Integrand size = 41, antiderivative size = 331 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{\left (a-b x^4\right )^4} \, dx=\frac {x \left (b c+a g+(b d+a h) x+(b e+a i) x^2+b f x^3\right )}{12 a b \left (a-b x^4\right )^3}+\frac {x \left (7 (11 b c-a g)+12 (5 b d-a h) x+15 (3 b e-a i) x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac {8 a f+x \left (11 b c-a g+2 (5 b d-a h) x+3 (3 b e-a i) x^2\right )}{96 a^2 b \left (a-b x^4\right )^2}+\frac {\left (\frac {7 \sqrt {b} (11 b c-a g)}{\sqrt {a}}-5 (3 b e-a i)\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{13/4} b^{7/4}}+\frac {\left (15 b e+\frac {7 \sqrt {b} (11 b c-a g)}{\sqrt {a}}-5 a i\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{13/4} b^{7/4}}+\frac {(5 b d-a h) \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} b^{3/2}} \]
1/12*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)^3+1/384* x*(-7*a*g+77*b*c+12*(-a*h+5*b*d)*x+15*(-a*i+3*b*e)*x^2)/a^3/b/(-b*x^4+a)+1 /96*(8*a*f+x*(11*b*c-a*g+2*(-a*h+5*b*d)*x+3*(-a*i+3*b*e)*x^2))/a^2/b/(-b*x ^4+a)^2+1/32*(-a*h+5*b*d)*arctanh(x^2*b^(1/2)/a^(1/2))/a^(7/2)/b^(3/2)+1/2 56*arctanh(b^(1/4)*x/a^(1/4))*(15*b*e-5*a*i+7*(-a*g+11*b*c)*b^(1/2)/a^(1/2 ))/a^(13/4)/b^(7/4)+1/256*arctan(b^(1/4)*x/a^(1/4))*(5*a*i-15*b*e+7*(-a*g+ 11*b*c)*b^(1/2)/a^(1/2))/a^(13/4)/b^(7/4)
Time = 0.33 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.27 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{\left (a-b x^4\right )^4} \, dx=\frac {-\frac {4 a b^{3/4} x (-77 b c+7 a g-15 b x (4 d+3 e x)+3 a x (4 h+5 i x))}{a-b x^4}-\frac {16 a^2 b^{3/4} x (-b (11 c+x (10 d+9 e x))+a (g+x (2 h+3 i x)))}{\left (a-b x^4\right )^2}+\frac {128 a^3 b^{3/4} (b x (c+x (d+e x))+a (f+x (g+x (h+i x))))}{\left (a-b x^4\right )^3}+6 \sqrt [4]{a} \left (77 b^{3/2} c-15 \sqrt {a} b e-7 a \sqrt {b} g+5 a^{3/2} i\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )+3 \sqrt [4]{a} \left (-77 b^{3/2} c-40 \sqrt [4]{a} b^{5/4} d-15 \sqrt {a} b e+7 a \sqrt {b} g+8 a^{5/4} \sqrt [4]{b} h+5 a^{3/2} i\right ) \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )-3 \sqrt [4]{a} \left (-77 b^{3/2} c+40 \sqrt [4]{a} b^{5/4} d-15 \sqrt {a} b e+7 a \sqrt {b} g-8 a^{5/4} \sqrt [4]{b} h+5 a^{3/2} i\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )-24 \sqrt {a} \sqrt [4]{b} (-5 b d+a h) \log \left (\sqrt {a}+\sqrt {b} x^2\right )}{1536 a^4 b^{7/4}} \]
((-4*a*b^(3/4)*x*(-77*b*c + 7*a*g - 15*b*x*(4*d + 3*e*x) + 3*a*x*(4*h + 5* i*x)))/(a - b*x^4) - (16*a^2*b^(3/4)*x*(-(b*(11*c + x*(10*d + 9*e*x))) + a *(g + x*(2*h + 3*i*x))))/(a - b*x^4)^2 + (128*a^3*b^(3/4)*(b*x*(c + x*(d + e*x)) + a*(f + x*(g + x*(h + i*x)))))/(a - b*x^4)^3 + 6*a^(1/4)*(77*b^(3/ 2)*c - 15*Sqrt[a]*b*e - 7*a*Sqrt[b]*g + 5*a^(3/2)*i)*ArcTan[(b^(1/4)*x)/a^ (1/4)] + 3*a^(1/4)*(-77*b^(3/2)*c - 40*a^(1/4)*b^(5/4)*d - 15*Sqrt[a]*b*e + 7*a*Sqrt[b]*g + 8*a^(5/4)*b^(1/4)*h + 5*a^(3/2)*i)*Log[a^(1/4) - b^(1/4) *x] - 3*a^(1/4)*(-77*b^(3/2)*c + 40*a^(1/4)*b^(5/4)*d - 15*Sqrt[a]*b*e + 7 *a*Sqrt[b]*g - 8*a^(5/4)*b^(1/4)*h + 5*a^(3/2)*i)*Log[a^(1/4) + b^(1/4)*x] - 24*Sqrt[a]*b^(1/4)*(-5*b*d + a*h)*Log[Sqrt[a] + Sqrt[b]*x^2])/(1536*a^4 *b^(7/4))
Time = 0.88 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {2397, 25, 2393, 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.
| \(\displaystyle \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{\left (a-b x^4\right )^4} \, dx\) |
| \(\Big \downarrow \) 2397 |
| \(\displaystyle \frac {x \left (x (a h+b d)+x^2 (a i+b e)+a g+b c+b f x^3\right )}{12 a b \left (a-b x^4\right )^3}-\frac {\int -\frac {8 b^2 f x^3+3 b (3 b e-a i) x^2+2 b (5 b d-a h) x+b (11 b c-a g)}{\left (a-b x^4\right )^3}dx}{12 a b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {8 b^2 f x^3+3 b (3 b e-a i) x^2+2 b (5 b d-a h) x+b (11 b c-a g)}{\left (a-b x^4\right )^3}dx}{12 a b^2}+\frac {x \left (x (a h+b d)+x^2 (a i+b e)+a g+b c+b f x^3\right )}{12 a b \left (a-b x^4\right )^3}\) |
| \(\Big \downarrow \) 2393 |
| \(\displaystyle \frac {\frac {x \left (b (11 b c-a g)+2 b x (5 b d-a h)+3 b x^2 (3 b e-a i)\right )+8 a b f}{8 a \left (a-b x^4\right )^2}-\frac {\int -\frac {15 b (3 b e-a i) x^2+12 b (5 b d-a h) x+7 b (11 b c-a g)}{\left (a-b x^4\right )^2}dx}{8 a}}{12 a b^2}+\frac {x \left (x (a h+b d)+x^2 (a i+b e)+a g+b c+b f x^3\right )}{12 a b \left (a-b x^4\right )^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {15 b (3 b e-a i) x^2+12 b (5 b d-a h) x+7 b (11 b c-a g)}{\left (a-b x^4\right )^2}dx}{8 a}+\frac {x \left (b (11 b c-a g)+2 b x (5 b d-a h)+3 b x^2 (3 b e-a i)\right )+8 a b f}{8 a \left (a-b x^4\right )^2}}{12 a b^2}+\frac {x \left (x (a h+b d)+x^2 (a i+b e)+a g+b c+b f x^3\right )}{12 a b \left (a-b x^4\right )^3}\) |
| \(\Big \downarrow \) 2394 |
| \(\displaystyle \frac {\frac {\frac {x \left (7 b (11 b c-a g)+12 b x (5 b d-a h)+15 b x^2 (3 b e-a i)\right )}{4 a \left (a-b x^4\right )}-\frac {\int -\frac {3 \left (5 b (3 b e-a i) x^2+8 b (5 b d-a h) x+7 b (11 b c-a g)\right )}{a-b x^4}dx}{4 a}}{8 a}+\frac {x \left (b (11 b c-a g)+2 b x (5 b d-a h)+3 b x^2 (3 b e-a i)\right )+8 a b f}{8 a \left (a-b x^4\right )^2}}{12 a b^2}+\frac {x \left (x (a h+b d)+x^2 (a i+b e)+a g+b c+b f x^3\right )}{12 a b \left (a-b x^4\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {5 b (3 b e-a i) x^2+8 b (5 b d-a h) x+7 b (11 b c-a g)}{a-b x^4}dx}{4 a}+\frac {x \left (7 b (11 b c-a g)+12 b x (5 b d-a h)+15 b x^2 (3 b e-a i)\right )}{4 a \left (a-b x^4\right )}}{8 a}+\frac {x \left (b (11 b c-a g)+2 b x (5 b d-a h)+3 b x^2 (3 b e-a i)\right )+8 a b f}{8 a \left (a-b x^4\right )^2}}{12 a b^2}+\frac {x \left (x (a h+b d)+x^2 (a i+b e)+a g+b c+b f x^3\right )}{12 a b \left (a-b x^4\right )^3}\) |
| \(\Big \downarrow \) 2415 |
| \(\displaystyle \frac {\frac {\frac {3 \int \left (\frac {8 b (5 b d-a h) x}{a-b x^4}+\frac {5 b (3 b e-a i) x^2+7 b (11 b c-a g)}{a-b x^4}\right )dx}{4 a}+\frac {x \left (7 b (11 b c-a g)+12 b x (5 b d-a h)+15 b x^2 (3 b e-a i)\right )}{4 a \left (a-b x^4\right )}}{8 a}+\frac {x \left (b (11 b c-a g)+2 b x (5 b d-a h)+3 b x^2 (3 b e-a i)\right )+8 a b f}{8 a \left (a-b x^4\right )^2}}{12 a b^2}+\frac {x \left (x (a h+b d)+x^2 (a i+b e)+a g+b c+b f x^3\right )}{12 a b \left (a-b x^4\right )^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac {7 \sqrt {b} (11 b c-a g)}{\sqrt {a}}-5 a i+15 b e\right )}{2 \sqrt [4]{a}}+\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac {7 \sqrt {b} (11 b c-a g)}{\sqrt {a}}-5 a i+15 b e\right )}{2 \sqrt [4]{a}}+\frac {4 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ) (5 b d-a h)}{\sqrt {a}}\right )}{4 a}+\frac {x \left (7 b (11 b c-a g)+12 b x (5 b d-a h)+15 b x^2 (3 b e-a i)\right )}{4 a \left (a-b x^4\right )}}{8 a}+\frac {x \left (b (11 b c-a g)+2 b x (5 b d-a h)+3 b x^2 (3 b e-a i)\right )+8 a b f}{8 a \left (a-b x^4\right )^2}}{12 a b^2}+\frac {x \left (x (a h+b d)+x^2 (a i+b e)+a g+b c+b f x^3\right )}{12 a b \left (a-b x^4\right )^3}\) |
(x*(b*c + a*g + (b*d + a*h)*x + (b*e + a*i)*x^2 + b*f*x^3))/(12*a*b*(a - b *x^4)^3) + ((8*a*b*f + x*(b*(11*b*c - a*g) + 2*b*(5*b*d - a*h)*x + 3*b*(3* b*e - a*i)*x^2))/(8*a*(a - b*x^4)^2) + ((x*(7*b*(11*b*c - a*g) + 12*b*(5*b *d - a*h)*x + 15*b*(3*b*e - a*i)*x^2))/(4*a*(a - b*x^4)) + (3*(-1/2*(b^(1/ 4)*(15*b*e - (7*Sqrt[b]*(11*b*c - a*g))/Sqrt[a] - 5*a*i)*ArcTan[(b^(1/4)*x )/a^(1/4)])/a^(1/4) + (b^(1/4)*(15*b*e + (7*Sqrt[b]*(11*b*c - a*g))/Sqrt[a ] - 5*a*i)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(2*a^(1/4)) + (4*Sqrt[b]*(5*b*d - a*h)*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/Sqrt[a]))/(4*a))/(8*a))/(12*a*b^2)
3.3.5.3.1 Defintions of rubi rules used
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] :> 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_.))^(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 = 1.57 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.73
| method | result | size |
| risch | \(\frac {-\frac {5 \left (a i -3 b e \right ) b \,x^{11}}{128 a^{3}}-\frac {\left (a h -5 b d \right ) b \,x^{10}}{32 a^{3}}-\frac {7 \left (a g -11 b c \right ) b \,x^{9}}{384 a^{3}}+\frac {7 \left (a i -3 b e \right ) x^{7}}{64 a^{2}}+\frac {\left (a h -5 b d \right ) x^{6}}{12 a^{2}}+\frac {3 \left (a g -11 b c \right ) x^{5}}{64 a^{2}}+\frac {\left (5 a i +113 b e \right ) x^{3}}{384 a b}+\frac {\left (a h +11 b d \right ) x^{2}}{32 a b}+\frac {\left (7 a g +51 b c \right ) x}{128 a b}+\frac {f}{12 b}}{\left (-b \,x^{4}+a \right )^{3}}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b -a \right )}{\sum }\frac {\left (-5 \left (a i -3 b e \right ) \textit {\_R}^{2}-8 \left (a h -5 b d \right ) \textit {\_R} -7 a g +77 b c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{512 a^{3} b^{2}}\) | \(240\) |
| default | \(\frac {-\frac {5 \left (a i -3 b e \right ) b \,x^{11}}{128 a^{3}}-\frac {\left (a h -5 b d \right ) b \,x^{10}}{32 a^{3}}-\frac {7 \left (a g -11 b c \right ) b \,x^{9}}{384 a^{3}}+\frac {7 \left (a i -3 b e \right ) x^{7}}{64 a^{2}}+\frac {\left (a h -5 b d \right ) x^{6}}{12 a^{2}}+\frac {3 \left (a g -11 b c \right ) x^{5}}{64 a^{2}}+\frac {\left (5 a i +113 b e \right ) x^{3}}{384 a b}+\frac {\left (a h +11 b d \right ) x^{2}}{32 a b}+\frac {\left (7 a g +51 b c \right ) x}{128 a b}+\frac {f}{12 b}}{\left (-b \,x^{4}+a \right )^{3}}+\frac {\frac {\left (-7 a g +77 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}+\frac {\left (-8 a h +40 b d \right ) \ln \left (\frac {a +x^{2} \sqrt {a b}}{a -x^{2} \sqrt {a b}}\right )}{4 \sqrt {a b}}-\frac {\left (-5 a i +15 b e \right ) \left (2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{128 a^{3} b}\) | \(346\) |
(-5/128*(a*i-3*b*e)/a^3*b*x^11-1/32*(a*h-5*b*d)/a^3*b*x^10-7/384*(a*g-11*b *c)/a^3*b*x^9+7/64*(a*i-3*b*e)/a^2*x^7+1/12/a^2*(a*h-5*b*d)*x^6+3/64/a^2*( a*g-11*b*c)*x^5+1/384*(5*a*i+113*b*e)/a/b*x^3+1/32*(a*h+11*b*d)/a/b*x^2+1/ 128*(7*a*g+51*b*c)/a/b*x+1/12*f/b)/(-b*x^4+a)^3-1/512/a^3/b^2*sum((-5*(a*i -3*b*e)*_R^2-8*(a*h-5*b*d)*_R-7*a*g+77*b*c)/_R^3*ln(x-_R),_R=RootOf(_Z^4*b -a))
Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{\left (a-b x^4\right )^4} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{\left (a-b x^4\right )^4} \, dx=\text {Timed out} \]
Time = 0.30 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.30 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{\left (a-b x^4\right )^4} \, dx=-\frac {15 \, {\left (3 \, b^{3} e - a b^{2} i\right )} x^{11} + 12 \, {\left (5 \, b^{3} d - a b^{2} h\right )} x^{10} + 7 \, {\left (11 \, b^{3} c - a b^{2} g\right )} x^{9} - 42 \, {\left (3 \, a b^{2} e - a^{2} b i\right )} x^{7} - 32 \, {\left (5 \, a b^{2} d - a^{2} b h\right )} x^{6} - 18 \, {\left (11 \, a b^{2} c - a^{2} b g\right )} x^{5} + 32 \, a^{3} f + {\left (113 \, a^{2} b e + 5 \, a^{3} i\right )} x^{3} + 12 \, {\left (11 \, a^{2} b d + a^{3} h\right )} x^{2} + 3 \, {\left (51 \, a^{2} b c + 7 \, a^{3} g\right )} x}{384 \, {\left (a^{3} b^{4} x^{12} - 3 \, a^{4} b^{3} x^{8} + 3 \, a^{5} b^{2} x^{4} - a^{6} b\right )}} + \frac {\frac {8 \, {\left (5 \, b d - a h\right )} \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} - \frac {8 \, {\left (5 \, b d - a h\right )} \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} + \frac {2 \, {\left (77 \, b^{\frac {3}{2}} c - 15 \, \sqrt {a} b e - 7 \, a \sqrt {b} g + 5 \, a^{\frac {3}{2}} i\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {{\left (77 \, b^{\frac {3}{2}} c + 15 \, \sqrt {a} b e - 7 \, a \sqrt {b} g - 5 \, a^{\frac {3}{2}} i\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}}{512 \, a^{3} b} \]
-1/384*(15*(3*b^3*e - a*b^2*i)*x^11 + 12*(5*b^3*d - a*b^2*h)*x^10 + 7*(11* b^3*c - a*b^2*g)*x^9 - 42*(3*a*b^2*e - a^2*b*i)*x^7 - 32*(5*a*b^2*d - a^2* b*h)*x^6 - 18*(11*a*b^2*c - a^2*b*g)*x^5 + 32*a^3*f + (113*a^2*b*e + 5*a^3 *i)*x^3 + 12*(11*a^2*b*d + a^3*h)*x^2 + 3*(51*a^2*b*c + 7*a^3*g)*x)/(a^3*b ^4*x^12 - 3*a^4*b^3*x^8 + 3*a^5*b^2*x^4 - a^6*b) + 1/512*(8*(5*b*d - a*h)* log(sqrt(b)*x^2 + sqrt(a))/(sqrt(a)*sqrt(b)) - 8*(5*b*d - a*h)*log(sqrt(b) *x^2 - sqrt(a))/(sqrt(a)*sqrt(b)) + 2*(77*b^(3/2)*c - 15*sqrt(a)*b*e - 7*a *sqrt(b)*g + 5*a^(3/2)*i)*arctan(sqrt(b)*x/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a) *sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - (77*b^(3/2)*c + 15*sqrt(a)*b*e - 7*a*sqr t(b)*g - 5*a^(3/2)*i)*log((sqrt(b)*x - sqrt(sqrt(a)*sqrt(b)))/(sqrt(b)*x + sqrt(sqrt(a)*sqrt(b))))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b)))/(a^3*b)
Leaf count of result is larger than twice the leaf count of optimal. 601 vs. \(2 (284) = 568\).
Time = 0.29 (sec) , antiderivative size = 601, normalized size of antiderivative = 1.82 \[ \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 )^4} \, dx=-\frac {\sqrt {2} {\left (77 \, b^{3} c - 7 \, a b^{2} g - 40 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b^{2} d + 8 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} a b h - 15 \, \sqrt {-a b} b^{2} e + 5 \, \sqrt {-a b} a b 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 )}{512 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{3} b} - \frac {\sqrt {2} {\left (77 \, b^{3} c - 7 \, a b^{2} g + 40 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b^{2} d - 8 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} a b h - 15 \, \sqrt {-a b} b^{2} e - 5 \, \sqrt {-a b} a b 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 )}{512 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{3} b} - \frac {\sqrt {2} {\left (77 \, b^{3} c - 7 \, a b^{2} g - 15 \, \sqrt {-a b} b^{2} e + 5 \, \sqrt {-a b} a b i\right )} \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{1024 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{3} b} + \frac {\sqrt {2} {\left (77 \, b^{3} c - 7 \, a b^{2} g - 15 \, \sqrt {-a b} b^{2} e + 5 \, \sqrt {-a b} a b i\right )} \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{1024 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{3} b} - \frac {45 \, b^{3} e x^{11} - 15 \, a b^{2} i x^{11} + 60 \, b^{3} d x^{10} - 12 \, a b^{2} h x^{10} + 77 \, b^{3} c x^{9} - 7 \, a b^{2} g x^{9} - 126 \, a b^{2} e x^{7} + 42 \, a^{2} b i x^{7} - 160 \, a b^{2} d x^{6} + 32 \, a^{2} b h x^{6} - 198 \, a b^{2} c x^{5} + 18 \, a^{2} b g x^{5} + 113 \, a^{2} b e x^{3} + 5 \, a^{3} i x^{3} + 132 \, a^{2} b d x^{2} + 12 \, a^{3} h x^{2} + 153 \, a^{2} b c x + 21 \, a^{3} g x + 32 \, a^{3} f}{384 \, {\left (b x^{4} - a\right )}^{3} a^{3} b} \]
-1/512*sqrt(2)*(77*b^3*c - 7*a*b^2*g - 40*sqrt(2)*(-a*b^3)^(1/4)*b^2*d + 8 *sqrt(2)*(-a*b^3)^(1/4)*a*b*h - 15*sqrt(-a*b)*b^2*e + 5*sqrt(-a*b)*a*b*i)* arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/((-a*b^3)^(3 /4)*a^3*b) - 1/512*sqrt(2)*(77*b^3*c - 7*a*b^2*g + 40*sqrt(2)*(-a*b^3)^(1/ 4)*b^2*d - 8*sqrt(2)*(-a*b^3)^(1/4)*a*b*h - 15*sqrt(-a*b)*b^2*e - 5*sqrt(- a*b)*a*b*i)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/ ((-a*b^3)^(3/4)*a^3*b) - 1/1024*sqrt(2)*(77*b^3*c - 7*a*b^2*g - 15*sqrt(-a *b)*b^2*e + 5*sqrt(-a*b)*a*b*i)*log(x^2 + sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a /b))/((-a*b^3)^(3/4)*a^3*b) + 1/1024*sqrt(2)*(77*b^3*c - 7*a*b^2*g - 15*sq rt(-a*b)*b^2*e + 5*sqrt(-a*b)*a*b*i)*log(x^2 - sqrt(2)*x*(-a/b)^(1/4) + sq rt(-a/b))/((-a*b^3)^(3/4)*a^3*b) - 1/384*(45*b^3*e*x^11 - 15*a*b^2*i*x^11 + 60*b^3*d*x^10 - 12*a*b^2*h*x^10 + 77*b^3*c*x^9 - 7*a*b^2*g*x^9 - 126*a*b ^2*e*x^7 + 42*a^2*b*i*x^7 - 160*a*b^2*d*x^6 + 32*a^2*b*h*x^6 - 198*a*b^2*c *x^5 + 18*a^2*b*g*x^5 + 113*a^2*b*e*x^3 + 5*a^3*i*x^3 + 132*a^2*b*d*x^2 + 12*a^3*h*x^2 + 153*a^2*b*c*x + 21*a^3*g*x + 32*a^3*f)/((b*x^4 - a)^3*a^3*b )
Time = 10.32 (sec) , antiderivative size = 2747, normalized size of antiderivative = 8.30 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{\left (a-b x^4\right )^4} \, dx=\text {Too large to display} \]
(f/(12*b) - (3*x^5*(11*b*c - a*g))/(64*a^2) - (x^6*(5*b*d - a*h))/(12*a^2) - (7*x^7*(3*b*e - a*i))/(64*a^2) + (7*b*x^9*(11*b*c - a*g))/(384*a^3) + ( x*(51*b*c + 7*a*g))/(128*a*b) + (b*x^10*(5*b*d - a*h))/(32*a^3) + (5*b*x^1 1*(3*b*e - a*i))/(128*a^3) + (x^2*(11*b*d + a*h))/(32*a*b) + (x^3*(113*b*e + 5*a*i))/(384*a*b))/(a^3 - b^3*x^12 - 3*a^2*b*x^4 + 3*a*b^2*x^8) + symsu m(log((125*a^4*i^3 - 3375*a*b^3*e^3 - 123200*b^4*c*d^2 + 88935*b^4*c^2*e - 4928*a^2*b^2*c*h^2 + 735*a^2*b^2*e*g^2 + 3375*a^2*b^2*e^2*i + 11200*a*b^3 *d^2*g - 29645*a*b^3*c^2*i - 1125*a^3*b*e*i^2 + 448*a^3*b*g*h^2 - 245*a^3* b*g^2*i + 5390*a^2*b^2*c*g*i - 4480*a^2*b^2*d*g*h + 49280*a*b^3*c*d*h - 16 170*a*b^3*c*e*g)/(2097152*a^9*b^2) - root(68719476736*a^15*b^7*z^4 - 12111 05280*a^8*b^6*c*e*z^2 + 403701760*a^9*b^5*c*i*z^2 + 335544320*a^9*b^5*d*h* z^2 + 110100480*a^9*b^5*e*g*z^2 - 36700160*a^10*b^4*g*i*z^2 - 838860800*a^ 8*b^6*d^2*z^2 - 33554432*a^10*b^4*h^2*z^2 + 2457600*a^7*b^3*e*h*i*z - 8830 9760*a^5*b^5*c*d*g*z + 17661952*a^6*b^4*c*g*h*z - 12288000*a^6*b^4*d*e*i*z + 485703680*a^4*b^6*c^2*d*z - 409600*a^8*b^2*h*i^2*z - 97140736*a^5*b^5*c ^2*h*z - 802816*a^7*b^3*g^2*h*z - 3686400*a^6*b^4*e^2*h*z + 2048000*a^7*b^ 3*d*i^2*z + 4014080*a^6*b^4*d*g^2*z + 18432000*a^5*b^5*d*e^2*z + 89600*a^4 *b^2*d*g*h*i - 985600*a^3*b^3*c*d*h*i + 323400*a^3*b^3*c*e*g*i - 268800*a^ 3*b^3*d*e*g*h + 2956800*a^2*b^4*c*d*e*h - 14700*a^4*b^2*e*g^2*i - 224000*a ^3*b^3*d^2*g*i + 98560*a^4*b^2*c*h^2*i + 26880*a^4*b^2*e*g*h^2 - 53900*...