Integrand size = 46, antiderivative size = 285 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left (a-b x^4\right )^3} \, dx=\frac {x \left (b c+a g+(b d+a h) x+(b e+a i) x^2+(b f+a j) x^3\right )}{8 a b \left (a-b x^4\right )^2}+\frac {4 a (b f-a j)+x \left (b (7 b c-a g)+2 b (3 b d-a h) x+b (5 b e-3 a i) x^2\right )}{32 a^2 b^2 \left (a-b x^4\right )}-\frac {\left (5 b e-\frac {3 \sqrt {b} (7 b c-a g)}{\sqrt {a}}-3 a i\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{9/4} b^{7/4}}+\frac {\left (5 b e+\frac {3 \sqrt {b} (7 b c-a g)}{\sqrt {a}}-3 a i\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{9/4} b^{7/4}}+\frac {(3 b d-a h) \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} b^{3/2}} \]
1/8*x*(b*c+a*g+(a*h+b*d)*x+(a*i+b*e)*x^2+(a*j+b*f)*x^3)/a/b/(-b*x^4+a)^2+1 /32*(4*a*(-a*j+b*f)+x*(b*(-a*g+7*b*c)+2*b*(-a*h+3*b*d)*x+b*(-3*a*i+5*b*e)* x^2))/a^2/b^2/(-b*x^4+a)+1/16*(-a*h+3*b*d)*arctanh(x^2*b^(1/2)/a^(1/2))/a^ (5/2)/b^(3/2)-1/64*arctan(b^(1/4)*x/a^(1/4))*(5*b*e-3*a*i-3*(-a*g+7*b*c)*b ^(1/2)/a^(1/2))/a^(9/4)/b^(7/4)+1/64*arctanh(b^(1/4)*x/a^(1/4))*(5*b*e-3*a *i+3*(-a*g+7*b*c)*b^(1/2)/a^(1/2))/a^(9/4)/b^(7/4)
Time = 0.28 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.33 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left (a-b x^4\right )^3} \, dx=\frac {-\frac {4 a^{3/4} \left (8 a^2 j-b^2 x (7 c+x (6 d+5 e x))+a b x (g+x (2 h+3 i x))\right )}{a-b x^4}+\frac {16 a^{7/4} \left (a^2 j+b^2 x (c+x (d+e x))+a b (f+x (g+x (h+i x)))\right )}{\left (a-b x^4\right )^2}+2 \sqrt [4]{b} \left (21 b^{3/2} c-5 \sqrt {a} b e-3 a \sqrt {b} g+3 a^{3/2} i\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )+\sqrt [4]{b} \left (-21 b^{3/2} c-12 \sqrt [4]{a} b^{5/4} d-5 \sqrt {a} b e+3 a \sqrt {b} g+4 a^{5/4} \sqrt [4]{b} h+3 a^{3/2} i\right ) \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )+\sqrt [4]{b} \left (21 b^{3/2} c-12 \sqrt [4]{a} b^{5/4} d+5 \sqrt {a} b e-3 a \sqrt {b} g+4 a^{5/4} \sqrt [4]{b} h-3 a^{3/2} i\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )-4 \sqrt [4]{a} \sqrt {b} (-3 b d+a h) \log \left (\sqrt {a}+\sqrt {b} x^2\right )}{128 a^{11/4} b^2} \]
((-4*a^(3/4)*(8*a^2*j - b^2*x*(7*c + x*(6*d + 5*e*x)) + a*b*x*(g + x*(2*h + 3*i*x))))/(a - b*x^4) + (16*a^(7/4)*(a^2*j + b^2*x*(c + x*(d + e*x)) + a *b*(f + x*(g + x*(h + i*x)))))/(a - b*x^4)^2 + 2*b^(1/4)*(21*b^(3/2)*c - 5 *Sqrt[a]*b*e - 3*a*Sqrt[b]*g + 3*a^(3/2)*i)*ArcTan[(b^(1/4)*x)/a^(1/4)] + b^(1/4)*(-21*b^(3/2)*c - 12*a^(1/4)*b^(5/4)*d - 5*Sqrt[a]*b*e + 3*a*Sqrt[b ]*g + 4*a^(5/4)*b^(1/4)*h + 3*a^(3/2)*i)*Log[a^(1/4) - b^(1/4)*x] + b^(1/4 )*(21*b^(3/2)*c - 12*a^(1/4)*b^(5/4)*d + 5*Sqrt[a]*b*e - 3*a*Sqrt[b]*g + 4 *a^(5/4)*b^(1/4)*h - 3*a^(3/2)*i)*Log[a^(1/4) + b^(1/4)*x] - 4*a^(1/4)*Sqr t[b]*(-3*b*d + a*h)*Log[Sqrt[a] + Sqrt[b]*x^2])/(128*a^(11/4)*b^2)
Time = 0.68 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, 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.
| \(\displaystyle \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left (a-b x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 2397 |
| \(\displaystyle \frac {x \left (x (a h+b d)+x^2 (a i+b e)+x^3 (a j+b f)+a g+b c\right )}{8 a b \left (a-b x^4\right )^2}-\frac {\int -\frac {4 b (b f-a j) x^3+b (5 b e-3 a i) x^2+2 b (3 b d-a h) x+b (7 b c-a g)}{\left (a-b x^4\right )^2}dx}{8 a b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {4 b (b f-a j) x^3+b (5 b e-3 a i) x^2+2 b (3 b d-a h) x+b (7 b c-a g)}{\left (a-b x^4\right )^2}dx}{8 a b^2}+\frac {x \left (x (a h+b d)+x^2 (a i+b e)+x^3 (a j+b f)+a g+b c\right )}{8 a b \left (a-b x^4\right )^2}\) |
| \(\Big \downarrow \) 2393 |
| \(\displaystyle \frac {\frac {x \left (b (7 b c-a g)+2 b x (3 b d-a h)+b x^2 (5 b e-3 a i)\right )+4 a (b f-a j)}{4 a \left (a-b x^4\right )}-\frac {\int -\frac {b (5 b e-3 a i) x^2+4 b (3 b d-a h) x+3 b (7 b c-a g)}{a-b x^4}dx}{4 a}}{8 a b^2}+\frac {x \left (x (a h+b d)+x^2 (a i+b e)+x^3 (a j+b f)+a g+b c\right )}{8 a b \left (a-b x^4\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {b (5 b e-3 a i) x^2+4 b (3 b d-a h) x+3 b (7 b c-a g)}{a-b x^4}dx}{4 a}+\frac {x \left (b (7 b c-a g)+2 b x (3 b d-a h)+b x^2 (5 b e-3 a i)\right )+4 a (b f-a j)}{4 a \left (a-b x^4\right )}}{8 a b^2}+\frac {x \left (x (a h+b d)+x^2 (a i+b e)+x^3 (a j+b f)+a g+b c\right )}{8 a b \left (a-b x^4\right )^2}\) |
| \(\Big \downarrow \) 2415 |
| \(\displaystyle \frac {\frac {\int \left (\frac {4 b (3 b d-a h) x}{a-b x^4}+\frac {b (5 b e-3 a i) x^2+3 b (7 b c-a g)}{a-b x^4}\right )dx}{4 a}+\frac {x \left (b (7 b c-a g)+2 b x (3 b d-a h)+b x^2 (5 b e-3 a i)\right )+4 a (b f-a j)}{4 a \left (a-b x^4\right )}}{8 a b^2}+\frac {x \left (x (a h+b d)+x^2 (a i+b e)+x^3 (a j+b f)+a g+b c\right )}{8 a b \left (a-b x^4\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {-\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac {3 \sqrt {b} (7 b c-a g)}{\sqrt {a}}-3 a i+5 b e\right )}{2 \sqrt [4]{a}}+\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac {3 \sqrt {b} (7 b c-a g)}{\sqrt {a}}-3 a i+5 b e\right )}{2 \sqrt [4]{a}}+\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ) (3 b d-a h)}{\sqrt {a}}}{4 a}+\frac {x \left (b (7 b c-a g)+2 b x (3 b d-a h)+b x^2 (5 b e-3 a i)\right )+4 a (b f-a j)}{4 a \left (a-b x^4\right )}}{8 a b^2}+\frac {x \left (x (a h+b d)+x^2 (a i+b e)+x^3 (a j+b f)+a g+b c\right )}{8 a b \left (a-b x^4\right )^2}\) |
(x*(b*c + a*g + (b*d + a*h)*x + (b*e + a*i)*x^2 + (b*f + a*j)*x^3))/(8*a*b *(a - b*x^4)^2) + ((4*a*(b*f - a*j) + x*(b*(7*b*c - a*g) + 2*b*(3*b*d - a* h)*x + b*(5*b*e - 3*a*i)*x^2))/(4*a*(a - b*x^4)) + (-1/2*(b^(1/4)*(5*b*e - (3*Sqrt[b]*(7*b*c - a*g))/Sqrt[a] - 3*a*i)*ArcTan[(b^(1/4)*x)/a^(1/4)])/a ^(1/4) + (b^(1/4)*(5*b*e + (3*Sqrt[b]*(7*b*c - a*g))/Sqrt[a] - 3*a*i)*ArcT anh[(b^(1/4)*x)/a^(1/4)])/(2*a^(1/4)) + (2*Sqrt[b]*(3*b*d - a*h)*ArcTanh[( Sqrt[b]*x^2)/Sqrt[a]])/Sqrt[a])/(4*a))/(8*a*b^2)
3.2.100.3.1 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 = 1.62 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.72
| method | result | size |
| risch | \(\frac {\frac {\left (3 a i -5 b e \right ) x^{7}}{32 a^{2}}+\frac {\left (a h -3 b d \right ) x^{6}}{16 a^{2}}+\frac {\left (a g -7 b c \right ) x^{5}}{32 a^{2}}+\frac {j \,x^{4}}{4 b}+\frac {\left (a i +9 b e \right ) x^{3}}{32 a b}+\frac {\left (a h +5 b d \right ) x^{2}}{16 a b}+\frac {\left (3 a g +11 b c \right ) x}{32 a b}-\frac {a j -b f}{8 b^{2}}}{\left (-b \,x^{4}+a \right )^{2}}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b -a \right )}{\sum }\frac {\left (-\left (3 a i -5 b e \right ) \textit {\_R}^{2}-4 \left (a h -3 b d \right ) \textit {\_R} -3 a g +21 b c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{128 a^{2} b^{2}}\) | \(206\) |
| default | \(\frac {\frac {\left (3 a i -5 b e \right ) x^{7}}{32 a^{2}}+\frac {\left (a h -3 b d \right ) x^{6}}{16 a^{2}}+\frac {\left (a g -7 b c \right ) x^{5}}{32 a^{2}}+\frac {j \,x^{4}}{4 b}+\frac {\left (a i +9 b e \right ) x^{3}}{32 a b}+\frac {\left (a h +5 b d \right ) x^{2}}{16 a b}+\frac {\left (3 a g +11 b c \right ) x}{32 a b}-\frac {a j -b f}{8 b^{2}}}{\left (-b \,x^{4}+a \right )^{2}}+\frac {\frac {\left (-3 a g +21 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}+\frac {\left (-4 a h +12 b d \right ) \ln \left (\frac {a +x^{2} \sqrt {a b}}{a -x^{2} \sqrt {a b}}\right )}{4 \sqrt {a b}}-\frac {\left (-3 a i +5 b e \right ) \left (2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{32 a^{2} b}\) | \(311\) |
(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+1/4*j*x^4/b+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*(a*j-b*f)/b^2)/(-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. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left (a-b x^4\right )^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left (a-b x^4\right )^3} \, dx=\text {Timed out} \]
Time = 0.31 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.32 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left (a-b x^4\right )^3} \, dx=\frac {8 \, a^{2} b j x^{4} - {\left (5 \, b^{3} e - 3 \, a b^{2} i\right )} x^{7} - 2 \, {\left (3 \, b^{3} d - a b^{2} h\right )} x^{6} - {\left (7 \, b^{3} c - a b^{2} g\right )} x^{5} + 4 \, a^{2} b f - 4 \, a^{3} j + {\left (9 \, a b^{2} e + a^{2} b i\right )} x^{3} + 2 \, {\left (5 \, a b^{2} d + a^{2} b h\right )} x^{2} + {\left (11 \, a b^{2} c + 3 \, a^{2} b g\right )} x}{32 \, {\left (a^{2} b^{4} x^{8} - 2 \, a^{3} b^{3} x^{4} + a^{4} b^{2}\right )}} + \frac {\frac {4 \, {\left (3 \, b d - a h\right )} \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} - \frac {4 \, {\left (3 \, b d - a h\right )} \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} + \frac {2 \, {\left (21 \, b^{\frac {3}{2}} c - 5 \, \sqrt {a} b e - 3 \, a \sqrt {b} g + 3 \, a^{\frac {3}{2}} i\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {{\left (21 \, b^{\frac {3}{2}} c + 5 \, \sqrt {a} b e - 3 \, a \sqrt {b} g - 3 \, a^{\frac {3}{2}} i\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}}{128 \, a^{2} b} \]
1/32*(8*a^2*b*j*x^4 - (5*b^3*e - 3*a*b^2*i)*x^7 - 2*(3*b^3*d - a*b^2*h)*x^ 6 - (7*b^3*c - a*b^2*g)*x^5 + 4*a^2*b*f - 4*a^3*j + (9*a*b^2*e + a^2*b*i)* x^3 + 2*(5*a*b^2*d + a^2*b*h)*x^2 + (11*a*b^2*c + 3*a^2*b*g)*x)/(a^2*b^4*x ^8 - 2*a^3*b^3*x^4 + a^4*b^2) + 1/128*(4*(3*b*d - a*h)*log(sqrt(b)*x^2 + s qrt(a))/(sqrt(a)*sqrt(b)) - 4*(3*b*d - a*h)*log(sqrt(b)*x^2 - sqrt(a))/(sq rt(a)*sqrt(b)) + 2*(21*b^(3/2)*c - 5*sqrt(a)*b*e - 3*a*sqrt(b)*g + 3*a^(3/ 2)*i)*arctan(sqrt(b)*x/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b ))*sqrt(b)) - (21*b^(3/2)*c + 5*sqrt(a)*b*e - 3*a*sqrt(b)*g - 3*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^2*b)
Leaf count of result is larger than twice the leaf count of optimal. 561 vs. \(2 (245) = 490\).
Time = 0.29 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.97 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left (a-b x^4\right )^3} \, dx=-\frac {\sqrt {2} {\left (21 \, b^{3} c - 3 \, a b^{2} g - 12 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b^{2} d + 4 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} a b h - 5 \, \sqrt {-a b} b^{2} e + 3 \, \sqrt {-a b} a b i\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2} b} - \frac {\sqrt {2} {\left (21 \, b^{3} c - 3 \, a b^{2} g + 12 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b^{2} d - 4 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} a b h - 5 \, \sqrt {-a b} b^{2} e - 3 \, \sqrt {-a b} a b i\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2} b} - \frac {\sqrt {2} {\left (21 \, b^{3} c - 3 \, a b^{2} g - 5 \, \sqrt {-a b} b^{2} e + 3 \, \sqrt {-a b} a b i\right )} \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{256 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2} b} + \frac {\sqrt {2} {\left (21 \, b^{3} c - 3 \, a b^{2} g - 5 \, \sqrt {-a b} b^{2} e + 3 \, \sqrt {-a b} a b i\right )} \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{256 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2} b} - \frac {5 \, b^{3} e x^{7} - 3 \, a b^{2} i x^{7} + 6 \, b^{3} d x^{6} - 2 \, a b^{2} h x^{6} + 7 \, b^{3} c x^{5} - a b^{2} g x^{5} - 8 \, a^{2} b j x^{4} - 9 \, a b^{2} e x^{3} - a^{2} b i x^{3} - 10 \, a b^{2} d x^{2} - 2 \, a^{2} b h x^{2} - 11 \, a b^{2} c x - 3 \, a^{2} b g x - 4 \, a^{2} b f + 4 \, a^{3} j}{32 \, {\left (b x^{4} - a\right )}^{2} a^{2} b^{2}} \]
-1/128*sqrt(2)*(21*b^3*c - 3*a*b^2*g - 12*sqrt(2)*(-a*b^3)^(1/4)*b^2*d + 4 *sqrt(2)*(-a*b^3)^(1/4)*a*b*h - 5*sqrt(-a*b)*b^2*e + 3*sqrt(-a*b)*a*b*i)*a rctan(1/2*sqrt(2)*(2*x + sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/((-a*b^3)^(3/ 4)*a^2*b) - 1/128*sqrt(2)*(21*b^3*c - 3*a*b^2*g + 12*sqrt(2)*(-a*b^3)^(1/4 )*b^2*d - 4*sqrt(2)*(-a*b^3)^(1/4)*a*b*h - 5*sqrt(-a*b)*b^2*e - 3*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^2*b) - 1/256*sqrt(2)*(21*b^3*c - 3*a*b^2*g - 5*sqrt(-a*b)* b^2*e + 3*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^2*b) + 1/256*sqrt(2)*(21*b^3*c - 3*a*b^2*g - 5*sqrt(-a* b)*b^2*e + 3*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^2*b) - 1/32*(5*b^3*e*x^7 - 3*a*b^2*i*x^7 + 6*b^3*d*x ^6 - 2*a*b^2*h*x^6 + 7*b^3*c*x^5 - a*b^2*g*x^5 - 8*a^2*b*j*x^4 - 9*a*b^2*e *x^3 - a^2*b*i*x^3 - 10*a*b^2*d*x^2 - 2*a^2*b*h*x^2 - 11*a*b^2*c*x - 3*a^2 *b*g*x - 4*a^2*b*f + 4*a^3*j)/((b*x^4 - a)^2*a^2*b^2)
Time = 10.02 (sec) , antiderivative size = 2696, normalized size of antiderivative = 9.46 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left (a-b x^4\right )^3} \, dx=\text {Too large to display} \]
symsum(log((27*a^4*i^3 - 125*a*b^3*e^3 - 3024*b^4*c*d^2 + 2205*b^4*c^2*e - 336*a^2*b^2*c*h^2 + 45*a^2*b^2*e*g^2 + 225*a^2*b^2*e^2*i + 432*a*b^3*d^2* g - 1323*a*b^3*c^2*i - 135*a^3*b*e*i^2 + 48*a^3*b*g*h^2 - 27*a^3*b*g^2*i + 378*a^2*b^2*c*g*i - 288*a^2*b^2*d*g*h + 2016*a*b^3*c*d*h - 630*a*b^3*c*e* g)/(32768*a^6*b^2) - 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 - 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 + 55296*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 + 55296*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 + 3456*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 - 51 84*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 - 60480*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...