Integrand size = 25, antiderivative size = 290 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a+b x^4\right )^3} \, dx=-\frac {f}{8 b \left (a+b x^4\right )^2}+\frac {x \left (c+d x+e x^2\right )}{8 a \left (a+b x^4\right )^2}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}+\frac {3 d \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}}-\frac {\left (21 \sqrt {b} c+5 \sqrt {a} e\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} b^{3/4}}+\frac {\left (21 \sqrt {b} c+5 \sqrt {a} e\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} b^{3/4}}+\frac {\left (21 \sqrt {b} c-5 \sqrt {a} e\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a}+\sqrt {b} x^2}\right )}{64 \sqrt {2} a^{11/4} b^{3/4}} \] Output:
-1/8*f/b/(b*x^4+a)^2+1/8*x*(e*x^2+d*x+c)/a/(b*x^4+a)^2+1/32*x*(5*e*x^2+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)+1 /128*(21*b^(1/2)*c+5*a^(1/2)*e)*arctan(-1+2^(1/2)*b^(1/4)*x/a^(1/4))*2^(1/ 2)/a^(11/4)/b^(3/4)+1/128*(21*b^(1/2)*c+5*a^(1/2)*e)*arctan(1+2^(1/2)*b^(1 /4)*x/a^(1/4))*2^(1/2)/a^(11/4)/b^(3/4)+1/128*(21*b^(1/2)*c-5*a^(1/2)*e)*a rctanh(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 ^(3/4)
Time = 0.27 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.20 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a+b x^4\right )^3} \, dx=\frac {\frac {8 a x (7 c+x (6 d+5 e x))}{a+b x^4}-\frac {32 a^2 (a f-b x (c+x (d+e x)))}{b \left (a+b x^4\right )^2}-\frac {2 \sqrt [4]{a} \left (21 \sqrt {2} \sqrt {b} c+24 \sqrt [4]{a} \sqrt [4]{b} d+5 \sqrt {2} \sqrt {a} e\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{b^{3/4}}+\frac {2 \sqrt [4]{a} \left (21 \sqrt {2} \sqrt {b} c-24 \sqrt [4]{a} \sqrt [4]{b} d+5 \sqrt {2} \sqrt {a} e\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{b^{3/4}}+\frac {\sqrt {2} \left (-21 \sqrt [4]{a} \sqrt {b} c+5 a^{3/4} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{b^{3/4}}+\frac {\sqrt {2} \left (21 \sqrt [4]{a} \sqrt {b} c-5 a^{3/4} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{b^{3/4}}}{256 a^3} \] Input:
Integrate[(c + d*x + e*x^2 + f*x^3)/(a + b*x^4)^3,x]
Output:
((8*a*x*(7*c + x*(6*d + 5*e*x)))/(a + b*x^4) - (32*a^2*(a*f - b*x*(c + x*( d + e*x))))/(b*(a + b*x^4)^2) - (2*a^(1/4)*(21*Sqrt[2]*Sqrt[b]*c + 24*a^(1 /4)*b^(1/4)*d + 5*Sqrt[2]*Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4 )])/b^(3/4) + (2*a^(1/4)*(21*Sqrt[2]*Sqrt[b]*c - 24*a^(1/4)*b^(1/4)*d + 5* Sqrt[2]*Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/b^(3/4) + (Sqr t[2]*(-21*a^(1/4)*Sqrt[b]*c + 5*a^(3/4)*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b ^(1/4)*x + Sqrt[b]*x^2])/b^(3/4) + (Sqrt[2]*(21*a^(1/4)*Sqrt[b]*c - 5*a^(3 /4)*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(3/4))/(2 56*a^3)
Time = 1.01 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.26, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2393, 25, 2394, 25, 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.
| \(\displaystyle \int \frac {c+d x+e x^2+f x^3}{\left (a+b x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 2393 |
| \(\displaystyle -\frac {\int -\frac {5 e x^2+6 d x+7 c}{\left (b x^4+a\right )^2}dx}{8 a}-\frac {a f-b x \left (c+d x+e x^2\right )}{8 a b \left (a+b x^4\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {5 e x^2+6 d x+7 c}{\left (b x^4+a\right )^2}dx}{8 a}-\frac {a f-b x \left (c+d x+e x^2\right )}{8 a b \left (a+b x^4\right )^2}\) |
| \(\Big \downarrow \) 2394 |
| \(\displaystyle \frac {\frac {x \left (7 c+6 d x+5 e x^2\right )}{4 a \left (a+b x^4\right )}-\frac {\int -\frac {5 e x^2+12 d x+21 c}{b x^4+a}dx}{4 a}}{8 a}-\frac {a f-b x \left (c+d x+e x^2\right )}{8 a b \left (a+b x^4\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {5 e x^2+12 d x+21 c}{b x^4+a}dx}{4 a}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{4 a \left (a+b x^4\right )}}{8 a}-\frac {a f-b x \left (c+d x+e x^2\right )}{8 a b \left (a+b x^4\right )^2}\) |
| \(\Big \downarrow \) 2415 |
| \(\displaystyle \frac {\frac {\int \left (\frac {12 d x}{b x^4+a}+\frac {5 e x^2+21 c}{b x^4+a}\right )dx}{4 a}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{4 a \left (a+b x^4\right )}}{8 a}-\frac {a f-b x \left (c+d x+e x^2\right )}{8 a b \left (a+b x^4\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (5 \sqrt {a} e+21 \sqrt {b} c\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (5 \sqrt {a} e+21 \sqrt {b} c\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}-\frac {\left (21 \sqrt {b} c-5 \sqrt {a} e\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}+\frac {\left (21 \sqrt {b} c-5 \sqrt {a} e\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}+\frac {6 d \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}}{4 a}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{4 a \left (a+b x^4\right )}}{8 a}-\frac {a f-b x \left (c+d x+e x^2\right )}{8 a b \left (a+b x^4\right )^2}\) |
Input:
Int[(c + d*x + e*x^2 + f*x^3)/(a + b*x^4)^3,x]
Output:
-1/8*(a*f - b*x*(c + d*x + e*x^2))/(a*b*(a + b*x^4)^2) + ((x*(7*c + 6*d*x + 5*e*x^2))/(4*a*(a + b*x^4)) + ((6*d*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(Sqrt [a]*Sqrt[b]) - ((21*Sqrt[b]*c + 5*Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x )/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*b^(3/4)) + ((21*Sqrt[b]*c + 5*Sqrt[a]*e)*Ar cTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*b^(3/4)) - ((21* Sqrt[b]*c - 5*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)) + ((21*Sqrt[b]*c - 5*Sqrt[a]*e)*Log[Sqr t[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)
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], i}, Simp[(a*Coeff[Pq, x, q] - b*x*ExpandToSum[Pq - Coeff[Pq, x, q]*x^q , x])*((a + b*x^n)^(p + 1)/(a*b*n*(p + 1))), x] + Simp[1/(a*n*(p + 1)) In t[Sum[(n*(p + 1) + i + 1)*Coeff[Pq, x, i]*x^i, {i, 0, q - 1}]*(a + b*x^n)^( p + 1), x], x] /; q == n - 1] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n , 0] && LtQ[p, -1]
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 = 117, normalized size of antiderivative = 0.40
| method | result | size |
| risch | \(\frac {\frac {5 b e \,x^{7}}{32 a^{2}}+\frac {3 b d \,x^{6}}{16 a^{2}}+\frac {7 b c \,x^{5}}{32 a^{2}}+\frac {9 e \,x^{3}}{32 a}+\frac {5 d \,x^{2}}{16 a}+\frac {11 c x}{32 a}-\frac {f}{8 b}}{\left (b \,x^{4}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\left (5 \textit {\_R}^{2} e +12 d \textit {\_R} +21 c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{128 b \,a^{2}}\) | \(117\) |
| default | \(c \left (\frac {x}{8 a \left (b \,x^{4}+a \right )^{2}}+\frac {\frac {7 x}{32 a \left (b \,x^{4}+a \right )}+\frac {21 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{256 a^{2}}}{a}\right )+d \left (\frac {x^{2}}{8 a \left (b \,x^{4}+a \right )^{2}}+\frac {\frac {3 x^{2}}{16 a \left (b \,x^{4}+a \right )}+\frac {3 \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right )}{16 a \sqrt {a b}}}{a}\right )+e \left (\frac {x^{3}}{8 a \left (b \,x^{4}+a \right )^{2}}+\frac {\frac {5 x^{3}}{32 a \left (b \,x^{4}+a \right )}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{256 a b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{a}\right )+f \left (\frac {x^{4}}{8 a \left (b \,x^{4}+a \right )^{2}}+\frac {x^{4}}{8 a^{2} \left (b \,x^{4}+a \right )}\right )\) | \(391\) |
Input:
int((f*x^3+e*x^2+d*x+c)/(b*x^4+a)^3,x,method=_RETURNVERBOSE)
Output:
(5/32*b*e/a^2*x^7+3/16*b*d/a^2*x^6+7/32*b/a^2*c*x^5+9/32*e/a*x^3+5/16*d/a* x^2+11/32*c/a*x-1/8*f/b)/(b*x^4+a)^2+1/128/b/a^2*sum((5*_R^2*e+12*_R*d+21* c)/_R^3*ln(x-_R),_R=RootOf(_Z^4*b+a))
Result contains complex when optimal does not.
Time = 7.47 (sec) , antiderivative size = 124838, normalized size of antiderivative = 430.48 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a+b x^4\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((f*x^3+e*x^2+d*x+c)/(b*x^4+a)^3,x, algorithm="fricas")
Output:
Too large to include
Leaf count of result is larger than twice the leaf count of optimal. 578 vs. \(2 (275) = 550\).
Time = 45.04 (sec) , antiderivative size = 578, normalized size of antiderivative = 1.99 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a+b x^4\right )^3} \, dx=\operatorname {RootSum} {\left (268435456 t^{4} a^{11} b^{3} + t^{2} \cdot \left (6881280 a^{6} b^{2} c e + 4718592 a^{6} b^{2} d^{2}\right ) + t \left (153600 a^{4} b d e^{2} - 2709504 a^{3} b^{2} c^{2} d\right ) + 625 a^{2} e^{4} + 22050 a b c^{2} e^{2} - 60480 a b c d^{2} e + 20736 a b d^{4} + 194481 b^{2} c^{4}, \left ( t \mapsto t \log {\left (x + \frac {262144000 t^{3} a^{10} b^{2} e^{3} - 4624220160 t^{3} a^{9} b^{3} c^{2} e + 12683575296 t^{3} a^{9} b^{3} c d^{2} + 309657600 t^{2} a^{7} b^{2} c d e^{2} - 283115520 t^{2} a^{7} b^{2} d^{3} e + 1820786688 t^{2} a^{6} b^{3} c^{3} d + 5040000 t a^{5} b c e^{4} + 6912000 t a^{5} b d^{2} e^{3} - 118540800 t a^{4} b^{2} c^{3} e^{2} + 365783040 t a^{4} b^{2} c^{2} d^{2} e + 111476736 t a^{4} b^{2} c d^{4} + 522764928 t a^{3} b^{3} c^{5} + 112500 a^{3} d e^{5} + 4536000 a^{2} b c d^{3} e^{2} - 2488320 a^{2} b d^{5} e + 58344300 a b^{2} c^{4} d e - 80015040 a b^{2} c^{3} d^{3}}{15625 a^{3} e^{6} - 275625 a^{2} b c^{2} e^{4} + 3024000 a^{2} b c d^{2} e^{3} - 2073600 a^{2} b d^{4} e^{2} - 4862025 a b^{2} c^{4} e^{2} + 53343360 a b^{2} c^{3} d^{2} e - 36578304 a b^{2} c^{2} d^{4} + 85766121 b^{3} c^{6}} \right )} \right )\right )} + \frac {- 4 a^{2} f + 11 a b c x + 10 a b d x^{2} + 9 a b e x^{3} + 7 b^{2} c x^{5} + 6 b^{2} d x^{6} + 5 b^{2} e x^{7}}{32 a^{4} b + 64 a^{3} b^{2} x^{4} + 32 a^{2} b^{3} x^{8}} \] Input:
integrate((f*x**3+e*x**2+d*x+c)/(b*x**4+a)**3,x)
Output:
RootSum(268435456*_t**4*a**11*b**3 + _t**2*(6881280*a**6*b**2*c*e + 471859 2*a**6*b**2*d**2) + _t*(153600*a**4*b*d*e**2 - 2709504*a**3*b**2*c**2*d) + 625*a**2*e**4 + 22050*a*b*c**2*e**2 - 60480*a*b*c*d**2*e + 20736*a*b*d**4 + 194481*b**2*c**4, Lambda(_t, _t*log(x + (262144000*_t**3*a**10*b**2*e** 3 - 4624220160*_t**3*a**9*b**3*c**2*e + 12683575296*_t**3*a**9*b**3*c*d**2 + 309657600*_t**2*a**7*b**2*c*d*e**2 - 283115520*_t**2*a**7*b**2*d**3*e + 1820786688*_t**2*a**6*b**3*c**3*d + 5040000*_t*a**5*b*c*e**4 + 6912000*_t *a**5*b*d**2*e**3 - 118540800*_t*a**4*b**2*c**3*e**2 + 365783040*_t*a**4*b **2*c**2*d**2*e + 111476736*_t*a**4*b**2*c*d**4 + 522764928*_t*a**3*b**3*c **5 + 112500*a**3*d*e**5 + 4536000*a**2*b*c*d**3*e**2 - 2488320*a**2*b*d** 5*e + 58344300*a*b**2*c**4*d*e - 80015040*a*b**2*c**3*d**3)/(15625*a**3*e* *6 - 275625*a**2*b*c**2*e**4 + 3024000*a**2*b*c*d**2*e**3 - 2073600*a**2*b *d**4*e**2 - 4862025*a*b**2*c**4*e**2 + 53343360*a*b**2*c**3*d**2*e - 3657 8304*a*b**2*c**2*d**4 + 85766121*b**3*c**6)))) + (-4*a**2*f + 11*a*b*c*x + 10*a*b*d*x**2 + 9*a*b*e*x**3 + 7*b**2*c*x**5 + 6*b**2*d*x**6 + 5*b**2*e*x **7)/(32*a**4*b + 64*a**3*b**2*x**4 + 32*a**2*b**3*x**8)
Time = 0.13 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.22 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a+b x^4\right )^3} \, dx=\frac {5 \, b^{2} e x^{7} + 6 \, b^{2} d x^{6} + 7 \, b^{2} c x^{5} + 9 \, a b e x^{3} + 10 \, a b d x^{2} + 11 \, a b c x - 4 \, a^{2} f}{32 \, {\left (a^{2} b^{3} x^{8} + 2 \, a^{3} b^{2} x^{4} + a^{4} b\right )}} + \frac {\frac {\sqrt {2} {\left (21 \, \sqrt {b} c - 5 \, \sqrt {a} e\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (21 \, \sqrt {b} c - 5 \, \sqrt {a} e\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} + \frac {2 \, {\left (21 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {3}{4}} c + 5 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} e - 24 \, \sqrt {a} \sqrt {b} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}} + \frac {2 \, {\left (21 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {3}{4}} c + 5 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} e + 24 \, \sqrt {a} \sqrt {b} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}}}{256 \, a^{2}} \] Input:
integrate((f*x^3+e*x^2+d*x+c)/(b*x^4+a)^3,x, algorithm="maxima")
Output:
1/32*(5*b^2*e*x^7 + 6*b^2*d*x^6 + 7*b^2*c*x^5 + 9*a*b*e*x^3 + 10*a*b*d*x^2 + 11*a*b*c*x - 4*a^2*f)/(a^2*b^3*x^8 + 2*a^3*b^2*x^4 + a^4*b) + 1/256*(sq rt(2)*(21*sqrt(b)*c - 5*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)*(21*sqrt(b)*c - 5*sqrt(a)*e)*l og(sqrt(b)*x^2 - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(3/4)) + 2*(21*sqrt(2)*a^(1/4)*b^(3/4)*c + 5*sqrt(2)*a^(3/4)*b^(1/4)*e - 24*sqrt(a) *sqrt(b)*d)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x + sqrt(2)*a^(1/4)*b^(1/4))/sqr t(sqrt(a)*sqrt(b)))/(a^(3/4)*sqrt(sqrt(a)*sqrt(b))*b^(3/4)) + 2*(21*sqrt(2 )*a^(1/4)*b^(3/4)*c + 5*sqrt(2)*a^(3/4)*b^(1/4)*e + 24*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)*sq rt(b)))/(a^(3/4)*sqrt(sqrt(a)*sqrt(b))*b^(3/4)))/a^2
Time = 0.14 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.20 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a+b x^4\right )^3} \, dx=\frac {\sqrt {2} {\left (12 \, \sqrt {2} \sqrt {a b} b^{2} d + 21 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 5 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, a^{3} b^{3}} + \frac {\sqrt {2} {\left (12 \, \sqrt {2} \sqrt {a b} b^{2} d + 21 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 5 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, a^{3} b^{3}} + \frac {\sqrt {2} {\left (21 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{256 \, a^{3} b^{3}} - \frac {\sqrt {2} {\left (21 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{256 \, a^{3} b^{3}} + \frac {5 \, b^{2} e x^{7} + 6 \, b^{2} d x^{6} + 7 \, b^{2} c x^{5} + 9 \, a b e x^{3} + 10 \, a b d x^{2} + 11 \, a b c x - 4 \, a^{2} f}{32 \, {\left (b x^{4} + a\right )}^{2} a^{2} b} \] Input:
integrate((f*x^3+e*x^2+d*x+c)/(b*x^4+a)^3,x, algorithm="giac")
Output:
1/128*sqrt(2)*(12*sqrt(2)*sqrt(a*b)*b^2*d + 21*(a*b^3)^(1/4)*b^2*c + 5*(a* b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/ (a^3*b^3) + 1/128*sqrt(2)*(12*sqrt(2)*sqrt(a*b)*b^2*d + 21*(a*b^3)^(1/4)*b ^2*c + 5*(a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4))/( a/b)^(1/4))/(a^3*b^3) + 1/256*sqrt(2)*(21*(a*b^3)^(1/4)*b^2*c - 5*(a*b^3)^ (3/4)*e)*log(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^3*b^3) - 1/256*sq rt(2)*(21*(a*b^3)^(1/4)*b^2*c - 5*(a*b^3)^(3/4)*e)*log(x^2 - sqrt(2)*x*(a/ b)^(1/4) + sqrt(a/b))/(a^3*b^3) + 1/32*(5*b^2*e*x^7 + 6*b^2*d*x^6 + 7*b^2* c*x^5 + 9*a*b*e*x^3 + 10*a*b*d*x^2 + 11*a*b*c*x - 4*a^2*f)/((b*x^4 + a)^2* a^2*b)
Time = 6.86 (sec) , antiderivative size = 832, normalized size of antiderivative = 2.87 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a+b x^4\right )^3} \, dx =\text {Too large to display} \] Input:
int((c + d*x + e*x^2 + f*x^3)/(a + b*x^4)^3,x)
Output:
symsum(log(-(b*(125*a*e^3 - 3024*b*c*d^2 + 2205*b*c^2*e - 1728*b*d^3*x + 3 44064*root(268435456*a^11*b^3*z^4 + 6881280*a^6*b^2*c*e*z^2 + 4718592*a^6* b^2*d^2*z^2 - 2709504*a^3*b^2*c^2*d*z + 153600*a^4*b*d*e^2*z - 60480*a*b*c *d^2*e + 22050*a*b*c^2*e^2 + 20736*a*b*d^4 + 625*a^2*e^4 + 194481*b^2*c^4, z, k)^2*a^5*b^2*c - 3200*root(268435456*a^11*b^3*z^4 + 6881280*a^6*b^2*c* e*z^2 + 4718592*a^6*b^2*d^2*z^2 - 2709504*a^3*b^2*c^2*d*z + 153600*a^4*b*d *e^2*z - 60480*a*b*c*d^2*e + 22050*a*b*c^2*e^2 + 20736*a*b*d^4 + 625*a^2*e ^4 + 194481*b^2*c^4, z, k)*a^3*b*e^2*x + 2520*b*c*d*e*x + 56448*root(26843 5456*a^11*b^3*z^4 + 6881280*a^6*b^2*c*e*z^2 + 4718592*a^6*b^2*d^2*z^2 - 27 09504*a^3*b^2*c^2*d*z + 153600*a^4*b*d*e^2*z - 60480*a*b*c*d^2*e + 22050*a *b*c^2*e^2 + 20736*a*b*d^4 + 625*a^2*e^4 + 194481*b^2*c^4, z, k)*a^2*b^2*c ^2*x - 196608*root(268435456*a^11*b^3*z^4 + 6881280*a^6*b^2*c*e*z^2 + 4718 592*a^6*b^2*d^2*z^2 - 2709504*a^3*b^2*c^2*d*z + 153600*a^4*b*d*e^2*z - 604 80*a*b*c*d^2*e + 22050*a*b*c^2*e^2 + 20736*a*b*d^4 + 625*a^2*e^4 + 194481* b^2*c^4, z, k)^2*a^5*b^2*d*x + 15360*root(268435456*a^11*b^3*z^4 + 6881280 *a^6*b^2*c*e*z^2 + 4718592*a^6*b^2*d^2*z^2 - 2709504*a^3*b^2*c^2*d*z + 153 600*a^4*b*d*e^2*z - 60480*a*b*c*d^2*e + 22050*a*b*c^2*e^2 + 20736*a*b*d^4 + 625*a^2*e^4 + 194481*b^2*c^4, z, k)*a^3*b*d*e))/(32768*a^6))*root(268435 456*a^11*b^3*z^4 + 6881280*a^6*b^2*c*e*z^2 + 4718592*a^6*b^2*d^2*z^2 - 270 9504*a^3*b^2*c^2*d*z + 153600*a^4*b*d*e^2*z - 60480*a*b*c*d^2*e + 22050...
Time = 0.19 (sec) , antiderivative size = 1207, normalized size of antiderivative = 4.16 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a+b x^4\right )^3} \, dx =\text {Too large to display} \] Input:
int((f*x^3+e*x^2+d*x+c)/(b*x^4+a)^3,x)
Output:
( - 10*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*e - 20*b**(1/4)*a**(3/4)*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*e*x**4 - 10*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**2*e*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)*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*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*s qrt(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)*s qrt(2) - 2*sqrt(b)*x)/(b**(1/4)*a**(1/4)*sqrt(2)))*b**2*d*x**8 + 10*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*e + 20*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*b*e*x**4 + 10*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**2*e*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...