Integrand size = 15, antiderivative size = 214 \[ \int \frac {d+e x}{\left (a+c x^4\right )^3} \, dx=\frac {x (d+e x)}{8 a \left (a+c x^4\right )^2}+\frac {x (7 d+6 e x)}{32 a^2 \left (a+c x^4\right )}+\frac {3 e \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {c}}-\frac {21 d \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} \sqrt [4]{c}}+\frac {21 d \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} \sqrt [4]{c}}+\frac {21 d \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x}{\sqrt {a}+\sqrt {c} x^2}\right )}{64 \sqrt {2} a^{11/4} \sqrt [4]{c}} \] Output:
1/8*x*(e*x+d)/a/(c*x^4+a)^2+1/32*x*(6*e*x+7*d)/a^2/(c*x^4+a)+3/16*e*arctan (c^(1/2)*x^2/a^(1/2))/a^(5/2)/c^(1/2)+21/128*d*arctan(-1+2^(1/2)*c^(1/4)*x /a^(1/4))*2^(1/2)/a^(11/4)/c^(1/4)+21/128*d*arctan(1+2^(1/2)*c^(1/4)*x/a^( 1/4))*2^(1/2)/a^(11/4)/c^(1/4)+21/128*d*arctanh(2^(1/2)*a^(1/4)*c^(1/4)*x/ (a^(1/2)+c^(1/2)*x^2))*2^(1/2)/a^(11/4)/c^(1/4)
Time = 0.21 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.16 \[ \int \frac {d+e x}{\left (a+c x^4\right )^3} \, dx=\frac {\frac {32 a^{7/4} x (d+e x)}{\left (a+c x^4\right )^2}+\frac {8 a^{3/4} x (7 d+6 e x)}{a+c x^4}-\frac {6 \left (7 \sqrt {2} \sqrt [4]{c} d+8 \sqrt [4]{a} e\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {c}}+\frac {6 \left (7 \sqrt {2} \sqrt [4]{c} d-8 \sqrt [4]{a} e\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {c}}-\frac {21 \sqrt {2} d \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{\sqrt [4]{c}}+\frac {21 \sqrt {2} d \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{\sqrt [4]{c}}}{256 a^{11/4}} \] Input:
Integrate[(d + e*x)/(a + c*x^4)^3,x]
Output:
((32*a^(7/4)*x*(d + e*x))/(a + c*x^4)^2 + (8*a^(3/4)*x*(7*d + 6*e*x))/(a + c*x^4) - (6*(7*Sqrt[2]*c^(1/4)*d + 8*a^(1/4)*e)*ArcTan[1 - (Sqrt[2]*c^(1/ 4)*x)/a^(1/4)])/Sqrt[c] + (6*(7*Sqrt[2]*c^(1/4)*d - 8*a^(1/4)*e)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/Sqrt[c] - (21*Sqrt[2]*d*Log[Sqrt[a] - Sqrt [2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/c^(1/4) + (21*Sqrt[2]*d*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/c^(1/4))/(256*a^(11/4))
Time = 0.76 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.31, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2394, 25, 2394, 27, 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.
| \(\displaystyle \int \frac {d+e x}{\left (a+c x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 2394 |
| \(\displaystyle \frac {x (d+e x)}{8 a \left (a+c x^4\right )^2}-\frac {\int -\frac {7 d+6 e x}{\left (c x^4+a\right )^2}dx}{8 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {7 d+6 e x}{\left (c x^4+a\right )^2}dx}{8 a}+\frac {x (d+e x)}{8 a \left (a+c x^4\right )^2}\) |
| \(\Big \downarrow \) 2394 |
| \(\displaystyle \frac {\frac {x (7 d+6 e x)}{4 a \left (a+c x^4\right )}-\frac {\int -\frac {3 (7 d+4 e x)}{c x^4+a}dx}{4 a}}{8 a}+\frac {x (d+e x)}{8 a \left (a+c x^4\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \int \frac {7 d+4 e x}{c x^4+a}dx}{4 a}+\frac {x (7 d+6 e x)}{4 a \left (a+c x^4\right )}}{8 a}+\frac {x (d+e x)}{8 a \left (a+c x^4\right )^2}\) |
| \(\Big \downarrow \) 2415 |
| \(\displaystyle \frac {\frac {3 \int \left (\frac {7 d}{c x^4+a}+\frac {4 e x}{c x^4+a}\right )dx}{4 a}+\frac {x (7 d+6 e x)}{4 a \left (a+c x^4\right )}}{8 a}+\frac {x (d+e x)}{8 a \left (a+c x^4\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {3 \left (-\frac {7 d \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{c}}+\frac {7 d \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{c}}-\frac {7 d \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{c}}+\frac {7 d \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{c}}+\frac {2 e \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}}\right )}{4 a}+\frac {x (7 d+6 e x)}{4 a \left (a+c x^4\right )}}{8 a}+\frac {x (d+e x)}{8 a \left (a+c x^4\right )^2}\) |
Input:
Int[(d + e*x)/(a + c*x^4)^3,x]
Output:
(x*(d + e*x))/(8*a*(a + c*x^4)^2) + ((x*(7*d + 6*e*x))/(4*a*(a + c*x^4)) + (3*((2*e*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(Sqrt[a]*Sqrt[c]) - (7*d*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*c^(1/4)) + (7*d*ArcTan [1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*c^(1/4)) - (7*d*Log[ Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(3/4)*c^( 1/4)) + (7*d*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sq rt[2]*a^(3/4)*c^(1/4))))/(4*a))/(8*a)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(-x)*Pq*((a + b *x^n)^(p + 1)/(a*n*(p + 1))), x] + Simp[1/(a*n*(p + 1)) Int[ExpandToSum[n *(p + 1)*Pq + D[x*Pq, x], x]*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x ] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[Expon[Pq, x], n - 1]
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 = 0.25 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.40
| method | result | size |
| risch | \(\frac {\frac {3 c e \,x^{6}}{16 a^{2}}+\frac {7 c d \,x^{5}}{32 a^{2}}+\frac {5 e \,x^{2}}{16 a}+\frac {11 d x}{32 a}}{\left (c \,x^{4}+a \right )^{2}}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\left (4 e \textit {\_R} +7 d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{128 a^{2} c}\) | \(86\) |
| default | \(d \left (\frac {x}{8 a \left (c \,x^{4}+a \right )^{2}}+\frac {\frac {7 x}{32 a \left (c \,x^{4}+a \right )}+\frac {21 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 a^{2}}}{a}\right )+e \left (\frac {x^{2}}{8 a \left (c \,x^{4}+a \right )^{2}}+\frac {\frac {3 x^{2}}{16 a \left (c \,x^{4}+a \right )}+\frac {3 \arctan \left (\sqrt {\frac {c}{a}}\, x^{2}\right )}{16 a \sqrt {a c}}}{a}\right )\) | \(207\) |
Input:
int((e*x+d)/(c*x^4+a)^3,x,method=_RETURNVERBOSE)
Output:
(3/16*c*e/a^2*x^6+7/32*c*d/a^2*x^5+5/16*e/a*x^2+11/32*d/a*x)/(c*x^4+a)^2+3 /128/a^2/c*sum((4*_R*e+7*d)/_R^3*ln(x-_R),_R=RootOf(_Z^4*c+a))
Result contains complex when optimal does not.
Time = 2.67 (sec) , antiderivative size = 43180, normalized size of antiderivative = 201.78 \[ \int \frac {d+e x}{\left (a+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)/(c*x^4+a)^3,x, algorithm="fricas")
Output:
Too large to include
Time = 0.98 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.90 \[ \int \frac {d+e x}{\left (a+c x^4\right )^3} \, dx=\operatorname {RootSum} {\left (268435456 t^{4} a^{11} c^{2} + 4718592 t^{2} a^{6} c e^{2} - 2709504 t a^{3} c d^{2} e + 20736 a e^{4} + 194481 c d^{4}, \left ( t \mapsto t \log {\left (x + \frac {- 67108864 t^{3} a^{9} c e^{2} - 9633792 t^{2} a^{6} c d^{2} e - 589824 t a^{4} e^{4} - 2765952 t a^{3} c d^{4} + 423360 a d^{2} e^{3}}{193536 a d e^{4} - 453789 c d^{5}} \right )} \right )\right )} + \frac {11 a d x + 10 a e x^{2} + 7 c d x^{5} + 6 c e x^{6}}{32 a^{4} + 64 a^{3} c x^{4} + 32 a^{2} c^{2} x^{8}} \] Input:
integrate((e*x+d)/(c*x**4+a)**3,x)
Output:
RootSum(268435456*_t**4*a**11*c**2 + 4718592*_t**2*a**6*c*e**2 - 2709504*_ t*a**3*c*d**2*e + 20736*a*e**4 + 194481*c*d**4, Lambda(_t, _t*log(x + (-67 108864*_t**3*a**9*c*e**2 - 9633792*_t**2*a**6*c*d**2*e - 589824*_t*a**4*e* *4 - 2765952*_t*a**3*c*d**4 + 423360*a*d**2*e**3)/(193536*a*d*e**4 - 45378 9*c*d**5)))) + (11*a*d*x + 10*a*e*x**2 + 7*c*d*x**5 + 6*c*e*x**6)/(32*a**4 + 64*a**3*c*x**4 + 32*a**2*c**2*x**8)
Time = 0.12 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.26 \[ \int \frac {d+e x}{\left (a+c x^4\right )^3} \, dx=\frac {6 \, c e x^{6} + 7 \, c d x^{5} + 10 \, a e x^{2} + 11 \, a d x}{32 \, {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )}} + \frac {3 \, {\left (\frac {7 \, \sqrt {2} d \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {1}{4}}} - \frac {7 \, \sqrt {2} d \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {1}{4}}} + \frac {2 \, {\left (7 \, \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} d - 8 \, \sqrt {a} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {1}{4}}} + \frac {2 \, {\left (7 \, \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} d + 8 \, \sqrt {a} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {1}{4}}}\right )}}{256 \, a^{2}} \] Input:
integrate((e*x+d)/(c*x^4+a)^3,x, algorithm="maxima")
Output:
1/32*(6*c*e*x^6 + 7*c*d*x^5 + 10*a*e*x^2 + 11*a*d*x)/(a^2*c^2*x^8 + 2*a^3* c*x^4 + a^4) + 3/256*(7*sqrt(2)*d*log(sqrt(c)*x^2 + sqrt(2)*a^(1/4)*c^(1/4 )*x + sqrt(a))/(a^(3/4)*c^(1/4)) - 7*sqrt(2)*d*log(sqrt(c)*x^2 - sqrt(2)*a ^(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(1/4)) + 2*(7*sqrt(2)*a^(1/4)*c^(1/ 4)*d - 8*sqrt(a)*e)*arctan(1/2*sqrt(2)*(2*sqrt(c)*x + sqrt(2)*a^(1/4)*c^(1 /4))/sqrt(sqrt(a)*sqrt(c)))/(a^(3/4)*sqrt(sqrt(a)*sqrt(c))*c^(1/4)) + 2*(7 *sqrt(2)*a^(1/4)*c^(1/4)*d + 8*sqrt(a)*e)*arctan(1/2*sqrt(2)*(2*sqrt(c)*x - sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a)*sqrt(c)))/(a^(3/4)*sqrt(sqrt(a)*sq rt(c))*c^(1/4)))/a^2
Time = 0.12 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.20 \[ \int \frac {d+e x}{\left (a+c x^4\right )^3} \, dx=\frac {21 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} d \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{256 \, a^{3} c} - \frac {21 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} d \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{256 \, a^{3} c} + \frac {3 \, \sqrt {2} {\left (4 \, \sqrt {2} \sqrt {a c} c e + 7 \, \left (a c^{3}\right )^{\frac {1}{4}} c d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{128 \, a^{3} c^{2}} + \frac {3 \, \sqrt {2} {\left (4 \, \sqrt {2} \sqrt {a c} c e + 7 \, \left (a c^{3}\right )^{\frac {1}{4}} c d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{128 \, a^{3} c^{2}} + \frac {6 \, c e x^{6} + 7 \, c d x^{5} + 10 \, a e x^{2} + 11 \, a d x}{32 \, {\left (c x^{4} + a\right )}^{2} a^{2}} \] Input:
integrate((e*x+d)/(c*x^4+a)^3,x, algorithm="giac")
Output:
21/256*sqrt(2)*(a*c^3)^(1/4)*d*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c) )/(a^3*c) - 21/256*sqrt(2)*(a*c^3)^(1/4)*d*log(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^3*c) + 3/128*sqrt(2)*(4*sqrt(2)*sqrt(a*c)*c*e + 7*(a*c^3) ^(1/4)*c*d)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a ^3*c^2) + 3/128*sqrt(2)*(4*sqrt(2)*sqrt(a*c)*c*e + 7*(a*c^3)^(1/4)*c*d)*ar ctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a^3*c^2) + 1/32 *(6*c*e*x^6 + 7*c*d*x^5 + 10*a*e*x^2 + 11*a*d*x)/((c*x^4 + a)^2*a^2)
Time = 0.31 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.47 \[ \int \frac {d+e x}{\left (a+c x^4\right )^3} \, dx=\frac {\frac {5\,e\,x^2}{16\,a}+\frac {11\,d\,x}{32\,a}+\frac {7\,c\,d\,x^5}{32\,a^2}+\frac {3\,c\,e\,x^6}{16\,a^2}}{a^2+2\,a\,c\,x^4+c^2\,x^8}+\left (\sum _{k=1}^4\ln \left (\frac {c^2\,\left (63\,d\,e^2+36\,e^3\,x-{\mathrm {root}\left (268435456\,a^{11}\,c^2\,z^4+4718592\,a^6\,c\,e^2\,z^2-2709504\,a^3\,c\,d^2\,e\,z+194481\,c\,d^4+20736\,a\,e^4,z,k\right )}^2\,a^5\,c\,d\,7168-\mathrm {root}\left (268435456\,a^{11}\,c^2\,z^4+4718592\,a^6\,c\,e^2\,z^2-2709504\,a^3\,c\,d^2\,e\,z+194481\,c\,d^4+20736\,a\,e^4,z,k\right )\,a^2\,c\,d^2\,x\,1176+{\mathrm {root}\left (268435456\,a^{11}\,c^2\,z^4+4718592\,a^6\,c\,e^2\,z^2-2709504\,a^3\,c\,d^2\,e\,z+194481\,c\,d^4+20736\,a\,e^4,z,k\right )}^2\,a^5\,c\,e\,x\,4096\right )\,3}{a^6\,2048}\right )\,\mathrm {root}\left (268435456\,a^{11}\,c^2\,z^4+4718592\,a^6\,c\,e^2\,z^2-2709504\,a^3\,c\,d^2\,e\,z+194481\,c\,d^4+20736\,a\,e^4,z,k\right )\right ) \] Input:
int((d + e*x)/(a + c*x^4)^3,x)
Output:
((5*e*x^2)/(16*a) + (11*d*x)/(32*a) + (7*c*d*x^5)/(32*a^2) + (3*c*e*x^6)/( 16*a^2))/(a^2 + c^2*x^8 + 2*a*c*x^4) + symsum(log((3*c^2*(63*d*e^2 + 36*e^ 3*x - 7168*root(268435456*a^11*c^2*z^4 + 4718592*a^6*c*e^2*z^2 - 2709504*a ^3*c*d^2*e*z + 194481*c*d^4 + 20736*a*e^4, z, k)^2*a^5*c*d - 1176*root(268 435456*a^11*c^2*z^4 + 4718592*a^6*c*e^2*z^2 - 2709504*a^3*c*d^2*e*z + 1944 81*c*d^4 + 20736*a*e^4, z, k)*a^2*c*d^2*x + 4096*root(268435456*a^11*c^2*z ^4 + 4718592*a^6*c*e^2*z^2 - 2709504*a^3*c*d^2*e*z + 194481*c*d^4 + 20736* a*e^4, z, k)^2*a^5*c*e*x))/(2048*a^6))*root(268435456*a^11*c^2*z^4 + 47185 92*a^6*c*e^2*z^2 - 2709504*a^3*c*d^2*e*z + 194481*c*d^4 + 20736*a*e^4, z, k), k, 1, 4)
Time = 0.16 (sec) , antiderivative size = 740, normalized size of antiderivative = 3.46 \[ \int \frac {d+e x}{\left (a+c x^4\right )^3} \, dx =\text {Too large to display} \] Input:
int((e*x+d)/(c*x^4+a)^3,x)
Output:
( - 42*c**(3/4)*a**(1/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt( c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a**2*d - 84*c**(3/4)*a**(1/4)*sqrt(2)*a tan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2))) *a*c*d*x**4 - 42*c**(3/4)*a**(1/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*c**2*d*x**8 - 48*sqrt(c)*sqrt (a)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt (2)))*a**2*e - 96*sqrt(c)*sqrt(a)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt (c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a*c*e*x**4 - 48*sqrt(c)*sqrt(a)*atan(( c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*c**2 *e*x**8 + 42*c**(3/4)*a**(1/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) + 2 *sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a**2*d + 84*c**(3/4)*a**(1/4)*sqr t(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqr t(2)))*a*c*d*x**4 + 42*c**(3/4)*a**(1/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*s qrt(2) + 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*c**2*d*x**8 - 48*sqrt(c )*sqrt(a)*atan((c**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(c)*x)/(c**(1/4)*a**(1/4 )*sqrt(2)))*a**2*e - 96*sqrt(c)*sqrt(a)*atan((c**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a*c*e*x**4 - 48*sqrt(c)*sqrt(a)* atan((c**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)) )*c**2*e*x**8 - 21*c**(3/4)*a**(1/4)*sqrt(2)*log( - c**(1/4)*a**(1/4)*sqrt (2)*x + sqrt(a) + sqrt(c)*x**2)*a**2*d - 42*c**(3/4)*a**(1/4)*sqrt(2)*l...