3.5.89 \(\int \frac {c+d x+e x^2+f x^3}{(a+b x^4)^2} \, dx\) [489]

Optimal. Leaf size=318 \[ -\frac {a f-b x \left (c+d x+e x^2\right )}{4 a b \left (a+b x^4\right )}+\frac {d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b}}-\frac {\left (3 \sqrt {b} c+\sqrt {a} e\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{3/4}}+\frac {\left (3 \sqrt {b} c+\sqrt {a} e\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{3/4}}-\frac {\left (3 \sqrt {b} c-\sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{3/4}}+\frac {\left (3 \sqrt {b} c-\sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{3/4}} \]

[Out]

1/4*(-a*f+b*x*(e*x^2+d*x+c))/a/b/(b*x^4+a)+1/4*d*arctan(x^2*b^(1/2)/a^(1/2))/a^(3/2)/b^(1/2)-1/32*ln(-a^(1/4)*
b^(1/4)*x*2^(1/2)+a^(1/2)+x^2*b^(1/2))*(-e*a^(1/2)+3*c*b^(1/2))/a^(7/4)/b^(3/4)*2^(1/2)+1/32*ln(a^(1/4)*b^(1/4
)*x*2^(1/2)+a^(1/2)+x^2*b^(1/2))*(-e*a^(1/2)+3*c*b^(1/2))/a^(7/4)/b^(3/4)*2^(1/2)+1/16*arctan(-1+b^(1/4)*x*2^(
1/2)/a^(1/4))*(e*a^(1/2)+3*c*b^(1/2))/a^(7/4)/b^(3/4)*2^(1/2)+1/16*arctan(1+b^(1/4)*x*2^(1/2)/a^(1/4))*(e*a^(1
/2)+3*c*b^(1/2))/a^(7/4)/b^(3/4)*2^(1/2)

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Rubi [A]
time = 0.18, antiderivative size = 318, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1868, 1890, 281, 211, 1182, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {\text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt {a} e+3 \sqrt {b} c\right )}{8 \sqrt {2} a^{7/4} b^{3/4}}+\frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {a} e+3 \sqrt {b} c\right )}{8 \sqrt {2} a^{7/4} b^{3/4}}+\frac {d \text {ArcTan}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b}}-\frac {\left (3 \sqrt {b} c-\sqrt {a} e\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{3/4}}+\frac {\left (3 \sqrt {b} c-\sqrt {a} e\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{3/4}}-\frac {a f-b x \left (c+d x+e x^2\right )}{4 a b \left (a+b x^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(a*f - b*x*(c + d*x + e*x^2))/(a*b*(a + b*x^4)) + (d*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(4*a^(3/2)*Sqrt[b]) -
((3*Sqrt[b]*c + Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*b^(3/4)) + ((3*Sqrt[b]
*c + Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*b^(3/4)) - ((3*Sqrt[b]*c - Sqrt[a]
*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(7/4)*b^(3/4)) + ((3*Sqrt[b]*c - Sqr
t[a]*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(7/4)*b^(3/4))

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

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Mathematica [A]
time = 0.23, size = 315, normalized size = 0.99 \begin {gather*} \frac {-\frac {8 a (a f-b x (c+x (d+e x)))}{a+b x^4}-2 \sqrt [4]{a} \sqrt [4]{b} \left (3 \sqrt {2} \sqrt {b} c+4 \sqrt [4]{a} \sqrt [4]{b} d+\sqrt {2} \sqrt {a} e\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+2 \sqrt [4]{a} \sqrt [4]{b} \left (3 \sqrt {2} \sqrt {b} c-4 \sqrt [4]{a} \sqrt [4]{b} d+\sqrt {2} \sqrt {a} e\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+\sqrt {2} \sqrt [4]{b} \left (-3 \sqrt [4]{a} \sqrt {b} c+a^{3/4} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )+\sqrt {2} \sqrt [4]{b} \left (3 \sqrt [4]{a} \sqrt {b} c-a^{3/4} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{32 a^2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-8*a*(a*f - b*x*(c + x*(d + e*x))))/(a + b*x^4) - 2*a^(1/4)*b^(1/4)*(3*Sqrt[2]*Sqrt[b]*c + 4*a^(1/4)*b^(1/4)
*d + Sqrt[2]*Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*a^(1/4)*b^(1/4)*(3*Sqrt[2]*Sqrt[b]*c - 4*a
^(1/4)*b^(1/4)*d + Sqrt[2]*Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + Sqrt[2]*b^(1/4)*(-3*a^(1/4)*Sq
rt[b]*c + a^(3/4)*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + Sqrt[2]*b^(1/4)*(3*a^(1/4)*Sqrt[
b]*c - a^(3/4)*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(32*a^2*b)

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Maple [A]
time = 0.37, size = 305, normalized size = 0.96

method	result	size
risch	\(\frac {\frac {e \,x^{3}}{4 a}+\frac {d \,x^{2}}{4 a}+\frac {c x}{4 a}-\frac {f}{4 b}}{b \,x^{4}+a}+\frac {\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\left (\textit {\_R}^{2} e +2 \textit {\_R} d +3 c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{16 b a}\)	\(86\)
default	\(c \left (\frac {x}{4 a \left (b \,x^{4}+a \right )}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a^{2}}\right )+d \left (\frac {x^{2}}{4 a \left (b \,x^{4}+a \right )}+\frac {\arctan \left (x^{2} \sqrt {\frac {b}{a}}\right )}{4 a \sqrt {a b}}\right )+e \left (\frac {x^{3}}{4 a \left (b \,x^{4}+a \right )}+\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+\frac {f \,x^{4}}{4 a \left (b \,x^{4}+a \right )}\)	\(305\)

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)^2,x,method=_RETURNVERBOSE)

[Out]

c*(1/4*x/a/(b*x^4+a)+3/32/a^2*(a/b)^(1/4)*2^(1/2)*(ln((x^2+(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2))/(x^2-(a/b)^(1/4)
*x*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+2*arctan(2^(1/2)/(a/b)^(1/4)*x-1)))+d*(1/4*x^2/a/(b
*x^4+a)+1/4/a/(a*b)^(1/2)*arctan(x^2*(b/a)^(1/2)))+e*(1/4*x^3/a/(b*x^4+a)+1/32/a/b/(a/b)^(1/4)*2^(1/2)*(ln((x^
2-(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2))/(x^2+(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x+1
)+2*arctan(2^(1/2)/(a/b)^(1/4)*x-1)))+1/4*f*x^4/a/(b*x^4+a)

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Maxima [A]
time = 0.48, size = 310, normalized size = 0.97 \begin {gather*} \frac {b x^{3} e + b d x^{2} + b c x - a f}{4 \, {\left (a b^{2} x^{4} + a^{2} b\right )}} + \frac {\frac {\sqrt {2} {\left (3 \, \sqrt {b} c - \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 (3 \, \sqrt {b} c - \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 (3 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {3}{4}} c + \sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} e - 4 \, \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 (3 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {3}{4}} c + \sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} e + 4 \, \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}}}}{32 \, a} \end {gather*}

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)^2,x, algorithm="maxima")

[Out]

1/4*(b*x^3*e + b*d*x^2 + b*c*x - a*f)/(a*b^2*x^4 + a^2*b) + 1/32*(sqrt(2)*(3*sqrt(b)*c - 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)*(3*sqrt(b)*c - 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*(3*sqrt(2)*a^(1/4)*b^(3/4)*c + sqrt(2)*a^(3/
4)*b^(1/4)*e - 4*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)*sq
rt(b)))/(a^(3/4)*sqrt(sqrt(a)*sqrt(b))*b^(3/4)) + 2*(3*sqrt(2)*a^(1/4)*b^(3/4)*c + sqrt(2)*a^(3/4)*b^(1/4)*e +
4*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

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Fricas [C] Result contains complex when optimal does not.
time = 3.67, size = 124301, normalized size = 390.88 \begin {gather*} \text {too large to display} \end {gather*}

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)^2,x, algorithm="fricas")

[Out]

1/9216*(2304*b*e*x^3 + 2304*b*d*x^2 + 2304*b*c*x + 2*(a*b^2*x^4 + a^2*b)*((-I*sqrt(3) + 1)*((a^2*b*sqrt(-1/(a*
b))*sqrt(-(6*a*b*c*e*sqrt(-1/(a*b)) + 9*b*c^2 - a*e^2)/(a^4*b^2*sqrt(-1/(a*b)))) - 2*d)^2/(a^3*b) - 3*(4*a^2*b
*d*sqrt(-(6*a*b*c*e*sqrt(-1/(a*b)) + 9*b*c^2 - a*e^2)/(a^4*b^2*sqrt(-1/(a*b)))) + 9*b*c^2 - (2*(2*d^2 + 3*c*e)
*b*sqrt(-1/(a*b)) + e^2)*a)/(a^4*b^2*sqrt(-1/(a*b))))/(-1/24576*(4*a^2*b*d*sqrt(-(6*a*b*c*e*sqrt(-1/(a*b)) + 9
*b*c^2 - a*e^2)/(a^4*b^2*sqrt(-1/(a*b)))) + 9*b*c^2 - (2*(2*d^2 + 3*c*e)*b*sqrt(-1/(a*b)) + e^2)*a)*(a^2*b*sqr
t(-1/(a*b))*sqrt(-(6*a*b*c*e*sqrt(-1/(a*b)) + 9*b*c^2 - a*e^2)/(a^4*b^2*sqrt(-1/(a*b)))) - 2*d)/(a^5*b^2) + 1/
8192*(a^5*b^2*sqrt(-1/(a*b))*(-(6*a*b*c*e*sqrt(-1/(a*b)) + 9*b*c^2 - a*e^2)/(a^4*b^2*sqrt(-1/(a*b))))^(3/2) -
4*(d^2*sqrt(-(6*a*b*c*e*sqrt(-1/(a*b)) + 9*b*c^2 - a*e^2)/(a^4*b^2*sqrt(-1/(a*b)))) - 3*c*e*sqrt(-(6*a*b*c*e*s
qrt(-1/(a*b)) + 9*b*c^2 - a*e^2)/(a^4*b^2*sqrt(-1/(a*b)))))*a^2*b*sqrt(-1/(a*b)) + 18*b*c^2*d*sqrt(-1/(a*b)) -
2*a*d*e^2 ...

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Sympy [A]
time = 43.84, size = 517, normalized size = 1.63 \begin {gather*} \operatorname {RootSum} {\left (65536 t^{4} a^{7} b^{3} + t^{2} \cdot \left (3072 a^{4} b^{2} c e + 2048 a^{4} b^{2} d^{2}\right ) + t \left (128 a^{3} b d e^{2} - 1152 a^{2} b^{2} c^{2} d\right ) + a^{2} e^{4} + 18 a b c^{2} e^{2} - 48 a b c d^{2} e + 16 a b d^{4} + 81 b^{2} c^{4}, \left ( t \mapsto t \log {\left (x + \frac {4096 t^{3} a^{7} b^{2} e^{3} - 36864 t^{3} a^{6} b^{3} c^{2} e + 98304 t^{3} a^{6} b^{3} c d^{2} + 4608 t^{2} a^{5} b^{2} c d e^{2} - 4096 t^{2} a^{5} b^{2} d^{3} e + 13824 t^{2} a^{4} b^{3} c^{3} d + 144 t a^{4} b c e^{4} + 192 t a^{4} b d^{2} e^{3} - 1728 t a^{3} b^{2} c^{3} e^{2} + 5184 t a^{3} b^{2} c^{2} d^{2} e + 1536 t a^{3} b^{2} c d^{4} + 3888 t a^{2} b^{3} c^{5} + 6 a^{3} d e^{5} + 120 a^{2} b c d^{3} e^{2} - 64 a^{2} b d^{5} e + 810 a b^{2} c^{4} d e - 1080 a b^{2} c^{3} d^{3}}{a^{3} e^{6} - 9 a^{2} b c^{2} e^{4} + 96 a^{2} b c d^{2} e^{3} - 64 a^{2} b d^{4} e^{2} - 81 a b^{2} c^{4} e^{2} + 864 a b^{2} c^{3} d^{2} e - 576 a b^{2} c^{2} d^{4} + 729 b^{3} c^{6}} \right )} \right )\right )} + \frac {- a f + b c x + b d x^{2} + b e x^{3}}{4 a^{2} b + 4 a b^{2} x^{4}} \end {gather*}

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)**2,x)

[Out]

RootSum(65536*_t**4*a**7*b**3 + _t**2*(3072*a**4*b**2*c*e + 2048*a**4*b**2*d**2) + _t*(128*a**3*b*d*e**2 - 115
2*a**2*b**2*c**2*d) + a**2*e**4 + 18*a*b*c**2*e**2 - 48*a*b*c*d**2*e + 16*a*b*d**4 + 81*b**2*c**4, Lambda(_t,
_t*log(x + (4096*_t**3*a**7*b**2*e**3 - 36864*_t**3*a**6*b**3*c**2*e + 98304*_t**3*a**6*b**3*c*d**2 + 4608*_t*
*2*a**5*b**2*c*d*e**2 - 4096*_t**2*a**5*b**2*d**3*e + 13824*_t**2*a**4*b**3*c**3*d + 144*_t*a**4*b*c*e**4 + 19
2*_t*a**4*b*d**2*e**3 - 1728*_t*a**3*b**2*c**3*e**2 + 5184*_t*a**3*b**2*c**2*d**2*e + 1536*_t*a**3*b**2*c*d**4
+ 3888*_t*a**2*b**3*c**5 + 6*a**3*d*e**5 + 120*a**2*b*c*d**3*e**2 - 64*a**2*b*d**5*e + 810*a*b**2*c**4*d*e -
1080*a*b**2*c**3*d**3)/(a**3*e**6 - 9*a**2*b*c**2*e**4 + 96*a**2*b*c*d**2*e**3 - 64*a**2*b*d**4*e**2 - 81*a*b*
*2*c**4*e**2 + 864*a*b**2*c**3*d**2*e - 576*a*b**2*c**2*d**4 + 729*b**3*c**6)))) + (-a*f + b*c*x + b*d*x**2 +
b*e*x**3)/(4*a**2*b + 4*a*b**2*x**4)

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Giac [A]
time = 0.60, size = 316, normalized size = 0.99 \begin {gather*} \frac {b x^{3} e + b d x^{2} + b c x - a f}{4 \, {\left (b x^{4} + a\right )} a b} + \frac {\sqrt {2} {\left (2 \, \sqrt {2} \sqrt {a b} b^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + \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 )}{16 \, a^{2} b^{3}} + \frac {\sqrt {2} {\left (2 \, \sqrt {2} \sqrt {a b} b^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + \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 )}{16 \, a^{2} b^{3}} + \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - \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 )}{32 \, a^{2} b^{3}} - \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - \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 )}{32 \, a^{2} b^{3}} \end {gather*}

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)^2,x, algorithm="giac")

[Out]

1/4*(b*x^3*e + b*d*x^2 + b*c*x - a*f)/((b*x^4 + a)*a*b) + 1/16*sqrt(2)*(2*sqrt(2)*sqrt(a*b)*b^2*d + 3*(a*b^3)^
(1/4)*b^2*c + (a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^2*b^3) + 1/16*sq
rt(2)*(2*sqrt(2)*sqrt(a*b)*b^2*d + 3*(a*b^3)^(1/4)*b^2*c + (a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*
(a/b)^(1/4))/(a/b)^(1/4))/(a^2*b^3) + 1/32*sqrt(2)*(3*(a*b^3)^(1/4)*b^2*c - (a*b^3)^(3/4)*e)*log(x^2 + sqrt(2)
*x*(a/b)^(1/4) + sqrt(a/b))/(a^2*b^3) - 1/32*sqrt(2)*(3*(a*b^3)^(1/4)*b^2*c - (a*b^3)^(3/4)*e)*log(x^2 - sqrt(
2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^2*b^3)

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Mupad [B]
time = 0.36, size = 478, normalized size = 1.50 \begin {gather*} \left (\sum _{k=1}^4\ln \left (-\mathrm {root}\left (65536\,a^7\,b^3\,z^4+3072\,a^4\,b^2\,c\,e\,z^2+2048\,a^4\,b^2\,d^2\,z^2-1152\,a^2\,b^2\,c^2\,d\,z+128\,a^3\,b\,d\,e^2\,z-48\,a\,b\,c\,d^2\,e+18\,a\,b\,c^2\,e^2+16\,a\,b\,d^4+81\,b^2\,c^4+a^2\,e^4,z,k\right )\,\left (\mathrm {root}\left (65536\,a^7\,b^3\,z^4+3072\,a^4\,b^2\,c\,e\,z^2+2048\,a^4\,b^2\,d^2\,z^2-1152\,a^2\,b^2\,c^2\,d\,z+128\,a^3\,b\,d\,e^2\,z-48\,a\,b\,c\,d^2\,e+18\,a\,b\,c^2\,e^2+16\,a\,b\,d^4+81\,b^2\,c^4+a^2\,e^4,z,k\right )\,\left (12\,b^3\,c-8\,b^3\,d\,x\right )+\frac {x\,\left (36\,a\,b^3\,c^2-4\,a^2\,b^2\,e^2\right )}{16\,a^3}+\frac {b^2\,d\,e}{a}\right )-\frac {9\,b^2\,c^2\,e-12\,b^2\,c\,d^2+a\,b\,e^3}{64\,a^3}+\frac {x\,\left (2\,b^2\,d^3-3\,b^2\,c\,d\,e\right )}{16\,a^3}\right )\,\mathrm {root}\left (65536\,a^7\,b^3\,z^4+3072\,a^4\,b^2\,c\,e\,z^2+2048\,a^4\,b^2\,d^2\,z^2-1152\,a^2\,b^2\,c^2\,d\,z+128\,a^3\,b\,d\,e^2\,z-48\,a\,b\,c\,d^2\,e+18\,a\,b\,c^2\,e^2+16\,a\,b\,d^4+81\,b^2\,c^4+a^2\,e^4,z,k\right )\right )+\frac {\frac {d\,x^2}{4\,a}-\frac {f}{4\,b}+\frac {e\,x^3}{4\,a}+\frac {c\,x}{4\,a}}{b\,x^4+a} \end {gather*}

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)^2,x)

[Out]

symsum(log((x*(2*b^2*d^3 - 3*b^2*c*d*e))/(16*a^3) - (9*b^2*c^2*e - 12*b^2*c*d^2 + a*b*e^3)/(64*a^3) - root(655
36*a^7*b^3*z^4 + 3072*a^4*b^2*c*e*z^2 + 2048*a^4*b^2*d^2*z^2 - 1152*a^2*b^2*c^2*d*z + 128*a^3*b*d*e^2*z - 48*a
*b*c*d^2*e + 18*a*b*c^2*e^2 + 16*a*b*d^4 + 81*b^2*c^4 + a^2*e^4, z, k)*(root(65536*a^7*b^3*z^4 + 3072*a^4*b^2*
c*e*z^2 + 2048*a^4*b^2*d^2*z^2 - 1152*a^2*b^2*c^2*d*z + 128*a^3*b*d*e^2*z - 48*a*b*c*d^2*e + 18*a*b*c^2*e^2 +
16*a*b*d^4 + 81*b^2*c^4 + a^2*e^4, z, k)*(12*b^3*c - 8*b^3*d*x) + (x*(36*a*b^3*c^2 - 4*a^2*b^2*e^2))/(16*a^3)
+ (b^2*d*e)/a))*root(65536*a^7*b^3*z^4 + 3072*a^4*b^2*c*e*z^2 + 2048*a^4*b^2*d^2*z^2 - 1152*a^2*b^2*c^2*d*z +
128*a^3*b*d*e^2*z - 48*a*b*c*d^2*e + 18*a*b*c^2*e^2 + 16*a*b*d^4 + 81*b^2*c^4 + a^2*e^4, z, k), k, 1, 4) + ((d
*x^2)/(4*a) - f/(4*b) + (e*x^3)/(4*a) + (c*x)/(4*a))/(a + b*x^4)

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