\(\int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{(a-b x^4)^4} \, dx\) [206]

Optimal result

Integrand size = 46, antiderivative size = 349 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left (a-b x^4\right )^4} \, dx=\frac {x \left (b c+a g+(b d+a h) x+(b e+a i) x^2+(b f+a j) x^3\right )}{12 a b \left (a-b x^4\right )^3}+\frac {x \left (7 (11 b c-a g)+12 (5 b d-a h) x+15 (3 b e-a i) x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac {4 a (2 b f-a j)+x \left (b (11 b c-a g)+2 b (5 b d-a h) x+3 b (3 b e-a i) x^2\right )}{96 a^2 b^2 \left (a-b x^4\right )^2}+\frac {\left (\frac {7 \sqrt {b} (11 b c-a g)}{\sqrt {a}}-5 (3 b e-a i)\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{13/4} b^{7/4}}+\frac {\left (15 b e+\frac {7 \sqrt {b} (11 b c-a g)}{\sqrt {a}}-5 a i\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{13/4} b^{7/4}}+\frac {(5 b d-a h) \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} b^{3/2}} \]

[Out]

1/12*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)^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*(4*a*(-a*j+2*b*f)+x*(b*(-a*g+11*b*c)+2*b*(-a*h+5*b*d)*x+3*b*(-
a*i+3*b*e)*x^2))/a^2/b^2/(-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)+1/256*arc
tanh(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)

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {1872, 1868, 1869, 1890, 281, 214, 1181, 211} \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left (a-b x^4\right )^4} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac {7 \sqrt {b} (11 b c-a g)}{\sqrt {a}}-5 (3 b e-a i)\right )}{256 a^{13/4} b^{7/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac {7 \sqrt {b} (11 b c-a g)}{\sqrt {a}}-5 a i+15 b e\right )}{256 a^{13/4} b^{7/4}}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ) (5 b d-a h)}{32 a^{7/2} b^{3/2}}+\frac {x \left (7 (11 b c-a g)+12 x (5 b d-a h)+15 x^2 (3 b e-a i)\right )}{384 a^3 b \left (a-b x^4\right )}+\frac {x \left (b (11 b c-a g)+2 b x (5 b d-a h)+3 b x^2 (3 b e-a i)\right )+4 a (2 b f-a j)}{96 a^2 b^2 \left (a-b x^4\right )^2}+\frac {x \left (x (a h+b d)+x^2 (a i+b e)+x^3 (a j+b f)+a g+b c\right )}{12 a b \left (a-b x^4\right )^3} \]

[In]

Int[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6 + j*x^7)/(a - b*x^4)^4,x]

[Out]

(x*(b*c + a*g + (b*d + a*h)*x + (b*e + a*i)*x^2 + (b*f + a*j)*x^3))/(12*a*b*(a - b*x^4)^3) + (x*(7*(11*b*c - a
*g) + 12*(5*b*d - a*h)*x + 15*(3*b*e - a*i)*x^2))/(384*a^3*b*(a - b*x^4)) + (4*a*(2*b*f - a*j) + x*(b*(11*b*c
- a*g) + 2*b*(5*b*d - a*h)*x + 3*b*(3*b*e - a*i)*x^2))/(96*a^2*b^2*(a - b*x^4)^2) + (((7*Sqrt[b]*(11*b*c - a*g
))/Sqrt[a] - 5*(3*b*e - a*i))*ArcTan[(b^(1/4)*x)/a^(1/4)])/(256*a^(13/4)*b^(7/4)) + ((15*b*e + (7*Sqrt[b]*(11*
b*c - a*g))/Sqrt[a] - 5*a*i)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(256*a^(13/4)*b^(7/4)) + ((5*b*d - a*h)*ArcTanh[(Sq
rt[b]*x^2)/Sqrt[a]])/(32*a^(7/2)*b^(3/2))

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 1181

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Dist[e/2 + c*(d/(2*q))
, Int[1/(-q + c*x^2), x], x] + Dist[e/2 - c*(d/(2*q)), Int[1/(q + c*x^2), x], x]] /; FreeQ[{a, c, d, e}, x] &&
NeQ[c*d^2 - a*e^2, 0] && PosQ[(-a)*c]

Rule 1868

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] + Dist[1/(a*n*(p + 1))
, Int[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]

Rule 1869

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(-x)*Pq*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] +
Dist[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]

Rule 1872

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
]}, Dist[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] + Simp[(-x)*R*((a + b*x^n)^(p + 1)/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))), x]] /
; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 1890

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

Rubi steps \begin{align*} \text {integral}& = \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 )}{12 a b \left (a-b x^4\right )^3}-\frac {\int \frac {-b (11 b c-a g)-2 b (5 b d-a h) x-3 b (3 b e-a i) x^2-4 b (2 b f-a j) x^3}{\left (a-b x^4\right )^3} \, dx}{12 a b^2} \\ & = \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 )}{12 a b \left (a-b x^4\right )^3}+\frac {4 a (2 b f-a j)+x \left (b (11 b c-a g)+2 b (5 b d-a h) x+3 b (3 b e-a i) x^2\right )}{96 a^2 b^2 \left (a-b x^4\right )^2}+\frac {\int \frac {7 b (11 b c-a g)+12 b (5 b d-a h) x+15 b (3 b e-a i) x^2}{\left (a-b x^4\right )^2} \, dx}{96 a^2 b^2} \\ & = \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 )}{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 {4 a (2 b f-a j)+x \left (b (11 b c-a g)+2 b (5 b d-a h) x+3 b (3 b e-a i) x^2\right )}{96 a^2 b^2 \left (a-b x^4\right )^2}-\frac {\int \frac {-21 b (11 b c-a g)-24 b (5 b d-a h) x-15 b (3 b e-a i) x^2}{a-b x^4} \, dx}{384 a^3 b^2} \\ & = \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 )}{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 {4 a (2 b f-a j)+x \left (b (11 b c-a g)+2 b (5 b d-a h) x+3 b (3 b e-a i) x^2\right )}{96 a^2 b^2 \left (a-b x^4\right )^2}-\frac {\int \left (-\frac {24 b (5 b d-a h) x}{a-b x^4}+\frac {-21 b (11 b c-a g)-15 b (3 b e-a i) x^2}{a-b x^4}\right ) \, dx}{384 a^3 b^2} \\ & = \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 )}{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 {4 a (2 b f-a j)+x \left (b (11 b c-a g)+2 b (5 b d-a h) x+3 b (3 b e-a i) x^2\right )}{96 a^2 b^2 \left (a-b x^4\right )^2}-\frac {\int \frac {-21 b (11 b c-a g)-15 b (3 b e-a i) x^2}{a-b x^4} \, dx}{384 a^3 b^2}+\frac {(5 b d-a h) \int \frac {x}{a-b x^4} \, dx}{16 a^3 b} \\ & = \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 )}{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 {4 a (2 b f-a j)+x \left (b (11 b c-a g)+2 b (5 b d-a h) x+3 b (3 b e-a i) x^2\right )}{96 a^2 b^2 \left (a-b x^4\right )^2}+\frac {(5 b d-a h) \text {Subst}\left (\int \frac {1}{a-b x^2} \, dx,x,x^2\right )}{32 a^3 b}+\frac {\left (15 b e+\frac {7 \sqrt {b} (11 b c-a g)}{\sqrt {a}}-5 a i\right ) \int \frac {1}{\sqrt {a} \sqrt {b}-b x^2} \, dx}{256 a^3 b}-\frac {\left (\frac {7 \sqrt {b} (11 b c-a g)}{\sqrt {a}}-5 (3 b e-a i)\right ) \int \frac {1}{-\sqrt {a} \sqrt {b}-b x^2} \, dx}{256 a^3 b} \\ & = \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 )}{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 {4 a (2 b f-a j)+x \left (b (11 b c-a g)+2 b (5 b d-a h) x+3 b (3 b e-a i) x^2\right )}{96 a^2 b^2 \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 ) \tan ^{-1}\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 ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{13/4} b^{7/4}}+\frac {(5 b d-a h) \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} b^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.26 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left (a-b x^4\right )^4} \, dx=\frac {-\frac {4 a b x (-77 b c+7 a g-15 b x (4 d+3 e x)+3 a x (4 h+5 i x))}{a-b x^4}-\frac {16 a^2 \left (12 a^2 j-b^2 x (11 c+x (10 d+9 e x))+a b x (g+x (2 h+3 i x))\right )}{\left (a-b x^4\right )^2}+\frac {128 a^3 \left (a^2 j+b^2 x (c+x (d+e x))+a b (f+x (g+x (h+i x)))\right )}{\left (a-b x^4\right )^3}+6 \sqrt [4]{a} \sqrt [4]{b} \left (77 b^{3/2} c-15 \sqrt {a} b e-7 a \sqrt {b} g+5 a^{3/2} i\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )+3 \sqrt [4]{a} \sqrt [4]{b} \left (-77 b^{3/2} c-40 \sqrt [4]{a} b^{5/4} d-15 \sqrt {a} b e+7 a \sqrt {b} g+8 a^{5/4} \sqrt [4]{b} h+5 a^{3/2} i\right ) \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )+3 \sqrt [4]{a} \sqrt [4]{b} \left (77 b^{3/2} c-40 \sqrt [4]{a} b^{5/4} d+15 \sqrt {a} b e-7 a \sqrt {b} g+8 a^{5/4} \sqrt [4]{b} h-5 a^{3/2} i\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )-24 \sqrt {a} \sqrt {b} (-5 b d+a h) \log \left (\sqrt {a}+\sqrt {b} x^2\right )}{1536 a^4 b^2} \]

[In]

Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6 + j*x^7)/(a - b*x^4)^4,x]

[Out]

((-4*a*b*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*(12*a^2*j - b
^2*x*(11*c + x*(10*d + 9*e*x)) + a*b*x*(g + x*(2*h + 3*i*x))))/(a - b*x^4)^2 + (128*a^3*(a^2*j + b^2*x*(c + x*
(d + e*x)) + a*b*(f + x*(g + x*(h + i*x)))))/(a - b*x^4)^3 + 6*a^(1/4)*b^(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)*b^(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)*b^(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]*Sqrt[b]*(-5*b*d + a*h)*Log[Sqrt[a] + Sqrt[b]*x^2])/(1536*a^4*b^
2)

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.60 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.73

method	result	size
risch	\(\frac {-\frac {5 \left (a i -3 b e \right ) b \,x^{11}}{128 a^{3}}-\frac {\left (a h -5 b d \right ) b \,x^{10}}{32 a^{3}}-\frac {7 \left (a g -11 b c \right ) b \,x^{9}}{384 a^{3}}+\frac {7 \left (a i -3 b e \right ) x^{7}}{64 a^{2}}+\frac {\left (a h -5 b d \right ) x^{6}}{12 a^{2}}+\frac {3 \left (a g -11 b c \right ) x^{5}}{64 a^{2}}+\frac {j \,x^{4}}{8 b}+\frac {\left (5 a i +113 b e \right ) x^{3}}{384 a b}+\frac {\left (a h +11 b d \right ) x^{2}}{32 a b}+\frac {\left (7 a g +51 b c \right ) x}{128 a b}-\frac {a j -2 b f}{24 b^{2}}}{\left (-b \,x^{4}+a \right )^{3}}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b -a \right )}{\sum }\frac {\left (-5 \left (a i -3 b e \right ) \textit {\_R}^{2}-8 \left (a h -5 b d \right ) \textit {\_R} -7 a g +77 b c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{512 a^{3} b^{2}}\)	\(256\)
default	\(\frac {-\frac {5 \left (a i -3 b e \right ) b \,x^{11}}{128 a^{3}}-\frac {\left (a h -5 b d \right ) b \,x^{10}}{32 a^{3}}-\frac {7 \left (a g -11 b c \right ) b \,x^{9}}{384 a^{3}}+\frac {7 \left (a i -3 b e \right ) x^{7}}{64 a^{2}}+\frac {\left (a h -5 b d \right ) x^{6}}{12 a^{2}}+\frac {3 \left (a g -11 b c \right ) x^{5}}{64 a^{2}}+\frac {j \,x^{4}}{8 b}+\frac {\left (5 a i +113 b e \right ) x^{3}}{384 a b}+\frac {\left (a h +11 b d \right ) x^{2}}{32 a b}+\frac {\left (7 a g +51 b c \right ) x}{128 a b}-\frac {a j -2 b f}{24 b^{2}}}{\left (-b \,x^{4}+a \right )^{3}}+\frac {\frac {\left (-7 a g +77 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}+\frac {\left (-8 a h +40 b d \right ) \ln \left (\frac {a +x^{2} \sqrt {a b}}{a -x^{2} \sqrt {a b}}\right )}{4 \sqrt {a b}}-\frac {\left (-5 a i +15 b e \right ) \left (2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{128 a^{3} b}\)	\(362\)

[In]

int((j*x^7+i*x^6+h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(-b*x^4+a)^4,x,method=_RETURNVERBOSE)

[Out]

(-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+1/8*j*x^4/b+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-1/24*(a*j-2*b*f)/b^2)/(-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))

Fricas [F(-1)]

Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left (a-b x^4\right )^4} \, dx=\text {Timed out} \]

[In]

integrate((j*x^7+i*x^6+h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(-b*x^4+a)^4,x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left (a-b x^4\right )^4} \, dx=\text {Timed out} \]

[In]

integrate((j*x**7+i*x**6+h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/(-b*x**4+a)**4,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.33 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left (a-b x^4\right )^4} \, dx=-\frac {15 \, {\left (3 \, b^{4} e - a b^{3} i\right )} x^{11} + 12 \, {\left (5 \, b^{4} d - a b^{3} h\right )} x^{10} + 7 \, {\left (11 \, b^{4} c - a b^{3} g\right )} x^{9} + 48 \, a^{3} b j x^{4} - 42 \, {\left (3 \, a b^{3} e - a^{2} b^{2} i\right )} x^{7} - 32 \, {\left (5 \, a b^{3} d - a^{2} b^{2} h\right )} x^{6} - 18 \, {\left (11 \, a b^{3} c - a^{2} b^{2} g\right )} x^{5} + 32 \, a^{3} b f - 16 \, a^{4} j + {\left (113 \, a^{2} b^{2} e + 5 \, a^{3} b i\right )} x^{3} + 12 \, {\left (11 \, a^{2} b^{2} d + a^{3} b h\right )} x^{2} + 3 \, {\left (51 \, a^{2} b^{2} c + 7 \, a^{3} b g\right )} x}{384 \, {\left (a^{3} b^{5} x^{12} - 3 \, a^{4} b^{4} x^{8} + 3 \, a^{5} b^{3} x^{4} - a^{6} b^{2}\right )}} + \frac {\frac {8 \, {\left (5 \, b d - a h\right )} \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} - \frac {8 \, {\left (5 \, b d - a h\right )} \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} + \frac {2 \, {\left (77 \, b^{\frac {3}{2}} c - 15 \, \sqrt {a} b e - 7 \, a \sqrt {b} g + 5 \, a^{\frac {3}{2}} i\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {{\left (77 \, b^{\frac {3}{2}} c + 15 \, \sqrt {a} b e - 7 \, a \sqrt {b} g - 5 \, a^{\frac {3}{2}} i\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}}{512 \, a^{3} b} \]

[In]

integrate((j*x^7+i*x^6+h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(-b*x^4+a)^4,x, algorithm="maxima")

[Out]

-1/384*(15*(3*b^4*e - a*b^3*i)*x^11 + 12*(5*b^4*d - a*b^3*h)*x^10 + 7*(11*b^4*c - a*b^3*g)*x^9 + 48*a^3*b*j*x^
4 - 42*(3*a*b^3*e - a^2*b^2*i)*x^7 - 32*(5*a*b^3*d - a^2*b^2*h)*x^6 - 18*(11*a*b^3*c - a^2*b^2*g)*x^5 + 32*a^3
*b*f - 16*a^4*j + (113*a^2*b^2*e + 5*a^3*b*i)*x^3 + 12*(11*a^2*b^2*d + a^3*b*h)*x^2 + 3*(51*a^2*b^2*c + 7*a^3*
b*g)*x)/(a^3*b^5*x^12 - 3*a^4*b^4*x^8 + 3*a^5*b^3*x^4 - a^6*b^2) + 1/512*(8*(5*b*d - a*h)*log(sqrt(b)*x^2 + sq
rt(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*sqrt(b)*g - 5*a^(3/2)*i)*log((sqrt(b)*x - sqrt(sqrt(a)*sq
rt(b)))/(sqrt(b)*x + sqrt(sqrt(a)*sqrt(b))))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b)))/(a^3*b)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 633 vs. \(2 (302) = 604\).

Time = 0.28 (sec) , antiderivative size = 633, normalized size of antiderivative = 1.81 \[ \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 )^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^{4} e x^{11} - 15 \, a b^{3} i x^{11} + 60 \, b^{4} d x^{10} - 12 \, a b^{3} h x^{10} + 77 \, b^{4} c x^{9} - 7 \, a b^{3} g x^{9} - 126 \, a b^{3} e x^{7} + 42 \, a^{2} b^{2} i x^{7} - 160 \, a b^{3} d x^{6} + 32 \, a^{2} b^{2} h x^{6} - 198 \, a b^{3} c x^{5} + 18 \, a^{2} b^{2} g x^{5} + 48 \, a^{3} b j x^{4} + 113 \, a^{2} b^{2} e x^{3} + 5 \, a^{3} b i x^{3} + 132 \, a^{2} b^{2} d x^{2} + 12 \, a^{3} b h x^{2} + 153 \, a^{2} b^{2} c x + 21 \, a^{3} b g x + 32 \, a^{3} b f - 16 \, a^{4} j}{384 \, {\left (b x^{4} - a\right )}^{3} a^{3} b^{2}} \]

[In]

integrate((j*x^7+i*x^6+h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(-b*x^4+a)^4,x, algorithm="giac")

[Out]

-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*s
qrt(-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*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) - 1/384*(45*b^4*e*x^11 - 15*a*b^3*i*x^11 + 60*b^4*d*x^10 - 12*a*b^3*h*x^10 + 77*b^4*c*x^9 - 7*a*b^3*g*x^9 -
126*a*b^3*e*x^7 + 42*a^2*b^2*i*x^7 - 160*a*b^3*d*x^6 + 32*a^2*b^2*h*x^6 - 198*a*b^3*c*x^5 + 18*a^2*b^2*g*x^5
+ 48*a^3*b*j*x^4 + 113*a^2*b^2*e*x^3 + 5*a^3*b*i*x^3 + 132*a^2*b^2*d*x^2 + 12*a^3*b*h*x^2 + 153*a^2*b^2*c*x +
21*a^3*b*g*x + 32*a^3*b*f - 16*a^4*j)/((b*x^4 - a)^3*a^3*b^2)

Mupad [B] (verification not implemented)

Time = 10.57 (sec) , antiderivative size = 2764, normalized size of antiderivative = 7.92 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left (a-b x^4\right )^4} \, dx=\text {Too large to display} \]

[In]

int((c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6 + j*x^7)/(a - b*x^4)^4,x)

[Out]

symsum(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 - 2
45*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 - 16170*a*b^3*c*e*g)/(2097152*a^9
*b^2) - root(68719476736*a^15*b^7*z^4 - 1211105280*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 - 88309760*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*a^4*b^2*c*g*i^2 - 1778700*a^2*b^4*c^2*e*i + 2464000*a^2*b^4*c*d^2*i + 672000*a^2*b^4*d^2*e*g - 295
680*a^3*b^3*c*e*h^2 - 485100*a^2*b^4*c*e^2*g - 8960*a^5*b*g*h^2*i - 7392000*a*b^5*c*d^2*e + 7500*a^5*b*e*i^3 +
12782924*a*b^5*c^3*g - 33750*a^4*b^2*e^2*i^2 + 614400*a^3*b^3*d^2*h^2 + 296450*a^3*b^3*c^2*i^2 + 22050*a^3*b^
3*e^2*g^2 - 1743126*a^2*b^4*c^2*g^2 + 2450*a^5*b*g^2*i^2 + 67500*a^3*b^3*e^3*i - 2048000*a^2*b^4*d^3*h - 81920
*a^4*b^2*d*h^3 + 105644*a^3*b^3*c*g^3 + 2668050*a*b^5*c^2*e^2 - 2401*a^4*b^2*g^4 - 50625*a^2*b^4*e^4 + 4096*a^
5*b*h^4 + 2560000*a*b^5*d^4 - 625*a^6*i^4 - 35153041*b^6*c^4, z, m)*(root(68719476736*a^15*b^7*z^4 - 121110528
0*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 - 367001
60*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 - 883097
60*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*a^4*b^2*c*g*i^2 - 1778700*a^2*b^4*c^2
*e*i + 2464000*a^2*b^4*c*d^2*i + 672000*a^2*b^4*d^2*e*g - 295680*a^3*b^3*c*e*h^2 - 485100*a^2*b^4*c*e^2*g - 89
60*a^5*b*g*h^2*i - 7392000*a*b^5*c*d^2*e + 7500*a^5*b*e*i^3 + 12782924*a*b^5*c^3*g - 33750*a^4*b^2*e^2*i^2 + 6
14400*a^3*b^3*d^2*h^2 + 296450*a^3*b^3*c^2*i^2 + 22050*a^3*b^3*e^2*g^2 - 1743126*a^2*b^4*c^2*g^2 + 2450*a^5*b*
g^2*i^2 + 67500*a^3*b^3*e^3*i - 2048000*a^2*b^4*d^3*h - 81920*a^4*b^2*d*h^3 + 105644*a^3*b^3*c*g^3 + 2668050*a
*b^5*c^2*e^2 - 2401*a^4*b^2*g^4 - 50625*a^2*b^4*e^4 + 4096*a^5*b*h^4 + 2560000*a*b^5*d^4 - 625*a^6*i^4 - 35153
041*b^6*c^4, z, m)*((20185088*a^7*b^5*c - 1835008*a^8*b^4*g)/(2097152*a^9*b^2) - (x*(655360*a^7*b^4*d - 131072
*a^8*b^3*h))/(131072*a^9*b)) - (614400*a^4*b^4*d*e - 204800*a^5*b^3*d*i - 122880*a^5*b^3*e*h + 40960*a^6*b^2*h
*i)/(2097152*a^9*b^2) + (x*(800*a^6*b*i^2 + 189728*a^3*b^4*c^2 + 7200*a^4*b^3*e^2 + 1568*a^5*b^2*g^2 - 34496*a
^4*b^3*c*g - 4800*a^5*b^2*e*i))/(131072*a^9*b)) - (x*(4000*b^3*d^3 - 32*a^3*h^3 - 5775*b^3*c*d*e + 35*a^3*g*h*
i - 2400*a*b^2*d^2*h + 480*a^2*b*d*h^2 + 1925*a*b^2*c*d*i + 1155*a*b^2*c*e*h + 525*a*b^2*d*e*g - 385*a^2*b*c*h
*i - 175*a^2*b*d*g*i - 105*a^2*b*e*g*h))/(131072*a^9*b))*root(68719476736*a^15*b^7*z^4 - 1211105280*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 - 88309760*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*a^4*b^2*c*g*i^2 - 1778700*a^2*b^4*c^2*e*i + 24640
00*a^2*b^4*c*d^2*i + 672000*a^2*b^4*d^2*e*g - 295680*a^3*b^3*c*e*h^2 - 485100*a^2*b^4*c*e^2*g - 8960*a^5*b*g*h
^2*i - 7392000*a*b^5*c*d^2*e + 7500*a^5*b*e*i^3 + 12782924*a*b^5*c^3*g - 33750*a^4*b^2*e^2*i^2 + 614400*a^3*b^
3*d^2*h^2 + 296450*a^3*b^3*c^2*i^2 + 22050*a^3*b^3*e^2*g^2 - 1743126*a^2*b^4*c^2*g^2 + 2450*a^5*b*g^2*i^2 + 67
500*a^3*b^3*e^3*i - 2048000*a^2*b^4*d^3*h - 81920*a^4*b^2*d*h^3 + 105644*a^3*b^3*c*g^3 + 2668050*a*b^5*c^2*e^2
- 2401*a^4*b^2*g^4 - 50625*a^2*b^4*e^4 + 4096*a^5*b*h^4 + 2560000*a*b^5*d^4 - 625*a^6*i^4 - 35153041*b^6*c^4,
z, m), m, 1, 4) + ((2*b*f - a*j)/(24*b^2) + (j*x^4)/(8*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))/(12
8*a*b) + (b*x^10*(5*b*d - a*h))/(32*a^3) + (5*b*x^11*(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)