3.1.66 \(\int \frac {d+e x^2+f x^4}{x^3 (a+b x^2+c x^4)^2} \, dx\) [66]

Optimal. Leaf size=234 \[ -\frac {d}{2 a^2 x^2}-\frac {b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)+c \left (b^2 d-a b e-2 a (c d-a f)\right ) x^2}{2 a^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\left (2 b^4 d-12 a b^2 c d-a b^3 e+6 a^2 b c e+4 a^2 c (3 c d-a f)\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^3 \left (b^2-4 a c\right )^{3/2}}-\frac {(2 b d-a e) \log (x)}{a^3}+\frac {(2 b d-a e) \log \left (a+b x^2+c x^4\right )}{4 a^3} \]

[Out]

-1/2*d/a^2/x^2+1/2*(-b^3*d+a*b^2*e-2*a^2*c*e+a*b*(-a*f+3*c*d)-c*(b^2*d-a*b*e-2*a*(-a*f+c*d))*x^2)/a^2/(-4*a*c+
b^2)/(c*x^4+b*x^2+a)-1/2*(2*b^4*d-12*a*b^2*c*d-a*b^3*e+6*a^2*b*c*e+4*a^2*c*(-a*f+3*c*d))*arctanh((2*c*x^2+b)/(
-4*a*c+b^2)^(1/2))/a^3/(-4*a*c+b^2)^(3/2)-(-a*e+2*b*d)*ln(x)/a^3+1/4*(-a*e+2*b*d)*ln(c*x^4+b*x^2+a)/a^3

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Rubi [A]
time = 0.48, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {1677, 1660, 1642, 648, 632, 212, 642} \begin {gather*} \frac {(2 b d-a e) \log \left (a+b x^2+c x^4\right )}{4 a^3}-\frac {\log (x) (2 b d-a e)}{a^3}-\frac {2 a^2 c e+c x^2 \left (-a b e-2 a (c d-a f)+b^2 d\right )-a b^2 e-a b (3 c d-a f)+b^3 d}{2 a^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {d}{2 a^2 x^2}-\frac {\tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (6 a^2 b c e+4 a^2 c (3 c d-a f)-a b^3 e-12 a b^2 c d+2 b^4 d\right )}{2 a^3 \left (b^2-4 a c\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*d/(a^2*x^2) - (b^3*d - a*b^2*e + 2*a^2*c*e - a*b*(3*c*d - a*f) + c*(b^2*d - a*b*e - 2*a*(c*d - a*f))*x^2)
/(2*a^2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - ((2*b^4*d - 12*a*b^2*c*d - a*b^3*e + 6*a^2*b*c*e + 4*a^2*c*(3*c*d
- a*f))*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*a^3*(b^2 - 4*a*c)^(3/2)) - ((2*b*d - a*e)*Log[x])/a^3 +
((2*b*d - a*e)*Log[a + b*x^2 + c*x^4])/(4*a^3)

Rule 212

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

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; 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 648

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

Rule 1642

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 1660

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomi
alQuotient[(d + e*x)^m*Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2,
x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*
c*f - b*g)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*f - b*
g))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1677

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)
*SubstFor[x^2, Pq, x]*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && Inte
gerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \frac {d+e x^2+f x^4}{x^3 \left (a+b x^2+c x^4\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {d+e x+f x^2}{x^2 \left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )\\ &=-\frac {b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)+c \left (b^2 d-a b e-2 a (c d-a f)\right ) x^2}{2 a^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\text {Subst}\left (\int \frac {-\left (\frac {b^2}{a}-4 c\right ) d+\frac {\left (b^2-4 a c\right ) (b d-a e) x}{a^2}+\frac {c \left (b^2 d-a b e-2 a (c d-a f)\right ) x^2}{a^2}}{x^2 \left (a+b x+c x^2\right )} \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )}\\ &=-\frac {b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)+c \left (b^2 d-a b e-2 a (c d-a f)\right ) x^2}{2 a^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\text {Subst}\left (\int \left (\frac {\left (-b^2+4 a c\right ) d}{a^2 x^2}+\frac {\left (-b^2+4 a c\right ) (-2 b d+a e)}{a^3 x}+\frac {-2 b^4 d+10 a b^2 c d+a b^3 e-5 a^2 b c e-2 a^2 c (3 c d-a f)-c \left (b^2-4 a c\right ) (2 b d-a e) x}{a^3 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )}\\ &=-\frac {d}{2 a^2 x^2}-\frac {b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)+c \left (b^2 d-a b e-2 a (c d-a f)\right ) x^2}{2 a^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {(2 b d-a e) \log (x)}{a^3}-\frac {\text {Subst}\left (\int \frac {-2 b^4 d+10 a b^2 c d+a b^3 e-5 a^2 b c e-2 a^2 c (3 c d-a f)-c \left (b^2-4 a c\right ) (2 b d-a e) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 a^3 \left (b^2-4 a c\right )}\\ &=-\frac {d}{2 a^2 x^2}-\frac {b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)+c \left (b^2 d-a b e-2 a (c d-a f)\right ) x^2}{2 a^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {(2 b d-a e) \log (x)}{a^3}+\frac {(2 b d-a e) \text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^3}+\frac {\left (2 b^4 d-12 a b^2 c d-a b^3 e+6 a^2 b c e+4 a^2 c (3 c d-a f)\right ) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^3 \left (b^2-4 a c\right )}\\ &=-\frac {d}{2 a^2 x^2}-\frac {b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)+c \left (b^2 d-a b e-2 a (c d-a f)\right ) x^2}{2 a^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {(2 b d-a e) \log (x)}{a^3}+\frac {(2 b d-a e) \log \left (a+b x^2+c x^4\right )}{4 a^3}-\frac {\left (2 b^4 d-12 a b^2 c d-a b^3 e+6 a^2 b c e+4 a^2 c (3 c d-a f)\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 a^3 \left (b^2-4 a c\right )}\\ &=-\frac {d}{2 a^2 x^2}-\frac {b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)+c \left (b^2 d-a b e-2 a (c d-a f)\right ) x^2}{2 a^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\left (2 b^4 d-12 a b^2 c d-a b^3 e+6 a^2 b c e+4 a^2 c (3 c d-a f)\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^3 \left (b^2-4 a c\right )^{3/2}}-\frac {(2 b d-a e) \log (x)}{a^3}+\frac {(2 b d-a e) \log \left (a+b x^2+c x^4\right )}{4 a^3}\\ \end {align*}

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Mathematica [A]
time = 0.42, size = 403, normalized size = 1.72 \begin {gather*} \frac {-\frac {2 a d}{x^2}-\frac {2 a \left (b^3 d+b^2 \left (-a e+c d x^2\right )+a b \left (a f-c \left (3 d+e x^2\right )\right )+2 a c \left (-c d x^2+a \left (e+f x^2\right )\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+4 (-2 b d+a e) \log (x)+\frac {\left (2 b^4 d+b^3 \left (2 \sqrt {b^2-4 a c} d-a e\right )+2 a b c \left (-4 \sqrt {b^2-4 a c} d+3 a e\right )-a b^2 \left (12 c d+\sqrt {b^2-4 a c} e\right )+4 a^2 c \left (3 c d+\sqrt {b^2-4 a c} e-a f\right )\right ) \log \left (b-\sqrt {b^2-4 a c}+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac {\left (-2 b^4 d+b^3 \left (2 \sqrt {b^2-4 a c} d+a e\right )-2 a b c \left (4 \sqrt {b^2-4 a c} d+3 a e\right )+a b^2 \left (12 c d-\sqrt {b^2-4 a c} e\right )+4 a^2 c \left (-3 c d+\sqrt {b^2-4 a c} e+a f\right )\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}}{4 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*a*d)/x^2 - (2*a*(b^3*d + b^2*(-(a*e) + c*d*x^2) + a*b*(a*f - c*(3*d + e*x^2)) + 2*a*c*(-(c*d*x^2) + a*(e
+ f*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + 4*(-2*b*d + a*e)*Log[x] + ((2*b^4*d + b^3*(2*Sqrt[b^2 - 4*a*
c]*d - a*e) + 2*a*b*c*(-4*Sqrt[b^2 - 4*a*c]*d + 3*a*e) - a*b^2*(12*c*d + Sqrt[b^2 - 4*a*c]*e) + 4*a^2*c*(3*c*d
+ Sqrt[b^2 - 4*a*c]*e - a*f))*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2) + ((-2*b^4*d + b^3*(2
*Sqrt[b^2 - 4*a*c]*d + a*e) - 2*a*b*c*(4*Sqrt[b^2 - 4*a*c]*d + 3*a*e) + a*b^2*(12*c*d - Sqrt[b^2 - 4*a*c]*e) +
4*a^2*c*(-3*c*d + Sqrt[b^2 - 4*a*c]*e + a*f))*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/(4*a
^3)

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Maple [A]
time = 0.09, size = 316, normalized size = 1.35

method	result	size
default	\(\frac {\frac {\frac {a c \left (2 a^{2} f -a b e -2 a c d +b^{2} d \right ) x^{2}}{4 a c -b^{2}}+\frac {a \left (a^{2} b f +2 a^{2} c e -a \,b^{2} e -3 a b c d +b^{3} d \right )}{4 a c -b^{2}}}{c \,x^{4}+b \,x^{2}+a}+\frac {\frac {\left (-4 a^{2} c^{2} e +a \,b^{2} c e +8 a b \,c^{2} d -2 b^{3} c d \right ) \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c}+\frac {2 \left (2 a^{3} c f -5 a^{2} b c e -6 a^{2} c^{2} d +a \,b^{3} e +10 a \,b^{2} c d -2 b^{4} d -\frac {\left (-4 a^{2} c^{2} e +a \,b^{2} c e +8 a b \,c^{2} d -2 b^{3} c d \right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{2 a^{3}}-\frac {d}{2 a^{2} x^{2}}+\frac {\left (a e -2 b d \right ) \ln \left (x \right )}{a^{3}}\)	\(316\)
risch	\(\frac {\frac {c \left (2 a^{2} f -a b e -6 a c d +2 b^{2} d \right ) x^{4}}{2 a^{2} \left (4 a c -b^{2}\right )}+\frac {\left (a^{2} b f +2 a^{2} c e -a \,b^{2} e -7 a b c d +2 b^{3} d \right ) x^{2}}{2 \left (4 a c -b^{2}\right ) a^{2}}-\frac {d}{2 a}}{x^{2} \left (c \,x^{4}+b \,x^{2}+a \right )}+\frac {\ln \left (x \right ) e}{a^{2}}-\frac {2 \ln \left (x \right ) b d}{a^{3}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (64 a^{6} c^{3}-48 a^{5} b^{2} c^{2}+12 a^{4} b^{4} c -a^{3} b^{6}\right ) \textit {\_Z}^{2}+\left (64 a^{4} c^{3} e -48 a^{3} b^{2} c^{2} e -128 a^{3} b \,c^{3} d +12 a^{2} b^{4} c e +96 a^{2} b^{3} c^{2} d -a \,b^{6} e -24 a \,b^{5} c d +2 b^{7} d \right ) \textit {\_Z} +4 a^{3} c^{2} f^{2}-12 a^{2} b \,c^{2} e f -24 a^{2} c^{3} d f +16 a^{2} c^{3} e^{2}+2 a \,b^{3} c e f +24 a \,b^{2} c^{2} d f -3 a \,b^{2} c^{2} e^{2}-28 a b \,c^{3} d e +36 a \,c^{4} d^{2}-4 b^{4} c d f +6 b^{3} c^{2} d e -8 b^{2} c^{3} d^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (160 a^{7} c^{3}-128 a^{6} b^{2} c^{2}+34 b^{4} a^{5} c -3 a^{4} b^{6}\right ) \textit {\_R}^{2}+\left (-8 a^{5} b \,c^{2} f +80 a^{5} c^{3} e +2 a^{4} b^{3} c f -36 a^{4} b^{2} c^{2} e -136 a^{4} b \,c^{3} d +4 a^{3} b^{4} c e +66 a^{3} b^{3} c^{2} d -8 a^{2} b^{5} c d \right ) \textit {\_R} +8 a^{4} c^{2} f^{2}-8 a^{3} b \,c^{2} e f -48 a^{3} c^{3} d f +16 a^{2} b^{2} c^{2} d f +2 a^{2} b^{2} c^{2} e^{2}+24 a^{2} b \,c^{3} d e +72 a^{2} c^{4} d^{2}-8 a \,b^{3} c^{2} d e -48 a \,b^{2} c^{3} d^{2}+8 b^{4} c^{2} d^{2}\right ) x^{2}+\left (-16 a^{7} b \,c^{2}+8 a^{6} b^{3} c -a^{5} b^{5}\right ) \textit {\_R}^{2}+\left (-8 a^{6} c^{2} f +2 a^{5} b^{2} c f +36 a^{5} b \,c^{2} e +24 a^{5} c^{3} d -17 a^{4} b^{3} c e -78 a^{4} b^{2} c^{2} d +2 a^{3} b^{5} e +34 a^{3} b^{4} c d -4 a^{2} b^{6} d \right ) \textit {\_R} +16 a^{4} c^{2} e f -4 a^{3} b^{2} c e f -32 a^{3} b \,c^{2} d f -8 a^{3} b \,c^{2} e^{2}-48 a^{3} c^{3} d e +8 a^{2} b^{3} c d f +2 a^{2} b^{3} c \,e^{2}+44 a^{2} b^{2} c^{2} d e +96 a^{2} b \,c^{3} d^{2}-8 a \,b^{4} c d e -56 a \,b^{3} c^{2} d^{2}+8 b^{5} c \,d^{2}\right )\right )}{2}\)	\(892\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/a^3*((a*c*(2*a^2*f-a*b*e-2*a*c*d+b^2*d)/(4*a*c-b^2)*x^2+a*(a^2*b*f+2*a^2*c*e-a*b^2*e-3*a*b*c*d+b^3*d)/(4*a
*c-b^2))/(c*x^4+b*x^2+a)+1/(4*a*c-b^2)*(1/2*(-4*a^2*c^2*e+a*b^2*c*e+8*a*b*c^2*d-2*b^3*c*d)/c*ln(c*x^4+b*x^2+a)
+2*(2*a^3*c*f-5*a^2*b*c*e-6*a^2*c^2*d+a*b^3*e+10*a*b^2*c*d-2*b^4*d-1/2*(-4*a^2*c^2*e+a*b^2*c*e+8*a*b*c^2*d-2*b
^3*c*d)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))))-1/2*d/a^2/x^2+(a*e-2*b*d)/a^3*ln(x)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 870 vs. \(2 (222) = 444\).
time = 2.02, size = 1764, normalized size = 7.54 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(2*(2*(a*b^4*c - 7*a^2*b^2*c^2 + 12*a^3*c^3)*d - (a^2*b^3*c - 4*a^3*b*c^2)*e + 2*(a^3*b^2*c - 4*a^4*c^2)
*f)*x^4 + 2*((2*a*b^5 - 15*a^2*b^3*c + 28*a^3*b*c^2)*d - (a^2*b^4 - 6*a^3*b^2*c + 8*a^4*c^2)*e + (a^3*b^3 - 4*
a^4*b*c)*f)*x^2 + ((4*a^3*c^2*f - 2*(b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*d + (a*b^3*c - 6*a^2*b*c^2)*e)*x^6 + (4*
a^3*b*c*f - 2*(b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*d + (a*b^4 - 6*a^2*b^2*c)*e)*x^4 + (4*a^4*c*f - 2*(a*b^4 - 6*a^2
*b^2*c + 6*a^3*c^2)*d + (a^2*b^3 - 6*a^3*b*c)*e)*x^2)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^4 + 2*b*c*x^2 + b^2 - 2*a
*c - (2*c*x^2 + b)*sqrt(b^2 - 4*a*c))/(c*x^4 + b*x^2 + a)) + 2*(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*d - ((2*(b
^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d - (a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3)*e)*x^6 + (2*(b^6 - 8*a*b^4*c + 1
6*a^2*b^2*c^2)*d - (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*e)*x^4 + (2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d - (
a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*e)*x^2)*log(c*x^4 + b*x^2 + a) + 4*((2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3
)*d - (a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3)*e)*x^6 + (2*(b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2)*d - (a*b^5 - 8*a^2
*b^3*c + 16*a^3*b*c^2)*e)*x^4 + (2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d - (a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^
2)*e)*x^2)*log(x))/((a^3*b^4*c - 8*a^4*b^2*c^2 + 16*a^5*c^3)*x^6 + (a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*x^4
+ (a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)*x^2), -1/4*(2*(2*(a*b^4*c - 7*a^2*b^2*c^2 + 12*a^3*c^3)*d - (a^2*b^3*c
- 4*a^3*b*c^2)*e + 2*(a^3*b^2*c - 4*a^4*c^2)*f)*x^4 + 2*((2*a*b^5 - 15*a^2*b^3*c + 28*a^3*b*c^2)*d - (a^2*b^4
- 6*a^3*b^2*c + 8*a^4*c^2)*e + (a^3*b^3 - 4*a^4*b*c)*f)*x^2 - 2*((4*a^3*c^2*f - 2*(b^4*c - 6*a*b^2*c^2 + 6*a^2
*c^3)*d + (a*b^3*c - 6*a^2*b*c^2)*e)*x^6 + (4*a^3*b*c*f - 2*(b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*d + (a*b^4 - 6*a^2
*b^2*c)*e)*x^4 + (4*a^4*c*f - 2*(a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2)*d + (a^2*b^3 - 6*a^3*b*c)*e)*x^2)*sqrt(-b^2
+ 4*a*c)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) + 2*(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*d -
((2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d - (a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3)*e)*x^6 + (2*(b^6 - 8*a*b^4
*c + 16*a^2*b^2*c^2)*d - (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*e)*x^4 + (2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)
*d - (a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*e)*x^2)*log(c*x^4 + b*x^2 + a) + 4*((2*(b^5*c - 8*a*b^3*c^2 + 16*a^2
*b*c^3)*d - (a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3)*e)*x^6 + (2*(b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2)*d - (a*b^5 -
8*a^2*b^3*c + 16*a^3*b*c^2)*e)*x^4 + (2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d - (a^2*b^4 - 8*a^3*b^2*c + 16*
a^4*c^2)*e)*x^2)*log(x))/((a^3*b^4*c - 8*a^4*b^2*c^2 + 16*a^5*c^3)*x^6 + (a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2
)*x^4 + (a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)*x^2)]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]
time = 5.06, size = 287, normalized size = 1.23 \begin {gather*} \frac {{\left (2 \, b^{4} d - 12 \, a b^{2} c d + 12 \, a^{2} c^{2} d - 4 \, a^{3} c f - a b^{3} e + 6 \, a^{2} b c e\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {2 \, b^{2} c d x^{4} - 6 \, a c^{2} d x^{4} + 2 \, a^{2} c f x^{4} - a b c x^{4} e + 2 \, b^{3} d x^{2} - 7 \, a b c d x^{2} + a^{2} b f x^{2} - a b^{2} x^{2} e + 2 \, a^{2} c x^{2} e + a b^{2} d - 4 \, a^{2} c d}{2 \, {\left (c x^{6} + b x^{4} + a x^{2}\right )} {\left (a^{2} b^{2} - 4 \, a^{3} c\right )}} + \frac {{\left (2 \, b d - a e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{3}} - \frac {{\left (2 \, b d - a e\right )} \log \left (x^{2}\right )}{2 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*b^4*d - 12*a*b^2*c*d + 12*a^2*c^2*d - 4*a^3*c*f - a*b^3*e + 6*a^2*b*c*e)*arctan((2*c*x^2 + b)/sqrt(-b^2
+ 4*a*c))/((a^3*b^2 - 4*a^4*c)*sqrt(-b^2 + 4*a*c)) - 1/2*(2*b^2*c*d*x^4 - 6*a*c^2*d*x^4 + 2*a^2*c*f*x^4 - a*b
*c*x^4*e + 2*b^3*d*x^2 - 7*a*b*c*d*x^2 + a^2*b*f*x^2 - a*b^2*x^2*e + 2*a^2*c*x^2*e + a*b^2*d - 4*a^2*c*d)/((c*
x^6 + b*x^4 + a*x^2)*(a^2*b^2 - 4*a^3*c)) + 1/4*(2*b*d - a*e)*log(c*x^4 + b*x^2 + a)/a^3 - 1/2*(2*b*d - a*e)*l
og(x^2)/a^3

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Mupad [B]
time = 12.98, size = 2500, normalized size = 10.68 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x^2*(2*b^3*d - a*b^2*e + a^2*b*f + 2*a^2*c*e - 7*a*b*c*d))/(2*a^2*(4*a*c - b^2)) - d/(2*a) + (c*x^4*(2*b^2*d
+ 2*a^2*f - a*b*e - 6*a*c*d))/(2*a^2*(4*a*c - b^2)))/(a*x^2 + b*x^4 + c*x^6) + (log(x)*(a*e - 2*b*d))/a^3 + (
log((((((a*e - 2*b*d + a^3*(-(2*b^4*d + 12*a^2*c^2*d - a*b^3*e - 4*a^3*c*f - 12*a*b^2*c*d + 6*a^2*b*c*e)^2/(a^
6*(4*a*c - b^2)^3))^(1/2))*((2*c^3*x^2*(2*b^4*d - 60*a^2*c^2*d - 8*a^2*b^2*f - a*b^3*e + 20*a^3*c*f + 4*a*b^2*
c*d + 10*a^2*b*c*e))/(a^2*(4*a*c - b^2)) + (4*b*c^2*(2*b^4*d + 6*a^2*c^2*d - a*b^3*e - 2*a^3*c*f - 10*a*b^2*c*
d + 5*a^2*b*c*e))/(a^2*(4*a*c - b^2)) + (b*c^2*(a*e - 2*b*d + a^3*(-(2*b^4*d + 12*a^2*c^2*d - a*b^3*e - 4*a^3*
c*f - 12*a*b^2*c*d + 6*a^2*b*c*e)^2/(a^6*(4*a*c - b^2)^3))^(1/2))*(a*b + 3*b^2*x^2 - 10*a*c*x^2))/a^3))/(4*a^3
) + (c^3*(4*a^5*c*f^2 - 16*b^6*d^2 - 4*a^2*b^4*e^2 + 36*a^3*c^3*d^2 + 17*a^3*b^2*c*e^2 + 16*a*b^5*d*e - 216*a^
2*b^2*c^2*d^2 + 116*a*b^4*c*d^2 - 16*a^2*b^4*d*f + 8*a^3*b^3*e*f - 24*a^4*c^2*d*f - 92*a^2*b^3*c*d*e + 108*a^3
*b*c^2*d*e + 72*a^3*b^2*c*d*f - 36*a^4*b*c*e*f))/(a^4*(4*a*c - b^2)^2) - (2*c^4*x^2*(12*b^5*d^2 + 2*a^4*b*f^2
+ 3*a^2*b^3*e^2 + 138*a^2*b*c^2*d^2 - 12*a*b^4*d*e + 20*a^4*c*e*f - 82*a*b^3*c*d^2 - 10*a^3*b*c*e^2 + 14*a^2*b
^3*d*f - 60*a^3*c^2*d*e - 7*a^3*b^2*e*f + 61*a^2*b^2*c*d*e - 52*a^3*b*c*d*f))/(a^4*(4*a*c - b^2)^2))*(a*e - 2*
b*d + a^3*(-(2*b^4*d + 12*a^2*c^2*d - a*b^3*e - 4*a^3*c*f - 12*a*b^2*c*d + 6*a^2*b*c*e)^2/(a^6*(4*a*c - b^2)^3
))^(1/2)))/(4*a^3) + (c^4*(a*e - 2*b*d)*(2*b^2*d + 2*a^2*f - a*b*e - 6*a*c*d)^2)/(a^6*(4*a*c - b^2)^2) + (c^5*
x^2*(2*b^2*d + 2*a^2*f - a*b*e - 6*a*c*d)^3)/(a^6*(4*a*c - b^2)^3))*(((((2*b*d - a*e + a^3*(-(2*b^4*d + 12*a^2
*c^2*d - a*b^3*e - 4*a^3*c*f - 12*a*b^2*c*d + 6*a^2*b*c*e)^2/(a^6*(4*a*c - b^2)^3))^(1/2))*((2*c^3*x^2*(2*b^4*
d - 60*a^2*c^2*d - 8*a^2*b^2*f - a*b^3*e + 20*a^3*c*f + 4*a*b^2*c*d + 10*a^2*b*c*e))/(a^2*(4*a*c - b^2)) + (4*
b*c^2*(2*b^4*d + 6*a^2*c^2*d - a*b^3*e - 2*a^3*c*f - 10*a*b^2*c*d + 5*a^2*b*c*e))/(a^2*(4*a*c - b^2)) - (b*c^2
*(2*b*d - a*e + a^3*(-(2*b^4*d + 12*a^2*c^2*d - a*b^3*e - 4*a^3*c*f - 12*a*b^2*c*d + 6*a^2*b*c*e)^2/(a^6*(4*a*
c - b^2)^3))^(1/2))*(a*b + 3*b^2*x^2 - 10*a*c*x^2))/a^3))/(4*a^3) - (c^3*(4*a^5*c*f^2 - 16*b^6*d^2 - 4*a^2*b^4
*e^2 + 36*a^3*c^3*d^2 + 17*a^3*b^2*c*e^2 + 16*a*b^5*d*e - 216*a^2*b^2*c^2*d^2 + 116*a*b^4*c*d^2 - 16*a^2*b^4*d
*f + 8*a^3*b^3*e*f - 24*a^4*c^2*d*f - 92*a^2*b^3*c*d*e + 108*a^3*b*c^2*d*e + 72*a^3*b^2*c*d*f - 36*a^4*b*c*e*f
))/(a^4*(4*a*c - b^2)^2) + (2*c^4*x^2*(12*b^5*d^2 + 2*a^4*b*f^2 + 3*a^2*b^3*e^2 + 138*a^2*b*c^2*d^2 - 12*a*b^4
*d*e + 20*a^4*c*e*f - 82*a*b^3*c*d^2 - 10*a^3*b*c*e^2 + 14*a^2*b^3*d*f - 60*a^3*c^2*d*e - 7*a^3*b^2*e*f + 61*a
^2*b^2*c*d*e - 52*a^3*b*c*d*f))/(a^4*(4*a*c - b^2)^2))*(2*b*d - a*e + a^3*(-(2*b^4*d + 12*a^2*c^2*d - a*b^3*e
- 4*a^3*c*f - 12*a*b^2*c*d + 6*a^2*b*c*e)^2/(a^6*(4*a*c - b^2)^3))^(1/2)))/(4*a^3) + (c^4*(a*e - 2*b*d)*(2*b^2
*d + 2*a^2*f - a*b*e - 6*a*c*d)^2)/(a^6*(4*a*c - b^2)^2) + (c^5*x^2*(2*b^2*d + 2*a^2*f - a*b*e - 6*a*c*d)^3)/(
a^6*(4*a*c - b^2)^3)))*(4*b^7*d + 128*a^4*c^3*e - 2*a*b^6*e + 192*a^2*b^3*c^2*d - 96*a^3*b^2*c^2*e - 48*a*b^5*
c*d - 256*a^3*b*c^3*d + 24*a^2*b^4*c*e))/(2*(4*a^3*b^6 - 256*a^6*c^3 - 48*a^4*b^4*c + 192*a^5*b^2*c^2)) + (ata
n((x^2*((((216*a^3*c^8*d^3 - 8*b^6*c^5*d^3 - 8*a^6*c^5*f^3 + 72*a*b^4*c^6*d^3 - 216*a^4*c^7*d^2*f + 72*a^5*c^6
*d*f^2 - 216*a^2*b^2*c^7*d^3 + a^3*b^3*c^5*e^3 + 12*a*b^5*c^5*d^2*e + 108*a^3*b*c^7*d^2*e + 12*a^5*b*c^5*e*f^2
- 72*a^2*b^3*c^6*d^2*e - 6*a^2*b^4*c^5*d*e^2 + 18*a^3*b^2*c^6*d*e^2 - 24*a^2*b^4*c^5*d^2*f + 144*a^3*b^2*c^6*
d^2*f - 24*a^4*b^2*c^5*d*f^2 - 6*a^4*b^2*c^5*e^2*f - 72*a^4*b*c^6*d*e*f + 24*a^3*b^3*c^5*d*e*f)/(a^6*b^6 - 64*
a^9*c^3 - 12*a^7*b^4*c + 48*a^8*b^2*c^2) + (((80*a^6*b*c^6*e^2 - 1104*a^5*b*c^7*d^2 - 16*a^7*b*c^5*f^2 + 24*a^
2*b^7*c^4*d^2 - 260*a^3*b^5*c^5*d^2 + 932*a^4*b^3*c^6*d^2 + 6*a^4*b^5*c^4*e^2 - 44*a^5*b^3*c^5*e^2 + 4*a^6*b^3
*c^4*f^2 + 480*a^6*c^7*d*e - 160*a^7*c^6*e*f + 416*a^6*b*c^6*d*f - 24*a^3*b^6*c^4*d*e + 218*a^4*b^4*c^5*d*e -
608*a^5*b^2*c^6*d*e + 28*a^4*b^5*c^4*d*f - 216*a^5*b^3*c^5*d*f - 14*a^5*b^4*c^4*e*f + 96*a^6*b^2*c^5*e*f)/(a^6
*b^6 - 64*a^9*c^3 - 12*a^7*b^4*c + 48*a^8*b^2*c^2) + (((1920*a^8*c^7*d - 640*a^9*c^6*f - 4*a^4*b^8*c^3*d + 24*
a^5*b^6*c^4*d + 120*a^6*b^4*c^5*d - 1088*a^7*b^2*c^6*d + 2*a^5*b^7*c^3*e - 36*a^6*b^5*c^4*e + 192*a^7*b^3*c^5*
e + 16*a^6*b^6*c^3*f - 168*a^7*b^4*c^4*f + 576*a^8*b^2*c^5*f - 320*a^8*b*c^6*e)/(a^6*b^6 - 64*a^9*c^3 - 12*a^7
*b^4*c + 48*a^8*b^2*c^2) - ((2560*a^10*b*c^6 + 12*a^6*b^9*c^2 - 184*a^7*b^7*c^3 + 1056*a^8*b^5*c^4 - 2688*a^9*
b^3*c^5)*(4*b^7*d + 128*a^4*c^3*e - 2*a*b^6*e + 192*a^2*b^3*c^2*d - 96*a^3*b^2*c^2*e - 48*a*b^5*c*d - 256*a^3*
b*c^3*d + 24*a^2*b^4*c*e))/(2*(a^6*b^6 - 64*a^9*c^3 - 12*a^7*b^4*c + 48*a^8*b^2*c^2)*(4*a^3*b^6 - 256*a^6*c^3
- 48*a^4*b^4*c + 192*a^5*b^2*c^2)))*(4*b^7*d + 128*a^4*c^3*e - 2*a*b^6*e + 192*a^2*b^3*c^2*d - 96*a^3*b^2*c^2*
e - 48*a*b^5*c*d - 256*a^3*b*c^3*d + 24*a^2*b^4*c*e))/(2*(4*a^3*b^6 - 256*a^6*c^3 - 48*a^4*b^4*c + 192*a^5*b^2
*c^2)))*(4*b^7*d + 128*a^4*c^3*e - 2*a*b^6*e + ...

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