Integrand size = 20, antiderivative size = 304 \[ \int \frac {c+d x+e x^2}{\left (a+b x^4\right )^4} \, dx=\frac {x \left (c+d x+e x^2\right )}{12 a \left (a+b x^4\right )^3}+\frac {x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a+b x^4\right )^2}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a+b x^4\right )}+\frac {5 d \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} \sqrt {b}}-\frac {\left (77 \sqrt {b} c+15 \sqrt {a} e\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{15/4} b^{3/4}}+\frac {\left (77 \sqrt {b} c+15 \sqrt {a} e\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{15/4} b^{3/4}}+\frac {\left (77 \sqrt {b} c-15 \sqrt {a} e\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a}+\sqrt {b} x^2}\right )}{256 \sqrt {2} a^{15/4} b^{3/4}} \] Output:
1/12*x*(e*x^2+d*x+c)/a/(b*x^4+a)^3+1/96*x*(9*e*x^2+10*d*x+11*c)/a^2/(b*x^4 +a)^2+1/384*x*(45*e*x^2+60*d*x+77*c)/a^3/(b*x^4+a)+5/32*d*arctan(b^(1/2)*x ^2/a^(1/2))/a^(7/2)/b^(1/2)+1/512*(77*b^(1/2)*c+15*a^(1/2)*e)*arctan(-1+2^ (1/2)*b^(1/4)*x/a^(1/4))*2^(1/2)/a^(15/4)/b^(3/4)+1/512*(77*b^(1/2)*c+15*a ^(1/2)*e)*arctan(1+2^(1/2)*b^(1/4)*x/a^(1/4))*2^(1/2)/a^(15/4)/b^(3/4)+1/5 12*(77*b^(1/2)*c-15*a^(1/2)*e)*arctanh(2^(1/2)*a^(1/4)*b^(1/4)*x/(a^(1/2)+ b^(1/2)*x^2))*2^(1/2)/a^(15/4)/b^(3/4)
Time = 0.30 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.21 \[ \int \frac {c+d x+e x^2}{\left (a+b x^4\right )^4} \, dx=\frac {\frac {256 a^3 x (c+x (d+e x))}{\left (a+b x^4\right )^3}+\frac {8 a x (77 c+15 x (4 d+3 e x))}{a+b x^4}+\frac {32 a^2 x (11 c+x (10 d+9 e x))}{\left (a+b x^4\right )^2}-\frac {6 \sqrt [4]{a} \left (77 \sqrt {2} \sqrt {b} c+80 \sqrt [4]{a} \sqrt [4]{b} d+15 \sqrt {2} \sqrt {a} e\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{b^{3/4}}+\frac {6 \sqrt [4]{a} \left (77 \sqrt {2} \sqrt {b} c-80 \sqrt [4]{a} \sqrt [4]{b} d+15 \sqrt {2} \sqrt {a} e\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{b^{3/4}}+\frac {3 \sqrt {2} \left (-77 \sqrt [4]{a} \sqrt {b} c+15 a^{3/4} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{b^{3/4}}+\frac {3 \sqrt {2} \left (77 \sqrt [4]{a} \sqrt {b} c-15 a^{3/4} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{b^{3/4}}}{3072 a^4} \] Input:
Integrate[(c + d*x + e*x^2)/(a + b*x^4)^4,x]
Output:
((256*a^3*x*(c + x*(d + e*x)))/(a + b*x^4)^3 + (8*a*x*(77*c + 15*x*(4*d + 3*e*x)))/(a + b*x^4) + (32*a^2*x*(11*c + x*(10*d + 9*e*x)))/(a + b*x^4)^2 - (6*a^(1/4)*(77*Sqrt[2]*Sqrt[b]*c + 80*a^(1/4)*b^(1/4)*d + 15*Sqrt[2]*Sqr t[a]*e)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/b^(3/4) + (6*a^(1/4)*(77* Sqrt[2]*Sqrt[b]*c - 80*a^(1/4)*b^(1/4)*d + 15*Sqrt[2]*Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/b^(3/4) + (3*Sqrt[2]*(-77*a^(1/4)*Sqrt[b]* c + 15*a^(3/4)*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/ b^(3/4) + (3*Sqrt[2]*(77*a^(1/4)*Sqrt[b]*c - 15*a^(3/4)*e)*Log[Sqrt[a] + S qrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(3/4))/(3072*a^4)
Time = 1.08 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.30, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2394, 25, 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.
| \(\displaystyle \int \frac {c+d x+e x^2}{\left (a+b x^4\right )^4} \, dx\) |
| \(\Big \downarrow \) 2394 |
| \(\displaystyle \frac {x \left (c+d x+e x^2\right )}{12 a \left (a+b x^4\right )^3}-\frac {\int -\frac {9 e x^2+10 d x+11 c}{\left (b x^4+a\right )^3}dx}{12 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {9 e x^2+10 d x+11 c}{\left (b x^4+a\right )^3}dx}{12 a}+\frac {x \left (c+d x+e x^2\right )}{12 a \left (a+b x^4\right )^3}\) |
| \(\Big \downarrow \) 2394 |
| \(\displaystyle \frac {\frac {x \left (11 c+10 d x+9 e x^2\right )}{8 a \left (a+b x^4\right )^2}-\frac {\int -\frac {45 e x^2+60 d x+77 c}{\left (b x^4+a\right )^2}dx}{8 a}}{12 a}+\frac {x \left (c+d x+e x^2\right )}{12 a \left (a+b x^4\right )^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {45 e x^2+60 d x+77 c}{\left (b x^4+a\right )^2}dx}{8 a}+\frac {x \left (11 c+10 d x+9 e x^2\right )}{8 a \left (a+b x^4\right )^2}}{12 a}+\frac {x \left (c+d x+e x^2\right )}{12 a \left (a+b x^4\right )^3}\) |
| \(\Big \downarrow \) 2394 |
| \(\displaystyle \frac {\frac {\frac {x \left (77 c+60 d x+45 e x^2\right )}{4 a \left (a+b x^4\right )}-\frac {\int -\frac {3 \left (15 e x^2+40 d x+77 c\right )}{b x^4+a}dx}{4 a}}{8 a}+\frac {x \left (11 c+10 d x+9 e x^2\right )}{8 a \left (a+b x^4\right )^2}}{12 a}+\frac {x \left (c+d x+e x^2\right )}{12 a \left (a+b x^4\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {15 e x^2+40 d x+77 c}{b x^4+a}dx}{4 a}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{4 a \left (a+b x^4\right )}}{8 a}+\frac {x \left (11 c+10 d x+9 e x^2\right )}{8 a \left (a+b x^4\right )^2}}{12 a}+\frac {x \left (c+d x+e x^2\right )}{12 a \left (a+b x^4\right )^3}\) |
| \(\Big \downarrow \) 2415 |
| \(\displaystyle \frac {\frac {\frac {3 \int \left (\frac {40 d x}{b x^4+a}+\frac {15 e x^2+77 c}{b x^4+a}\right )dx}{4 a}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{4 a \left (a+b x^4\right )}}{8 a}+\frac {x \left (11 c+10 d x+9 e x^2\right )}{8 a \left (a+b x^4\right )^2}}{12 a}+\frac {x \left (c+d x+e x^2\right )}{12 a \left (a+b x^4\right )^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (15 \sqrt {a} e+77 \sqrt {b} c\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (15 \sqrt {a} e+77 \sqrt {b} c\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}-\frac {\left (77 \sqrt {b} c-15 \sqrt {a} e\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}+\frac {\left (77 \sqrt {b} c-15 \sqrt {a} e\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}+\frac {20 d \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}\right )}{4 a}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{4 a \left (a+b x^4\right )}}{8 a}+\frac {x \left (11 c+10 d x+9 e x^2\right )}{8 a \left (a+b x^4\right )^2}}{12 a}+\frac {x \left (c+d x+e x^2\right )}{12 a \left (a+b x^4\right )^3}\) |
Input:
Int[(c + d*x + e*x^2)/(a + b*x^4)^4,x]
Output:
(x*(c + d*x + e*x^2))/(12*a*(a + b*x^4)^3) + ((x*(11*c + 10*d*x + 9*e*x^2) )/(8*a*(a + b*x^4)^2) + ((x*(77*c + 60*d*x + 45*e*x^2))/(4*a*(a + b*x^4)) + (3*((20*d*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]) - ((77*Sqrt[b ]*c + 15*Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^ (3/4)*b^(3/4)) + ((77*Sqrt[b]*c + 15*Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*b^(1/4 )*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*b^(3/4)) - ((77*Sqrt[b]*c - 15*Sqrt[a]*e )*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(3/ 4)*b^(3/4)) + ((77*Sqrt[b]*c - 15*Sqrt[a]*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4) *b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(3/4)*b^(3/4))))/(4*a))/(8*a))/(12 *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 = 0.14 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.48
| method | result | size |
| risch | \(\frac {\frac {15 e \,b^{2} x^{11}}{128 a^{3}}+\frac {5 d \,b^{2} x^{10}}{32 a^{3}}+\frac {77 c \,b^{2} x^{9}}{384 a^{3}}+\frac {21 b e \,x^{7}}{64 a^{2}}+\frac {5 b d \,x^{6}}{12 a^{2}}+\frac {33 b c \,x^{5}}{64 a^{2}}+\frac {113 e \,x^{3}}{384 a}+\frac {11 d \,x^{2}}{32 a}+\frac {51 c x}{128 a}}{\left (b \,x^{4}+a \right )^{3}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\left (15 \textit {\_R}^{2} e +40 d \textit {\_R} +77 c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{512 b \,a^{3}}\) | \(147\) |
| default | \(\frac {\frac {15 e \,b^{2} x^{11}}{128 a^{3}}+\frac {5 d \,b^{2} x^{10}}{32 a^{3}}+\frac {77 c \,b^{2} x^{9}}{384 a^{3}}+\frac {21 b e \,x^{7}}{64 a^{2}}+\frac {5 b d \,x^{6}}{12 a^{2}}+\frac {33 b c \,x^{5}}{64 a^{2}}+\frac {113 e \,x^{3}}{384 a}+\frac {11 d \,x^{2}}{32 a}+\frac {51 c x}{128 a}}{\left (b \,x^{4}+a \right )^{3}}+\frac {\frac {77 c \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {20 d \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right )}{\sqrt {a b}}+\frac {15 e \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{128 a^{3}}\) | \(334\) |
Input:
int((e*x^2+d*x+c)/(b*x^4+a)^4,x,method=_RETURNVERBOSE)
Output:
(15/128*e/a^3*b^2*x^11+5/32/a^3*d*b^2*x^10+77/384*c/a^3*b^2*x^9+21/64*b*e/ a^2*x^7+5/12*b*d/a^2*x^6+33/64*b/a^2*c*x^5+113/384*e/a*x^3+11/32*d/a*x^2+5 1/128*c/a*x)/(b*x^4+a)^3+1/512/b/a^3*sum((15*_R^2*e+40*_R*d+77*c)/_R^3*ln( x-_R),_R=RootOf(_Z^4*b+a))
Result contains complex when optimal does not.
Time = 14.33 (sec) , antiderivative size = 124960, normalized size of antiderivative = 411.05 \[ \int \frac {c+d x+e x^2}{\left (a+b x^4\right )^4} \, dx=\text {Too large to display} \] Input:
integrate((e*x^2+d*x+c)/(b*x^4+a)^4,x, algorithm="fricas")
Output:
Too large to include
Leaf count of result is larger than twice the leaf count of optimal. 610 vs. \(2 (292) = 584\).
Time = 30.23 (sec) , antiderivative size = 610, normalized size of antiderivative = 2.01 \[ \int \frac {c+d x+e x^2}{\left (a+b x^4\right )^4} \, dx=\operatorname {RootSum} {\left (68719476736 t^{4} a^{15} b^{3} + t^{2} \cdot \left (1211105280 a^{8} b^{2} c e + 838860800 a^{8} b^{2} d^{2}\right ) + t \left (18432000 a^{5} b d e^{2} - 485703680 a^{4} b^{2} c^{2} d\right ) + 50625 a^{2} e^{4} + 2668050 a b c^{2} e^{2} - 7392000 a b c d^{2} e + 2560000 a b d^{4} + 35153041 b^{2} c^{4}, \left ( t \mapsto t \log {\left (x + \frac {452984832000 t^{3} a^{13} b^{2} e^{3} - 11936653639680 t^{3} a^{12} b^{3} c^{2} e + 33071248179200 t^{3} a^{12} b^{3} c d^{2} + 544997376000 t^{2} a^{9} b^{2} c d e^{2} - 503316480000 t^{2} a^{9} b^{2} d^{3} e + 4787095470080 t^{2} a^{8} b^{3} c^{3} d + 5987520000 t a^{6} b c e^{4} + 8294400000 t a^{6} b d^{2} e^{3} - 210370406400 t a^{5} b^{2} c^{3} e^{2} + 655699968000 t a^{5} b^{2} c^{2} d^{2} e + 201850880000 t a^{5} b^{2} c d^{4} + 1385873488384 t a^{4} b^{3} c^{5} + 91125000 a^{3} d e^{5} + 5544000000 a^{2} b c d^{3} e^{2} - 3072000000 a^{2} b d^{5} e + 105459123000 a b^{2} c^{4} d e - 146090560000 a b^{2} c^{3} d^{3}}{11390625 a^{3} e^{6} - 300155625 a^{2} b c^{2} e^{4} + 3326400000 a^{2} b c d^{2} e^{3} - 2304000000 a^{2} b d^{4} e^{2} - 7909434225 a b^{2} c^{4} e^{2} + 87654336000 a b^{2} c^{3} d^{2} e - 60712960000 a b^{2} c^{2} d^{4} + 208422380089 b^{3} c^{6}} \right )} \right )\right )} + \frac {153 a^{2} c x + 132 a^{2} d x^{2} + 113 a^{2} e x^{3} + 198 a b c x^{5} + 160 a b d x^{6} + 126 a b e x^{7} + 77 b^{2} c x^{9} + 60 b^{2} d x^{10} + 45 b^{2} e x^{11}}{384 a^{6} + 1152 a^{5} b x^{4} + 1152 a^{4} b^{2} x^{8} + 384 a^{3} b^{3} x^{12}} \] Input:
integrate((e*x**2+d*x+c)/(b*x**4+a)**4,x)
Output:
RootSum(68719476736*_t**4*a**15*b**3 + _t**2*(1211105280*a**8*b**2*c*e + 8 38860800*a**8*b**2*d**2) + _t*(18432000*a**5*b*d*e**2 - 485703680*a**4*b** 2*c**2*d) + 50625*a**2*e**4 + 2668050*a*b*c**2*e**2 - 7392000*a*b*c*d**2*e + 2560000*a*b*d**4 + 35153041*b**2*c**4, Lambda(_t, _t*log(x + (452984832 000*_t**3*a**13*b**2*e**3 - 11936653639680*_t**3*a**12*b**3*c**2*e + 33071 248179200*_t**3*a**12*b**3*c*d**2 + 544997376000*_t**2*a**9*b**2*c*d*e**2 - 503316480000*_t**2*a**9*b**2*d**3*e + 4787095470080*_t**2*a**8*b**3*c**3 *d + 5987520000*_t*a**6*b*c*e**4 + 8294400000*_t*a**6*b*d**2*e**3 - 210370 406400*_t*a**5*b**2*c**3*e**2 + 655699968000*_t*a**5*b**2*c**2*d**2*e + 20 1850880000*_t*a**5*b**2*c*d**4 + 1385873488384*_t*a**4*b**3*c**5 + 9112500 0*a**3*d*e**5 + 5544000000*a**2*b*c*d**3*e**2 - 3072000000*a**2*b*d**5*e + 105459123000*a*b**2*c**4*d*e - 146090560000*a*b**2*c**3*d**3)/(11390625*a **3*e**6 - 300155625*a**2*b*c**2*e**4 + 3326400000*a**2*b*c*d**2*e**3 - 23 04000000*a**2*b*d**4*e**2 - 7909434225*a*b**2*c**4*e**2 + 87654336000*a*b* *2*c**3*d**2*e - 60712960000*a*b**2*c**2*d**4 + 208422380089*b**3*c**6)))) + (153*a**2*c*x + 132*a**2*d*x**2 + 113*a**2*e*x**3 + 198*a*b*c*x**5 + 16 0*a*b*d*x**6 + 126*a*b*e*x**7 + 77*b**2*c*x**9 + 60*b**2*d*x**10 + 45*b**2 *e*x**11)/(384*a**6 + 1152*a**5*b*x**4 + 1152*a**4*b**2*x**8 + 384*a**3*b* *3*x**12)
Time = 0.11 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.26 \[ \int \frac {c+d x+e x^2}{\left (a+b x^4\right )^4} \, dx=\frac {45 \, b^{2} e x^{11} + 60 \, b^{2} d x^{10} + 77 \, b^{2} c x^{9} + 126 \, a b e x^{7} + 160 \, a b d x^{6} + 198 \, a b c x^{5} + 113 \, a^{2} e x^{3} + 132 \, a^{2} d x^{2} + 153 \, a^{2} c x}{384 \, {\left (a^{3} b^{3} x^{12} + 3 \, a^{4} b^{2} x^{8} + 3 \, a^{5} b x^{4} + a^{6}\right )}} + \frac {\frac {\sqrt {2} {\left (77 \, \sqrt {b} c - 15 \, \sqrt {a} e\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (77 \, \sqrt {b} c - 15 \, \sqrt {a} e\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} + \frac {2 \, {\left (77 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {3}{4}} c + 15 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} e - 80 \, \sqrt {a} \sqrt {b} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}} + \frac {2 \, {\left (77 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {3}{4}} c + 15 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} e + 80 \, \sqrt {a} \sqrt {b} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}}}{1024 \, a^{3}} \] Input:
integrate((e*x^2+d*x+c)/(b*x^4+a)^4,x, algorithm="maxima")
Output:
1/384*(45*b^2*e*x^11 + 60*b^2*d*x^10 + 77*b^2*c*x^9 + 126*a*b*e*x^7 + 160* a*b*d*x^6 + 198*a*b*c*x^5 + 113*a^2*e*x^3 + 132*a^2*d*x^2 + 153*a^2*c*x)/( a^3*b^3*x^12 + 3*a^4*b^2*x^8 + 3*a^5*b*x^4 + a^6) + 1/1024*(sqrt(2)*(77*sq rt(b)*c - 15*sqrt(a)*e)*log(sqrt(b)*x^2 + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt (a))/(a^(3/4)*b^(3/4)) - sqrt(2)*(77*sqrt(b)*c - 15*sqrt(a)*e)*log(sqrt(b) *x^2 - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(3/4)) + 2*(77*sqrt (2)*a^(1/4)*b^(3/4)*c + 15*sqrt(2)*a^(3/4)*b^(1/4)*e - 80*sqrt(a)*sqrt(b)* 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^(3/4)) + 2*(77*sqrt(2)*a^(1/4) *b^(3/4)*c + 15*sqrt(2)*a^(3/4)*b^(1/4)*e + 80*sqrt(a)*sqrt(b)*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^(3/4)))/a^3
Time = 0.14 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.20 \[ \int \frac {c+d x+e x^2}{\left (a+b x^4\right )^4} \, dx=\frac {\sqrt {2} {\left (40 \, \sqrt {2} \sqrt {a b} b^{2} d + 77 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 15 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 \, a^{4} b^{3}} + \frac {\sqrt {2} {\left (40 \, \sqrt {2} \sqrt {a b} b^{2} d + 77 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 15 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 \, a^{4} b^{3}} + \frac {\sqrt {2} {\left (77 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - 15 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{1024 \, a^{4} b^{3}} - \frac {\sqrt {2} {\left (77 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - 15 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{1024 \, a^{4} b^{3}} + \frac {45 \, b^{2} e x^{11} + 60 \, b^{2} d x^{10} + 77 \, b^{2} c x^{9} + 126 \, a b e x^{7} + 160 \, a b d x^{6} + 198 \, a b c x^{5} + 113 \, a^{2} e x^{3} + 132 \, a^{2} d x^{2} + 153 \, a^{2} c x}{384 \, {\left (b x^{4} + a\right )}^{3} a^{3}} \] Input:
integrate((e*x^2+d*x+c)/(b*x^4+a)^4,x, algorithm="giac")
Output:
1/512*sqrt(2)*(40*sqrt(2)*sqrt(a*b)*b^2*d + 77*(a*b^3)^(1/4)*b^2*c + 15*(a *b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4)) /(a^4*b^3) + 1/512*sqrt(2)*(40*sqrt(2)*sqrt(a*b)*b^2*d + 77*(a*b^3)^(1/4)* b^2*c + 15*(a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4)) /(a/b)^(1/4))/(a^4*b^3) + 1/1024*sqrt(2)*(77*(a*b^3)^(1/4)*b^2*c - 15*(a*b ^3)^(3/4)*e)*log(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^4*b^3) - 1/10 24*sqrt(2)*(77*(a*b^3)^(1/4)*b^2*c - 15*(a*b^3)^(3/4)*e)*log(x^2 - sqrt(2) *x*(a/b)^(1/4) + sqrt(a/b))/(a^4*b^3) + 1/384*(45*b^2*e*x^11 + 60*b^2*d*x^ 10 + 77*b^2*c*x^9 + 126*a*b*e*x^7 + 160*a*b*d*x^6 + 198*a*b*c*x^5 + 113*a^ 2*e*x^3 + 132*a^2*d*x^2 + 153*a^2*c*x)/((b*x^4 + a)^3*a^3)
Time = 6.37 (sec) , antiderivative size = 873, normalized size of antiderivative = 2.87 \[ \int \frac {c+d x+e x^2}{\left (a+b x^4\right )^4} \, dx =\text {Too large to display} \] Input:
int((c + d*x + e*x^2)/(a + b*x^4)^4,x)
Output:
((11*d*x^2)/(32*a) + (113*e*x^3)/(384*a) + (51*c*x)/(128*a) + (77*b^2*c*x^ 9)/(384*a^3) + (5*b^2*d*x^10)/(32*a^3) + (15*b^2*e*x^11)/(128*a^3) + (33*b *c*x^5)/(64*a^2) + (5*b*d*x^6)/(12*a^2) + (21*b*e*x^7)/(64*a^2))/(a^3 + b^ 3*x^12 + 3*a^2*b*x^4 + 3*a*b^2*x^8) + symsum(log(-(b*(3375*a*e^3 - 123200* b*c*d^2 + 88935*b*c^2*e - 64000*b*d^3*x + 20185088*root(68719476736*a^15*b ^3*z^4 + 1211105280*a^8*b^2*c*e*z^2 + 838860800*a^8*b^2*d^2*z^2 - 48570368 0*a^4*b^2*c^2*d*z + 18432000*a^5*b*d*e^2*z - 7392000*a*b*c*d^2*e + 2668050 *a*b*c^2*e^2 + 2560000*a*b*d^4 + 35153041*b^2*c^4 + 50625*a^2*e^4, z, k)^2 *a^7*b^2*c - 115200*root(68719476736*a^15*b^3*z^4 + 1211105280*a^8*b^2*c*e *z^2 + 838860800*a^8*b^2*d^2*z^2 - 485703680*a^4*b^2*c^2*d*z + 18432000*a^ 5*b*d*e^2*z - 7392000*a*b*c*d^2*e + 2668050*a*b*c^2*e^2 + 2560000*a*b*d^4 + 35153041*b^2*c^4 + 50625*a^2*e^4, z, k)*a^4*b*e^2*x + 92400*b*c*d*e*x + 3035648*root(68719476736*a^15*b^3*z^4 + 1211105280*a^8*b^2*c*e*z^2 + 83886 0800*a^8*b^2*d^2*z^2 - 485703680*a^4*b^2*c^2*d*z + 18432000*a^5*b*d*e^2*z - 7392000*a*b*c*d^2*e + 2668050*a*b*c^2*e^2 + 2560000*a*b*d^4 + 35153041*b ^2*c^4 + 50625*a^2*e^4, z, k)*a^3*b^2*c^2*x - 10485760*root(68719476736*a^ 15*b^3*z^4 + 1211105280*a^8*b^2*c*e*z^2 + 838860800*a^8*b^2*d^2*z^2 - 4857 03680*a^4*b^2*c^2*d*z + 18432000*a^5*b*d*e^2*z - 7392000*a*b*c*d^2*e + 266 8050*a*b*c^2*e^2 + 2560000*a*b*d^4 + 35153041*b^2*c^4 + 50625*a^2*e^4, z, k)^2*a^7*b^2*d*x + 614400*root(68719476736*a^15*b^3*z^4 + 1211105280*a^...
Time = 0.18 (sec) , antiderivative size = 1636, normalized size of antiderivative = 5.38 \[ \int \frac {c+d x+e x^2}{\left (a+b x^4\right )^4} \, dx =\text {Too large to display} \] Input:
int((e*x^2+d*x+c)/(b*x^4+a)^4,x)
Output:
( - 90*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt( b)*x)/(b**(1/4)*a**(1/4)*sqrt(2)))*a**3*e - 270*b**(1/4)*a**(3/4)*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*b*e*x**4 - 270*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sq rt(2) - 2*sqrt(b)*x)/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b**2*e*x**8 - 90*b**(1 /4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(b)*x)/(b**(1 /4)*a**(1/4)*sqrt(2)))*b**3*e*x**12 - 462*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**3 *c - 1386*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sq rt(b)*x)/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*b*c*x**4 - 1386*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*b**2*c*x**8 - 462*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**3*c*x**12 - 480*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**3*d - 1440*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*b*d*x**4 - 14 40*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**2*d*x**8 - 480*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**3*d*x**12 + 90*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(b...