Integrand size = 15, antiderivative size = 214 \[ \int \frac {c+d x}{\left (a+b x^4\right )^3} \, dx=\frac {x (c+d x)}{8 a \left (a+b x^4\right )^2}+\frac {x (7 c+6 d x)}{32 a^2 \left (a+b x^4\right )}+\frac {3 d \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}}-\frac {21 c \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} \sqrt [4]{b}}+\frac {21 c \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} \sqrt [4]{b}}+\frac {21 c \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a}+\sqrt {b} x^2}\right )}{64 \sqrt {2} a^{11/4} \sqrt [4]{b}} \] Output:
1/8*x*(d*x+c)/a/(b*x^4+a)^2+1/32*x*(6*d*x+7*c)/a^2/(b*x^4+a)+3/16*d*arctan (b^(1/2)*x^2/a^(1/2))/a^(5/2)/b^(1/2)+21/128*c*arctan(-1+2^(1/2)*b^(1/4)*x /a^(1/4))*2^(1/2)/a^(11/4)/b^(1/4)+21/128*c*arctan(1+2^(1/2)*b^(1/4)*x/a^( 1/4))*2^(1/2)/a^(11/4)/b^(1/4)+21/128*c*arctanh(2^(1/2)*a^(1/4)*b^(1/4)*x/ (a^(1/2)+b^(1/2)*x^2))*2^(1/2)/a^(11/4)/b^(1/4)
Time = 0.14 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.16 \[ \int \frac {c+d x}{\left (a+b x^4\right )^3} \, dx=\frac {\frac {32 a^{7/4} x (c+d x)}{\left (a+b x^4\right )^2}+\frac {8 a^{3/4} x (7 c+6 d x)}{a+b x^4}-\frac {6 \left (7 \sqrt {2} \sqrt [4]{b} c+8 \sqrt [4]{a} d\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {b}}+\frac {6 \left (7 \sqrt {2} \sqrt [4]{b} c-8 \sqrt [4]{a} d\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {b}}-\frac {21 \sqrt {2} c \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{\sqrt [4]{b}}+\frac {21 \sqrt {2} c \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{\sqrt [4]{b}}}{256 a^{11/4}} \] Input:
Integrate[(c + d*x)/(a + b*x^4)^3,x]
Output:
((32*a^(7/4)*x*(c + d*x))/(a + b*x^4)^2 + (8*a^(3/4)*x*(7*c + 6*d*x))/(a + b*x^4) - (6*(7*Sqrt[2]*b^(1/4)*c + 8*a^(1/4)*d)*ArcTan[1 - (Sqrt[2]*b^(1/ 4)*x)/a^(1/4)])/Sqrt[b] + (6*(7*Sqrt[2]*b^(1/4)*c - 8*a^(1/4)*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/Sqrt[b] - (21*Sqrt[2]*c*Log[Sqrt[a] - Sqrt [2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(1/4) + (21*Sqrt[2]*c*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(1/4))/(256*a^(11/4))
Time = 0.77 (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.
| \(\displaystyle \int \frac {c+d x}{\left (a+b x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 2394 |
| \(\displaystyle \frac {x (c+d x)}{8 a \left (a+b x^4\right )^2}-\frac {\int -\frac {7 c+6 d x}{\left (b x^4+a\right )^2}dx}{8 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {7 c+6 d x}{\left (b x^4+a\right )^2}dx}{8 a}+\frac {x (c+d x)}{8 a \left (a+b x^4\right )^2}\) |
| \(\Big \downarrow \) 2394 |
| \(\displaystyle \frac {\frac {x (7 c+6 d x)}{4 a \left (a+b x^4\right )}-\frac {\int -\frac {3 (7 c+4 d x)}{b x^4+a}dx}{4 a}}{8 a}+\frac {x (c+d x)}{8 a \left (a+b x^4\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \int \frac {7 c+4 d x}{b x^4+a}dx}{4 a}+\frac {x (7 c+6 d x)}{4 a \left (a+b x^4\right )}}{8 a}+\frac {x (c+d x)}{8 a \left (a+b x^4\right )^2}\) |
| \(\Big \downarrow \) 2415 |
| \(\displaystyle \frac {\frac {3 \int \left (\frac {7 c}{b x^4+a}+\frac {4 d x}{b x^4+a}\right )dx}{4 a}+\frac {x (7 c+6 d x)}{4 a \left (a+b x^4\right )}}{8 a}+\frac {x (c+d x)}{8 a \left (a+b x^4\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {3 \left (-\frac {7 c \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}+\frac {7 c \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {7 c \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}}+\frac {7 c \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}}+\frac {2 d \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}\right )}{4 a}+\frac {x (7 c+6 d x)}{4 a \left (a+b x^4\right )}}{8 a}+\frac {x (c+d x)}{8 a \left (a+b x^4\right )^2}\) |
Input:
Int[(c + d*x)/(a + b*x^4)^3,x]
Output:
(x*(c + d*x))/(8*a*(a + b*x^4)^2) + ((x*(7*c + 6*d*x))/(4*a*(a + b*x^4)) + (3*((2*d*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]) - (7*c*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*b^(1/4)) + (7*c*ArcTan [1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*b^(1/4)) - (7*c*Log[ Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(3/4)*b^( 1/4)) + (7*c*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sq rt[2]*a^(3/4)*b^(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 = 0.13 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.40
| method | result | size |
| risch | \(\frac {\frac {3 b d \,x^{6}}{16 a^{2}}+\frac {7 b c \,x^{5}}{32 a^{2}}+\frac {5 d \,x^{2}}{16 a}+\frac {11 c x}{32 a}}{\left (b \,x^{4}+a \right )^{2}}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\left (4 d \textit {\_R} +7 c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{128 b \,a^{2}}\) | \(86\) |
| default | \(c \left (\frac {x}{8 a \left (b \,x^{4}+a \right )^{2}}+\frac {\frac {7 x}{32 a \left (b \,x^{4}+a \right )}+\frac {21 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{256 a^{2}}}{a}\right )+d \left (\frac {x^{2}}{8 a \left (b \,x^{4}+a \right )^{2}}+\frac {\frac {3 x^{2}}{16 a \left (b \,x^{4}+a \right )}+\frac {3 \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right )}{16 a \sqrt {a b}}}{a}\right )\) | \(207\) |
Input:
int((d*x+c)/(b*x^4+a)^3,x,method=_RETURNVERBOSE)
Output:
(3/16*b*d/a^2*x^6+7/32*b/a^2*c*x^5+5/16*d/a*x^2+11/32*c/a*x)/(b*x^4+a)^2+3 /128/b/a^2*sum((4*_R*d+7*c)/_R^3*ln(x-_R),_R=RootOf(_Z^4*b+a))
Result contains complex when optimal does not.
Time = 2.05 (sec) , antiderivative size = 43180, normalized size of antiderivative = 201.78 \[ \int \frac {c+d x}{\left (a+b x^4\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)/(b*x^4+a)^3,x, algorithm="fricas")
Output:
Too large to include
Time = 1.20 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.90 \[ \int \frac {c+d x}{\left (a+b x^4\right )^3} \, dx=\operatorname {RootSum} {\left (268435456 t^{4} a^{11} b^{2} + 4718592 t^{2} a^{6} b d^{2} - 2709504 t a^{3} b c^{2} d + 20736 a d^{4} + 194481 b c^{4}, \left ( t \mapsto t \log {\left (x + \frac {- 67108864 t^{3} a^{9} b d^{2} - 9633792 t^{2} a^{6} b c^{2} d - 589824 t a^{4} d^{4} - 2765952 t a^{3} b c^{4} + 423360 a c^{2} d^{3}}{193536 a c d^{4} - 453789 b c^{5}} \right )} \right )\right )} + \frac {11 a c x + 10 a d x^{2} + 7 b c x^{5} + 6 b d x^{6}}{32 a^{4} + 64 a^{3} b x^{4} + 32 a^{2} b^{2} x^{8}} \] Input:
integrate((d*x+c)/(b*x**4+a)**3,x)
Output:
RootSum(268435456*_t**4*a**11*b**2 + 4718592*_t**2*a**6*b*d**2 - 2709504*_ t*a**3*b*c**2*d + 20736*a*d**4 + 194481*b*c**4, Lambda(_t, _t*log(x + (-67 108864*_t**3*a**9*b*d**2 - 9633792*_t**2*a**6*b*c**2*d - 589824*_t*a**4*d* *4 - 2765952*_t*a**3*b*c**4 + 423360*a*c**2*d**3)/(193536*a*c*d**4 - 45378 9*b*c**5)))) + (11*a*c*x + 10*a*d*x**2 + 7*b*c*x**5 + 6*b*d*x**6)/(32*a**4 + 64*a**3*b*x**4 + 32*a**2*b**2*x**8)
Time = 0.11 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.26 \[ \int \frac {c+d x}{\left (a+b x^4\right )^3} \, dx=\frac {6 \, b d x^{6} + 7 \, b c x^{5} + 10 \, a d x^{2} + 11 \, a c x}{32 \, {\left (a^{2} b^{2} x^{8} + 2 \, a^{3} b x^{4} + a^{4}\right )}} + \frac {3 \, {\left (\frac {7 \, \sqrt {2} c \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {7 \, \sqrt {2} c \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, {\left (7 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} c - 8 \, \sqrt {a} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {1}{4}}} + \frac {2 \, {\left (7 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} c + 8 \, \sqrt {a} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {1}{4}}}\right )}}{256 \, a^{2}} \] Input:
integrate((d*x+c)/(b*x^4+a)^3,x, algorithm="maxima")
Output:
1/32*(6*b*d*x^6 + 7*b*c*x^5 + 10*a*d*x^2 + 11*a*c*x)/(a^2*b^2*x^8 + 2*a^3* b*x^4 + a^4) + 3/256*(7*sqrt(2)*c*log(sqrt(b)*x^2 + sqrt(2)*a^(1/4)*b^(1/4 )*x + sqrt(a))/(a^(3/4)*b^(1/4)) - 7*sqrt(2)*c*log(sqrt(b)*x^2 - sqrt(2)*a ^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(1/4)) + 2*(7*sqrt(2)*a^(1/4)*b^(1/ 4)*c - 8*sqrt(a)*d)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x + sqrt(2)*a^(1/4)*b^(1 /4))/sqrt(sqrt(a)*sqrt(b)))/(a^(3/4)*sqrt(sqrt(a)*sqrt(b))*b^(1/4)) + 2*(7 *sqrt(2)*a^(1/4)*b^(1/4)*c + 8*sqrt(a)*d)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x - sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/(a^(3/4)*sqrt(sqrt(a)*sq rt(b))*b^(1/4)))/a^2
Time = 0.13 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.20 \[ \int \frac {c+d x}{\left (a+b x^4\right )^3} \, dx=\frac {21 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} c \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{256 \, a^{3} b} - \frac {21 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} c \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{256 \, a^{3} b} + \frac {3 \, \sqrt {2} {\left (4 \, \sqrt {2} \sqrt {a b} b d + 7 \, \left (a b^{3}\right )^{\frac {1}{4}} b c\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, a^{3} b^{2}} + \frac {3 \, \sqrt {2} {\left (4 \, \sqrt {2} \sqrt {a b} b d + 7 \, \left (a b^{3}\right )^{\frac {1}{4}} b c\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, a^{3} b^{2}} + \frac {6 \, b d x^{6} + 7 \, b c x^{5} + 10 \, a d x^{2} + 11 \, a c x}{32 \, {\left (b x^{4} + a\right )}^{2} a^{2}} \] Input:
integrate((d*x+c)/(b*x^4+a)^3,x, algorithm="giac")
Output:
21/256*sqrt(2)*(a*b^3)^(1/4)*c*log(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b) )/(a^3*b) - 21/256*sqrt(2)*(a*b^3)^(1/4)*c*log(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^3*b) + 3/128*sqrt(2)*(4*sqrt(2)*sqrt(a*b)*b*d + 7*(a*b^3) ^(1/4)*b*c)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a ^3*b^2) + 3/128*sqrt(2)*(4*sqrt(2)*sqrt(a*b)*b*d + 7*(a*b^3)^(1/4)*b*c)*ar ctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^3*b^2) + 1/32 *(6*b*d*x^6 + 7*b*c*x^5 + 10*a*d*x^2 + 11*a*c*x)/((b*x^4 + a)^2*a^2)
Time = 6.12 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.47 \[ \int \frac {c+d x}{\left (a+b x^4\right )^3} \, dx=\frac {\frac {5\,d\,x^2}{16\,a}+\frac {11\,c\,x}{32\,a}+\frac {7\,b\,c\,x^5}{32\,a^2}+\frac {3\,b\,d\,x^6}{16\,a^2}}{a^2+2\,a\,b\,x^4+b^2\,x^8}+\left (\sum _{k=1}^4\ln \left (\frac {b^2\,\left (63\,c\,d^2+36\,d^3\,x-{\mathrm {root}\left (268435456\,a^{11}\,b^2\,z^4+4718592\,a^6\,b\,d^2\,z^2-2709504\,a^3\,b\,c^2\,d\,z+194481\,b\,c^4+20736\,a\,d^4,z,k\right )}^2\,a^5\,b\,c\,7168-\mathrm {root}\left (268435456\,a^{11}\,b^2\,z^4+4718592\,a^6\,b\,d^2\,z^2-2709504\,a^3\,b\,c^2\,d\,z+194481\,b\,c^4+20736\,a\,d^4,z,k\right )\,a^2\,b\,c^2\,x\,1176+{\mathrm {root}\left (268435456\,a^{11}\,b^2\,z^4+4718592\,a^6\,b\,d^2\,z^2-2709504\,a^3\,b\,c^2\,d\,z+194481\,b\,c^4+20736\,a\,d^4,z,k\right )}^2\,a^5\,b\,d\,x\,4096\right )\,3}{a^6\,2048}\right )\,\mathrm {root}\left (268435456\,a^{11}\,b^2\,z^4+4718592\,a^6\,b\,d^2\,z^2-2709504\,a^3\,b\,c^2\,d\,z+194481\,b\,c^4+20736\,a\,d^4,z,k\right )\right ) \] Input:
int((c + d*x)/(a + b*x^4)^3,x)
Output:
((5*d*x^2)/(16*a) + (11*c*x)/(32*a) + (7*b*c*x^5)/(32*a^2) + (3*b*d*x^6)/( 16*a^2))/(a^2 + b^2*x^8 + 2*a*b*x^4) + symsum(log((3*b^2*(63*c*d^2 + 36*d^ 3*x - 7168*root(268435456*a^11*b^2*z^4 + 4718592*a^6*b*d^2*z^2 - 2709504*a ^3*b*c^2*d*z + 194481*b*c^4 + 20736*a*d^4, z, k)^2*a^5*b*c - 1176*root(268 435456*a^11*b^2*z^4 + 4718592*a^6*b*d^2*z^2 - 2709504*a^3*b*c^2*d*z + 1944 81*b*c^4 + 20736*a*d^4, z, k)*a^2*b*c^2*x + 4096*root(268435456*a^11*b^2*z ^4 + 4718592*a^6*b*d^2*z^2 - 2709504*a^3*b*c^2*d*z + 194481*b*c^4 + 20736* a*d^4, z, k)^2*a^5*b*d*x))/(2048*a^6))*root(268435456*a^11*b^2*z^4 + 47185 92*a^6*b*d^2*z^2 - 2709504*a^3*b*c^2*d*z + 194481*b*c^4 + 20736*a*d^4, z, k), k, 1, 4)
Time = 0.18 (sec) , antiderivative size = 740, normalized size of antiderivative = 3.46 \[ \int \frac {c+d x}{\left (a+b x^4\right )^3} \, dx =\text {Too large to display} \] Input:
int((d*x+c)/(b*x^4+a)^3,x)
Output:
( - 42*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt( b)*x)/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*c - 84*b**(3/4)*a**(1/4)*sqrt(2)*a tan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(b)*x)/(b**(1/4)*a**(1/4)*sqrt(2))) *a*b*c*x**4 - 42*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(b)*x)/(b**(1/4)*a**(1/4)*sqrt(2)))*b**2*c*x**8 - 48*sqrt(b)*sqrt (a)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(b)*x)/(b**(1/4)*a**(1/4)*sqrt (2)))*a**2*d - 96*sqrt(b)*sqrt(a)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt (b)*x)/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b*d*x**4 - 48*sqrt(b)*sqrt(a)*atan(( b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(b)*x)/(b**(1/4)*a**(1/4)*sqrt(2)))*b**2 *d*x**8 + 42*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2 *sqrt(b)*x)/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*c + 84*b**(3/4)*a**(1/4)*sqr t(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(b)*x)/(b**(1/4)*a**(1/4)*sqr t(2)))*a*b*c*x**4 + 42*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*s qrt(2) + 2*sqrt(b)*x)/(b**(1/4)*a**(1/4)*sqrt(2)))*b**2*c*x**8 - 48*sqrt(b )*sqrt(a)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(b)*x)/(b**(1/4)*a**(1/4 )*sqrt(2)))*a**2*d - 96*sqrt(b)*sqrt(a)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(b)*x)/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b*d*x**4 - 48*sqrt(b)*sqrt(a)* atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(b)*x)/(b**(1/4)*a**(1/4)*sqrt(2)) )*b**2*d*x**8 - 21*b**(3/4)*a**(1/4)*sqrt(2)*log( - b**(1/4)*a**(1/4)*sqrt (2)*x + sqrt(a) + sqrt(b)*x**2)*a**2*c - 42*b**(3/4)*a**(1/4)*sqrt(2)*l...