*b^5*c^8*d^3 + 70*a^5*b^4*c^7*d^4 + 56*a^6*b^3*c^6*d^5 + 28*a^7*b^2*c^5* d^6 + 8*a^8*b*c^4*d^7 + a^9*c^3*d^8))^(1/4)) + (-I*(a*b*c + a^2*d)*x^4 + I *b*c^2 + I*a*c*d)*((81*b^4*c^4 - 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 - 12 *a^3*b*c*d^3 + a^4*d^4)/(a*b^8*c^11 + 8*a^2*b^7*c^10*d + 28*a^3*b^6*c^9*d^ 2 + 56*a^4*b^5*c^8*d^3 + 70*a^5*b^4*c^7*d^4 + 56*a^6*b^3*c^6*d^5 + 28*a...
Timed out. \[ \int \frac {x^4}{\left (-c+a x^4\right )^2 \left (d+b x^4\right )} \, dx=\text {Timed out} \] Input:
integrate(x**4/(a*x**4-c)**2/(b*x**4+d),x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.01 \[ \int \frac {x^4}{\left (-c+a x^4\right )^2 \left (d+b x^4\right )} \, dx=-\frac {{\left (\frac {2 \, \sqrt {2} \sqrt {d} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} b^{\frac {1}{4}} d^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {d}}}\right )}{\sqrt {\sqrt {b} \sqrt {d}}} + \frac {2 \, \sqrt {2} \sqrt {d} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} b^{\frac {1}{4}} d^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {d}}}\right )}{\sqrt {\sqrt {b} \sqrt {d}}} + \frac {\sqrt {2} d^{\frac {1}{4}} \log \left (\sqrt {b} x^{2} + \sqrt {2} b^{\frac {1}{4}} d^{\frac {1}{4}} x + \sqrt {d}\right )}{b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{\frac {1}{4}} \log \left (\sqrt {b} x^{2} - \sqrt {2} b^{\frac {1}{4}} d^{\frac {1}{4}} x + \sqrt {d}\right )}{b^{\frac {1}{4}}}\right )} b}{8 \, {\left (b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )}} - \frac {x}{4 \, {\left ({\left (a b c + a^{2} d\right )} x^{4} - b c^{2} - a c d\right )}} + \frac {\frac {2 \, {\left (3 \, b c - a d\right )} \arctan \left (\frac {\sqrt {a} x}{\sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} - \frac {{\left (3 \, b c - a d\right )} \log \left (\frac {\sqrt {a} x - \sqrt {\sqrt {a} \sqrt {c}}}{\sqrt {a} x + \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}}}{16 \, {\left (b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )}} \] Input:
integrate(x^4/(a*x^4-c)^2/(b*x^4+d),x, algorithm="maxima")
Output:
-1/8*(2*sqrt(2)*sqrt(d)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x + sqrt(2)*b^(1/4)* d^(1/4))/sqrt(sqrt(b)*sqrt(d)))/sqrt(sqrt(b)*sqrt(d)) + 2*sqrt(2)*sqrt(d)* arctan(1/2*sqrt(2)*(2*sqrt(b)*x - sqrt(2)*b^(1/4)*d^(1/4))/sqrt(sqrt(b)*sq rt(d)))/sqrt(sqrt(b)*sqrt(d)) + sqrt(2)*d^(1/4)*log(sqrt(b)*x^2 + sqrt(2)* b^(1/4)*d^(1/4)*x + sqrt(d))/b^(1/4) - sqrt(2)*d^(1/4)*log(sqrt(b)*x^2 - s qrt(2)*b^(1/4)*d^(1/4)*x + sqrt(d))/b^(1/4))*b/(b^2*c^2 + 2*a*b*c*d + a^2* d^2) - 1/4*x/((a*b*c + a^2*d)*x^4 - b*c^2 - a*c*d) + 1/16*(2*(3*b*c - a*d) *arctan(sqrt(a)*x/sqrt(sqrt(a)*sqrt(c)))/(sqrt(sqrt(a)*sqrt(c))*sqrt(c)) - (3*b*c - a*d)*log((sqrt(a)*x - sqrt(sqrt(a)*sqrt(c)))/(sqrt(a)*x + sqrt(s qrt(a)*sqrt(c))))/(sqrt(sqrt(a)*sqrt(c))*sqrt(c)))/(b^2*c^2 + 2*a*b*c*d + a^2*d^2)
Leaf count of result is larger than twice the leaf count of optimal. 668 vs. \(2 (258) = 516\).
Time = 0.14 (sec) , antiderivative size = 668, normalized size of antiderivative = 1.94 \[ \int \frac {x^4}{\left (-c+a x^4\right )^2 \left (d+b x^4\right )} \, dx =\text {Too large to display} \] Input:
integrate(x^4/(a*x^4-c)^2/(b*x^4+d),x, algorithm="giac")
Output:
1/8*(3*(-a^3*c)^(1/4)*b*c - (-a^3*c)^(1/4)*a*d)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(-c/a)^(1/4))/(-c/a)^(1/4))/(sqrt(2)*a*b^2*c^3 + 2*sqrt(2)*a^2*b*c ^2*d + sqrt(2)*a^3*c*d^2) + 1/8*(3*(-a^3*c)^(1/4)*b*c - (-a^3*c)^(1/4)*a*d )*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(-c/a)^(1/4))/(-c/a)^(1/4))/(sqrt(2)*a *b^2*c^3 + 2*sqrt(2)*a^2*b*c^2*d + sqrt(2)*a^3*c*d^2) + 1/16*(3*(-a^3*c)^( 1/4)*b*c - (-a^3*c)^(1/4)*a*d)*log(x^2 + sqrt(2)*x*(-c/a)^(1/4) + sqrt(-c/ a))/(sqrt(2)*a*b^2*c^3 + 2*sqrt(2)*a^2*b*c^2*d + sqrt(2)*a^3*c*d^2) - 1/16 *(3*(-a^3*c)^(1/4)*b*c - (-a^3*c)^(1/4)*a*d)*log(x^2 - sqrt(2)*x*(-c/a)^(1 /4) + sqrt(-c/a))/(sqrt(2)*a*b^2*c^3 + 2*sqrt(2)*a^2*b*c^2*d + sqrt(2)*a^3 *c*d^2) - 1/2*(b^3*d)^(1/4)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(d/b)^(1/4)) /(d/b)^(1/4))/(sqrt(2)*b^2*c^2 + 2*sqrt(2)*a*b*c*d + sqrt(2)*a^2*d^2) - 1/ 2*(b^3*d)^(1/4)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(d/b)^(1/4))/(d/b)^(1/4) )/(sqrt(2)*b^2*c^2 + 2*sqrt(2)*a*b*c*d + sqrt(2)*a^2*d^2) - 1/4*(b^3*d)^(1 /4)*log(x^2 + sqrt(2)*x*(d/b)^(1/4) + sqrt(d/b))/(sqrt(2)*b^2*c^2 + 2*sqrt (2)*a*b*c*d + sqrt(2)*a^2*d^2) + 1/4*(b^3*d)^(1/4)*log(x^2 - sqrt(2)*x*(d/ b)^(1/4) + sqrt(d/b))/(sqrt(2)*b^2*c^2 + 2*sqrt(2)*a*b*c*d + sqrt(2)*a^2*d ^2) - 1/4*x/((a*x^4 - c)*(b*c + a*d))
Time = 11.82 (sec) , antiderivative size = 20619, normalized size of antiderivative = 59.77 \[ \int \frac {x^4}{\left (-c+a x^4\right )^2 \left (d+b x^4\right )} \, dx=\text {Too large to display} \] Input:
int(x^4/((c - a*x^4)^2*(d + b*x^4)),x)
Output:
2*atan(((((((a^6*b^6*d^5)/16 + (51*a^5*b^7*c*d^4)/16 + (81*a^3*b^9*c^3*d^2 )/16 - (189*a^4*b^8*c^2*d^3)/16)*1i)/(a^3*d^3 + b^3*c^3 + 3*a*b^2*c^2*d + 3*a^2*b*c*d^2) - ((((a^4*d^4 + 81*b^4*c^4 + 54*a^2*b^2*c^2*d^2 - 108*a*b^3 *c^3*d - 12*a^3*b*c*d^3)/(65536*a*b^8*c^11 + 65536*a^9*c^3*d^8 + 524288*a^ 2*b^7*c^10*d + 524288*a^8*b*c^4*d^7 + 1835008*a^3*b^6*c^9*d^2 + 3670016*a^ 4*b^5*c^8*d^3 + 4587520*a^5*b^4*c^7*d^4 + 3670016*a^6*b^3*c^6*d^5 + 183500 8*a^7*b^2*c^5*d^6))^(1/4)*(4096*a^4*b^13*c^10*d^2 - 1024*a^13*b^4*c*d^11 + 31744*a^5*b^12*c^9*d^3 + 106496*a^6*b^11*c^8*d^4 + 200704*a^7*b^10*c^7*d^ 5 + 229376*a^8*b^9*c^6*d^6 + 157696*a^9*b^8*c^5*d^7 + 57344*a^10*b^7*c^4*d ^8 + 4096*a^11*b^6*c^3*d^9 - 4096*a^12*b^5*c^2*d^10))/(a^3*d^3 + b^3*c^3 + 3*a*b^2*c^2*d + 3*a^2*b*c*d^2) - (x*(65536*a^4*b^15*c^11*d^2 - 8192*a^14* b^5*c*d^12 - 4096*a^15*b^4*d^13 + 487424*a^5*b^14*c^10*d^3 + 1564672*a^6*b ^13*c^9*d^4 + 2830336*a^7*b^12*c^8*d^5 + 3178496*a^8*b^11*c^7*d^6 + 235110 4*a^9*b^10*c^6*d^7 + 1261568*a^10*b^9*c^5*d^8 + 581632*a^11*b^8*c^4*d^9 + 229376*a^12*b^7*c^3*d^10 + 45056*a^13*b^6*c^2*d^11)*1i)/(64*(a^6*d^6 + b^6 *c^6 + 15*a^2*b^4*c^4*d^2 + 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 + 6*a* b^5*c^5*d + 6*a^5*b*c*d^5)))*((a^4*d^4 + 81*b^4*c^4 + 54*a^2*b^2*c^2*d^2 - 108*a*b^3*c^3*d - 12*a^3*b*c*d^3)/(65536*a*b^8*c^11 + 65536*a^9*c^3*d^8 + 524288*a^2*b^7*c^10*d + 524288*a^8*b*c^4*d^7 + 1835008*a^3*b^6*c^9*d^2 + 3670016*a^4*b^5*c^8*d^3 + 4587520*a^5*b^4*c^7*d^4 + 3670016*a^6*b^3*c^6...
Time = 0.28 (sec) , antiderivative size = 655, normalized size of antiderivative = 1.90 \[ \int \frac {x^4}{\left (-c+a x^4\right )^2 \left (d+b x^4\right )} \, dx =\text {Too large to display} \] Input:
int(x^4/(a*x^4-c)^2/(b*x^4+d),x)
Output:
(4*d**(1/4)*b**(3/4)*sqrt(2)*atan((d**(1/4)*b**(1/4)*sqrt(2) - 2*sqrt(b)*x )/(d**(1/4)*b**(1/4)*sqrt(2)))*a**2*c*x**4 - 4*d**(1/4)*b**(3/4)*sqrt(2)*a tan((d**(1/4)*b**(1/4)*sqrt(2) - 2*sqrt(b)*x)/(d**(1/4)*b**(1/4)*sqrt(2))) *a*c**2 - 4*d**(1/4)*b**(3/4)*sqrt(2)*atan((d**(1/4)*b**(1/4)*sqrt(2) + 2* sqrt(b)*x)/(d**(1/4)*b**(1/4)*sqrt(2)))*a**2*c*x**4 + 4*d**(1/4)*b**(3/4)* sqrt(2)*atan((d**(1/4)*b**(1/4)*sqrt(2) + 2*sqrt(b)*x)/(d**(1/4)*b**(1/4)* sqrt(2)))*a*c**2 - 2*c**(1/4)*a**(3/4)*atan((sqrt(a)*x)/(c**(1/4)*a**(1/4) ))*a**2*d*x**4 + 6*c**(1/4)*a**(3/4)*atan((sqrt(a)*x)/(c**(1/4)*a**(1/4))) *a*b*c*x**4 + 2*c**(1/4)*a**(3/4)*atan((sqrt(a)*x)/(c**(1/4)*a**(1/4)))*a* c*d - 6*c**(1/4)*a**(3/4)*atan((sqrt(a)*x)/(c**(1/4)*a**(1/4)))*b*c**2 - c **(1/4)*a**(3/4)*log(a**(1/4)*x + c**(1/4))*a**2*d*x**4 + 3*c**(1/4)*a**(3 /4)*log(a**(1/4)*x + c**(1/4))*a*b*c*x**4 + c**(1/4)*a**(3/4)*log(a**(1/4) *x + c**(1/4))*a*c*d - 3*c**(1/4)*a**(3/4)*log(a**(1/4)*x + c**(1/4))*b*c* *2 + c**(1/4)*a**(3/4)*log(a**(1/4)*x - c**(1/4))*a**2*d*x**4 - 3*c**(1/4) *a**(3/4)*log(a**(1/4)*x - c**(1/4))*a*b*c*x**4 - c**(1/4)*a**(3/4)*log(a* *(1/4)*x - c**(1/4))*a*c*d + 3*c**(1/4)*a**(3/4)*log(a**(1/4)*x - c**(1/4) )*b*c**2 + 2*d**(1/4)*b**(3/4)*sqrt(2)*log( - d**(1/4)*b**(1/4)*sqrt(2)*x + sqrt(b)*x**2 + sqrt(d))*a**2*c*x**4 - 2*d**(1/4)*b**(3/4)*sqrt(2)*log( - d**(1/4)*b**(1/4)*sqrt(2)*x + sqrt(b)*x**2 + sqrt(d))*a*c**2 - 2*d**(1/4) *b**(3/4)*sqrt(2)*log(d**(1/4)*b**(1/4)*sqrt(2)*x + sqrt(b)*x**2 + sqrt...