\(\int \frac {c+d x+e x^2+f x^3+g x^4}{(a-b x^4)^3} \, dx\) [173]

Optimal result

Integrand size = 31, antiderivative size = 221 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a-b x^4\right )^3} \, dx=\frac {x \left (b c+a g+b d x+b e x^2+b f x^3\right )}{8 a b \left (a-b x^4\right )^2}+\frac {4 a f+x \left (7 b c-a g+6 b d x+5 b e x^2\right )}{32 a^2 b \left (a-b x^4\right )}+\frac {\left (21 b c-5 \sqrt {a} \sqrt {b} e-3 a g\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{5/4}}+\frac {\left (21 b c+5 \sqrt {a} \sqrt {b} e-3 a g\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{5/4}}+\frac {3 d \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}} \]

[Out]

1/8*x*(b*f*x^3+b*e*x^2+b*d*x+a*g+b*c)/a/b/(-b*x^4+a)^2+1/32*(4*a*f+x*(5*b*e*x^2+6*b*d*x-a*g+7*b*c))/a^2/b/(-b*
x^4+a)+3/16*d*arctanh(x^2*b^(1/2)/a^(1/2))/a^(5/2)/b^(1/2)+1/64*arctan(b^(1/4)*x/a^(1/4))*(21*b*c-3*a*g-5*e*a^
(1/2)*b^(1/2))/a^(11/4)/b^(5/4)+1/64*arctanh(b^(1/4)*x/a^(1/4))*(21*b*c-3*a*g+5*e*a^(1/2)*b^(1/2))/a^(11/4)/b^
(5/4)

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {1872, 1868, 1890, 281, 214, 1181, 211} \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a-b x^4\right )^3} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-5 \sqrt {a} \sqrt {b} e-3 a g+21 b c\right )}{64 a^{11/4} b^{5/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (5 \sqrt {a} \sqrt {b} e-3 a g+21 b c\right )}{64 a^{11/4} b^{5/4}}+\frac {3 d \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}}+\frac {x \left (-a g+7 b c+6 b d x+5 b e x^2\right )+4 a f}{32 a^2 b \left (a-b x^4\right )}+\frac {x \left (a g+b c+b d x+b e x^2+b f x^3\right )}{8 a b \left (a-b x^4\right )^2} \]

[In]

Int[(c + d*x + e*x^2 + f*x^3 + g*x^4)/(a - b*x^4)^3,x]

[Out]

(x*(b*c + a*g + b*d*x + b*e*x^2 + b*f*x^3))/(8*a*b*(a - b*x^4)^2) + (4*a*f + x*(7*b*c - a*g + 6*b*d*x + 5*b*e*
x^2))/(32*a^2*b*(a - b*x^4)) + ((21*b*c - 5*Sqrt[a]*Sqrt[b]*e - 3*a*g)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(64*a^(11/
4)*b^(5/4)) + ((21*b*c + 5*Sqrt[a]*Sqrt[b]*e - 3*a*g)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(64*a^(11/4)*b^(5/4)) + (3
*d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(16*a^(5/2)*Sqrt[b])

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 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 x+b e x^2+b f x^3\right )}{8 a b \left (a-b x^4\right )^2}+\frac {\int \frac {7 b c-a g+6 b d x+5 b e x^2+4 b f x^3}{\left (a-b x^4\right )^2} \, dx}{8 a b} \\ & = \frac {x \left (b c+a g+b d x+b e 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+6 b d x+5 b e x^2\right )}{32 a^2 b \left (a-b x^4\right )}-\frac {\int \frac {-3 (7 b c-a g)-12 b d x-5 b e x^2}{a-b x^4} \, dx}{32 a^2 b} \\ & = \frac {x \left (b c+a g+b d x+b e 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+6 b d x+5 b e x^2\right )}{32 a^2 b \left (a-b x^4\right )}-\frac {\int \left (-\frac {12 b d x}{a-b x^4}+\frac {-3 (7 b c-a g)-5 b e x^2}{a-b x^4}\right ) \, dx}{32 a^2 b} \\ & = \frac {x \left (b c+a g+b d x+b e 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+6 b d x+5 b e x^2\right )}{32 a^2 b \left (a-b x^4\right )}-\frac {\int \frac {-3 (7 b c-a g)-5 b e x^2}{a-b x^4} \, dx}{32 a^2 b}+\frac {(3 d) \int \frac {x}{a-b x^4} \, dx}{8 a^2} \\ & = \frac {x \left (b c+a g+b d x+b e 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+6 b d x+5 b e x^2\right )}{32 a^2 b \left (a-b x^4\right )}+\frac {(3 d) \text {Subst}\left (\int \frac {1}{a-b x^2} \, dx,x,x^2\right )}{16 a^2}-\frac {\left (21 b c-5 \sqrt {a} \sqrt {b} e-3 a g\right ) \int \frac {1}{-\sqrt {a} \sqrt {b}-b x^2} \, dx}{64 a^{5/2} \sqrt {b}}+\frac {\left (21 b c+5 \sqrt {a} \sqrt {b} e-3 a g\right ) \int \frac {1}{\sqrt {a} \sqrt {b}-b x^2} \, dx}{64 a^{5/2} \sqrt {b}} \\ & = \frac {x \left (b c+a g+b d x+b e 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+6 b d x+5 b e x^2\right )}{32 a^2 b \left (a-b x^4\right )}+\frac {\left (21 b c-5 \sqrt {a} \sqrt {b} e-3 a g\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{5/4}}+\frac {\left (21 b c+5 \sqrt {a} \sqrt {b} e-3 a g\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{5/4}}+\frac {3 d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.19 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a-b x^4\right )^3} \, dx=\frac {\frac {4 a^{3/4} \sqrt [4]{b} \left (a^2 (4 f+3 g x)-b^2 x^5 (7 c+x (6 d+5 e x))+a b x \left (11 c+x \left (10 d+9 e x+g x^3\right )\right )\right )}{\left (a-b x^4\right )^2}+2 \left (21 b c-5 \sqrt {a} \sqrt {b} e-3 a g\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )-\left (21 b c+12 \sqrt [4]{a} b^{3/4} d+5 \sqrt {a} \sqrt {b} e-3 a g\right ) \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )+\left (21 b c-12 \sqrt [4]{a} b^{3/4} d+5 \sqrt {a} \sqrt {b} e-3 a g\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )+12 \sqrt [4]{a} b^{3/4} d \log \left (\sqrt {a}+\sqrt {b} x^2\right )}{128 a^{11/4} b^{5/4}} \]

[In]

Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4)/(a - b*x^4)^3,x]

[Out]

((4*a^(3/4)*b^(1/4)*(a^2*(4*f + 3*g*x) - b^2*x^5*(7*c + x*(6*d + 5*e*x)) + a*b*x*(11*c + x*(10*d + 9*e*x + g*x
^3))))/(a - b*x^4)^2 + 2*(21*b*c - 5*Sqrt[a]*Sqrt[b]*e - 3*a*g)*ArcTan[(b^(1/4)*x)/a^(1/4)] - (21*b*c + 12*a^(
1/4)*b^(3/4)*d + 5*Sqrt[a]*Sqrt[b]*e - 3*a*g)*Log[a^(1/4) - b^(1/4)*x] + (21*b*c - 12*a^(1/4)*b^(3/4)*d + 5*Sq
rt[a]*Sqrt[b]*e - 3*a*g)*Log[a^(1/4) + b^(1/4)*x] + 12*a^(1/4)*b^(3/4)*d*Log[Sqrt[a] + Sqrt[b]*x^2])/(128*a^(1
1/4)*b^(5/4))

Maple [C] (verified)

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

Time = 1.51 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.67

method	result	size
risch	\(\frac {-\frac {5 b e \,x^{7}}{32 a^{2}}-\frac {3 b d \,x^{6}}{16 a^{2}}+\frac {\left (a g -7 b c \right ) x^{5}}{32 a^{2}}+\frac {9 e \,x^{3}}{32 a}+\frac {5 d \,x^{2}}{16 a}+\frac {\left (3 a g +11 b c \right ) x}{32 a b}+\frac {f}{8 b}}{\left (-b \,x^{4}+a \right )^{2}}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b -a \right )}{\sum }\frac {\left (5 \textit {\_R}^{2} e +12 \textit {\_R} d -\frac {3 \left (a g -7 b c \right )}{b}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{128 a^{2} b}\)	\(147\)
default	\(\frac {-\frac {5 b e \,x^{7}}{32 a^{2}}-\frac {3 b d \,x^{6}}{16 a^{2}}+\frac {\left (a g -7 b c \right ) x^{5}}{32 a^{2}}+\frac {9 e \,x^{3}}{32 a}+\frac {5 d \,x^{2}}{16 a}+\frac {\left (3 a g +11 b c \right ) x}{32 a b}+\frac {f}{8 b}}{\left (-b \,x^{4}+a \right )^{2}}+\frac {\frac {\left (-3 a g +21 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}+\frac {3 b d \ln \left (\frac {a +x^{2} \sqrt {a b}}{a -x^{2} \sqrt {a b}}\right )}{\sqrt {a b}}-\frac {5 e \left (2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{32 a^{2} b}\)	\(244\)

[In]

int((g*x^4+f*x^3+e*x^2+d*x+c)/(-b*x^4+a)^3,x,method=_RETURNVERBOSE)

[Out]

(-5/32*b*e/a^2*x^7-3/16*b*d/a^2*x^6+1/32*(a*g-7*b*c)/a^2*x^5+9/32/a*e*x^3+5/16*d/a*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*sum((5*_R^2*e+12*_R*d-3/b*(a*g-7*b*c))/_R^3*ln(x-_R),_R=RootOf(_Z^4*b-a))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 42.83 (sec) , antiderivative size = 343626, normalized size of antiderivative = 1554.87 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a-b x^4\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

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

[In]

integrate((g*x**4+f*x**3+e*x**2+d*x+c)/(-b*x**4+a)**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.29 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a-b x^4\right )^3} \, dx=-\frac {5 \, b^{2} e x^{7} + 6 \, b^{2} d x^{6} - 9 \, a b e x^{3} + {\left (7 \, b^{2} c - a b g\right )} x^{5} - 10 \, a b d x^{2} - 4 \, a^{2} f - {\left (11 \, a b c + 3 \, a^{2} g\right )} x}{32 \, {\left (a^{2} b^{3} x^{8} - 2 \, a^{3} b^{2} x^{4} + a^{4} b\right )}} + \frac {\frac {12 \, \sqrt {b} d \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{\sqrt {a}} - \frac {12 \, \sqrt {b} d \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{\sqrt {a}} + \frac {2 \, {\left (21 \, b^{\frac {3}{2}} c - 5 \, \sqrt {a} b e - 3 \, a \sqrt {b} g\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {{\left (21 \, b^{\frac {3}{2}} c + 5 \, \sqrt {a} b e - 3 \, a \sqrt {b} g\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}}{128 \, a^{2} b} \]

[In]

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

[Out]

-1/32*(5*b^2*e*x^7 + 6*b^2*d*x^6 - 9*a*b*e*x^3 + (7*b^2*c - a*b*g)*x^5 - 10*a*b*d*x^2 - 4*a^2*f - (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) + 1/128*(12*sqrt(b)*d*log(sqrt(b)*x^2 + sqrt(a))/sqrt(a) - 1
2*sqrt(b)*d*log(sqrt(b)*x^2 - sqrt(a))/sqrt(a) + 2*(21*b^(3/2)*c - 5*sqrt(a)*b*e - 3*a*sqrt(b)*g)*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)*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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (181) = 362\).

Time = 0.43 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.75 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a-b x^4\right )^3} \, dx=-\frac {\sqrt {2} {\left (21 \, b^{2} c - 3 \, a b g - 12 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d + 5 \, \sqrt {-a b} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2}} - \frac {\sqrt {2} {\left (21 \, b^{2} c - 3 \, a b g + 12 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d - 5 \, \sqrt {-a b} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2}} - \frac {\sqrt {2} {\left (21 \, b^{2} c - 3 \, a b g - 5 \, \sqrt {-a b} b e\right )} \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{256 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2}} + \frac {\sqrt {2} {\left (21 \, b^{2} c - 3 \, a b g - 5 \, \sqrt {-a b} b e\right )} \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{256 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2}} - \frac {5 \, b^{2} e x^{7} + 6 \, b^{2} d x^{6} + 7 \, b^{2} c x^{5} - a b g x^{5} - 9 \, a b e x^{3} - 10 \, a b d x^{2} - 11 \, a b c x - 3 \, a^{2} g x - 4 \, a^{2} f}{32 \, {\left (b x^{4} - a\right )}^{2} a^{2} b} \]

[In]

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

[Out]

-1/128*sqrt(2)*(21*b^2*c - 3*a*b*g - 12*sqrt(2)*(-a*b^3)^(1/4)*b*d + 5*sqrt(-a*b)*b*e)*arctan(1/2*sqrt(2)*(2*x
+ sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/((-a*b^3)^(3/4)*a^2) - 1/128*sqrt(2)*(21*b^2*c - 3*a*b*g + 12*sqrt(2)*(
-a*b^3)^(1/4)*b*d - 5*sqrt(-a*b)*b*e)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/((-a*b^3)^
(3/4)*a^2) - 1/256*sqrt(2)*(21*b^2*c - 3*a*b*g - 5*sqrt(-a*b)*b*e)*log(x^2 + sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/
b))/((-a*b^3)^(3/4)*a^2) + 1/256*sqrt(2)*(21*b^2*c - 3*a*b*g - 5*sqrt(-a*b)*b*e)*log(x^2 - sqrt(2)*x*(-a/b)^(1
/4) + sqrt(-a/b))/((-a*b^3)^(3/4)*a^2) - 1/32*(5*b^2*e*x^7 + 6*b^2*d*x^6 + 7*b^2*c*x^5 - a*b*g*x^5 - 9*a*b*e*x
^3 - 10*a*b*d*x^2 - 11*a*b*c*x - 3*a^2*g*x - 4*a^2*f)/((b*x^4 - a)^2*a^2*b)

Mupad [B] (verification not implemented)

Time = 9.66 (sec) , antiderivative size = 1002, normalized size of antiderivative = 4.53 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a-b x^4\right )^3} \, dx=\frac {\frac {f}{8\,b}+\frac {5\,d\,x^2}{16\,a}+\frac {9\,e\,x^3}{32\,a}-\frac {x^5\,\left (7\,b\,c-a\,g\right )}{32\,a^2}+\frac {x\,\left (11\,b\,c+3\,a\,g\right )}{32\,a\,b}-\frac {3\,b\,d\,x^6}{16\,a^2}-\frac {5\,b\,e\,x^7}{32\,a^2}}{a^2-2\,a\,b\,x^4+b^2\,x^8}+\left (\sum _{k=1}^4\ln \left (-\mathrm {root}\left (268435456\,a^{11}\,b^5\,z^4+983040\,a^7\,b^3\,e\,g\,z^2-6881280\,a^6\,b^4\,c\,e\,z^2-4718592\,a^6\,b^4\,d^2\,z^2-774144\,a^4\,b^3\,c\,d\,g\,z+55296\,a^5\,b^2\,d\,g^2\,z+153600\,a^4\,b^3\,d\,e^2\,z+2709504\,a^3\,b^4\,c^2\,d\,z+8640\,a^2\,b^2\,d^2\,e\,g-6300\,a^2\,b^2\,c\,e^2\,g-60480\,a\,b^3\,c\,d^2\,e+111132\,a\,b^3\,c^3\,g+2268\,a^3\,b\,c\,g^3-23814\,a^2\,b^2\,c^2\,g^2+450\,a^3\,b\,e^2\,g^2+22050\,a\,b^3\,c^2\,e^2-625\,a^2\,b^2\,e^4+20736\,a\,b^3\,d^4-81\,a^4\,g^4-194481\,b^4\,c^4,z,k\right )\,\left (\mathrm {root}\left (268435456\,a^{11}\,b^5\,z^4+983040\,a^7\,b^3\,e\,g\,z^2-6881280\,a^6\,b^4\,c\,e\,z^2-4718592\,a^6\,b^4\,d^2\,z^2-774144\,a^4\,b^3\,c\,d\,g\,z+55296\,a^5\,b^2\,d\,g^2\,z+153600\,a^4\,b^3\,d\,e^2\,z+2709504\,a^3\,b^4\,c^2\,d\,z+8640\,a^2\,b^2\,d^2\,e\,g-6300\,a^2\,b^2\,c\,e^2\,g-60480\,a\,b^3\,c\,d^2\,e+111132\,a\,b^3\,c^3\,g+2268\,a^3\,b\,c\,g^3-23814\,a^2\,b^2\,c^2\,g^2+450\,a^3\,b\,e^2\,g^2+22050\,a\,b^3\,c^2\,e^2-625\,a^2\,b^2\,e^4+20736\,a\,b^3\,d^4-81\,a^4\,g^4-194481\,b^4\,c^4,z,k\right )\,\left (\frac {344064\,a^5\,b^3\,c-49152\,a^6\,b^2\,g}{32768\,a^6}-\frac {6\,b^3\,d\,x}{a}\right )+\frac {x\,\left (144\,a^4\,b\,g^2-2016\,a^3\,b^2\,c\,g+400\,a^3\,b^2\,e^2+7056\,a^2\,b^3\,c^2\right )}{4096\,a^6}-\frac {15\,b^2\,d\,e}{32\,a^3}\right )-\frac {-45\,a^2\,e\,g^2+630\,a\,b\,c\,e\,g-432\,a\,b\,d^2\,g+125\,a\,b\,e^3-2205\,b^2\,c^2\,e+3024\,b^2\,c\,d^2}{32768\,a^6}-\frac {x\,\left (216\,b^2\,d^3-315\,c\,e\,b^2\,d+45\,a\,e\,g\,b\,d\right )}{4096\,a^6}\right )\,\mathrm {root}\left (268435456\,a^{11}\,b^5\,z^4+983040\,a^7\,b^3\,e\,g\,z^2-6881280\,a^6\,b^4\,c\,e\,z^2-4718592\,a^6\,b^4\,d^2\,z^2-774144\,a^4\,b^3\,c\,d\,g\,z+55296\,a^5\,b^2\,d\,g^2\,z+153600\,a^4\,b^3\,d\,e^2\,z+2709504\,a^3\,b^4\,c^2\,d\,z+8640\,a^2\,b^2\,d^2\,e\,g-6300\,a^2\,b^2\,c\,e^2\,g-60480\,a\,b^3\,c\,d^2\,e+111132\,a\,b^3\,c^3\,g+2268\,a^3\,b\,c\,g^3-23814\,a^2\,b^2\,c^2\,g^2+450\,a^3\,b\,e^2\,g^2+22050\,a\,b^3\,c^2\,e^2-625\,a^2\,b^2\,e^4+20736\,a\,b^3\,d^4-81\,a^4\,g^4-194481\,b^4\,c^4,z,k\right )\right ) \]

[In]

int((c + d*x + e*x^2 + f*x^3 + g*x^4)/(a - b*x^4)^3,x)

[Out]

(f/(8*b) + (5*d*x^2)/(16*a) + (9*e*x^3)/(32*a) - (x^5*(7*b*c - a*g))/(32*a^2) + (x*(11*b*c + 3*a*g))/(32*a*b)
- (3*b*d*x^6)/(16*a^2) - (5*b*e*x^7)/(32*a^2))/(a^2 + b^2*x^8 - 2*a*b*x^4) + symsum(log(- root(268435456*a^11*
b^5*z^4 + 983040*a^7*b^3*e*g*z^2 - 6881280*a^6*b^4*c*e*z^2 - 4718592*a^6*b^4*d^2*z^2 - 774144*a^4*b^3*c*d*g*z
+ 55296*a^5*b^2*d*g^2*z + 153600*a^4*b^3*d*e^2*z + 2709504*a^3*b^4*c^2*d*z + 8640*a^2*b^2*d^2*e*g - 6300*a^2*b
^2*c*e^2*g - 60480*a*b^3*c*d^2*e + 111132*a*b^3*c^3*g + 2268*a^3*b*c*g^3 - 23814*a^2*b^2*c^2*g^2 + 450*a^3*b*e
^2*g^2 + 22050*a*b^3*c^2*e^2 - 625*a^2*b^2*e^4 + 20736*a*b^3*d^4 - 81*a^4*g^4 - 194481*b^4*c^4, z, k)*(root(26
8435456*a^11*b^5*z^4 + 983040*a^7*b^3*e*g*z^2 - 6881280*a^6*b^4*c*e*z^2 - 4718592*a^6*b^4*d^2*z^2 - 774144*a^4
*b^3*c*d*g*z + 55296*a^5*b^2*d*g^2*z + 153600*a^4*b^3*d*e^2*z + 2709504*a^3*b^4*c^2*d*z + 8640*a^2*b^2*d^2*e*g
- 6300*a^2*b^2*c*e^2*g - 60480*a*b^3*c*d^2*e + 111132*a*b^3*c^3*g + 2268*a^3*b*c*g^3 - 23814*a^2*b^2*c^2*g^2
+ 450*a^3*b*e^2*g^2 + 22050*a*b^3*c^2*e^2 - 625*a^2*b^2*e^4 + 20736*a*b^3*d^4 - 81*a^4*g^4 - 194481*b^4*c^4, z
, k)*((344064*a^5*b^3*c - 49152*a^6*b^2*g)/(32768*a^6) - (6*b^3*d*x)/a) + (x*(144*a^4*b*g^2 + 7056*a^2*b^3*c^2
+ 400*a^3*b^2*e^2 - 2016*a^3*b^2*c*g))/(4096*a^6) - (15*b^2*d*e)/(32*a^3)) - (3024*b^2*c*d^2 - 2205*b^2*c^2*e
- 45*a^2*e*g^2 + 125*a*b*e^3 - 432*a*b*d^2*g + 630*a*b*c*e*g)/(32768*a^6) - (x*(216*b^2*d^3 - 315*b^2*c*d*e +
45*a*b*d*e*g))/(4096*a^6))*root(268435456*a^11*b^5*z^4 + 983040*a^7*b^3*e*g*z^2 - 6881280*a^6*b^4*c*e*z^2 - 4
718592*a^6*b^4*d^2*z^2 - 774144*a^4*b^3*c*d*g*z + 55296*a^5*b^2*d*g^2*z + 153600*a^4*b^3*d*e^2*z + 2709504*a^3
*b^4*c^2*d*z + 8640*a^2*b^2*d^2*e*g - 6300*a^2*b^2*c*e^2*g - 60480*a*b^3*c*d^2*e + 111132*a*b^3*c^3*g + 2268*a
^3*b*c*g^3 - 23814*a^2*b^2*c^2*g^2 + 450*a^3*b*e^2*g^2 + 22050*a*b^3*c^2*e^2 - 625*a^2*b^2*e^4 + 20736*a*b^3*d
^4 - 81*a^4*g^4 - 194481*b^4*c^4, z, k), k, 1, 4)