3.1.52 \(\int \frac {d+e x^2+f x^4}{x^7 (a+b x^2+c x^4)} \, dx\)

Optimal. Leaf size=244 \[ -\frac {-a b e-a (c d-a f)+b^2 d}{2 a^3 x^2}+\frac {b d-a e}{4 a^2 x^4}+\frac {\log \left (a+b x^2+c x^4\right ) \left (a^2 c e-a b^2 e-a b (2 c d-a f)+b^3 d\right )}{4 a^4}-\frac {\log (x) \left (a^2 c e-a b^2 e-a b (2 c d-a f)+b^3 d\right )}{a^4}-\frac {\tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (3 a^2 b c e+2 a^2 c (c d-a f)-a b^3 e-a b^2 (4 c d-a f)+b^4 d\right )}{2 a^4 \sqrt {b^2-4 a c}}-\frac {d}{6 a x^6} \]

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Rubi [A]  time = 0.57, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1663, 1628, 634, 618, 206, 628} \begin {gather*} -\frac {-a b e-a (c d-a f)+b^2 d}{2 a^3 x^2}+\frac {\log \left (a+b x^2+c x^4\right ) \left (a^2 c e-a b^2 e-a b (2 c d-a f)+b^3 d\right )}{4 a^4}-\frac {\tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (3 a^2 b c e+2 a^2 c (c d-a f)-a b^2 (4 c d-a f)-a b^3 e+b^4 d\right )}{2 a^4 \sqrt {b^2-4 a c}}-\frac {\log (x) \left (a^2 c e-a b^2 e-a b (2 c d-a f)+b^3 d\right )}{a^4}+\frac {b d-a e}{4 a^2 x^4}-\frac {d}{6 a x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2 + f*x^4)/(x^7*(a + b*x^2 + c*x^4)),x]

[Out]

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

Rule 206

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

Rule 618

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 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 634

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 1628

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 1663

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^7 \left (a+b x^2+c x^4\right )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {d+e x+f x^2}{x^4 \left (a+b x+c x^2\right )} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {d}{a x^4}+\frac {-b d+a e}{a^2 x^3}+\frac {b^2 d-a b e-a (c d-a f)}{a^3 x^2}+\frac {-b^3 d+a b^2 e-a^2 c e+a b (2 c d-a f)}{a^4 x}+\frac {b^4 d-a b^3 e+2 a^2 b c e+a^2 c (c d-a f)-a b^2 (3 c d-a f)+c \left (b^3 d-a b^2 e+a^2 c e-a b (2 c d-a f)\right ) x}{a^4 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac {d}{6 a x^6}+\frac {b d-a e}{4 a^2 x^4}-\frac {b^2 d-a b e-a (c d-a f)}{2 a^3 x^2}-\frac {\left (b^3 d-a b^2 e+a^2 c e-a b (2 c d-a f)\right ) \log (x)}{a^4}+\frac {\operatorname {Subst}\left (\int \frac {b^4 d-a b^3 e+2 a^2 b c e+a^2 c (c d-a f)-a b^2 (3 c d-a f)+c \left (b^3 d-a b^2 e+a^2 c e-a b (2 c d-a f)\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 a^4}\\ &=-\frac {d}{6 a x^6}+\frac {b d-a e}{4 a^2 x^4}-\frac {b^2 d-a b e-a (c d-a f)}{2 a^3 x^2}-\frac {\left (b^3 d-a b^2 e+a^2 c e-a b (2 c d-a f)\right ) \log (x)}{a^4}+\frac {\left (b^3 d-a b^2 e+a^2 c e-a b (2 c d-a f)\right ) \operatorname {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^4}+\frac {\left (b^4 d-a b^3 e+3 a^2 b c e+2 a^2 c (c d-a f)-a b^2 (4 c d-a f)\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^4}\\ &=-\frac {d}{6 a x^6}+\frac {b d-a e}{4 a^2 x^4}-\frac {b^2 d-a b e-a (c d-a f)}{2 a^3 x^2}-\frac {\left (b^3 d-a b^2 e+a^2 c e-a b (2 c d-a f)\right ) \log (x)}{a^4}+\frac {\left (b^3 d-a b^2 e+a^2 c e-a b (2 c d-a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 a^4}-\frac {\left (b^4 d-a b^3 e+3 a^2 b c e+2 a^2 c (c d-a f)-a b^2 (4 c d-a f)\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 a^4}\\ &=-\frac {d}{6 a x^6}+\frac {b d-a e}{4 a^2 x^4}-\frac {b^2 d-a b e-a (c d-a f)}{2 a^3 x^2}-\frac {\left (b^4 d-a b^3 e+3 a^2 b c e+2 a^2 c (c d-a f)-a b^2 (4 c d-a f)\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^4 \sqrt {b^2-4 a c}}-\frac {\left (b^3 d-a b^2 e+a^2 c e-a b (2 c d-a f)\right ) \log (x)}{a^4}+\frac {\left (b^3 d-a b^2 e+a^2 c e-a b (2 c d-a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 a^4}\\ \end {align*}

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Mathematica [A]  time = 0.35, size = 416, normalized size = 1.70 \begin {gather*} \frac {-\frac {2 a^3 d}{x^6}-12 \log (x) \left (a^2 c e-a b^2 e+a b (a f-2 c d)+b^3 d\right )+\frac {3 \log \left (-\sqrt {b^2-4 a c}+b+2 c x^2\right ) \left (a^2 c \left (e \sqrt {b^2-4 a c}-2 a f+2 c d\right )+a b^2 \left (-e \sqrt {b^2-4 a c}+a f-4 c d\right )+a b \left (-2 c d \sqrt {b^2-4 a c}+a f \sqrt {b^2-4 a c}+3 a c e\right )+b^3 \left (d \sqrt {b^2-4 a c}-a e\right )+b^4 d\right )}{\sqrt {b^2-4 a c}}+\frac {3 \log \left (\sqrt {b^2-4 a c}+b+2 c x^2\right ) \left (a^2 c \left (e \sqrt {b^2-4 a c}+2 a f-2 c d\right )-a b^2 \left (e \sqrt {b^2-4 a c}+a f-4 c d\right )+a b \left (-2 c d \sqrt {b^2-4 a c}+a f \sqrt {b^2-4 a c}-3 a c e\right )+b^3 \left (d \sqrt {b^2-4 a c}+a e\right )+b^4 (-d)\right )}{\sqrt {b^2-4 a c}}+\frac {3 a^2 (b d-a e)}{x^4}+\frac {6 a \left (a b e+a (c d-a f)+b^2 (-d)\right )}{x^2}}{12 a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x^2 + f*x^4)/(x^7*(a + b*x^2 + c*x^4)),x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d+e x^2+f x^4}{x^7 \left (a+b x^2+c x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x^2 + f*x^4)/(x^7*(a + b*x^2 + c*x^4)),x]

[Out]

IntegrateAlgebraic[(d + e*x^2 + f*x^4)/(x^7*(a + b*x^2 + c*x^4)), x]

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fricas [A]  time = 6.64, size = 834, normalized size = 3.42 \begin {gather*} \left [-\frac {3 \, \sqrt {b^{2} - 4 \, a c} {\left ({\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} d - {\left (a b^{3} - 3 \, a^{2} b c\right )} e + {\left (a^{2} b^{2} - 2 \, a^{3} c\right )} f\right )} x^{6} \log \left (\frac {2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c + {\left (2 \, c x^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) - 3 \, {\left ({\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} d - {\left (a b^{4} - 5 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} e + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} f\right )} x^{6} \log \left (c x^{4} + b x^{2} + a\right ) + 12 \, {\left ({\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} d - {\left (a b^{4} - 5 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} e + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} f\right )} x^{6} \log \relax (x) + 6 \, {\left ({\left (a b^{4} - 5 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} d - {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} e + {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} f\right )} x^{4} - 3 \, {\left ({\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} d - {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} e\right )} x^{2} + 2 \, {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} d}{12 \, {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x^{6}}, -\frac {6 \, \sqrt {-b^{2} + 4 \, a c} {\left ({\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} d - {\left (a b^{3} - 3 \, a^{2} b c\right )} e + {\left (a^{2} b^{2} - 2 \, a^{3} c\right )} f\right )} x^{6} \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - 3 \, {\left ({\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} d - {\left (a b^{4} - 5 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} e + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} f\right )} x^{6} \log \left (c x^{4} + b x^{2} + a\right ) + 12 \, {\left ({\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} d - {\left (a b^{4} - 5 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} e + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} f\right )} x^{6} \log \relax (x) + 6 \, {\left ({\left (a b^{4} - 5 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} d - {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} e + {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} f\right )} x^{4} - 3 \, {\left ({\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} d - {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} e\right )} x^{2} + 2 \, {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} d}{12 \, {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x^{6}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^4+e*x^2+d)/x^7/(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 1.94, size = 313, normalized size = 1.28 \begin {gather*} \frac {{\left (b^{3} d - 2 \, a b c d + a^{2} b f - a b^{2} e + a^{2} c e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{4}} - \frac {{\left (b^{3} d - 2 \, a b c d + a^{2} b f - a b^{2} e + a^{2} c e\right )} \log \left (x^{2}\right )}{2 \, a^{4}} + \frac {{\left (b^{4} d - 4 \, a b^{2} c d + 2 \, a^{2} c^{2} d + a^{2} b^{2} f - 2 \, a^{3} c f - a b^{3} e + 3 \, a^{2} b c e\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt {-b^{2} + 4 \, a c} a^{4}} + \frac {11 \, b^{3} d x^{6} - 22 \, a b c d x^{6} + 11 \, a^{2} b f x^{6} - 11 \, a b^{2} x^{6} e + 11 \, a^{2} c x^{6} e - 6 \, a b^{2} d x^{4} + 6 \, a^{2} c d x^{4} - 6 \, a^{3} f x^{4} + 6 \, a^{2} b x^{4} e + 3 \, a^{2} b d x^{2} - 3 \, a^{3} x^{2} e - 2 \, a^{3} d}{12 \, a^{4} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^4+e*x^2+d)/x^7/(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

1/4*(b^3*d - 2*a*b*c*d + a^2*b*f - a*b^2*e + a^2*c*e)*log(c*x^4 + b*x^2 + a)/a^4 - 1/2*(b^3*d - 2*a*b*c*d + a^
2*b*f - a*b^2*e + a^2*c*e)*log(x^2)/a^4 + 1/2*(b^4*d - 4*a*b^2*c*d + 2*a^2*c^2*d + a^2*b^2*f - 2*a^3*c*f - a*b
^3*e + 3*a^2*b*c*e)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*a^4) + 1/12*(11*b^3*d*x^6 - 2
2*a*b*c*d*x^6 + 11*a^2*b*f*x^6 - 11*a*b^2*x^6*e + 11*a^2*c*x^6*e - 6*a*b^2*d*x^4 + 6*a^2*c*d*x^4 - 6*a^3*f*x^4
+ 6*a^2*b*x^4*e + 3*a^2*b*d*x^2 - 3*a^3*x^2*e - 2*a^3*d)/(a^4*x^6)

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maple [B]  time = 0.01, size = 523, normalized size = 2.14 \begin {gather*} -\frac {c f \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, a}+\frac {b^{2} f \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, a^{2}}+\frac {3 b c e \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, a^{2}}+\frac {c^{2} d \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, a^{2}}-\frac {b^{3} e \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, a^{3}}-\frac {2 b^{2} c d \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, a^{3}}+\frac {b^{4} d \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, a^{4}}-\frac {b f \ln \relax (x )}{a^{2}}+\frac {b f \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 a^{2}}-\frac {c e \ln \relax (x )}{a^{2}}+\frac {c e \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 a^{2}}+\frac {b^{2} e \ln \relax (x )}{a^{3}}-\frac {b^{2} e \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 a^{3}}+\frac {2 b c d \ln \relax (x )}{a^{3}}-\frac {b c d \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 a^{3}}-\frac {b^{3} d \ln \relax (x )}{a^{4}}+\frac {b^{3} d \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 a^{4}}-\frac {f}{2 a \,x^{2}}+\frac {b e}{2 a^{2} x^{2}}+\frac {c d}{2 a^{2} x^{2}}-\frac {b^{2} d}{2 a^{3} x^{2}}-\frac {e}{4 a \,x^{4}}+\frac {b d}{4 a^{2} x^{4}}-\frac {d}{6 a \,x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^4+e*x^2+d)/x^7/(c*x^4+b*x^2+a),x)

[Out]

-1/6*d/a/x^6-1/4/a/x^4*e+1/4/a^2/x^4*b*d-1/2/a/x^2*f+1/2/a^2/x^2*b*e+1/2/a^2/x^2*c*d-1/2/a^3/x^2*b^2*d-1/a^2*l
n(x)*b*f-1/a^2*ln(x)*c*e+1/a^3*ln(x)*b^2*e+2/a^3*ln(x)*b*c*d-1/a^4*ln(x)*b^3*d+1/4/a^2*ln(c*x^4+b*x^2+a)*b*f+1
/4/a^2*c*ln(c*x^4+b*x^2+a)*e-1/4/a^3*ln(c*x^4+b*x^2+a)*b^2*e-1/2/a^3*c*ln(c*x^4+b*x^2+a)*b*d+1/4/a^4*ln(c*x^4+
b*x^2+a)*b^3*d-1/a/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*c*f+1/2/a^2/(4*a*c-b^2)^(1/2)*arcta
n((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b^2*f+3/2/a^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b*c*e+1
/a^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*c^2*d-1/2/a^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b
)/(4*a*c-b^2)^(1/2))*b^3*e-2/a^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b^2*c*d+1/2/a^4/(4*a*
c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b^4*d

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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^7/(c*x^4+b*x^2+a),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 details)Is 4*a*c-b^2 positive or negative?

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mupad [B]  time = 13.83, size = 9141, normalized size = 37.46

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x^2 + f*x^4)/(x^7*(a + b*x^2 + c*x^4)),x)

[Out]

(atan((16*a^12*(4*a*c - b^2)^(3/2)*(x^2*((((a^3*c^8*d^3 - b^6*c^5*d^3 - a^6*c^5*f^3 + 3*a*b^4*c^6*d^3 - 3*a^4*
c^7*d^2*f + 3*a^5*c^6*d*f^2 - 3*a^2*b^2*c^7*d^3 + a^3*b^3*c^5*e^3 + 3*a*b^5*c^5*d^2*e + 3*a^3*b*c^7*d^2*e + 3*
a^5*b*c^5*e*f^2 - 6*a^2*b^3*c^6*d^2*e - 3*a^2*b^4*c^5*d*e^2 + 3*a^3*b^2*c^6*d*e^2 - 3*a^2*b^4*c^5*d^2*f + 6*a^
3*b^2*c^6*d^2*f - 3*a^4*b^2*c^5*d*f^2 - 3*a^4*b^2*c^5*e^2*f - 6*a^4*b*c^6*d*e*f + 6*a^3*b^3*c^5*d*e*f)/a^9 - (
((11*a^5*b*c^6*d^2 - 5*a^6*b*c^5*e^2 + 6*a^7*b*c^4*f^2 + 6*a^3*b^5*c^4*d^2 - 17*a^4*b^3*c^5*d^2 + 6*a^5*b^3*c^
4*e^2 - 5*a^6*c^6*d*e + 5*a^7*c^5*e*f - 17*a^6*b*c^5*d*f - 12*a^4*b^4*c^4*d*e + 22*a^5*b^2*c^5*d*e + 12*a^5*b^
3*c^4*d*f - 12*a^6*b^2*c^4*e*f)/a^9 + (((20*a^9*c^4*f - 20*a^8*c^5*d + 2*a^6*b^4*c^3*d + 8*a^7*b^2*c^4*d - 2*a
^7*b^3*c^3*e + 2*a^8*b^2*c^3*f - 10*a^8*b*c^4*e)/a^9 + ((40*a^10*b*c^3 - 12*a^9*b^3*c^2)*(2*b^5*d + 2*a^2*b^3*
f - 8*a^3*c^2*e - 2*a*b^4*e - 12*a*b^3*c*d - 8*a^3*b*c*f + 16*a^2*b*c^2*d + 10*a^2*b^2*c*e))/(2*a^9*(16*a^5*c
- 4*a^4*b^2)))*(2*b^5*d + 2*a^2*b^3*f - 8*a^3*c^2*e - 2*a*b^4*e - 12*a*b^3*c*d - 8*a^3*b*c*f + 16*a^2*b*c^2*d
+ 10*a^2*b^2*c*e))/(2*(16*a^5*c - 4*a^4*b^2)))*(2*b^5*d + 2*a^2*b^3*f - 8*a^3*c^2*e - 2*a*b^4*e - 12*a*b^3*c*d
- 8*a^3*b*c*f + 16*a^2*b*c^2*d + 10*a^2*b^2*c*e))/(2*(16*a^5*c - 4*a^4*b^2)) + (((((20*a^9*c^4*f - 20*a^8*c^5
*d + 2*a^6*b^4*c^3*d + 8*a^7*b^2*c^4*d - 2*a^7*b^3*c^3*e + 2*a^8*b^2*c^3*f - 10*a^8*b*c^4*e)/a^9 + ((40*a^10*b
*c^3 - 12*a^9*b^3*c^2)*(2*b^5*d + 2*a^2*b^3*f - 8*a^3*c^2*e - 2*a*b^4*e - 12*a*b^3*c*d - 8*a^3*b*c*f + 16*a^2*
b*c^2*d + 10*a^2*b^2*c*e))/(2*a^9*(16*a^5*c - 4*a^4*b^2)))*(b^4*d + 2*a^2*c^2*d + a^2*b^2*f - a*b^3*e - 2*a^3*
c*f - 4*a*b^2*c*d + 3*a^2*b*c*e))/(4*a^4*(4*a*c - b^2)^(1/2)) + ((40*a^10*b*c^3 - 12*a^9*b^3*c^2)*(b^4*d + 2*a
^2*c^2*d + a^2*b^2*f - a*b^3*e - 2*a^3*c*f - 4*a*b^2*c*d + 3*a^2*b*c*e)*(2*b^5*d + 2*a^2*b^3*f - 8*a^3*c^2*e -
2*a*b^4*e - 12*a*b^3*c*d - 8*a^3*b*c*f + 16*a^2*b*c^2*d + 10*a^2*b^2*c*e))/(8*a^13*(4*a*c - b^2)^(1/2)*(16*a^
5*c - 4*a^4*b^2)))*(b^4*d + 2*a^2*c^2*d + a^2*b^2*f - a*b^3*e - 2*a^3*c*f - 4*a*b^2*c*d + 3*a^2*b*c*e))/(4*a^4
*(4*a*c - b^2)^(1/2)) + ((40*a^10*b*c^3 - 12*a^9*b^3*c^2)*(b^4*d + 2*a^2*c^2*d + a^2*b^2*f - a*b^3*e - 2*a^3*c
*f - 4*a*b^2*c*d + 3*a^2*b*c*e)^2*(2*b^5*d + 2*a^2*b^3*f - 8*a^3*c^2*e - 2*a*b^4*e - 12*a*b^3*c*d - 8*a^3*b*c*
f + 16*a^2*b*c^2*d + 10*a^2*b^2*c*e))/(32*a^17*(4*a*c - b^2)*(16*a^5*c - 4*a^4*b^2)))*(3*b^6*d - a^3*c^3*d + 3
*a^2*b^4*f + a^4*c^2*f - 3*a*b^5*e + 18*a^2*b^2*c^2*d - 15*a*b^4*c*d + 12*a^2*b^3*c*e - 9*a^3*b*c^2*e - 9*a^3*
b^2*c*f))/(8*a^3*c^2*(a^4*c^4*d^2 - 6*a^2*b^6*e^2 - 6*b^8*d^2 - 6*a^4*b^4*f^2 + 25*a^5*c^3*e^2 + a^6*c^2*f^2 +
36*a^3*b^4*c*e^2 + 24*a^5*b^2*c*f^2 + 12*a*b^7*d*e - 120*a^2*b^4*c^2*d^2 + 96*a^3*b^2*c^3*d^2 - 54*a^4*b^2*c^
2*e^2 + 48*a*b^6*c*d^2 - 12*a^2*b^6*d*f + 12*a^3*b^5*e*f - 2*a^5*c^3*d*f - 84*a^2*b^5*c*d*e - 97*a^4*b*c^3*d*e
+ 72*a^3*b^4*c*d*f - 60*a^4*b^3*c*e*f + 47*a^5*b*c^2*e*f + 168*a^3*b^3*c^2*d*e - 95*a^4*b^2*c^2*d*f)) + (((((
((20*a^9*c^4*f - 20*a^8*c^5*d + 2*a^6*b^4*c^3*d + 8*a^7*b^2*c^4*d - 2*a^7*b^3*c^3*e + 2*a^8*b^2*c^3*f - 10*a^8
*b*c^4*e)/a^9 + ((40*a^10*b*c^3 - 12*a^9*b^3*c^2)*(2*b^5*d + 2*a^2*b^3*f - 8*a^3*c^2*e - 2*a*b^4*e - 12*a*b^3*
c*d - 8*a^3*b*c*f + 16*a^2*b*c^2*d + 10*a^2*b^2*c*e))/(2*a^9*(16*a^5*c - 4*a^4*b^2)))*(b^4*d + 2*a^2*c^2*d + a
^2*b^2*f - a*b^3*e - 2*a^3*c*f - 4*a*b^2*c*d + 3*a^2*b*c*e))/(4*a^4*(4*a*c - b^2)^(1/2)) + ((40*a^10*b*c^3 - 1
2*a^9*b^3*c^2)*(b^4*d + 2*a^2*c^2*d + a^2*b^2*f - a*b^3*e - 2*a^3*c*f - 4*a*b^2*c*d + 3*a^2*b*c*e)*(2*b^5*d +
2*a^2*b^3*f - 8*a^3*c^2*e - 2*a*b^4*e - 12*a*b^3*c*d - 8*a^3*b*c*f + 16*a^2*b*c^2*d + 10*a^2*b^2*c*e))/(8*a^13
*(4*a*c - b^2)^(1/2)*(16*a^5*c - 4*a^4*b^2)))*(2*b^5*d + 2*a^2*b^3*f - 8*a^3*c^2*e - 2*a*b^4*e - 12*a*b^3*c*d
- 8*a^3*b*c*f + 16*a^2*b*c^2*d + 10*a^2*b^2*c*e))/(2*(16*a^5*c - 4*a^4*b^2)) + (((11*a^5*b*c^6*d^2 - 5*a^6*b*c
^5*e^2 + 6*a^7*b*c^4*f^2 + 6*a^3*b^5*c^4*d^2 - 17*a^4*b^3*c^5*d^2 + 6*a^5*b^3*c^4*e^2 - 5*a^6*c^6*d*e + 5*a^7*
c^5*e*f - 17*a^6*b*c^5*d*f - 12*a^4*b^4*c^4*d*e + 22*a^5*b^2*c^5*d*e + 12*a^5*b^3*c^4*d*f - 12*a^6*b^2*c^4*e*f
)/a^9 + (((20*a^9*c^4*f - 20*a^8*c^5*d + 2*a^6*b^4*c^3*d + 8*a^7*b^2*c^4*d - 2*a^7*b^3*c^3*e + 2*a^8*b^2*c^3*f
- 10*a^8*b*c^4*e)/a^9 + ((40*a^10*b*c^3 - 12*a^9*b^3*c^2)*(2*b^5*d + 2*a^2*b^3*f - 8*a^3*c^2*e - 2*a*b^4*e -
12*a*b^3*c*d - 8*a^3*b*c*f + 16*a^2*b*c^2*d + 10*a^2*b^2*c*e))/(2*a^9*(16*a^5*c - 4*a^4*b^2)))*(2*b^5*d + 2*a^
2*b^3*f - 8*a^3*c^2*e - 2*a*b^4*e - 12*a*b^3*c*d - 8*a^3*b*c*f + 16*a^2*b*c^2*d + 10*a^2*b^2*c*e))/(2*(16*a^5*
c - 4*a^4*b^2)))*(b^4*d + 2*a^2*c^2*d + a^2*b^2*f - a*b^3*e - 2*a^3*c*f - 4*a*b^2*c*d + 3*a^2*b*c*e))/(4*a^4*(
4*a*c - b^2)^(1/2)) - ((40*a^10*b*c^3 - 12*a^9*b^3*c^2)*(b^4*d + 2*a^2*c^2*d + a^2*b^2*f - a*b^3*e - 2*a^3*c*f
- 4*a*b^2*c*d + 3*a^2*b*c*e)^3)/(64*a^21*(4*a*c - b^2)^(3/2)))*(6*b^7*d + 6*a^2*b^5*f + 20*a^4*c^3*e - 6*a*b^
6*e + 84*a^2*b^3*c^2*d - 54*a^3*b^2*c^2*e - 42*a*b^5*c*d - 46*a^3*b*c^3*d + 36*a^2*b^4*c*e - 30*a^3*b^3*c*f +
26*a^4*b*c^2*f))/(16*a^3*c^2*(4*a*c - b^2)^(1/2)*(a^4*c^4*d^2 - 6*a^2*b^6*e^2 - 6*b^8*d^2 - 6*a^4*b^4*f^2 + 25
*a^5*c^3*e^2 + a^6*c^2*f^2 + 36*a^3*b^4*c*e^2 + 24*a^5*b^2*c*f^2 + 12*a*b^7*d*e - 120*a^2*b^4*c^2*d^2 + 96*a^3
*b^2*c^3*d^2 - 54*a^4*b^2*c^2*e^2 + 48*a*b^6*c*d^2 - 12*a^2*b^6*d*f + 12*a^3*b^5*e*f - 2*a^5*c^3*d*f - 84*a^2*
b^5*c*d*e - 97*a^4*b*c^3*d*e + 72*a^3*b^4*c*d*f - 60*a^4*b^3*c*e*f + 47*a^5*b*c^2*e*f + 168*a^3*b^3*c^2*d*e -
95*a^4*b^2*c^2*d*f))) - (((b^7*c^4*d^3 - 4*a*b^5*c^5*d^3 - 2*a^3*b*c^7*d^3 + a^6*b*c^4*f^3 + a^4*c^7*d^2*e + a
^6*c^5*e*f^2 + 5*a^2*b^3*c^6*d^3 - a^3*b^4*c^4*e^3 + a^4*b^2*c^5*e^3 - 2*a^5*c^6*d*e*f - 3*a*b^6*c^4*d^2*e + 2
*a^4*b*c^6*d*e^2 + 5*a^4*b*c^6*d^2*f - 4*a^5*b*c^5*d*f^2 - 2*a^5*b*c^5*e^2*f + 9*a^2*b^4*c^5*d^2*e + 3*a^2*b^5
*c^4*d*e^2 - 7*a^3*b^2*c^6*d^2*e - 6*a^3*b^3*c^5*d*e^2 + 3*a^2*b^5*c^4*d^2*f - 8*a^3*b^3*c^5*d^2*f + 3*a^4*b^3
*c^4*d*f^2 + 3*a^4*b^3*c^4*e^2*f - 3*a^5*b^2*c^4*e*f^2 - 6*a^3*b^4*c^4*d*e*f + 10*a^4*b^2*c^5*d*e*f)/a^9 - (((
a^6*c^6*d^2 + a^8*c^4*f^2 - 4*a^3*b^6*c^3*d^2 + 13*a^4*b^4*c^4*d^2 - 10*a^5*b^2*c^5*d^2 - 4*a^5*b^4*c^3*e^2 +
5*a^6*b^2*c^4*e^2 - 4*a^7*b^2*c^3*f^2 - 2*a^7*c^5*d*f + 6*a^6*b*c^5*d*e - 6*a^7*b*c^4*e*f + 8*a^4*b^5*c^3*d*e
- 18*a^5*b^3*c^4*d*e - 8*a^5*b^4*c^3*d*f + 14*a^6*b^2*c^4*d*f + 8*a^6*b^3*c^3*e*f)/a^9 - (((4*a^6*b^5*c^2*d -
12*a^7*b^3*c^3*d - 4*a^7*b^4*c^2*e + 8*a^8*b^2*c^3*e + 4*a^8*b^3*c^2*f + 4*a^8*b*c^4*d - 4*a^9*b*c^3*f)/a^9 -
(2*a*b^2*c^2*(2*b^5*d + 2*a^2*b^3*f - 8*a^3*c^2*e - 2*a*b^4*e - 12*a*b^3*c*d - 8*a^3*b*c*f + 16*a^2*b*c^2*d +
10*a^2*b^2*c*e))/(16*a^5*c - 4*a^4*b^2))*(2*b^5*d + 2*a^2*b^3*f - 8*a^3*c^2*e - 2*a*b^4*e - 12*a*b^3*c*d - 8*a
^3*b*c*f + 16*a^2*b*c^2*d + 10*a^2*b^2*c*e))/(2*(16*a^5*c - 4*a^4*b^2)))*(2*b^5*d + 2*a^2*b^3*f - 8*a^3*c^2*e
- 2*a*b^4*e - 12*a*b^3*c*d - 8*a^3*b*c*f + 16*a^2*b*c^2*d + 10*a^2*b^2*c*e))/(2*(16*a^5*c - 4*a^4*b^2)) - ((((
(4*a^6*b^5*c^2*d - 12*a^7*b^3*c^3*d - 4*a^7*b^4*c^2*e + 8*a^8*b^2*c^3*e + 4*a^8*b^3*c^2*f + 4*a^8*b*c^4*d - 4*
a^9*b*c^3*f)/a^9 - (2*a*b^2*c^2*(2*b^5*d + 2*a^2*b^3*f - 8*a^3*c^2*e - 2*a*b^4*e - 12*a*b^3*c*d - 8*a^3*b*c*f
+ 16*a^2*b*c^2*d + 10*a^2*b^2*c*e))/(16*a^5*c - 4*a^4*b^2))*(b^4*d + 2*a^2*c^2*d + a^2*b^2*f - a*b^3*e - 2*a^3
*c*f - 4*a*b^2*c*d + 3*a^2*b*c*e))/(4*a^4*(4*a*c - b^2)^(1/2)) - (b^2*c^2*(b^4*d + 2*a^2*c^2*d + a^2*b^2*f - a
*b^3*e - 2*a^3*c*f - 4*a*b^2*c*d + 3*a^2*b*c*e)*(2*b^5*d + 2*a^2*b^3*f - 8*a^3*c^2*e - 2*a*b^4*e - 12*a*b^3*c*
d - 8*a^3*b*c*f + 16*a^2*b*c^2*d + 10*a^2*b^2*c*e))/(2*a^3*(4*a*c - b^2)^(1/2)*(16*a^5*c - 4*a^4*b^2)))*(b^4*d
+ 2*a^2*c^2*d + a^2*b^2*f - a*b^3*e - 2*a^3*c*f - 4*a*b^2*c*d + 3*a^2*b*c*e))/(4*a^4*(4*a*c - b^2)^(1/2)) + (
b^2*c^2*(b^4*d + 2*a^2*c^2*d + a^2*b^2*f - a*b^3*e - 2*a^3*c*f - 4*a*b^2*c*d + 3*a^2*b*c*e)^2*(2*b^5*d + 2*a^2
*b^3*f - 8*a^3*c^2*e - 2*a*b^4*e - 12*a*b^3*c*d - 8*a^3*b*c*f + 16*a^2*b*c^2*d + 10*a^2*b^2*c*e))/(8*a^7*(4*a*
c - b^2)*(16*a^5*c - 4*a^4*b^2)))*(3*b^6*d - a^3*c^3*d + 3*a^2*b^4*f + a^4*c^2*f - 3*a*b^5*e + 18*a^2*b^2*c^2*
d - 15*a*b^4*c*d + 12*a^2*b^3*c*e - 9*a^3*b*c^2*e - 9*a^3*b^2*c*f))/(8*a^3*c^2*(a^4*c^4*d^2 - 6*a^2*b^6*e^2 -
6*b^8*d^2 - 6*a^4*b^4*f^2 + 25*a^5*c^3*e^2 + a^6*c^2*f^2 + 36*a^3*b^4*c*e^2 + 24*a^5*b^2*c*f^2 + 12*a*b^7*d*e
- 120*a^2*b^4*c^2*d^2 + 96*a^3*b^2*c^3*d^2 - 54*a^4*b^2*c^2*e^2 + 48*a*b^6*c*d^2 - 12*a^2*b^6*d*f + 12*a^3*b^5
*e*f - 2*a^5*c^3*d*f - 84*a^2*b^5*c*d*e - 97*a^4*b*c^3*d*e + 72*a^3*b^4*c*d*f - 60*a^4*b^3*c*e*f + 47*a^5*b*c^
2*e*f + 168*a^3*b^3*c^2*d*e - 95*a^4*b^2*c^2*d*f)) + (((((((4*a^6*b^5*c^2*d - 12*a^7*b^3*c^3*d - 4*a^7*b^4*c^2
*e + 8*a^8*b^2*c^3*e + 4*a^8*b^3*c^2*f + 4*a^8*b*c^4*d - 4*a^9*b*c^3*f)/a^9 - (2*a*b^2*c^2*(2*b^5*d + 2*a^2*b^
3*f - 8*a^3*c^2*e - 2*a*b^4*e - 12*a*b^3*c*d - 8*a^3*b*c*f + 16*a^2*b*c^2*d + 10*a^2*b^2*c*e))/(16*a^5*c - 4*a
^4*b^2))*(b^4*d + 2*a^2*c^2*d + a^2*b^2*f - a*b^3*e - 2*a^3*c*f - 4*a*b^2*c*d + 3*a^2*b*c*e))/(4*a^4*(4*a*c -
b^2)^(1/2)) - (b^2*c^2*(b^4*d + 2*a^2*c^2*d + a^2*b^2*f - a*b^3*e - 2*a^3*c*f - 4*a*b^2*c*d + 3*a^2*b*c*e)*(2*
b^5*d + 2*a^2*b^3*f - 8*a^3*c^2*e - 2*a*b^4*e - 12*a*b^3*c*d - 8*a^3*b*c*f + 16*a^2*b*c^2*d + 10*a^2*b^2*c*e))
/(2*a^3*(4*a*c - b^2)^(1/2)*(16*a^5*c - 4*a^4*b^2)))*(2*b^5*d + 2*a^2*b^3*f - 8*a^3*c^2*e - 2*a*b^4*e - 12*a*b
^3*c*d - 8*a^3*b*c*f + 16*a^2*b*c^2*d + 10*a^2*b^2*c*e))/(2*(16*a^5*c - 4*a^4*b^2)) - (((a^6*c^6*d^2 + a^8*c^4
*f^2 - 4*a^3*b^6*c^3*d^2 + 13*a^4*b^4*c^4*d^2 - 10*a^5*b^2*c^5*d^2 - 4*a^5*b^4*c^3*e^2 + 5*a^6*b^2*c^4*e^2 - 4
*a^7*b^2*c^3*f^2 - 2*a^7*c^5*d*f + 6*a^6*b*c^5*d*e - 6*a^7*b*c^4*e*f + 8*a^4*b^5*c^3*d*e - 18*a^5*b^3*c^4*d*e
- 8*a^5*b^4*c^3*d*f + 14*a^6*b^2*c^4*d*f + 8*a^6*b^3*c^3*e*f)/a^9 - (((4*a^6*b^5*c^2*d - 12*a^7*b^3*c^3*d - 4*
a^7*b^4*c^2*e + 8*a^8*b^2*c^3*e + 4*a^8*b^3*c^2*f + 4*a^8*b*c^4*d - 4*a^9*b*c^3*f)/a^9 - (2*a*b^2*c^2*(2*b^5*d
+ 2*a^2*b^3*f - 8*a^3*c^2*e - 2*a*b^4*e - 12*a*b^3*c*d - 8*a^3*b*c*f + 16*a^2*b*c^2*d + 10*a^2*b^2*c*e))/(16*
a^5*c - 4*a^4*b^2))*(2*b^5*d + 2*a^2*b^3*f - 8*a^3*c^2*e - 2*a*b^4*e - 12*a*b^3*c*d - 8*a^3*b*c*f + 16*a^2*b*c
^2*d + 10*a^2*b^2*c*e))/(2*(16*a^5*c - 4*a^4*b^2)))*(b^4*d + 2*a^2*c^2*d + a^2*b^2*f - a*b^3*e - 2*a^3*c*f - 4
*a*b^2*c*d + 3*a^2*b*c*e))/(4*a^4*(4*a*c - b^2)^(1/2)) + (b^2*c^2*(b^4*d + 2*a^2*c^2*d + a^2*b^2*f - a*b^3*e -
2*a^3*c*f - 4*a*b^2*c*d + 3*a^2*b*c*e)^3)/(16*a^11*(4*a*c - b^2)^(3/2)))*(6*b^7*d + 6*a^2*b^5*f + 20*a^4*c^3*
e - 6*a*b^6*e + 84*a^2*b^3*c^2*d - 54*a^3*b^2*c^2*e - 42*a*b^5*c*d - 46*a^3*b*c^3*d + 36*a^2*b^4*c*e - 30*a^3*
b^3*c*f + 26*a^4*b*c^2*f))/(16*a^3*c^2*(4*a*c - b^2)^(1/2)*(a^4*c^4*d^2 - 6*a^2*b^6*e^2 - 6*b^8*d^2 - 6*a^4*b^
4*f^2 + 25*a^5*c^3*e^2 + a^6*c^2*f^2 + 36*a^3*b^4*c*e^2 + 24*a^5*b^2*c*f^2 + 12*a*b^7*d*e - 120*a^2*b^4*c^2*d^
2 + 96*a^3*b^2*c^3*d^2 - 54*a^4*b^2*c^2*e^2 + 48*a*b^6*c*d^2 - 12*a^2*b^6*d*f + 12*a^3*b^5*e*f - 2*a^5*c^3*d*f
- 84*a^2*b^5*c*d*e - 97*a^4*b*c^3*d*e + 72*a^3*b^4*c*d*f - 60*a^4*b^3*c*e*f + 47*a^5*b*c^2*e*f + 168*a^3*b^3*
c^2*d*e - 95*a^4*b^2*c^2*d*f))))/(4*a^4*c^6*d^2 + b^8*c^2*d^2 + 4*a^6*c^4*f^2 - 8*a*b^6*c^3*d^2 + 20*a^2*b^4*c
^4*d^2 - 16*a^3*b^2*c^5*d^2 + a^2*b^6*c^2*e^2 - 6*a^3*b^4*c^3*e^2 + 9*a^4*b^2*c^4*e^2 + a^4*b^4*c^2*f^2 - 4*a^
5*b^2*c^3*f^2 - 8*a^5*c^5*d*f - 2*a*b^7*c^2*d*e + 12*a^4*b*c^5*d*e - 12*a^5*b*c^4*e*f + 14*a^2*b^5*c^3*d*e - 2
8*a^3*b^3*c^4*d*e + 2*a^2*b^6*c^2*d*f - 12*a^3*b^4*c^3*d*f + 20*a^4*b^2*c^4*d*f - 2*a^3*b^5*c^2*e*f + 10*a^4*b
^3*c^3*e*f))*(b^4*d + 2*a^2*c^2*d + a^2*b^2*f - a*b^3*e - 2*a^3*c*f - 4*a*b^2*c*d + 3*a^2*b*c*e))/(2*a^4*(4*a*
c - b^2)^(1/2)) - (log(((c^4*(b^2*d + a^2*f - a*b*e - a*c*d)^2*(b^3*d - a*b^2*e + a^2*b*f + a^2*c*e - 2*a*b*c*
d))/a^9 - (((c^3*(4*b^6*d^2 - a^5*c*f^2 + 4*a^2*b^4*e^2 - a^3*c^3*d^2 + 4*a^4*b^2*f^2 - 5*a^3*b^2*c*e^2 - 8*a*
b^5*d*e + 10*a^2*b^2*c^2*d^2 - 13*a*b^4*c*d^2 + 8*a^2*b^4*d*f - 8*a^3*b^3*e*f + 2*a^4*c^2*d*f + 18*a^2*b^3*c*d
*e - 6*a^3*b*c^2*d*e - 14*a^3*b^2*c*d*f + 6*a^4*b*c*e*f))/a^6 - (((4*b*c^2*(b^4*d + a^2*c^2*d + a^2*b^2*f - a*
b^3*e - a^3*c*f - 3*a*b^2*c*d + 2*a^2*b*c*e))/a^3 + (2*c^3*x^2*(b^4*d - 10*a^2*c^2*d + a^2*b^2*f - a*b^3*e + 1
0*a^3*c*f + 4*a*b^2*c*d - 5*a^2*b*c*e))/a^3 + (b*c^2*(a*b + 3*b^2*x^2 - 10*a*c*x^2)*(b^3*d + a^4*(-(b^4*d + 2*
a^2*c^2*d + a^2*b^2*f - a*b^3*e - 2*a^3*c*f - 4*a*b^2*c*d + 3*a^2*b*c*e)^2/(a^8*(4*a*c - b^2)))^(1/2) - a*b^2*
e + a^2*b*f + a^2*c*e - 2*a*b*c*d))/a^4)*(b^3*d + a^4*(-(b^4*d + 2*a^2*c^2*d + a^2*b^2*f - a*b^3*e - 2*a^3*c*f
- 4*a*b^2*c*d + 3*a^2*b*c*e)^2/(a^8*(4*a*c - b^2)))^(1/2) - a*b^2*e + a^2*b*f + a^2*c*e - 2*a*b*c*d))/(4*a^4)
+ (c^4*x^2*(6*b^5*d^2 + 6*a^4*b*f^2 + 6*a^2*b^3*e^2 + 11*a^2*b*c^2*d^2 - 12*a*b^4*d*e + 5*a^4*c*e*f - 17*a*b^
3*c*d^2 - 5*a^3*b*c*e^2 + 12*a^2*b^3*d*f - 5*a^3*c^2*d*e - 12*a^3*b^2*e*f + 22*a^2*b^2*c*d*e - 17*a^3*b*c*d*f)
)/a^6)*(b^3*d + a^4*(-(b^4*d + 2*a^2*c^2*d + a^2*b^2*f - a*b^3*e - 2*a^3*c*f - 4*a*b^2*c*d + 3*a^2*b*c*e)^2/(a
^8*(4*a*c - b^2)))^(1/2) - a*b^2*e + a^2*b*f + a^2*c*e - 2*a*b*c*d))/(4*a^4) + (c^5*x^2*(b^2*d + a^2*f - a*b*e
- a*c*d)^3)/a^9)*((c^4*(b^2*d + a^2*f - a*b*e - a*c*d)^2*(b^3*d - a*b^2*e + a^2*b*f + a^2*c*e - 2*a*b*c*d))/a
^9 - (((c^3*(4*b^6*d^2 - a^5*c*f^2 + 4*a^2*b^4*e^2 - a^3*c^3*d^2 + 4*a^4*b^2*f^2 - 5*a^3*b^2*c*e^2 - 8*a*b^5*d
*e + 10*a^2*b^2*c^2*d^2 - 13*a*b^4*c*d^2 + 8*a^2*b^4*d*f - 8*a^3*b^3*e*f + 2*a^4*c^2*d*f + 18*a^2*b^3*c*d*e -
6*a^3*b*c^2*d*e - 14*a^3*b^2*c*d*f + 6*a^4*b*c*e*f))/a^6 - (((4*b*c^2*(b^4*d + a^2*c^2*d + a^2*b^2*f - a*b^3*e
- a^3*c*f - 3*a*b^2*c*d + 2*a^2*b*c*e))/a^3 + (2*c^3*x^2*(b^4*d - 10*a^2*c^2*d + a^2*b^2*f - a*b^3*e + 10*a^3
*c*f + 4*a*b^2*c*d - 5*a^2*b*c*e))/a^3 + (b*c^2*(a*b + 3*b^2*x^2 - 10*a*c*x^2)*(b^3*d - a^4*(-(b^4*d + 2*a^2*c
^2*d + a^2*b^2*f - a*b^3*e - 2*a^3*c*f - 4*a*b^2*c*d + 3*a^2*b*c*e)^2/(a^8*(4*a*c - b^2)))^(1/2) - a*b^2*e + a
^2*b*f + a^2*c*e - 2*a*b*c*d))/a^4)*(b^3*d - a^4*(-(b^4*d + 2*a^2*c^2*d + a^2*b^2*f - a*b^3*e - 2*a^3*c*f - 4*
a*b^2*c*d + 3*a^2*b*c*e)^2/(a^8*(4*a*c - b^2)))^(1/2) - a*b^2*e + a^2*b*f + a^2*c*e - 2*a*b*c*d))/(4*a^4) + (c
^4*x^2*(6*b^5*d^2 + 6*a^4*b*f^2 + 6*a^2*b^3*e^2 + 11*a^2*b*c^2*d^2 - 12*a*b^4*d*e + 5*a^4*c*e*f - 17*a*b^3*c*d
^2 - 5*a^3*b*c*e^2 + 12*a^2*b^3*d*f - 5*a^3*c^2*d*e - 12*a^3*b^2*e*f + 22*a^2*b^2*c*d*e - 17*a^3*b*c*d*f))/a^6
)*(b^3*d - a^4*(-(b^4*d + 2*a^2*c^2*d + a^2*b^2*f - a*b^3*e - 2*a^3*c*f - 4*a*b^2*c*d + 3*a^2*b*c*e)^2/(a^8*(4
*a*c - b^2)))^(1/2) - a*b^2*e + a^2*b*f + a^2*c*e - 2*a*b*c*d))/(4*a^4) + (c^5*x^2*(b^2*d + a^2*f - a*b*e - a*
c*d)^3)/a^9))*(2*b^5*d + 2*a^2*b^3*f - 8*a^3*c^2*e - 2*a*b^4*e - 12*a*b^3*c*d - 8*a^3*b*c*f + 16*a^2*b*c^2*d +
10*a^2*b^2*c*e))/(2*(16*a^5*c - 4*a^4*b^2)) - (log(x)*(b^3*d - a*b^2*e + a^2*b*f + a^2*c*e - 2*a*b*c*d))/a^4
- (d/(6*a) + (x^4*(b^2*d + a^2*f - a*b*e - a*c*d))/(2*a^3) + (x^2*(a*e - b*d))/(4*a^2))/x^6

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sympy [F(-1)]  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**7/(c*x**4+b*x**2+a),x)

[Out]

Timed out

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