\(\int \frac {c+d x+e x^2+f x^3+g x^4}{(a+b x^4)^3} \, dx\) [177]

Optimal result

Integrand size = 30, antiderivative size = 394 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^4\right )^3} \, dx=\frac {x \left (b c-a g+b d x+b e x^2+b f x^3\right )}{8 a b \left (a+b x^4\right )^2}-\frac {4 a f-x \left (7 b c+a g+6 b d x+5 b e x^2\right )}{32 a^2 b \left (a+b x^4\right )}+\frac {3 d \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}}-\frac {\left (21 b c+5 \sqrt {a} \sqrt {b} e+3 a g\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} b^{5/4}}+\frac {\left (21 b c+5 \sqrt {a} \sqrt {b} e+3 a g\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} b^{5/4}}-\frac {\left (21 b c-5 \sqrt {a} \sqrt {b} e+3 a g\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{11/4} b^{5/4}}+\frac {\left (21 b c-5 \sqrt {a} \sqrt {b} e+3 a g\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{11/4} b^{5/4}} \]

[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*arctan(x^2*b^(1/2)/a^(1/2))/a^(5/2)/b^(1/2)-1/256*ln(-a^(1/4)*b^(1/4)*x*2^(1/2)+a^(1/2)+x^2*b^(1/
2))*(21*b*c+3*a*g-5*e*a^(1/2)*b^(1/2))/a^(11/4)/b^(5/4)*2^(1/2)+1/256*ln(a^(1/4)*b^(1/4)*x*2^(1/2)+a^(1/2)+x^2
*b^(1/2))*(21*b*c+3*a*g-5*e*a^(1/2)*b^(1/2))/a^(11/4)/b^(5/4)*2^(1/2)+1/128*arctan(-1+b^(1/4)*x*2^(1/2)/a^(1/4
))*(21*b*c+3*a*g+5*e*a^(1/2)*b^(1/2))/a^(11/4)/b^(5/4)*2^(1/2)+1/128*arctan(1+b^(1/4)*x*2^(1/2)/a^(1/4))*(21*b
*c+3*a*g+5*e*a^(1/2)*b^(1/2))/a^(11/4)/b^(5/4)*2^(1/2)

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {1872, 1868, 1890, 281, 211, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^4\right )^3} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (5 \sqrt {a} \sqrt {b} e+3 a g+21 b c\right )}{64 \sqrt {2} a^{11/4} b^{5/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (5 \sqrt {a} \sqrt {b} e+3 a g+21 b c\right )}{64 \sqrt {2} a^{11/4} b^{5/4}}+\frac {3 d \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (-5 \sqrt {a} \sqrt {b} e+3 a g+21 b c\right )}{128 \sqrt {2} a^{11/4} b^{5/4}}+\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (-5 \sqrt {a} \sqrt {b} e+3 a g+21 b c\right )}{128 \sqrt {2} a^{11/4} b^{5/4}}-\frac {4 a f-x \left (a g+7 b c+6 b d x+5 b e x^2\right )}{32 a^2 b \left (a+b x^4\right )}+\frac {x \left (-a g+b c+b d x+b e x^2+b f x^3\right )}{8 a b \left (a+b x^4\right )^2} \]

[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)) + (3*d*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(16*a^(5/2)*Sqrt[b]) - ((21*b*c + 5*Sqrt[a]
*Sqrt[b]*e + 3*a*g)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(11/4)*b^(5/4)) + ((21*b*c + 5*Sqrt
[a]*Sqrt[b]*e + 3*a*g)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(11/4)*b^(5/4)) - ((21*b*c - 5*S
qrt[a]*Sqrt[b]*e + 3*a*g)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(128*Sqrt[2]*a^(11/4)*b^(5/4
)) + ((21*b*c - 5*Sqrt[a]*Sqrt[b]*e + 3*a*g)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(128*Sqrt
[2]*a^(11/4)*b^(5/4))

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

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 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 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

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

Rule 1179

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

Rule 1182

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(-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 {\sqrt {a} \sqrt {b}-b x^2}{a+b x^4} \, dx}{64 a^{5/2} b^{3/2}}+\frac {\left (21 b c+5 \sqrt {a} \sqrt {b} e+3 a g\right ) \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx}{64 a^{5/2} b^{3/2}} \\ & = \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 \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}}-\frac {\left (21 b c-5 \sqrt {a} \sqrt {b} e+3 a g\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{128 \sqrt {2} a^{11/4} b^{5/4}}-\frac {\left (21 b c-5 \sqrt {a} \sqrt {b} e+3 a g\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{128 \sqrt {2} a^{11/4} b^{5/4}}+\frac {\left (21 b c+5 \sqrt {a} \sqrt {b} e+3 a g\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{128 a^{5/2} b^{3/2}}+\frac {\left (21 b c+5 \sqrt {a} \sqrt {b} e+3 a g\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{128 a^{5/2} b^{3/2}} \\ & = \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 \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}}-\frac {\left (21 b c-5 \sqrt {a} \sqrt {b} e+3 a g\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^{5/4}}+\frac {\left (21 b c-5 \sqrt {a} \sqrt {b} e+3 a g\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^{5/4}}+\frac {\left (21 b c+5 \sqrt {a} \sqrt {b} e+3 a g\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} b^{5/4}}-\frac {\left (21 b c+5 \sqrt {a} \sqrt {b} e+3 a g\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} b^{5/4}} \\ & = \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 \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}}-\frac {\left (21 b c+5 \sqrt {a} \sqrt {b} e+3 a g\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} b^{5/4}}+\frac {\left (21 b c+5 \sqrt {a} \sqrt {b} e+3 a g\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} b^{5/4}}-\frac {\left (21 b c-5 \sqrt {a} \sqrt {b} e+3 a g\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^{5/4}}+\frac {\left (21 b c-5 \sqrt {a} \sqrt {b} e+3 a g\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^{5/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 366, normalized size of antiderivative = 0.93 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^4\right )^3} \, dx=\frac {\frac {8 a^{3/4} \sqrt [4]{b} x (7 b c+a g+b x (6 d+5 e x))}{a+b x^4}-\frac {32 a^{7/4} \sqrt [4]{b} (a (f+g x)-b x (c+x (d+e x)))}{\left (a+b x^4\right )^2}-2 \left (21 \sqrt {2} b c+24 \sqrt [4]{a} b^{3/4} d+5 \sqrt {2} \sqrt {a} \sqrt {b} e+3 \sqrt {2} a g\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+2 \left (21 \sqrt {2} b c-24 \sqrt [4]{a} b^{3/4} d+5 \sqrt {2} \sqrt {a} \sqrt {b} e+3 \sqrt {2} a g\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+\sqrt {2} \left (-21 b c+5 \sqrt {a} \sqrt {b} e-3 a g\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )+\sqrt {2} \left (21 b c-5 \sqrt {a} \sqrt {b} e+3 a g\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{256 a^{11/4} b^{5/4}} \]

[In]

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

[Out]

((8*a^(3/4)*b^(1/4)*x*(7*b*c + a*g + b*x*(6*d + 5*e*x)))/(a + b*x^4) - (32*a^(7/4)*b^(1/4)*(a*(f + g*x) - b*x*
(c + x*(d + e*x))))/(a + b*x^4)^2 - 2*(21*Sqrt[2]*b*c + 24*a^(1/4)*b^(3/4)*d + 5*Sqrt[2]*Sqrt[a]*Sqrt[b]*e + 3
*Sqrt[2]*a*g)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*(21*Sqrt[2]*b*c - 24*a^(1/4)*b^(3/4)*d + 5*Sqrt[2]*S
qrt[a]*Sqrt[b]*e + 3*Sqrt[2]*a*g)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + Sqrt[2]*(-21*b*c + 5*Sqrt[a]*Sqrt[
b]*e - 3*a*g)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + Sqrt[2]*(21*b*c - 5*Sqrt[a]*Sqrt[b]*e +
3*a*g)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(256*a^(11/4)*b^(5/4))

Maple [C] (verified)

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

Time = 1.53 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.37

method	result	size
risch	\(\frac {\frac {5 b e \,x^{7}}{32 a^{2}}+\frac {3 b d \,x^{6}}{16 a^{2}}+\frac {\left (a g +7 b c \right ) x^{5}}{32 a^{2}}+\frac {9 e \,x^{3}}{32 a}+\frac {5 d \,x^{2}}{16 a}-\frac {\left (3 a g -11 b c \right ) x}{32 a b}-\frac {f}{8 b}}{\left (b \,x^{4}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (5 \textit {\_R}^{2} e +12 \textit {\_R} d +\frac {3 a g +21 b c}{b}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{128 a^{2} b}\)	\(144\)
default	\(\frac {\frac {5 b e \,x^{7}}{32 a^{2}}+\frac {3 b d \,x^{6}}{16 a^{2}}+\frac {\left (a g +7 b c \right ) x^{5}}{32 a^{2}}+\frac {9 e \,x^{3}}{32 a}+\frac {5 d \,x^{2}}{16 a}-\frac {\left (3 a g -11 b c \right ) x}{32 a b}-\frac {f}{8 b}}{\left (b \,x^{4}+a \right )^{2}}+\frac {\frac {\left (3 a g +21 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {6 b d \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right )}{\sqrt {a b}}+\frac {5 e \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{32 a^{2} b}\)	\(330\)

[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 = 58.34 (sec) , antiderivative size = 358509, normalized size of antiderivative = 909.92 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^4\right )^3} \, dx=\text {Too large to display} \]

[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 = 0.31 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.05 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^4\right )^3} \, dx=\frac {5 \, b^{2} e x^{7} + 6 \, b^{2} d x^{6} + 9 \, a b e x^{3} + {\left (7 \, b^{2} c + a b g\right )} x^{5} + 10 \, a b d x^{2} - 4 \, a^{2} f + {\left (11 \, a b c - 3 \, a^{2} g\right )} x}{32 \, {\left (a^{2} b^{3} x^{8} + 2 \, a^{3} b^{2} x^{4} + a^{4} b\right )}} + \frac {\frac {\sqrt {2} {\left (21 \, b^{\frac {3}{2}} c - 5 \, \sqrt {a} b e + 3 \, a \sqrt {b} g\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (21 \, b^{\frac {3}{2}} c - 5 \, \sqrt {a} b e + 3 \, a \sqrt {b} g\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} + \frac {2 \, {\left (21 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {7}{4}} c + 5 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} e + 3 \, \sqrt {2} a^{\frac {5}{4}} b^{\frac {3}{4}} g - 24 \, \sqrt {a} b^{\frac {3}{2}} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}} + \frac {2 \, {\left (21 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {7}{4}} c + 5 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} e + 3 \, \sqrt {2} a^{\frac {5}{4}} b^{\frac {3}{4}} g + 24 \, \sqrt {a} b^{\frac {3}{2}} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}}}{256 \, a^{2} b} \]

[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) + 1/256*(sqrt(2)*(21*b^(3/2)*c - 5*sqrt(a)*b*e + 3*a*sqrt(b)*
g)*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)*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 - 24*sqrt(a)*b^(3/2)*d)*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 + 24*sqrt(
a)*b^(3/2)*d)*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)))/(a^2*b)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.04 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^4\right )^3} \, dx=\frac {\sqrt {2} {\left (12 \, \sqrt {2} \sqrt {a b} b^{2} d + 21 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b g + 5 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, a^{3} b^{3}} + \frac {\sqrt {2} {\left (12 \, \sqrt {2} \sqrt {a b} b^{2} d + 21 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b g + 5 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, a^{3} b^{3}} + \frac {\sqrt {2} {\left (21 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b g - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{256 \, a^{3} b^{3}} - \frac {\sqrt {2} {\left (21 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b g - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{256 \, a^{3} b^{3}} + \frac {5 \, b^{2} e x^{7} + 6 \, b^{2} d x^{6} + 7 \, b^{2} c x^{5} + a b g x^{5} + 9 \, a b e x^{3} + 10 \, a b d x^{2} + 11 \, a b c x - 3 \, a^{2} g x - 4 \, a^{2} f}{32 \, {\left (b x^{4} + a\right )}^{2} a^{2} b} \]

[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)*(12*sqrt(2)*sqrt(a*b)*b^2*d + 21*(a*b^3)^(1/4)*b^2*c + 3*(a*b^3)^(1/4)*a*b*g + 5*(a*b^3)^(3/4)*e
)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^3*b^3) + 1/128*sqrt(2)*(12*sqrt(2)*sqrt(a*b)*
b^2*d + 21*(a*b^3)^(1/4)*b^2*c + 3*(a*b^3)^(1/4)*a*b*g + 5*(a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*
(a/b)^(1/4))/(a/b)^(1/4))/(a^3*b^3) + 1/256*sqrt(2)*(21*(a*b^3)^(1/4)*b^2*c + 3*(a*b^3)^(1/4)*a*b*g - 5*(a*b^3
)^(3/4)*e)*log(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^3*b^3) - 1/256*sqrt(2)*(21*(a*b^3)^(1/4)*b^2*c + 3*
(a*b^3)^(1/4)*a*b*g - 5*(a*b^3)^(3/4)*e)*log(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^3*b^3) + 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 = 9.75 (sec) , antiderivative size = 1001, normalized size of antiderivative = 2.54 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^4\right )^3} \, dx=\frac {\frac {5\,d\,x^2}{16\,a}-\frac {f}{8\,b}+\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 {49152\,g\,a^6\,b^2+344064\,c\,a^5\,b^3}{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]

((5*d*x^2)/(16*a) - f/(8*b) + (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)) - (2205*b^2*c^2*e - 3024*b^2*c*d^2
+ 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*(315*b^2*c*d*e - 216*b^2*d^3 +
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)