3.53 \(\int \frac {d+e x^2+f x^4}{x^5 (a+b x^2+c x^4)} \, dx\)

Optimal. Leaf size=174 \[ -\frac {\log \left (a+b x^2+c x^4\right ) \left (-a b e-a (c d-a f)+b^2 d\right )}{4 a^3}+\frac {\log (x) \left (-a b e-a (c d-a f)+b^2 d\right )}{a^3}+\frac {b d-a e}{2 a^2 x^2}+\frac {\tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (2 a^2 c e-a b^2 e-a b (3 c d-a f)+b^3 d\right )}{2 a^3 \sqrt {b^2-4 a c}}-\frac {d}{4 a x^4} \]

[Out]

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

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Rubi [A]  time = 0.41, antiderivative size = 174, 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} \[ -\frac {\log \left (a+b x^2+c x^4\right ) \left (-a b e-a (c d-a f)+b^2 d\right )}{4 a^3}+\frac {\tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (2 a^2 c e-a b^2 e-a b (3 c d-a f)+b^3 d\right )}{2 a^3 \sqrt {b^2-4 a c}}+\frac {\log (x) \left (-a b e-a (c d-a f)+b^2 d\right )}{a^3}+\frac {b d-a e}{2 a^2 x^2}-\frac {d}{4 a x^4} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2 + f*x^4)/(x^5*(a + b*x^2 + c*x^4)),x]

[Out]

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

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

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Mathematica [A]  time = 0.35, size = 314, normalized size = 1.80 \[ -\frac {\frac {a^2 d}{x^4}-4 \log (x) \left (-a b e+a (a f-c d)+b^2 d\right )+\frac {\log \left (-\sqrt {b^2-4 a c}+b+2 c x^2\right ) \left (a b \left (-e \sqrt {b^2-4 a c}+a f-3 c d\right )+a \left (-c d \sqrt {b^2-4 a c}+a f \sqrt {b^2-4 a c}+2 a c e\right )+b^2 \left (d \sqrt {b^2-4 a c}-a e\right )+b^3 d\right )}{\sqrt {b^2-4 a c}}+\frac {\log \left (\sqrt {b^2-4 a c}+b+2 c x^2\right ) \left (-a b \left (e \sqrt {b^2-4 a c}+a f-3 c d\right )+a \left (a f \sqrt {b^2-4 a c}-c \left (d \sqrt {b^2-4 a c}+2 a e\right )\right )+b^2 \left (d \sqrt {b^2-4 a c}+a e\right )+b^3 (-d)\right )}{\sqrt {b^2-4 a c}}+\frac {2 a (a e-b d)}{x^2}}{4 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x^2 + f*x^4)/(x^5*(a + b*x^2 + c*x^4)),x]

[Out]

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

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fricas [A]  time = 2.54, size = 609, normalized size = 3.50 \[ \left [\frac {{\left (a^{2} b f + {\left (b^{3} - 3 \, a b c\right )} d - {\left (a b^{2} - 2 \, a^{2} c\right )} e\right )} \sqrt {b^{2} - 4 \, a c} x^{4} \log \left (\frac {2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c + {\left (2 \, c x^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) - {\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d - {\left (a b^{3} - 4 \, a^{2} b c\right )} e + {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} f\right )} x^{4} \log \left (c x^{4} + b x^{2} + a\right ) + 4 \, {\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d - {\left (a b^{3} - 4 \, a^{2} b c\right )} e + {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} f\right )} x^{4} \log \relax (x) + 2 \, {\left ({\left (a b^{3} - 4 \, a^{2} b c\right )} d - {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} e\right )} x^{2} - {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d}{4 \, {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x^{4}}, \frac {2 \, {\left (a^{2} b f + {\left (b^{3} - 3 \, a b c\right )} d - {\left (a b^{2} - 2 \, a^{2} c\right )} e\right )} \sqrt {-b^{2} + 4 \, a c} x^{4} \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - {\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d - {\left (a b^{3} - 4 \, a^{2} b c\right )} e + {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} f\right )} x^{4} \log \left (c x^{4} + b x^{2} + a\right ) + 4 \, {\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d - {\left (a b^{3} - 4 \, a^{2} b c\right )} e + {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} f\right )} x^{4} \log \relax (x) + 2 \, {\left ({\left (a b^{3} - 4 \, a^{2} b c\right )} d - {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} e\right )} x^{2} - {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d}{4 \, {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^4+e*x^2+d)/x^5/(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 1.72, size = 212, normalized size = 1.22 \[ -\frac {{\left (b^{2} d - a c d + a^{2} f - a b e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{3}} + \frac {{\left (b^{2} d - a c d + a^{2} f - a b e\right )} \log \left (x^{2}\right )}{2 \, a^{3}} - \frac {{\left (b^{3} d - 3 \, a b c d + a^{2} b f - a b^{2} e + 2 \, a^{2} c e\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt {-b^{2} + 4 \, a c} a^{3}} - \frac {3 \, b^{2} d x^{4} - 3 \, a c d x^{4} + 3 \, a^{2} f x^{4} - 3 \, a b x^{4} e - 2 \, a b d x^{2} + 2 \, a^{2} x^{2} e + a^{2} d}{4 \, a^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^4+e*x^2+d)/x^5/(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.01, size = 356, normalized size = 2.05 \[ -\frac {b f \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, a}-\frac {c e \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, a}+\frac {b^{2} e \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, a^{2}}+\frac {3 b c d \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, a^{2}}-\frac {b^{3} d \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, a^{3}}+\frac {f \ln \relax (x )}{a}-\frac {f \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 a}-\frac {b e \ln \relax (x )}{a^{2}}+\frac {b e \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 a^{2}}-\frac {c d \ln \relax (x )}{a^{2}}+\frac {c d \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 a^{2}}+\frac {b^{2} d \ln \relax (x )}{a^{3}}-\frac {b^{2} d \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 a^{3}}-\frac {e}{2 a \,x^{2}}+\frac {b d}{2 a^{2} x^{2}}-\frac {d}{4 a \,x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^4+e*x^2+d)/x^5/(c*x^4+b*x^2+a),x)

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^4+e*x^2+d)/x^5/(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 = 9.92, size = 6187, normalized size = 35.56 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x^2 + f*x^4)/(x^5*(a + b*x^2 + c*x^4)),x)

[Out]

(log(x)*(b^2*d + a^2*f - a*b*e - a*c*d))/a^3 - (d/(4*a) + (x^2*(a*e - b*d))/(2*a^2))/x^4 + (log(((((((2*c^3*x^
2*(b^3*d - a*b^2*e + 5*a^2*b*f - 10*a^2*c*e + 5*a*b*c*d))/a^2 + (4*b*c^2*(b^3*d - a*b^2*e + a^2*b*f + a^2*c*e
- 2*a*b*c*d))/a^2 + (b*c^2*(a*b + 3*b^2*x^2 - 10*a*c*x^2)*(b^2*d + a^2*f + a^3*(-(b^3*d - a*b^2*e + a^2*b*f +
2*a^2*c*e - 3*a*b*c*d)^2/(a^6*(4*a*c - b^2)))^(1/2) - a*b*e - a*c*d))/a^3)*(b^2*d + a^2*f + a^3*(-(b^3*d - a*b
^2*e + a^2*b*f + 2*a^2*c*e - 3*a*b*c*d)^2/(a^6*(4*a*c - b^2)))^(1/2) - a*b*e - a*c*d))/(4*a^3) + (c^3*(a*e - b
*d)*(4*b^3*d - 4*a*b^2*e + 4*a^2*b*f + a^2*c*e - 5*a*b*c*d))/a^4 + (c^4*x^2*(a*e - b*d)*(6*b^2*d + 5*a^2*f - 6
*a*b*e - 5*a*c*d))/a^4)*(b^2*d + a^2*f + a^3*(-(b^3*d - a*b^2*e + a^2*b*f + 2*a^2*c*e - 3*a*b*c*d)^2/(a^6*(4*a
*c - b^2)))^(1/2) - a*b*e - a*c*d))/(4*a^3) + (c^4*(a*e - b*d)^2*(b^2*d + a^2*f - a*b*e - a*c*d))/a^6 - (c^5*x
^2*(a*e - b*d)^3)/a^6)*((((c^3*(a*e - b*d)*(4*b^3*d - 4*a*b^2*e + 4*a^2*b*f + a^2*c*e - 5*a*b*c*d))/a^4 - (((2
*c^3*x^2*(b^3*d - a*b^2*e + 5*a^2*b*f - 10*a^2*c*e + 5*a*b*c*d))/a^2 + (4*b*c^2*(b^3*d - a*b^2*e + a^2*b*f + a
^2*c*e - 2*a*b*c*d))/a^2 - (b*c^2*(a*b + 3*b^2*x^2 - 10*a*c*x^2)*(a^3*(-(b^3*d - a*b^2*e + a^2*b*f + 2*a^2*c*e
- 3*a*b*c*d)^2/(a^6*(4*a*c - b^2)))^(1/2) - a^2*f - b^2*d + a*b*e + a*c*d))/a^3)*(a^3*(-(b^3*d - a*b^2*e + a^
2*b*f + 2*a^2*c*e - 3*a*b*c*d)^2/(a^6*(4*a*c - b^2)))^(1/2) - a^2*f - b^2*d + a*b*e + a*c*d))/(4*a^3) + (c^4*x
^2*(a*e - b*d)*(6*b^2*d + 5*a^2*f - 6*a*b*e - 5*a*c*d))/a^4)*(a^3*(-(b^3*d - a*b^2*e + a^2*b*f + 2*a^2*c*e - 3
*a*b*c*d)^2/(a^6*(4*a*c - b^2)))^(1/2) - a^2*f - b^2*d + a*b*e + a*c*d))/(4*a^3) - (c^4*(a*e - b*d)^2*(b^2*d +
a^2*f - a*b*e - a*c*d))/a^6 + (c^5*x^2*(a*e - b*d)^3)/a^6))*(2*b^4*d + 8*a^2*c^2*d + 2*a^2*b^2*f - 2*a*b^3*e
- 8*a^3*c*f - 10*a*b^2*c*d + 8*a^2*b*c*e))/(2*(16*a^4*c - 4*a^3*b^2)) - (atan((16*a^9*(4*a*c - b^2)^(3/2)*(x^2
*((((a^3*c^5*e^3 - b^3*c^5*d^3 + 3*a*b^2*c^5*d^2*e - 3*a^2*b*c^5*d*e^2)/a^6 - (((6*a^4*b*c^4*e^2 - 5*a^3*b*c^5
*d^2 + 6*a^2*b^3*c^4*d^2 + 5*a^4*c^5*d*e - 5*a^5*c^4*e*f + 5*a^4*b*c^4*d*f - 12*a^3*b^2*c^4*d*e)/a^6 + (((2*a^
4*b^3*c^3*d - 20*a^6*c^4*e - 2*a^5*b^2*c^3*e + 10*a^5*b*c^4*d + 10*a^6*b*c^3*f)/a^6 + ((40*a^7*b*c^3 - 12*a^6*
b^3*c^2)*(2*b^4*d + 8*a^2*c^2*d + 2*a^2*b^2*f - 2*a*b^3*e - 8*a^3*c*f - 10*a*b^2*c*d + 8*a^2*b*c*e))/(2*a^6*(1
6*a^4*c - 4*a^3*b^2)))*(2*b^4*d + 8*a^2*c^2*d + 2*a^2*b^2*f - 2*a*b^3*e - 8*a^3*c*f - 10*a*b^2*c*d + 8*a^2*b*c
*e))/(2*(16*a^4*c - 4*a^3*b^2)))*(2*b^4*d + 8*a^2*c^2*d + 2*a^2*b^2*f - 2*a*b^3*e - 8*a^3*c*f - 10*a*b^2*c*d +
8*a^2*b*c*e))/(2*(16*a^4*c - 4*a^3*b^2)) + (((((2*a^4*b^3*c^3*d - 20*a^6*c^4*e - 2*a^5*b^2*c^3*e + 10*a^5*b*c
^4*d + 10*a^6*b*c^3*f)/a^6 + ((40*a^7*b*c^3 - 12*a^6*b^3*c^2)*(2*b^4*d + 8*a^2*c^2*d + 2*a^2*b^2*f - 2*a*b^3*e
- 8*a^3*c*f - 10*a*b^2*c*d + 8*a^2*b*c*e))/(2*a^6*(16*a^4*c - 4*a^3*b^2)))*(b^3*d - a*b^2*e + a^2*b*f + 2*a^2
*c*e - 3*a*b*c*d))/(4*a^3*(4*a*c - b^2)^(1/2)) + ((40*a^7*b*c^3 - 12*a^6*b^3*c^2)*(b^3*d - a*b^2*e + a^2*b*f +
2*a^2*c*e - 3*a*b*c*d)*(2*b^4*d + 8*a^2*c^2*d + 2*a^2*b^2*f - 2*a*b^3*e - 8*a^3*c*f - 10*a*b^2*c*d + 8*a^2*b*
c*e))/(8*a^9*(4*a*c - b^2)^(1/2)*(16*a^4*c - 4*a^3*b^2)))*(b^3*d - a*b^2*e + a^2*b*f + 2*a^2*c*e - 3*a*b*c*d))
/(4*a^3*(4*a*c - b^2)^(1/2)) + ((40*a^7*b*c^3 - 12*a^6*b^3*c^2)*(b^3*d - a*b^2*e + a^2*b*f + 2*a^2*c*e - 3*a*b
*c*d)^2*(2*b^4*d + 8*a^2*c^2*d + 2*a^2*b^2*f - 2*a*b^3*e - 8*a^3*c*f - 10*a*b^2*c*d + 8*a^2*b*c*e))/(32*a^12*(
4*a*c - b^2)*(16*a^4*c - 4*a^3*b^2)))*(3*b^5*d + 3*a^2*b^3*f - a^3*c^2*e - 3*a*b^4*e - 12*a*b^3*c*d - 8*a^3*b*
c*f + 9*a^2*b*c^2*d + 9*a^2*b^2*c*e))/(8*a^3*c^2*(25*a^5*c*f^2 - 6*b^6*d^2 - 6*a^2*b^4*e^2 + 25*a^3*c^3*d^2 -
6*a^4*b^2*f^2 + a^4*c^2*e^2 + 24*a^3*b^2*c*e^2 + 12*a*b^5*d*e - 54*a^2*b^2*c^2*d^2 + 36*a*b^4*c*d^2 - 12*a^2*b
^4*d*f + 12*a^3*b^3*e*f - 50*a^4*c^2*d*f - 60*a^2*b^3*c*d*e + 47*a^3*b*c^2*d*e + 61*a^3*b^2*c*d*f - 49*a^4*b*c
*e*f)) + (((((((2*a^4*b^3*c^3*d - 20*a^6*c^4*e - 2*a^5*b^2*c^3*e + 10*a^5*b*c^4*d + 10*a^6*b*c^3*f)/a^6 + ((40
*a^7*b*c^3 - 12*a^6*b^3*c^2)*(2*b^4*d + 8*a^2*c^2*d + 2*a^2*b^2*f - 2*a*b^3*e - 8*a^3*c*f - 10*a*b^2*c*d + 8*a
^2*b*c*e))/(2*a^6*(16*a^4*c - 4*a^3*b^2)))*(b^3*d - a*b^2*e + a^2*b*f + 2*a^2*c*e - 3*a*b*c*d))/(4*a^3*(4*a*c
- b^2)^(1/2)) + ((40*a^7*b*c^3 - 12*a^6*b^3*c^2)*(b^3*d - a*b^2*e + a^2*b*f + 2*a^2*c*e - 3*a*b*c*d)*(2*b^4*d
+ 8*a^2*c^2*d + 2*a^2*b^2*f - 2*a*b^3*e - 8*a^3*c*f - 10*a*b^2*c*d + 8*a^2*b*c*e))/(8*a^9*(4*a*c - b^2)^(1/2)*
(16*a^4*c - 4*a^3*b^2)))*(2*b^4*d + 8*a^2*c^2*d + 2*a^2*b^2*f - 2*a*b^3*e - 8*a^3*c*f - 10*a*b^2*c*d + 8*a^2*b
*c*e))/(2*(16*a^4*c - 4*a^3*b^2)) + (((6*a^4*b*c^4*e^2 - 5*a^3*b*c^5*d^2 + 6*a^2*b^3*c^4*d^2 + 5*a^4*c^5*d*e -
5*a^5*c^4*e*f + 5*a^4*b*c^4*d*f - 12*a^3*b^2*c^4*d*e)/a^6 + (((2*a^4*b^3*c^3*d - 20*a^6*c^4*e - 2*a^5*b^2*c^3
*e + 10*a^5*b*c^4*d + 10*a^6*b*c^3*f)/a^6 + ((40*a^7*b*c^3 - 12*a^6*b^3*c^2)*(2*b^4*d + 8*a^2*c^2*d + 2*a^2*b^
2*f - 2*a*b^3*e - 8*a^3*c*f - 10*a*b^2*c*d + 8*a^2*b*c*e))/(2*a^6*(16*a^4*c - 4*a^3*b^2)))*(2*b^4*d + 8*a^2*c^
2*d + 2*a^2*b^2*f - 2*a*b^3*e - 8*a^3*c*f - 10*a*b^2*c*d + 8*a^2*b*c*e))/(2*(16*a^4*c - 4*a^3*b^2)))*(b^3*d -
a*b^2*e + a^2*b*f + 2*a^2*c*e - 3*a*b*c*d))/(4*a^3*(4*a*c - b^2)^(1/2)) - ((40*a^7*b*c^3 - 12*a^6*b^3*c^2)*(b^
3*d - a*b^2*e + a^2*b*f + 2*a^2*c*e - 3*a*b*c*d)^3)/(64*a^15*(4*a*c - b^2)^(3/2)))*(6*b^6*d - 20*a^3*c^3*d + 6
*a^2*b^4*f + 20*a^4*c^2*f - 6*a*b^5*e + 54*a^2*b^2*c^2*d - 36*a*b^4*c*d + 30*a^2*b^3*c*e - 26*a^3*b*c^2*e - 28
*a^3*b^2*c*f))/(16*a^3*c^2*(4*a*c - b^2)^(1/2)*(25*a^5*c*f^2 - 6*b^6*d^2 - 6*a^2*b^4*e^2 + 25*a^3*c^3*d^2 - 6*
a^4*b^2*f^2 + a^4*c^2*e^2 + 24*a^3*b^2*c*e^2 + 12*a*b^5*d*e - 54*a^2*b^2*c^2*d^2 + 36*a*b^4*c*d^2 - 12*a^2*b^4
*d*f + 12*a^3*b^3*e*f - 50*a^4*c^2*d*f - 60*a^2*b^3*c*d*e + 47*a^3*b*c^2*d*e + 61*a^3*b^2*c*d*f - 49*a^4*b*c*e
*f))) - (((b^4*c^4*d^3 - a*b^2*c^5*d^3 - a^3*b*c^4*e^3 - a^3*c^5*d*e^2 + a^4*c^4*e^2*f - 3*a*b^3*c^4*d^2*e + 2
*a^2*b*c^5*d^2*e + 3*a^2*b^2*c^4*d*e^2 + a^2*b^2*c^4*d^2*f - 2*a^3*b*c^4*d*e*f)/a^6 - (((a^5*c^4*e^2 - 4*a^2*b
^4*c^3*d^2 + 5*a^3*b^2*c^4*d^2 - 4*a^4*b^2*c^3*e^2 - 6*a^4*b*c^4*d*e + 4*a^5*b*c^3*e*f + 8*a^3*b^3*c^3*d*e - 4
*a^4*b^2*c^3*d*f)/a^6 - (((4*a^4*b^4*c^2*d - 8*a^5*b^2*c^3*d - 4*a^5*b^3*c^2*e + 4*a^6*b^2*c^2*f + 4*a^6*b*c^3
*e)/a^6 - (2*a*b^2*c^2*(2*b^4*d + 8*a^2*c^2*d + 2*a^2*b^2*f - 2*a*b^3*e - 8*a^3*c*f - 10*a*b^2*c*d + 8*a^2*b*c
*e))/(16*a^4*c - 4*a^3*b^2))*(2*b^4*d + 8*a^2*c^2*d + 2*a^2*b^2*f - 2*a*b^3*e - 8*a^3*c*f - 10*a*b^2*c*d + 8*a
^2*b*c*e))/(2*(16*a^4*c - 4*a^3*b^2)))*(2*b^4*d + 8*a^2*c^2*d + 2*a^2*b^2*f - 2*a*b^3*e - 8*a^3*c*f - 10*a*b^2
*c*d + 8*a^2*b*c*e))/(2*(16*a^4*c - 4*a^3*b^2)) - (((((4*a^4*b^4*c^2*d - 8*a^5*b^2*c^3*d - 4*a^5*b^3*c^2*e + 4
*a^6*b^2*c^2*f + 4*a^6*b*c^3*e)/a^6 - (2*a*b^2*c^2*(2*b^4*d + 8*a^2*c^2*d + 2*a^2*b^2*f - 2*a*b^3*e - 8*a^3*c*
f - 10*a*b^2*c*d + 8*a^2*b*c*e))/(16*a^4*c - 4*a^3*b^2))*(b^3*d - a*b^2*e + a^2*b*f + 2*a^2*c*e - 3*a*b*c*d))/
(4*a^3*(4*a*c - b^2)^(1/2)) - (b^2*c^2*(b^3*d - a*b^2*e + a^2*b*f + 2*a^2*c*e - 3*a*b*c*d)*(2*b^4*d + 8*a^2*c^
2*d + 2*a^2*b^2*f - 2*a*b^3*e - 8*a^3*c*f - 10*a*b^2*c*d + 8*a^2*b*c*e))/(2*a^2*(4*a*c - b^2)^(1/2)*(16*a^4*c
- 4*a^3*b^2)))*(b^3*d - a*b^2*e + a^2*b*f + 2*a^2*c*e - 3*a*b*c*d))/(4*a^3*(4*a*c - b^2)^(1/2)) + (b^2*c^2*(b^
3*d - a*b^2*e + a^2*b*f + 2*a^2*c*e - 3*a*b*c*d)^2*(2*b^4*d + 8*a^2*c^2*d + 2*a^2*b^2*f - 2*a*b^3*e - 8*a^3*c*
f - 10*a*b^2*c*d + 8*a^2*b*c*e))/(8*a^5*(4*a*c - b^2)*(16*a^4*c - 4*a^3*b^2)))*(3*b^5*d + 3*a^2*b^3*f - a^3*c^
2*e - 3*a*b^4*e - 12*a*b^3*c*d - 8*a^3*b*c*f + 9*a^2*b*c^2*d + 9*a^2*b^2*c*e))/(8*a^3*c^2*(25*a^5*c*f^2 - 6*b^
6*d^2 - 6*a^2*b^4*e^2 + 25*a^3*c^3*d^2 - 6*a^4*b^2*f^2 + a^4*c^2*e^2 + 24*a^3*b^2*c*e^2 + 12*a*b^5*d*e - 54*a^
2*b^2*c^2*d^2 + 36*a*b^4*c*d^2 - 12*a^2*b^4*d*f + 12*a^3*b^3*e*f - 50*a^4*c^2*d*f - 60*a^2*b^3*c*d*e + 47*a^3*
b*c^2*d*e + 61*a^3*b^2*c*d*f - 49*a^4*b*c*e*f)) + (((((((4*a^4*b^4*c^2*d - 8*a^5*b^2*c^3*d - 4*a^5*b^3*c^2*e +
4*a^6*b^2*c^2*f + 4*a^6*b*c^3*e)/a^6 - (2*a*b^2*c^2*(2*b^4*d + 8*a^2*c^2*d + 2*a^2*b^2*f - 2*a*b^3*e - 8*a^3*
c*f - 10*a*b^2*c*d + 8*a^2*b*c*e))/(16*a^4*c - 4*a^3*b^2))*(b^3*d - a*b^2*e + a^2*b*f + 2*a^2*c*e - 3*a*b*c*d)
)/(4*a^3*(4*a*c - b^2)^(1/2)) - (b^2*c^2*(b^3*d - a*b^2*e + a^2*b*f + 2*a^2*c*e - 3*a*b*c*d)*(2*b^4*d + 8*a^2*
c^2*d + 2*a^2*b^2*f - 2*a*b^3*e - 8*a^3*c*f - 10*a*b^2*c*d + 8*a^2*b*c*e))/(2*a^2*(4*a*c - b^2)^(1/2)*(16*a^4*
c - 4*a^3*b^2)))*(2*b^4*d + 8*a^2*c^2*d + 2*a^2*b^2*f - 2*a*b^3*e - 8*a^3*c*f - 10*a*b^2*c*d + 8*a^2*b*c*e))/(
2*(16*a^4*c - 4*a^3*b^2)) - (((a^5*c^4*e^2 - 4*a^2*b^4*c^3*d^2 + 5*a^3*b^2*c^4*d^2 - 4*a^4*b^2*c^3*e^2 - 6*a^4
*b*c^4*d*e + 4*a^5*b*c^3*e*f + 8*a^3*b^3*c^3*d*e - 4*a^4*b^2*c^3*d*f)/a^6 - (((4*a^4*b^4*c^2*d - 8*a^5*b^2*c^3
*d - 4*a^5*b^3*c^2*e + 4*a^6*b^2*c^2*f + 4*a^6*b*c^3*e)/a^6 - (2*a*b^2*c^2*(2*b^4*d + 8*a^2*c^2*d + 2*a^2*b^2*
f - 2*a*b^3*e - 8*a^3*c*f - 10*a*b^2*c*d + 8*a^2*b*c*e))/(16*a^4*c - 4*a^3*b^2))*(2*b^4*d + 8*a^2*c^2*d + 2*a^
2*b^2*f - 2*a*b^3*e - 8*a^3*c*f - 10*a*b^2*c*d + 8*a^2*b*c*e))/(2*(16*a^4*c - 4*a^3*b^2)))*(b^3*d - a*b^2*e +
a^2*b*f + 2*a^2*c*e - 3*a*b*c*d))/(4*a^3*(4*a*c - b^2)^(1/2)) + (b^2*c^2*(b^3*d - a*b^2*e + a^2*b*f + 2*a^2*c*
e - 3*a*b*c*d)^3)/(16*a^8*(4*a*c - b^2)^(3/2)))*(6*b^6*d - 20*a^3*c^3*d + 6*a^2*b^4*f + 20*a^4*c^2*f - 6*a*b^5
*e + 54*a^2*b^2*c^2*d - 36*a*b^4*c*d + 30*a^2*b^3*c*e - 26*a^3*b*c^2*e - 28*a^3*b^2*c*f))/(16*a^3*c^2*(4*a*c -
b^2)^(1/2)*(25*a^5*c*f^2 - 6*b^6*d^2 - 6*a^2*b^4*e^2 + 25*a^3*c^3*d^2 - 6*a^4*b^2*f^2 + a^4*c^2*e^2 + 24*a^3*
b^2*c*e^2 + 12*a*b^5*d*e - 54*a^2*b^2*c^2*d^2 + 36*a*b^4*c*d^2 - 12*a^2*b^4*d*f + 12*a^3*b^3*e*f - 50*a^4*c^2*
d*f - 60*a^2*b^3*c*d*e + 47*a^3*b*c^2*d*e + 61*a^3*b^2*c*d*f - 49*a^4*b*c*e*f))))/(4*a^4*c^4*e^2 + b^6*c^2*d^2
- 6*a*b^4*c^3*d^2 + 9*a^2*b^2*c^4*d^2 + a^2*b^4*c^2*e^2 - 4*a^3*b^2*c^3*e^2 + a^4*b^2*c^2*f^2 - 2*a*b^5*c^2*d
*e - 12*a^3*b*c^4*d*e + 4*a^4*b*c^3*e*f + 10*a^2*b^3*c^3*d*e + 2*a^2*b^4*c^2*d*f - 6*a^3*b^2*c^3*d*f - 2*a^3*b
^3*c^2*e*f))*(b^3*d - a*b^2*e + a^2*b*f + 2*a^2*c*e - 3*a*b*c*d))/(2*a^3*(4*a*c - b^2)^(1/2))

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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**4+e*x**2+d)/x**5/(c*x**4+b*x**2+a),x)

[Out]

Timed out

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