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

Optimal. Leaf size=382 \[ -\frac {\left (77 \sqrt {b} c-15 \sqrt {a} e\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{15/4} b^{3/4}}+\frac {\left (77 \sqrt {b} c-15 \sqrt {a} e\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{15/4} b^{3/4}}-\frac {\left (15 \sqrt {a} e+77 \sqrt {b} c\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{15/4} b^{3/4}}+\frac {\left (15 \sqrt {a} e+77 \sqrt {b} c\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{256 \sqrt {2} a^{15/4} b^{3/4}}+\frac {5 d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} \sqrt {b}}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a+b x^4\right )}+\frac {x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a+b x^4\right )^2}-\frac {a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3} \]

[Out]

1/96*x*(9*e*x^2+10*d*x+11*c)/a^2/(b*x^4+a)^2+1/384*x*(45*e*x^2+60*d*x+77*c)/a^3/(b*x^4+a)+1/12*(-a*f+b*x*(e*x^
2+d*x+c))/a/b/(b*x^4+a)^3+5/32*d*arctan(x^2*b^(1/2)/a^(1/2))/a^(7/2)/b^(1/2)-1/1024*ln(-a^(1/4)*b^(1/4)*x*2^(1
/2)+a^(1/2)+x^2*b^(1/2))*(-15*e*a^(1/2)+77*c*b^(1/2))/a^(15/4)/b^(3/4)*2^(1/2)+1/1024*ln(a^(1/4)*b^(1/4)*x*2^(
1/2)+a^(1/2)+x^2*b^(1/2))*(-15*e*a^(1/2)+77*c*b^(1/2))/a^(15/4)/b^(3/4)*2^(1/2)+1/512*arctan(-1+b^(1/4)*x*2^(1
/2)/a^(1/4))*(15*e*a^(1/2)+77*c*b^(1/2))/a^(15/4)/b^(3/4)*2^(1/2)+1/512*arctan(1+b^(1/4)*x*2^(1/2)/a^(1/4))*(1
5*e*a^(1/2)+77*c*b^(1/2))/a^(15/4)/b^(3/4)*2^(1/2)

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Rubi [A]  time = 0.41, antiderivative size = 382, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {1854, 1855, 1876, 275, 205, 1168, 1162, 617, 204, 1165, 628} \[ -\frac {\left (77 \sqrt {b} c-15 \sqrt {a} e\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{15/4} b^{3/4}}+\frac {\left (77 \sqrt {b} c-15 \sqrt {a} e\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{15/4} b^{3/4}}-\frac {\left (15 \sqrt {a} e+77 \sqrt {b} c\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{15/4} b^{3/4}}+\frac {\left (15 \sqrt {a} e+77 \sqrt {b} c\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{256 \sqrt {2} a^{15/4} b^{3/4}}+\frac {x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a+b x^4\right )^2}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a+b x^4\right )}+\frac {5 d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} \sqrt {b}}-\frac {a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(11*c + 10*d*x + 9*e*x^2))/(96*a^2*(a + b*x^4)^2) + (x*(77*c + 60*d*x + 45*e*x^2))/(384*a^3*(a + b*x^4)) -
(a*f - b*x*(c + d*x + e*x^2))/(12*a*b*(a + b*x^4)^3) + (5*d*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(32*a^(7/2)*Sqrt[b]
) - ((77*Sqrt[b]*c + 15*Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(256*Sqrt[2]*a^(15/4)*b^(3/4)) + (
(77*Sqrt[b]*c + 15*Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(256*Sqrt[2]*a^(15/4)*b^(3/4)) - ((77*S
qrt[b]*c - 15*Sqrt[a]*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(512*Sqrt[2]*a^(15/4)*b^(3/4)
) + ((77*Sqrt[b]*c - 15*Sqrt[a]*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(512*Sqrt[2]*a^(15/
4)*b^(3/4))

Rule 204

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

Rule 205

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

Rule 275

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 617

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 628

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 1162

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 1165

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 1168

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 1854

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 1855

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(x*Pq*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Di
st[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 1876

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*} \int \frac {c+d x+e x^2+f x^3}{\left (a+b x^4\right )^4} \, dx &=-\frac {a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3}-\frac {\int \frac {-11 c-10 d x-9 e x^2}{\left (a+b x^4\right )^3} \, dx}{12 a}\\ &=\frac {x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a+b x^4\right )^2}-\frac {a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3}+\frac {\int \frac {77 c+60 d x+45 e x^2}{\left (a+b x^4\right )^2} \, dx}{96 a^2}\\ &=\frac {x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a+b x^4\right )^2}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a+b x^4\right )}-\frac {a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3}-\frac {\int \frac {-231 c-120 d x-45 e x^2}{a+b x^4} \, dx}{384 a^3}\\ &=\frac {x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a+b x^4\right )^2}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a+b x^4\right )}-\frac {a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3}-\frac {\int \left (-\frac {120 d x}{a+b x^4}+\frac {-231 c-45 e x^2}{a+b x^4}\right ) \, dx}{384 a^3}\\ &=\frac {x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a+b x^4\right )^2}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a+b x^4\right )}-\frac {a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3}-\frac {\int \frac {-231 c-45 e x^2}{a+b x^4} \, dx}{384 a^3}+\frac {(5 d) \int \frac {x}{a+b x^4} \, dx}{16 a^3}\\ &=\frac {x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a+b x^4\right )^2}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a+b x^4\right )}-\frac {a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3}+\frac {(5 d) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^2\right )}{32 a^3}+\frac {\left (\frac {77 \sqrt {b} c}{\sqrt {a}}-15 e\right ) \int \frac {\sqrt {a} \sqrt {b}-b x^2}{a+b x^4} \, dx}{256 a^3 b}+\frac {\left (\frac {77 \sqrt {b} c}{\sqrt {a}}+15 e\right ) \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx}{256 a^3 b}\\ &=\frac {x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a+b x^4\right )^2}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a+b x^4\right )}-\frac {a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3}+\frac {5 d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} \sqrt {b}}+\frac {\left (\frac {77 \sqrt {b} c}{\sqrt {a}}+15 e\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{512 a^3 b}+\frac {\left (\frac {77 \sqrt {b} c}{\sqrt {a}}+15 e\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{512 a^3 b}-\frac {\left (77 \sqrt {b} c-15 \sqrt {a} e\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}{512 \sqrt {2} a^{15/4} b^{3/4}}-\frac {\left (77 \sqrt {b} c-15 \sqrt {a} e\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}{512 \sqrt {2} a^{15/4} b^{3/4}}\\ &=\frac {x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a+b x^4\right )^2}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a+b x^4\right )}-\frac {a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3}+\frac {5 d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} \sqrt {b}}-\frac {\left (77 \sqrt {b} c-15 \sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{15/4} b^{3/4}}+\frac {\left (77 \sqrt {b} c-15 \sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{15/4} b^{3/4}}+\frac {\left (77 \sqrt {b} c+15 \sqrt {a} e\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{15/4} b^{3/4}}-\frac {\left (77 \sqrt {b} c+15 \sqrt {a} e\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{15/4} b^{3/4}}\\ &=\frac {x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a+b x^4\right )^2}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a+b x^4\right )}-\frac {a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3}+\frac {5 d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} \sqrt {b}}-\frac {\left (77 \sqrt {b} c+15 \sqrt {a} e\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{15/4} b^{3/4}}+\frac {\left (77 \sqrt {b} c+15 \sqrt {a} e\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{15/4} b^{3/4}}-\frac {\left (77 \sqrt {b} c-15 \sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{15/4} b^{3/4}}+\frac {\left (77 \sqrt {b} c-15 \sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{15/4} b^{3/4}}\\ \end {align*}

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Mathematica [A]  time = 0.44, size = 379, normalized size = 0.99 \[ \frac {\frac {3 \sqrt {2} \left (15 a^{3/4} e-77 \sqrt [4]{a} \sqrt {b} c\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{b^{3/4}}+\frac {3 \sqrt {2} \left (77 \sqrt [4]{a} \sqrt {b} c-15 a^{3/4} e\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{b^{3/4}}-\frac {256 a^3 (a f-b x (c+x (d+e x)))}{b \left (a+b x^4\right )^3}+\frac {32 a^2 x (11 c+x (10 d+9 e x))}{\left (a+b x^4\right )^2}-\frac {6 \sqrt [4]{a} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (80 \sqrt [4]{a} \sqrt [4]{b} d+15 \sqrt {2} \sqrt {a} e+77 \sqrt {2} \sqrt {b} c\right )}{b^{3/4}}+\frac {6 \sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-80 \sqrt [4]{a} \sqrt [4]{b} d+15 \sqrt {2} \sqrt {a} e+77 \sqrt {2} \sqrt {b} c\right )}{b^{3/4}}+\frac {8 a x (77 c+15 x (4 d+3 e x))}{a+b x^4}}{3072 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((8*a*x*(77*c + 15*x*(4*d + 3*e*x)))/(a + b*x^4) + (32*a^2*x*(11*c + x*(10*d + 9*e*x)))/(a + b*x^4)^2 - (256*a
^3*(a*f - b*x*(c + x*(d + e*x))))/(b*(a + b*x^4)^3) - (6*a^(1/4)*(77*Sqrt[2]*Sqrt[b]*c + 80*a^(1/4)*b^(1/4)*d
+ 15*Sqrt[2]*Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/b^(3/4) + (6*a^(1/4)*(77*Sqrt[2]*Sqrt[b]*c -
80*a^(1/4)*b^(1/4)*d + 15*Sqrt[2]*Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/b^(3/4) + (3*Sqrt[2]*(-7
7*a^(1/4)*Sqrt[b]*c + 15*a^(3/4)*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(3/4) + (3*Sqrt[
2]*(77*a^(1/4)*Sqrt[b]*c - 15*a^(3/4)*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(3/4))/(307
2*a^4)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [A]  time = 0.33, size = 391, normalized size = 1.02 \[ \frac {\sqrt {2} {\left (40 \, \sqrt {2} \sqrt {a b} b^{2} d + 77 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 15 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 \, a^{4} b^{3}} + \frac {\sqrt {2} {\left (40 \, \sqrt {2} \sqrt {a b} b^{2} d + 77 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 15 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 \, a^{4} b^{3}} + \frac {\sqrt {2} {\left (77 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - 15 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{1024 \, a^{4} b^{3}} - \frac {\sqrt {2} {\left (77 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - 15 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{1024 \, a^{4} b^{3}} + \frac {45 \, b^{3} x^{11} e + 60 \, b^{3} d x^{10} + 77 \, b^{3} c x^{9} + 126 \, a b^{2} x^{7} e + 160 \, a b^{2} d x^{6} + 198 \, a b^{2} c x^{5} + 113 \, a^{2} b x^{3} e + 132 \, a^{2} b d x^{2} + 153 \, a^{2} b c x - 32 \, a^{3} f}{384 \, {\left (b x^{4} + a\right )}^{3} a^{3} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/512*sqrt(2)*(40*sqrt(2)*sqrt(a*b)*b^2*d + 77*(a*b^3)^(1/4)*b^2*c + 15*(a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2
*x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^4*b^3) + 1/512*sqrt(2)*(40*sqrt(2)*sqrt(a*b)*b^2*d + 77*(a*b^3)^(1/4
)*b^2*c + 15*(a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^4*b^3) + 1/1024*s
qrt(2)*(77*(a*b^3)^(1/4)*b^2*c - 15*(a*b^3)^(3/4)*e)*log(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^4*b^3) -
1/1024*sqrt(2)*(77*(a*b^3)^(1/4)*b^2*c - 15*(a*b^3)^(3/4)*e)*log(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^4
*b^3) + 1/384*(45*b^3*x^11*e + 60*b^3*d*x^10 + 77*b^3*c*x^9 + 126*a*b^2*x^7*e + 160*a*b^2*d*x^6 + 198*a*b^2*c*
x^5 + 113*a^2*b*x^3*e + 132*a^2*b*d*x^2 + 153*a^2*b*c*x - 32*a^3*f)/((b*x^4 + a)^3*a^3*b)

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maple [A]  time = 0.06, size = 400, normalized size = 1.05 \[ \frac {5 d \arctan \left (\sqrt {\frac {b}{a}}\, x^{2}\right )}{32 \sqrt {a b}\, a^{3}}+\frac {15 \sqrt {2}\, e \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{512 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{3} b}+\frac {15 \sqrt {2}\, e \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{512 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{3} b}+\frac {15 \sqrt {2}\, e \ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}\right )}{1024 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{3} b}+\frac {77 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{512 a^{4}}+\frac {77 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{512 a^{4}}+\frac {77 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c \ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}\right )}{1024 a^{4}}+\frac {\frac {15 b^{2} e \,x^{11}}{128 a^{3}}+\frac {5 b^{2} d \,x^{10}}{32 a^{3}}+\frac {77 b^{2} c \,x^{9}}{384 a^{3}}+\frac {21 b e \,x^{7}}{64 a^{2}}+\frac {5 b d \,x^{6}}{12 a^{2}}+\frac {33 b c \,x^{5}}{64 a^{2}}+\frac {113 e \,x^{3}}{384 a}+\frac {11 d \,x^{2}}{32 a}+\frac {51 c x}{128 a}-\frac {f}{12 b}}{\left (b \,x^{4}+a \right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(15/128/a^3*b^2*e*x^11+5/32*d/a^3*b^2*x^10+77/384*c/a^3*b^2*x^9+21/64/a^2*b*e*x^7+5/12/a^2*d*b*x^6+33/64/a^2*c
*b*x^5+113/384/a*e*x^3+11/32*d/a*x^2+51/128*c/a*x-1/12/b*f)/(b*x^4+a)^3+77/1024/a^4*c*(a/b)^(1/4)*2^(1/2)*ln((
x^2+(a/b)^(1/4)*2^(1/2)*x+(a/b)^(1/2))/(x^2-(a/b)^(1/4)*2^(1/2)*x+(a/b)^(1/2)))+77/512/a^4*c*(a/b)^(1/4)*2^(1/
2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+77/512/a^4*c*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)+5/32/a^3*d
/(a*b)^(1/2)*arctan((1/a*b)^(1/2)*x^2)+15/1024/a^3*e/b/(a/b)^(1/4)*2^(1/2)*ln((x^2-(a/b)^(1/4)*2^(1/2)*x+(a/b)
^(1/2))/(x^2+(a/b)^(1/4)*2^(1/2)*x+(a/b)^(1/2)))+15/512/a^3*e/b/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)
*x+1)+15/512/a^3*e/b/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)

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maxima [A]  time = 3.09, size = 402, normalized size = 1.05 \[ \frac {45 \, b^{3} e x^{11} + 60 \, b^{3} d x^{10} + 77 \, b^{3} c x^{9} + 126 \, a b^{2} e x^{7} + 160 \, a b^{2} d x^{6} + 198 \, a b^{2} c x^{5} + 113 \, a^{2} b e x^{3} + 132 \, a^{2} b d x^{2} + 153 \, a^{2} b c x - 32 \, a^{3} f}{384 \, {\left (a^{3} b^{4} x^{12} + 3 \, a^{4} b^{3} x^{8} + 3 \, a^{5} b^{2} x^{4} + a^{6} b\right )}} + \frac {\frac {\sqrt {2} {\left (77 \, \sqrt {b} c - 15 \, \sqrt {a} e\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (77 \, \sqrt {b} c - 15 \, \sqrt {a} e\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} + \frac {2 \, {\left (77 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {3}{4}} c + 15 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} e - 80 \, \sqrt {a} \sqrt {b} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}} + \frac {2 \, {\left (77 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {3}{4}} c + 15 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} e + 80 \, \sqrt {a} \sqrt {b} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}}}{1024 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*(45*b^3*e*x^11 + 60*b^3*d*x^10 + 77*b^3*c*x^9 + 126*a*b^2*e*x^7 + 160*a*b^2*d*x^6 + 198*a*b^2*c*x^5 + 11
3*a^2*b*e*x^3 + 132*a^2*b*d*x^2 + 153*a^2*b*c*x - 32*a^3*f)/(a^3*b^4*x^12 + 3*a^4*b^3*x^8 + 3*a^5*b^2*x^4 + a^
6*b) + 1/1024*(sqrt(2)*(77*sqrt(b)*c - 15*sqrt(a)*e)*log(sqrt(b)*x^2 + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a
^(3/4)*b^(3/4)) - sqrt(2)*(77*sqrt(b)*c - 15*sqrt(a)*e)*log(sqrt(b)*x^2 - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))
/(a^(3/4)*b^(3/4)) + 2*(77*sqrt(2)*a^(1/4)*b^(3/4)*c + 15*sqrt(2)*a^(3/4)*b^(1/4)*e - 80*sqrt(a)*sqrt(b)*d)*ar
ctan(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*(77*sqrt(2)*a^(1/4)*b^(3/4)*c + 15*sqrt(2)*a^(3/4)*b^(1/4)*e + 80*sqrt(a)*sqrt(b)*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^3

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mupad [B]  time = 5.25, size = 879, normalized size = 2.30 \[ \left (\sum _{k=1}^4\ln \left (-\frac {b\,\left (3375\,a\,e^3-123200\,b\,c\,d^2+88935\,b\,c^2\,e-64000\,b\,d^3\,x+{\mathrm {root}\left (68719476736\,a^{15}\,b^3\,z^4+1211105280\,a^8\,b^2\,c\,e\,z^2+838860800\,a^8\,b^2\,d^2\,z^2-485703680\,a^4\,b^2\,c^2\,d\,z+18432000\,a^5\,b\,d\,e^2\,z-7392000\,a\,b\,c\,d^2\,e+2668050\,a\,b\,c^2\,e^2+2560000\,a\,b\,d^4+35153041\,b^2\,c^4+50625\,a^2\,e^4,z,k\right )}^2\,a^7\,b^2\,c\,20185088-\mathrm {root}\left (68719476736\,a^{15}\,b^3\,z^4+1211105280\,a^8\,b^2\,c\,e\,z^2+838860800\,a^8\,b^2\,d^2\,z^2-485703680\,a^4\,b^2\,c^2\,d\,z+18432000\,a^5\,b\,d\,e^2\,z-7392000\,a\,b\,c\,d^2\,e+2668050\,a\,b\,c^2\,e^2+2560000\,a\,b\,d^4+35153041\,b^2\,c^4+50625\,a^2\,e^4,z,k\right )\,a^4\,b\,e^2\,x\,115200+92400\,b\,c\,d\,e\,x+\mathrm {root}\left (68719476736\,a^{15}\,b^3\,z^4+1211105280\,a^8\,b^2\,c\,e\,z^2+838860800\,a^8\,b^2\,d^2\,z^2-485703680\,a^4\,b^2\,c^2\,d\,z+18432000\,a^5\,b\,d\,e^2\,z-7392000\,a\,b\,c\,d^2\,e+2668050\,a\,b\,c^2\,e^2+2560000\,a\,b\,d^4+35153041\,b^2\,c^4+50625\,a^2\,e^4,z,k\right )\,a^3\,b^2\,c^2\,x\,3035648-{\mathrm {root}\left (68719476736\,a^{15}\,b^3\,z^4+1211105280\,a^8\,b^2\,c\,e\,z^2+838860800\,a^8\,b^2\,d^2\,z^2-485703680\,a^4\,b^2\,c^2\,d\,z+18432000\,a^5\,b\,d\,e^2\,z-7392000\,a\,b\,c\,d^2\,e+2668050\,a\,b\,c^2\,e^2+2560000\,a\,b\,d^4+35153041\,b^2\,c^4+50625\,a^2\,e^4,z,k\right )}^2\,a^7\,b^2\,d\,x\,10485760+\mathrm {root}\left (68719476736\,a^{15}\,b^3\,z^4+1211105280\,a^8\,b^2\,c\,e\,z^2+838860800\,a^8\,b^2\,d^2\,z^2-485703680\,a^4\,b^2\,c^2\,d\,z+18432000\,a^5\,b\,d\,e^2\,z-7392000\,a\,b\,c\,d^2\,e+2668050\,a\,b\,c^2\,e^2+2560000\,a\,b\,d^4+35153041\,b^2\,c^4+50625\,a^2\,e^4,z,k\right )\,a^4\,b\,d\,e\,614400\right )}{a^9\,2097152}\right )\,\mathrm {root}\left (68719476736\,a^{15}\,b^3\,z^4+1211105280\,a^8\,b^2\,c\,e\,z^2+838860800\,a^8\,b^2\,d^2\,z^2-485703680\,a^4\,b^2\,c^2\,d\,z+18432000\,a^5\,b\,d\,e^2\,z-7392000\,a\,b\,c\,d^2\,e+2668050\,a\,b\,c^2\,e^2+2560000\,a\,b\,d^4+35153041\,b^2\,c^4+50625\,a^2\,e^4,z,k\right )\right )+\frac {\frac {11\,d\,x^2}{32\,a}-\frac {f}{12\,b}+\frac {113\,e\,x^3}{384\,a}+\frac {51\,c\,x}{128\,a}+\frac {77\,b^2\,c\,x^9}{384\,a^3}+\frac {5\,b^2\,d\,x^{10}}{32\,a^3}+\frac {15\,b^2\,e\,x^{11}}{128\,a^3}+\frac {33\,b\,c\,x^5}{64\,a^2}+\frac {5\,b\,d\,x^6}{12\,a^2}+\frac {21\,b\,e\,x^7}{64\,a^2}}{a^3+3\,a^2\,b\,x^4+3\,a\,b^2\,x^8+b^3\,x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

symsum(log(-(b*(3375*a*e^3 - 123200*b*c*d^2 + 88935*b*c^2*e - 64000*b*d^3*x + 20185088*root(68719476736*a^15*b
^3*z^4 + 1211105280*a^8*b^2*c*e*z^2 + 838860800*a^8*b^2*d^2*z^2 - 485703680*a^4*b^2*c^2*d*z + 18432000*a^5*b*d
*e^2*z - 7392000*a*b*c*d^2*e + 2668050*a*b*c^2*e^2 + 2560000*a*b*d^4 + 35153041*b^2*c^4 + 50625*a^2*e^4, z, k)
^2*a^7*b^2*c - 115200*root(68719476736*a^15*b^3*z^4 + 1211105280*a^8*b^2*c*e*z^2 + 838860800*a^8*b^2*d^2*z^2 -
485703680*a^4*b^2*c^2*d*z + 18432000*a^5*b*d*e^2*z - 7392000*a*b*c*d^2*e + 2668050*a*b*c^2*e^2 + 2560000*a*b*
d^4 + 35153041*b^2*c^4 + 50625*a^2*e^4, z, k)*a^4*b*e^2*x + 92400*b*c*d*e*x + 3035648*root(68719476736*a^15*b^
3*z^4 + 1211105280*a^8*b^2*c*e*z^2 + 838860800*a^8*b^2*d^2*z^2 - 485703680*a^4*b^2*c^2*d*z + 18432000*a^5*b*d*
e^2*z - 7392000*a*b*c*d^2*e + 2668050*a*b*c^2*e^2 + 2560000*a*b*d^4 + 35153041*b^2*c^4 + 50625*a^2*e^4, z, k)*
a^3*b^2*c^2*x - 10485760*root(68719476736*a^15*b^3*z^4 + 1211105280*a^8*b^2*c*e*z^2 + 838860800*a^8*b^2*d^2*z^
2 - 485703680*a^4*b^2*c^2*d*z + 18432000*a^5*b*d*e^2*z - 7392000*a*b*c*d^2*e + 2668050*a*b*c^2*e^2 + 2560000*a
*b*d^4 + 35153041*b^2*c^4 + 50625*a^2*e^4, z, k)^2*a^7*b^2*d*x + 614400*root(68719476736*a^15*b^3*z^4 + 121110
5280*a^8*b^2*c*e*z^2 + 838860800*a^8*b^2*d^2*z^2 - 485703680*a^4*b^2*c^2*d*z + 18432000*a^5*b*d*e^2*z - 739200
0*a*b*c*d^2*e + 2668050*a*b*c^2*e^2 + 2560000*a*b*d^4 + 35153041*b^2*c^4 + 50625*a^2*e^4, z, k)*a^4*b*d*e))/(2
097152*a^9))*root(68719476736*a^15*b^3*z^4 + 1211105280*a^8*b^2*c*e*z^2 + 838860800*a^8*b^2*d^2*z^2 - 48570368
0*a^4*b^2*c^2*d*z + 18432000*a^5*b*d*e^2*z - 7392000*a*b*c*d^2*e + 2668050*a*b*c^2*e^2 + 2560000*a*b*d^4 + 351
53041*b^2*c^4 + 50625*a^2*e^4, z, k), k, 1, 4) + ((11*d*x^2)/(32*a) - f/(12*b) + (113*e*x^3)/(384*a) + (51*c*x
)/(128*a) + (77*b^2*c*x^9)/(384*a^3) + (5*b^2*d*x^10)/(32*a^3) + (15*b^2*e*x^11)/(128*a^3) + (33*b*c*x^5)/(64*
a^2) + (5*b*d*x^6)/(12*a^2) + (21*b*e*x^7)/(64*a^2))/(a^3 + b^3*x^12 + 3*a^2*b*x^4 + 3*a*b^2*x^8)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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