Integrand size = 40, antiderivative size = 463 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{\left (a+b x^4\right )^3} \, dx=\frac {x \left (b c-a g+(b d-a h) x+(b e-a i) x^2+b f x^3\right )}{8 a b \left (a+b x^4\right )^2}-\frac {4 a f-x \left (7 b c+a g+2 (3 b d+a h) x+(5 b e+3 a i) x^2\right )}{32 a^2 b \left (a+b x^4\right )}+\frac {(3 b d+a h) \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} b^{3/2}}-\frac {\left (3 \sqrt {b} (7 b c+a g)+\sqrt {a} (5 b e+3 a i)\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} b^{7/4}}+\frac {\left (3 \sqrt {b} (7 b c+a g)+\sqrt {a} (5 b e+3 a i)\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} b^{7/4}}-\frac {\left (3 \sqrt {b} (7 b c+a g)-\sqrt {a} (5 b e+3 a i)\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{11/4} b^{7/4}}+\frac {\left (3 \sqrt {b} (7 b c+a g)-\sqrt {a} (5 b e+3 a i)\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{11/4} b^{7/4}} \]
1/8*x*(b*c-a*g+(-a*h+b*d)*x+(-a*i+b*e)*x^2+b*f*x^3)/a/b/(b*x^4+a)^2+1/32*( -4*a*f+x*(7*b*c+a*g+2*(a*h+3*b*d)*x+(3*a*i+5*b*e)*x^2))/a^2/b/(b*x^4+a)+1/ 16*(a*h+3*b*d)*arctan(x^2*b^(1/2)/a^(1/2))/a^(5/2)/b^(3/2)-1/256*ln(-a^(1/ 4)*b^(1/4)*x*2^(1/2)+a^(1/2)+x^2*b^(1/2))*(-(3*a*i+5*b*e)*a^(1/2)+3*(a*g+7 *b*c)*b^(1/2))/a^(11/4)/b^(7/4)*2^(1/2)+1/256*ln(a^(1/4)*b^(1/4)*x*2^(1/2) +a^(1/2)+x^2*b^(1/2))*(-(3*a*i+5*b*e)*a^(1/2)+3*(a*g+7*b*c)*b^(1/2))/a^(11 /4)/b^(7/4)*2^(1/2)+1/128*arctan(-1+b^(1/4)*x*2^(1/2)/a^(1/4))*((3*a*i+5*b *e)*a^(1/2)+3*(a*g+7*b*c)*b^(1/2))/a^(11/4)/b^(7/4)*2^(1/2)+1/128*arctan(1 +b^(1/4)*x*2^(1/2)/a^(1/4))*((3*a*i+5*b*e)*a^(1/2)+3*(a*g+7*b*c)*b^(1/2))/ a^(11/4)/b^(7/4)*2^(1/2)
Time = 0.52 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.02 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{\left (a+b x^4\right )^3} \, dx=\frac {\frac {8 a^{3/4} b^{3/4} x (7 b c+a g+b x (6 d+5 e x)+a x (2 h+3 i x))}{a+b x^4}-\frac {32 a^{7/4} b^{3/4} (-b x (c+x (d+e x))+a (f+x (g+x (h+i x))))}{\left (a+b x^4\right )^2}-2 \left (21 \sqrt {2} b^{3/2} c+24 \sqrt [4]{a} b^{5/4} d+5 \sqrt {2} \sqrt {a} b e+3 \sqrt {2} a \sqrt {b} g+8 a^{5/4} \sqrt [4]{b} h+3 \sqrt {2} a^{3/2} i\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+2 \left (21 \sqrt {2} b^{3/2} c-24 \sqrt [4]{a} b^{5/4} d+5 \sqrt {2} \sqrt {a} b e+3 \sqrt {2} a \sqrt {b} g-8 a^{5/4} \sqrt [4]{b} h+3 \sqrt {2} a^{3/2} i\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+\sqrt {2} \left (-21 b^{3/2} c+5 \sqrt {a} b e-3 a \sqrt {b} g+3 a^{3/2} i\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )+\sqrt {2} \left (21 b^{3/2} c-5 \sqrt {a} b e+3 a \sqrt {b} g-3 a^{3/2} i\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{256 a^{11/4} b^{7/4}} \]
((8*a^(3/4)*b^(3/4)*x*(7*b*c + a*g + b*x*(6*d + 5*e*x) + a*x*(2*h + 3*i*x) ))/(a + b*x^4) - (32*a^(7/4)*b^(3/4)*(-(b*x*(c + x*(d + e*x))) + a*(f + x* (g + x*(h + i*x)))))/(a + b*x^4)^2 - 2*(21*Sqrt[2]*b^(3/2)*c + 24*a^(1/4)* b^(5/4)*d + 5*Sqrt[2]*Sqrt[a]*b*e + 3*Sqrt[2]*a*Sqrt[b]*g + 8*a^(5/4)*b^(1 /4)*h + 3*Sqrt[2]*a^(3/2)*i)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*( 21*Sqrt[2]*b^(3/2)*c - 24*a^(1/4)*b^(5/4)*d + 5*Sqrt[2]*Sqrt[a]*b*e + 3*Sq rt[2]*a*Sqrt[b]*g - 8*a^(5/4)*b^(1/4)*h + 3*Sqrt[2]*a^(3/2)*i)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + Sqrt[2]*(-21*b^(3/2)*c + 5*Sqrt[a]*b*e - 3* a*Sqrt[b]*g + 3*a^(3/2)*i)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[ b]*x^2] + Sqrt[2]*(21*b^(3/2)*c - 5*Sqrt[a]*b*e + 3*a*Sqrt[b]*g - 3*a^(3/2 )*i)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(256*a^(11/4) *b^(7/4))
Time = 0.88 (sec) , antiderivative size = 483, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2397, 25, 2393, 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.
| \(\displaystyle \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{\left (a+b x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 2397 |
| \(\displaystyle \frac {x \left (x (b d-a h)+x^2 (b e-a i)-a g+b c+b f x^3\right )}{8 a b \left (a+b x^4\right )^2}-\frac {\int -\frac {4 b^2 f x^3+b (5 b e+3 a i) x^2+2 b (3 b d+a h) x+b (7 b c+a g)}{\left (b x^4+a\right )^2}dx}{8 a b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {4 b^2 f x^3+b (5 b e+3 a i) x^2+2 b (3 b d+a h) x+b (7 b c+a g)}{\left (b x^4+a\right )^2}dx}{8 a b^2}+\frac {x \left (x (b d-a h)+x^2 (b e-a i)-a g+b c+b f x^3\right )}{8 a b \left (a+b x^4\right )^2}\) |
| \(\Big \downarrow \) 2393 |
| \(\displaystyle \frac {-\frac {\int -\frac {b (5 b e+3 a i) x^2+4 b (3 b d+a h) x+3 b (7 b c+a g)}{b x^4+a}dx}{4 a}-\frac {4 a b f-x \left (b (a g+7 b c)+2 b x (a h+3 b d)+b x^2 (3 a i+5 b e)\right )}{4 a \left (a+b x^4\right )}}{8 a b^2}+\frac {x \left (x (b d-a h)+x^2 (b e-a i)-a g+b c+b f x^3\right )}{8 a b \left (a+b x^4\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {b (5 b e+3 a i) x^2+4 b (3 b d+a h) x+3 b (7 b c+a g)}{b x^4+a}dx}{4 a}-\frac {4 a b f-x \left (b (a g+7 b c)+2 b x (a h+3 b d)+b x^2 (3 a i+5 b e)\right )}{4 a \left (a+b x^4\right )}}{8 a b^2}+\frac {x \left (x (b d-a h)+x^2 (b e-a i)-a g+b c+b f x^3\right )}{8 a b \left (a+b x^4\right )^2}\) |
| \(\Big \downarrow \) 2415 |
| \(\displaystyle \frac {\frac {\int \left (\frac {4 b (3 b d+a h) x}{b x^4+a}+\frac {b (5 b e+3 a i) x^2+3 b (7 b c+a g)}{b x^4+a}\right )dx}{4 a}-\frac {4 a b f-x \left (b (a g+7 b c)+2 b x (a h+3 b d)+b x^2 (3 a i+5 b e)\right )}{4 a \left (a+b x^4\right )}}{8 a b^2}+\frac {x \left (x (b d-a h)+x^2 (b e-a i)-a g+b c+b f x^3\right )}{8 a b \left (a+b x^4\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {-\frac {\sqrt [4]{b} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (3 \sqrt {b} (a g+7 b c)+\sqrt {a} (3 a i+5 b e)\right )}{2 \sqrt {2} a^{3/4}}+\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (3 \sqrt {b} (a g+7 b c)+\sqrt {a} (3 a i+5 b e)\right )}{2 \sqrt {2} a^{3/4}}-\frac {\sqrt [4]{b} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (3 \sqrt {b} (a g+7 b c)-\sqrt {a} (3 a i+5 b e)\right )}{4 \sqrt {2} a^{3/4}}+\frac {\sqrt [4]{b} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (3 \sqrt {b} (a g+7 b c)-\sqrt {a} (3 a i+5 b e)\right )}{4 \sqrt {2} a^{3/4}}+\frac {2 \sqrt {b} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ) (a h+3 b d)}{\sqrt {a}}}{4 a}-\frac {4 a b f-x \left (b (a g+7 b c)+2 b x (a h+3 b d)+b x^2 (3 a i+5 b e)\right )}{4 a \left (a+b x^4\right )}}{8 a b^2}+\frac {x \left (x (b d-a h)+x^2 (b e-a i)-a g+b c+b f x^3\right )}{8 a b \left (a+b x^4\right )^2}\) |
(x*(b*c - a*g + (b*d - a*h)*x + (b*e - a*i)*x^2 + b*f*x^3))/(8*a*b*(a + b* x^4)^2) + (-1/4*(4*a*b*f - x*(b*(7*b*c + a*g) + 2*b*(3*b*d + a*h)*x + b*(5 *b*e + 3*a*i)*x^2))/(a*(a + b*x^4)) + ((2*Sqrt[b]*(3*b*d + a*h)*ArcTan[(Sq rt[b]*x^2)/Sqrt[a]])/Sqrt[a] - (b^(1/4)*(3*Sqrt[b]*(7*b*c + a*g) + Sqrt[a] *(5*b*e + 3*a*i))*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3 /4)) + (b^(1/4)*(3*Sqrt[b]*(7*b*c + a*g) + Sqrt[a]*(5*b*e + 3*a*i))*ArcTan [1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)) - (b^(1/4)*(3*Sqrt[ b]*(7*b*c + a*g) - Sqrt[a]*(5*b*e + 3*a*i))*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)*(3*Sqrt[b]*(7*b*c + a*g) - Sqrt[a]*(5*b*e + 3*a*i))*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(3/4)))/(4*a))/(8*a*b^2)
3.3.2.3.1 Defintions of rubi rules used
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] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x]}, S imp[(-x)*R*((a + b*x^n)^(p + 1)/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))), x] + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)) Int[(a + b*x^n)^(p + 1)* ExpandToSum[a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], x], x], x]] /; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -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 = 1.56 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.40
| method | result | size |
| risch | \(\frac {\frac {\left (3 a i +5 b e \right ) x^{7}}{32 a^{2}}+\frac {\left (a h +3 b d \right ) x^{6}}{16 a^{2}}+\frac {\left (a g +7 b c \right ) x^{5}}{32 a^{2}}-\frac {\left (a i -9 b e \right ) x^{3}}{32 a b}-\frac {\left (a h -5 b d \right ) x^{2}}{16 a b}-\frac {\left (3 a g -11 b c \right ) x}{32 a b}-\frac {f}{8 b}}{\left (b \,x^{4}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (\left (3 a i +5 b e \right ) \textit {\_R}^{2}+4 \left (a h +3 b d \right ) \textit {\_R} +3 a g +21 b c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{128 a^{2} b^{2}}\) | \(186\) |
| default | \(\frac {\frac {\left (3 a i +5 b e \right ) x^{7}}{32 a^{2}}+\frac {\left (a h +3 b d \right ) x^{6}}{16 a^{2}}+\frac {\left (a g +7 b c \right ) x^{5}}{32 a^{2}}-\frac {\left (a i -9 b e \right ) x^{3}}{32 a b}-\frac {\left (a h -5 b d \right ) x^{2}}{16 a b}-\frac {\left (3 a g -11 b c \right ) x}{32 a b}-\frac {f}{8 b}}{\left (b \,x^{4}+a \right )^{2}}+\frac {\frac {\left (3 a g +21 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {\left (4 a h +12 b d \right ) \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right )}{2 \sqrt {a b}}+\frac {\left (3 a i +5 b e \right ) \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{32 a^{2} b}\) | \(381\) |
(1/32*(3*a*i+5*b*e)/a^2*x^7+1/16*(a*h+3*b*d)/a^2*x^6+1/32*(a*g+7*b*c)/a^2* x^5-1/32*(a*i-9*b*e)/a/b*x^3-1/16*(a*h-5*b*d)/a/b*x^2-1/32*(3*a*g-11*b*c)/ a/b*x-1/8*f/b)/(b*x^4+a)^2+1/128/a^2/b^2*sum(((3*a*i+5*b*e)*_R^2+4*(a*h+3* b*d)*_R+3*a*g+21*b*c)/_R^3*ln(x-_R),_R=RootOf(_Z^4*b+a))
Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{\left (a+b x^4\right )^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{\left (a+b x^4\right )^3} \, dx=\text {Timed out} \]
Time = 0.32 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.07 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{\left (a+b x^4\right )^3} \, dx=\frac {{\left (5 \, b^{2} e + 3 \, a b i\right )} x^{7} + 2 \, {\left (3 \, b^{2} d + a b h\right )} x^{6} + {\left (7 \, b^{2} c + a b g\right )} x^{5} + {\left (9 \, a b e - a^{2} i\right )} x^{3} - 4 \, a^{2} f + 2 \, {\left (5 \, a b d - a^{2} h\right )} x^{2} + {\left (11 \, a b c - 3 \, a^{2} g\right )} x}{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 \, b^{\frac {3}{2}} c - 5 \, \sqrt {a} b e + 3 \, a \sqrt {b} g - 3 \, a^{\frac {3}{2}} i\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 \, b^{\frac {3}{2}} c - 5 \, \sqrt {a} b e + 3 \, a \sqrt {b} g - 3 \, a^{\frac {3}{2}} i\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 {7}{4}} c + 5 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} e + 3 \, \sqrt {2} a^{\frac {5}{4}} b^{\frac {3}{4}} g + 3 \, \sqrt {2} a^{\frac {7}{4}} b^{\frac {1}{4}} i - 24 \, \sqrt {a} b^{\frac {3}{2}} d - 8 \, a^{\frac {3}{2}} \sqrt {b} h\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 {7}{4}} c + 5 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} e + 3 \, \sqrt {2} a^{\frac {5}{4}} b^{\frac {3}{4}} g + 3 \, \sqrt {2} a^{\frac {7}{4}} b^{\frac {1}{4}} i + 24 \, \sqrt {a} b^{\frac {3}{2}} d + 8 \, a^{\frac {3}{2}} \sqrt {b} h\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} b} \]
1/32*((5*b^2*e + 3*a*b*i)*x^7 + 2*(3*b^2*d + a*b*h)*x^6 + (7*b^2*c + a*b*g )*x^5 + (9*a*b*e - a^2*i)*x^3 - 4*a^2*f + 2*(5*a*b*d - a^2*h)*x^2 + (11*a* b*c - 3*a^2*g)*x)/(a^2*b^3*x^8 + 2*a^3*b^2*x^4 + a^4*b) + 1/256*(sqrt(2)*( 21*b^(3/2)*c - 5*sqrt(a)*b*e + 3*a*sqrt(b)*g - 3*a^(3/2)*i)*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*b ^(3/2)*c - 5*sqrt(a)*b*e + 3*a*sqrt(b)*g - 3*a^(3/2)*i)*log(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^(7/4)*c + 5*sqrt(2)*a^(3/4)*b^(5/4)*e + 3*sqrt(2)*a^(5/4)*b^(3/4)*g + 3*sqrt(2)*a^(7/4)*b^(1/4)*i - 24*sqrt(a)*b^(3/2)*d - 8*a^(3/2)*sqrt(b)* h)*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*(21*sqrt(2)*a^(1/4) *b^(7/4)*c + 5*sqrt(2)*a^(3/4)*b^(5/4)*e + 3*sqrt(2)*a^(5/4)*b^(3/4)*g + 3 *sqrt(2)*a^(7/4)*b^(1/4)*i + 24*sqrt(a)*b^(3/2)*d + 8*a^(3/2)*sqrt(b)*h)*a rctan(1/2*sqrt(2)*(2*sqrt(b)*x - sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqr t(b)))/(a^(3/4)*sqrt(sqrt(a)*sqrt(b))*b^(3/4)))/(a^2*b)
Time = 0.28 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.14 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{\left (a+b x^4\right )^3} \, dx=\frac {5 \, b^{2} e x^{7} + 3 \, a b i x^{7} + 6 \, b^{2} d x^{6} + 2 \, a b h x^{6} + 7 \, b^{2} c x^{5} + a b g x^{5} + 9 \, a b e x^{3} - a^{2} i x^{3} + 10 \, a b d x^{2} - 2 \, a^{2} h x^{2} + 11 \, a b c x - 3 \, a^{2} g x - 4 \, a^{2} f}{32 \, {\left (b x^{4} + a\right )}^{2} a^{2} b} + \frac {\sqrt {2} {\left (12 \, \sqrt {2} \sqrt {a b} b^{3} d + 4 \, \sqrt {2} \sqrt {a b} a b^{2} h + 21 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} g + 5 \, \left (a b^{3}\right )^{\frac {3}{4}} b e + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a i\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^{4}} + \frac {\sqrt {2} {\left (12 \, \sqrt {2} \sqrt {a b} b^{3} d + 4 \, \sqrt {2} \sqrt {a b} a b^{2} h + 21 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} g + 5 \, \left (a b^{3}\right )^{\frac {3}{4}} b e + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a i\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^{4}} + \frac {\sqrt {2} {\left (21 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} g - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} b e - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a i\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^{4}} - \frac {\sqrt {2} {\left (21 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} g - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} b e - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a i\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^{4}} \]
1/32*(5*b^2*e*x^7 + 3*a*b*i*x^7 + 6*b^2*d*x^6 + 2*a*b*h*x^6 + 7*b^2*c*x^5 + a*b*g*x^5 + 9*a*b*e*x^3 - a^2*i*x^3 + 10*a*b*d*x^2 - 2*a^2*h*x^2 + 11*a* b*c*x - 3*a^2*g*x - 4*a^2*f)/((b*x^4 + a)^2*a^2*b) + 1/128*sqrt(2)*(12*sqr t(2)*sqrt(a*b)*b^3*d + 4*sqrt(2)*sqrt(a*b)*a*b^2*h + 21*(a*b^3)^(1/4)*b^3* c + 3*(a*b^3)^(1/4)*a*b^2*g + 5*(a*b^3)^(3/4)*b*e + 3*(a*b^3)^(3/4)*a*i)*a rctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^3*b^4) + 1/1 28*sqrt(2)*(12*sqrt(2)*sqrt(a*b)*b^3*d + 4*sqrt(2)*sqrt(a*b)*a*b^2*h + 21* (a*b^3)^(1/4)*b^3*c + 3*(a*b^3)^(1/4)*a*b^2*g + 5*(a*b^3)^(3/4)*b*e + 3*(a *b^3)^(3/4)*a*i)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4 ))/(a^3*b^4) + 1/256*sqrt(2)*(21*(a*b^3)^(1/4)*b^3*c + 3*(a*b^3)^(1/4)*a*b ^2*g - 5*(a*b^3)^(3/4)*b*e - 3*(a*b^3)^(3/4)*a*i)*log(x^2 + sqrt(2)*x*(a/b )^(1/4) + sqrt(a/b))/(a^3*b^4) - 1/256*sqrt(2)*(21*(a*b^3)^(1/4)*b^3*c + 3 *(a*b^3)^(1/4)*a*b^2*g - 5*(a*b^3)^(3/4)*b*e - 3*(a*b^3)^(3/4)*a*i)*log(x^ 2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^3*b^4)
Time = 10.48 (sec) , antiderivative size = 2680, normalized size of antiderivative = 5.79 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{\left (a+b x^4\right )^3} \, dx=\text {Too large to display} \]
symsum(log(- root(268435456*a^11*b^7*z^4 + 589824*a^8*b^4*g*i*z^2 + 412876 8*a^7*b^5*c*i*z^2 + 3145728*a^7*b^5*d*h*z^2 + 983040*a^7*b^5*e*g*z^2 + 688 1280*a^6*b^6*c*e*z^2 + 524288*a^8*b^4*h^2*z^2 + 4718592*a^6*b^6*d^2*z^2 + 61440*a^6*b^3*e*h*i*z - 258048*a^5*b^4*c*g*h*z + 184320*a^5*b^4*d*e*i*z - 774144*a^4*b^5*c*d*g*z + 18432*a^7*b^2*h*i^2*z - 18432*a^6*b^3*g^2*h*z + 5 5296*a^6*b^3*d*i^2*z + 51200*a^5*b^4*e^2*h*z - 903168*a^4*b^5*c^2*h*z - 55 296*a^5*b^4*d*g^2*z + 153600*a^4*b^5*d*e^2*z - 2709504*a^3*b^6*c^2*d*z - 3 456*a^4*b^2*d*g*h*i - 24192*a^3*b^3*c*d*h*i + 7560*a^3*b^3*c*e*g*i - 5760* a^3*b^3*d*e*g*h - 40320*a^2*b^4*c*d*e*h + 540*a^4*b^2*e*g^2*i - 5184*a^3*b ^3*d^2*g*i - 4032*a^4*b^2*c*h^2*i - 960*a^4*b^2*e*g*h^2 + 2268*a^4*b^2*c*g *i^2 + 26460*a^2*b^4*c^2*e*i - 36288*a^2*b^4*c*d^2*i - 8640*a^2*b^4*d^2*e* g - 6720*a^3*b^3*c*e*h^2 + 6300*a^2*b^4*c*e^2*g - 576*a^5*b*g*h^2*i - 6048 0*a*b^5*c*d^2*e + 540*a^5*b*e*i^3 + 111132*a*b^5*c^3*g + 1350*a^4*b^2*e^2* i^2 + 13824*a^3*b^3*d^2*h^2 + 7938*a^3*b^3*c^2*i^2 + 450*a^3*b^3*e^2*g^2 + 23814*a^2*b^4*c^2*g^2 + 162*a^5*b*g^2*i^2 + 1500*a^3*b^3*e^3*i + 27648*a^ 2*b^4*d^3*h + 3072*a^4*b^2*d*h^3 + 2268*a^3*b^3*c*g^3 + 22050*a*b^5*c^2*e^ 2 + 81*a^4*b^2*g^4 + 625*a^2*b^4*e^4 + 256*a^5*b*h^4 + 20736*a*b^5*d^4 + 8 1*a^6*i^4 + 194481*b^6*c^4, z, l)*(root(268435456*a^11*b^7*z^4 + 589824*a^ 8*b^4*g*i*z^2 + 4128768*a^7*b^5*c*i*z^2 + 3145728*a^7*b^5*d*h*z^2 + 983040 *a^7*b^5*e*g*z^2 + 6881280*a^6*b^6*c*e*z^2 + 524288*a^8*b^4*h^2*z^2 + 4...