Integrand size = 21, antiderivative size = 115 \[ \int \frac {c+d x+e x^2}{a-b x^4} \, dx=\frac {\left (c-\frac {\sqrt {a} e}{\sqrt {b}}\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt [4]{b}}+\frac {\left (\sqrt {b} c+\sqrt {a} e\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac {d \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b}} \] Output:
1/2*(c-a^(1/2)*e/b^(1/2))*arctan(b^(1/4)*x/a^(1/4))/a^(3/4)/b^(1/4)+1/2*(b ^(1/2)*c+a^(1/2)*e)*arctanh(b^(1/4)*x/a^(1/4))/a^(3/4)/b^(3/4)+1/2*d*arcta nh(b^(1/2)*x^2/a^(1/2))/a^(1/2)/b^(1/2)
Time = 0.03 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.63 \[ \int \frac {c+d x+e x^2}{a-b x^4} \, dx=\frac {2 \left (\sqrt {b} c-\sqrt {a} e\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )-\left (\sqrt {b} c+\sqrt [4]{a} \sqrt [4]{b} d+\sqrt {a} e\right ) \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )+\sqrt {b} c \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )-\sqrt [4]{a} \sqrt [4]{b} d \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )+\sqrt {a} e \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )+\sqrt [4]{a} \sqrt [4]{b} d \log \left (\sqrt {a}+\sqrt {b} x^2\right )}{4 a^{3/4} b^{3/4}} \] Input:
Integrate[(c + d*x + e*x^2)/(a - b*x^4),x]
Output:
(2*(Sqrt[b]*c - Sqrt[a]*e)*ArcTan[(b^(1/4)*x)/a^(1/4)] - (Sqrt[b]*c + a^(1 /4)*b^(1/4)*d + Sqrt[a]*e)*Log[a^(1/4) - b^(1/4)*x] + Sqrt[b]*c*Log[a^(1/4 ) + b^(1/4)*x] - a^(1/4)*b^(1/4)*d*Log[a^(1/4) + b^(1/4)*x] + Sqrt[a]*e*Lo g[a^(1/4) + b^(1/4)*x] + a^(1/4)*b^(1/4)*d*Log[Sqrt[a] + Sqrt[b]*x^2])/(4* a^(3/4)*b^(3/4))
Time = 0.45 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.01, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2415, 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.
\(\displaystyle \int \frac {c+d x+e x^2}{a-b x^4} \, dx\) |
\(\Big \downarrow \) 2415 |
\(\displaystyle \int \left (\frac {c+e x^2}{a-b x^4}+\frac {d x}{a-b x^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt {b} c-\sqrt {a} e\right )}{2 a^{3/4} b^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt {a} e+\sqrt {b} c\right )}{2 a^{3/4} b^{3/4}}+\frac {d \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b}}\) |
Input:
Int[(c + d*x + e*x^2)/(a - b*x^4),x]
Output:
((Sqrt[b]*c - Sqrt[a]*e)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(2*a^(3/4)*b^(3/4)) + ((Sqrt[b]*c + Sqrt[a]*e)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(2*a^(3/4)*b^(3/4 )) + (d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(2*Sqrt[a]*Sqrt[b])
Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[x^ii*((Coeff [Pq, x, ii] + Coeff[Pq, x, n/2 + ii]*x^(n/2))/(a + b*x^n)), {ii, 0, n/2 - 1 }]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && Expon[Pq, x] < n
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.12 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.34
method | result | size |
risch | \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-a \right )}{\sum }\frac {\left (\textit {\_R}^{2} e +d \textit {\_R} +c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 b}\) | \(39\) |
default | \(\frac {c \left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}+\frac {d \ln \left (\frac {a +x^{2} \sqrt {a b}}{a -x^{2} \sqrt {a b}}\right )}{4 \sqrt {a b}}-\frac {e \left (2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(139\) |
Input:
int((e*x^2+d*x+c)/(-b*x^4+a),x,method=_RETURNVERBOSE)
Output:
-1/4/b*sum((_R^2*e+_R*d+c)/_R^3*ln(x-_R),_R=RootOf(_Z^4*b-a))
Result contains complex when optimal does not.
Time = 2.43 (sec) , antiderivative size = 120560, normalized size of antiderivative = 1048.35 \[ \int \frac {c+d x+e x^2}{a-b x^4} \, dx=\text {Too large to display} \] Input:
integrate((e*x^2+d*x+c)/(-b*x^4+a),x, algorithm="fricas")
Output:
Too large to include
Leaf count of result is larger than twice the leaf count of optimal. 471 vs. \(2 (109) = 218\).
Time = 4.22 (sec) , antiderivative size = 471, normalized size of antiderivative = 4.10 \[ \int \frac {c+d x+e x^2}{a-b x^4} \, dx=- \operatorname {RootSum} {\left (256 t^{4} a^{3} b^{3} + t^{2} \left (- 64 a^{2} b^{2} c e - 32 a^{2} b^{2} d^{2}\right ) + t \left (- 16 a^{2} b d e^{2} - 16 a b^{2} c^{2} d\right ) - a^{2} e^{4} + 2 a b c^{2} e^{2} - 4 a b c d^{2} e + a b d^{4} - b^{2} c^{4}, \left ( t \mapsto t \log {\left (x + \frac {- 64 t^{3} a^{4} b^{2} e^{3} - 64 t^{3} a^{3} b^{3} c^{2} e + 128 t^{3} a^{3} b^{3} c d^{2} + 48 t^{2} a^{3} b^{2} c d e^{2} - 32 t^{2} a^{3} b^{2} d^{3} e - 16 t^{2} a^{2} b^{3} c^{3} d + 12 t a^{3} b c e^{4} + 12 t a^{3} b d^{2} e^{3} + 16 t a^{2} b^{2} c^{3} e^{2} - 36 t a^{2} b^{2} c^{2} d^{2} e - 8 t a^{2} b^{2} c d^{4} + 4 t a b^{3} c^{5} + 3 a^{3} d e^{5} - 5 a^{2} b c d^{3} e^{2} + 2 a^{2} b d^{5} e + 5 a b^{2} c^{4} d e - 5 a b^{2} c^{3} d^{3}}{a^{3} e^{6} + a^{2} b c^{2} e^{4} - 8 a^{2} b c d^{2} e^{3} + 4 a^{2} b d^{4} e^{2} - a b^{2} c^{4} e^{2} + 8 a b^{2} c^{3} d^{2} e - 4 a b^{2} c^{2} d^{4} - b^{3} c^{6}} \right )} \right )\right )} \] Input:
integrate((e*x**2+d*x+c)/(-b*x**4+a),x)
Output:
-RootSum(256*_t**4*a**3*b**3 + _t**2*(-64*a**2*b**2*c*e - 32*a**2*b**2*d** 2) + _t*(-16*a**2*b*d*e**2 - 16*a*b**2*c**2*d) - a**2*e**4 + 2*a*b*c**2*e* *2 - 4*a*b*c*d**2*e + a*b*d**4 - b**2*c**4, Lambda(_t, _t*log(x + (-64*_t* *3*a**4*b**2*e**3 - 64*_t**3*a**3*b**3*c**2*e + 128*_t**3*a**3*b**3*c*d**2 + 48*_t**2*a**3*b**2*c*d*e**2 - 32*_t**2*a**3*b**2*d**3*e - 16*_t**2*a**2 *b**3*c**3*d + 12*_t*a**3*b*c*e**4 + 12*_t*a**3*b*d**2*e**3 + 16*_t*a**2*b **2*c**3*e**2 - 36*_t*a**2*b**2*c**2*d**2*e - 8*_t*a**2*b**2*c*d**4 + 4*_t *a*b**3*c**5 + 3*a**3*d*e**5 - 5*a**2*b*c*d**3*e**2 + 2*a**2*b*d**5*e + 5* a*b**2*c**4*d*e - 5*a*b**2*c**3*d**3)/(a**3*e**6 + a**2*b*c**2*e**4 - 8*a* *2*b*c*d**2*e**3 + 4*a**2*b*d**4*e**2 - a*b**2*c**4*e**2 + 8*a*b**2*c**3*d **2*e - 4*a*b**2*c**2*d**4 - b**3*c**6))))
Time = 0.11 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.33 \[ \int \frac {c+d x+e x^2}{a-b x^4} \, dx=\frac {d \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{4 \, \sqrt {a} \sqrt {b}} - \frac {d \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{4 \, \sqrt {a} \sqrt {b}} + \frac {{\left (\sqrt {b} c - \sqrt {a} e\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{2 \, \sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {{\left (\sqrt {b} c + \sqrt {a} e\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{4 \, \sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} \] Input:
integrate((e*x^2+d*x+c)/(-b*x^4+a),x, algorithm="maxima")
Output:
1/4*d*log(sqrt(b)*x^2 + sqrt(a))/(sqrt(a)*sqrt(b)) - 1/4*d*log(sqrt(b)*x^2 - sqrt(a))/(sqrt(a)*sqrt(b)) + 1/2*(sqrt(b)*c - sqrt(a)*e)*arctan(sqrt(b) *x/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - 1/4*(s qrt(b)*c + sqrt(a)*e)*log((sqrt(b)*x - sqrt(sqrt(a)*sqrt(b)))/(sqrt(b)*x + sqrt(sqrt(a)*sqrt(b))))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b))
Leaf count of result is larger than twice the leaf count of optimal. 259 vs. \(2 (77) = 154\).
Time = 0.13 (sec) , antiderivative size = 259, normalized size of antiderivative = 2.25 \[ \int \frac {c+d x+e x^2}{a-b x^4} \, dx=-\frac {\sqrt {2} {\left (b^{2} c - \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d + \sqrt {-a b} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, \left (-a b^{3}\right )^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (b^{2} c + \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d - \sqrt {-a b} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, \left (-a b^{3}\right )^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (b^{2} c - \sqrt {-a b} b e\right )} \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{8 \, \left (-a b^{3}\right )^{\frac {3}{4}}} + \frac {\sqrt {2} {\left (b^{2} c - \sqrt {-a b} b e\right )} \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{8 \, \left (-a b^{3}\right )^{\frac {3}{4}}} \] Input:
integrate((e*x^2+d*x+c)/(-b*x^4+a),x, algorithm="giac")
Output:
-1/4*sqrt(2)*(b^2*c - sqrt(2)*(-a*b^3)^(1/4)*b*d + sqrt(-a*b)*b*e)*arctan( 1/2*sqrt(2)*(2*x + sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/(-a*b^3)^(3/4) - 1/ 4*sqrt(2)*(b^2*c + sqrt(2)*(-a*b^3)^(1/4)*b*d - sqrt(-a*b)*b*e)*arctan(1/2 *sqrt(2)*(2*x - sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/(-a*b^3)^(3/4) - 1/8*s qrt(2)*(b^2*c - sqrt(-a*b)*b*e)*log(x^2 + sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a /b))/(-a*b^3)^(3/4) + 1/8*sqrt(2)*(b^2*c - sqrt(-a*b)*b*e)*log(x^2 - sqrt( 2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(-a*b^3)^(3/4)
Time = 6.27 (sec) , antiderivative size = 725, normalized size of antiderivative = 6.30 \[ \int \frac {c+d x+e x^2}{a-b x^4} \, dx=\sum _{k=1}^4\ln \left (-b^2\,c\,d^2+b^2\,c^2\,e-b^2\,d^3\,x-a\,b\,e^3-{\mathrm {root}\left (256\,a^3\,b^3\,z^4-64\,a^2\,b^2\,c\,e\,z^2-32\,a^2\,b^2\,d^2\,z^2+16\,a^2\,b\,d\,e^2\,z+16\,a\,b^2\,c^2\,d\,z-4\,a\,b\,c\,d^2\,e+2\,a\,b\,c^2\,e^2+a\,b\,d^4-a^2\,e^4-b^2\,c^4,z,k\right )}^2\,a\,b^3\,c\,16-\mathrm {root}\left (256\,a^3\,b^3\,z^4-64\,a^2\,b^2\,c\,e\,z^2-32\,a^2\,b^2\,d^2\,z^2+16\,a^2\,b\,d\,e^2\,z+16\,a\,b^2\,c^2\,d\,z-4\,a\,b\,c\,d^2\,e+2\,a\,b\,c^2\,e^2+a\,b\,d^4-a^2\,e^4-b^2\,c^4,z,k\right )\,b^3\,c^2\,x\,4+{\mathrm {root}\left (256\,a^3\,b^3\,z^4-64\,a^2\,b^2\,c\,e\,z^2-32\,a^2\,b^2\,d^2\,z^2+16\,a^2\,b\,d\,e^2\,z+16\,a\,b^2\,c^2\,d\,z-4\,a\,b\,c\,d^2\,e+2\,a\,b\,c^2\,e^2+a\,b\,d^4-a^2\,e^4-b^2\,c^4,z,k\right )}^2\,a\,b^3\,d\,x\,16-\mathrm {root}\left (256\,a^3\,b^3\,z^4-64\,a^2\,b^2\,c\,e\,z^2-32\,a^2\,b^2\,d^2\,z^2+16\,a^2\,b\,d\,e^2\,z+16\,a\,b^2\,c^2\,d\,z-4\,a\,b\,c\,d^2\,e+2\,a\,b\,c^2\,e^2+a\,b\,d^4-a^2\,e^4-b^2\,c^4,z,k\right )\,a\,b^2\,e^2\,x\,4+\mathrm {root}\left (256\,a^3\,b^3\,z^4-64\,a^2\,b^2\,c\,e\,z^2-32\,a^2\,b^2\,d^2\,z^2+16\,a^2\,b\,d\,e^2\,z+16\,a\,b^2\,c^2\,d\,z-4\,a\,b\,c\,d^2\,e+2\,a\,b\,c^2\,e^2+a\,b\,d^4-a^2\,e^4-b^2\,c^4,z,k\right )\,a\,b^2\,d\,e\,8+2\,b^2\,c\,d\,e\,x\right )\,\mathrm {root}\left (256\,a^3\,b^3\,z^4-64\,a^2\,b^2\,c\,e\,z^2-32\,a^2\,b^2\,d^2\,z^2+16\,a^2\,b\,d\,e^2\,z+16\,a\,b^2\,c^2\,d\,z-4\,a\,b\,c\,d^2\,e+2\,a\,b\,c^2\,e^2+a\,b\,d^4-a^2\,e^4-b^2\,c^4,z,k\right ) \] Input:
int((c + d*x + e*x^2)/(a - b*x^4),x)
Output:
symsum(log(b^2*c^2*e - b^2*c*d^2 - b^2*d^3*x - a*b*e^3 - 16*root(256*a^3*b ^3*z^4 - 64*a^2*b^2*c*e*z^2 - 32*a^2*b^2*d^2*z^2 + 16*a^2*b*d*e^2*z + 16*a *b^2*c^2*d*z - 4*a*b*c*d^2*e + 2*a*b*c^2*e^2 + a*b*d^4 - a^2*e^4 - b^2*c^4 , z, k)^2*a*b^3*c - 4*root(256*a^3*b^3*z^4 - 64*a^2*b^2*c*e*z^2 - 32*a^2*b ^2*d^2*z^2 + 16*a^2*b*d*e^2*z + 16*a*b^2*c^2*d*z - 4*a*b*c*d^2*e + 2*a*b*c ^2*e^2 + a*b*d^4 - a^2*e^4 - b^2*c^4, z, k)*b^3*c^2*x + 16*root(256*a^3*b^ 3*z^4 - 64*a^2*b^2*c*e*z^2 - 32*a^2*b^2*d^2*z^2 + 16*a^2*b*d*e^2*z + 16*a* b^2*c^2*d*z - 4*a*b*c*d^2*e + 2*a*b*c^2*e^2 + a*b*d^4 - a^2*e^4 - b^2*c^4, z, k)^2*a*b^3*d*x - 4*root(256*a^3*b^3*z^4 - 64*a^2*b^2*c*e*z^2 - 32*a^2* b^2*d^2*z^2 + 16*a^2*b*d*e^2*z + 16*a*b^2*c^2*d*z - 4*a*b*c*d^2*e + 2*a*b* c^2*e^2 + a*b*d^4 - a^2*e^4 - b^2*c^4, z, k)*a*b^2*e^2*x + 8*root(256*a^3* b^3*z^4 - 64*a^2*b^2*c*e*z^2 - 32*a^2*b^2*d^2*z^2 + 16*a^2*b*d*e^2*z + 16* a*b^2*c^2*d*z - 4*a*b*c*d^2*e + 2*a*b*c^2*e^2 + a*b*d^4 - a^2*e^4 - b^2*c^ 4, z, k)*a*b^2*d*e + 2*b^2*c*d*e*x)*root(256*a^3*b^3*z^4 - 64*a^2*b^2*c*e* z^2 - 32*a^2*b^2*d^2*z^2 + 16*a^2*b*d*e^2*z + 16*a*b^2*c^2*d*z - 4*a*b*c*d ^2*e + 2*a*b*c^2*e^2 + a*b*d^4 - a^2*e^4 - b^2*c^4, z, k), k, 1, 4)
Time = 0.16 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.53 \[ \int \frac {c+d x+e x^2}{a-b x^4} \, dx=\frac {-2 b^{\frac {1}{4}} a^{\frac {3}{4}} \mathit {atan} \left (\frac {\sqrt {b}\, x}{b^{\frac {1}{4}} a^{\frac {1}{4}}}\right ) e +2 b^{\frac {3}{4}} a^{\frac {1}{4}} \mathit {atan} \left (\frac {\sqrt {b}\, x}{b^{\frac {1}{4}} a^{\frac {1}{4}}}\right ) c -b^{\frac {1}{4}} a^{\frac {3}{4}} \mathrm {log}\left (a^{\frac {1}{4}}-b^{\frac {1}{4}} x \right ) e +b^{\frac {1}{4}} a^{\frac {3}{4}} \mathrm {log}\left (a^{\frac {1}{4}}+b^{\frac {1}{4}} x \right ) e -b^{\frac {3}{4}} a^{\frac {1}{4}} \mathrm {log}\left (a^{\frac {1}{4}}-b^{\frac {1}{4}} x \right ) c +b^{\frac {3}{4}} a^{\frac {1}{4}} \mathrm {log}\left (a^{\frac {1}{4}}+b^{\frac {1}{4}} x \right ) c -\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (a^{\frac {1}{4}}-b^{\frac {1}{4}} x \right ) d -\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (a^{\frac {1}{4}}+b^{\frac {1}{4}} x \right ) d +\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}+\sqrt {b}\, x^{2}\right ) d}{4 a b} \] Input:
int((e*x^2+d*x+c)/(-b*x^4+a),x)
Output:
( - 2*b**(1/4)*a**(3/4)*atan((sqrt(b)*x)/(b**(1/4)*a**(1/4)))*e + 2*b**(3/ 4)*a**(1/4)*atan((sqrt(b)*x)/(b**(1/4)*a**(1/4)))*c - b**(1/4)*a**(3/4)*lo g(a**(1/4) - b**(1/4)*x)*e + b**(1/4)*a**(3/4)*log(a**(1/4) + b**(1/4)*x)* e - b**(3/4)*a**(1/4)*log(a**(1/4) - b**(1/4)*x)*c + b**(3/4)*a**(1/4)*log (a**(1/4) + b**(1/4)*x)*c - sqrt(b)*sqrt(a)*log(a**(1/4) - b**(1/4)*x)*d - sqrt(b)*sqrt(a)*log(a**(1/4) + b**(1/4)*x)*d + sqrt(b)*sqrt(a)*log(sqrt(a ) + sqrt(b)*x**2)*d)/(4*a*b)