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

Optimal. Leaf size=203 \[ \frac {x^2 \left (-c (a f+b e)+b^2 f+c^2 d\right )}{2 c^3}+\frac {\log \left (a+b x^2+c x^4\right ) \left (-b c (c d-2 a f)-a c^2 e+b^3 (-f)+b^2 c e\right )}{4 c^4}+\frac {\tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-b^2 c (c d-4 a f)-3 a b c^2 e+2 a c^2 (c d-a f)+b^4 (-f)+b^3 c e\right )}{2 c^4 \sqrt {b^2-4 a c}}+\frac {x^4 (c e-b f)}{4 c^2}+\frac {f x^6}{6 c} \]

[Out]

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

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Rubi [A]  time = 0.42, antiderivative size = 203, 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} \[ \frac {x^2 \left (-c (a f+b e)+b^2 f+c^2 d\right )}{2 c^3}+\frac {\log \left (a+b x^2+c x^4\right ) \left (-b c (c d-2 a f)-a c^2 e+b^2 c e+b^3 (-f)\right )}{4 c^4}+\frac {\tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-b^2 c (c d-4 a f)-3 a b c^2 e+2 a c^2 (c d-a f)+b^3 c e+b^4 (-f)\right )}{2 c^4 \sqrt {b^2-4 a c}}+\frac {x^4 (c e-b f)}{4 c^2}+\frac {f x^6}{6 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((c^2*d + b^2*f - c*(b*e + a*f))*x^2)/(2*c^3) + ((c*e - b*f)*x^4)/(4*c^2) + (f*x^6)/(6*c) + ((b^3*c*e - 3*a*b*
c^2*e - b^4*f - b^2*c*(c*d - 4*a*f) + 2*a*c^2*(c*d - a*f))*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^4*Sq
rt[b^2 - 4*a*c]) + ((b^2*c*e - a*c^2*e - b^3*f - b*c*(c*d - 2*a*f))*Log[a + b*x^2 + c*x^4])/(4*c^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 {x^5 \left (d+e x^2+f x^4\right )}{a+b x^2+c x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2 \left (d+e x+f x^2\right )}{a+b x+c x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {c^2 d+b^2 f-c (b e+a f)}{c^3}+\frac {(c e-b f) x}{c^2}+\frac {f x^2}{c}-\frac {a \left (c^2 d+b^2 f-c (b e+a f)\right )-\left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right ) x}{c^3 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {\left (c^2 d+b^2 f-c (b e+a f)\right ) x^2}{2 c^3}+\frac {(c e-b f) x^4}{4 c^2}+\frac {f x^6}{6 c}-\frac {\operatorname {Subst}\left (\int \frac {a \left (c^2 d+b^2 f-c (b e+a f)\right )-\left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 c^3}\\ &=\frac {\left (c^2 d+b^2 f-c (b e+a f)\right ) x^2}{2 c^3}+\frac {(c e-b f) x^4}{4 c^2}+\frac {f x^6}{6 c}+\frac {\left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right ) \operatorname {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^4}-\frac {\left (b^3 c e-3 a b c^2 e-b^4 f-b^2 c (c d-4 a f)+2 a c^2 (c d-a f)\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^4}\\ &=\frac {\left (c^2 d+b^2 f-c (b e+a f)\right ) x^2}{2 c^3}+\frac {(c e-b f) x^4}{4 c^2}+\frac {f x^6}{6 c}+\frac {\left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 c^4}+\frac {\left (b^3 c e-3 a b c^2 e-b^4 f-b^2 c (c d-4 a f)+2 a c^2 (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 c^4}\\ &=\frac {\left (c^2 d+b^2 f-c (b e+a f)\right ) x^2}{2 c^3}+\frac {(c e-b f) x^4}{4 c^2}+\frac {f x^6}{6 c}+\frac {\left (b^3 c e-3 a b c^2 e-b^4 f-b^2 c (c d-4 a f)+2 a c^2 (c d-a f)\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^4 \sqrt {b^2-4 a c}}+\frac {\left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 c^4}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 193, normalized size = 0.95 \[ \frac {6 c x^2 \left (-c (a f+b e)+b^2 f+c^2 d\right )-3 \log \left (a+b x^2+c x^4\right ) \left (b c (c d-2 a f)+a c^2 e+b^3 f-b^2 c e\right )+\frac {6 \tan ^{-1}\left (\frac {b+2 c x^2}{\sqrt {4 a c-b^2}}\right ) \left (b^2 c (c d-4 a f)+3 a b c^2 e+2 a c^2 (a f-c d)+b^4 f-b^3 c e\right )}{\sqrt {4 a c-b^2}}+3 c^2 x^4 (c e-b f)+2 c^3 f x^6}{12 c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.96, size = 677, normalized size = 3.33 \[ \left [\frac {2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} f x^{6} + 3 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} e - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} f\right )} x^{4} + 6 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} e + {\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} f\right )} x^{2} + 3 \, \sqrt {b^{2} - 4 \, a c} {\left ({\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d - {\left (b^{3} c - 3 \, a b c^{2}\right )} e + {\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} f\right )} \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^{3} c^{2} - 4 \, a b c^{3}\right )} d - {\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} e + {\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} f\right )} \log \left (c x^{4} + b x^{2} + a\right )}{12 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )}}, \frac {2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} f x^{6} + 3 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} e - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} f\right )} x^{4} + 6 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} e + {\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} f\right )} x^{2} - 6 \, \sqrt {-b^{2} + 4 \, a c} {\left ({\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d - {\left (b^{3} c - 3 \, a b c^{2}\right )} e + {\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} f\right )} \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^{3} c^{2} - 4 \, a b c^{3}\right )} d - {\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} e + {\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} f\right )} \log \left (c x^{4} + b x^{2} + a\right )}{12 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 2.00, size = 214, normalized size = 1.05 \[ \frac {2 \, c^{2} f x^{6} - 3 \, b c f x^{4} + 3 \, c^{2} x^{4} e + 6 \, c^{2} d x^{2} + 6 \, b^{2} f x^{2} - 6 \, a c f x^{2} - 6 \, b c x^{2} e}{12 \, c^{3}} - \frac {{\left (b c^{2} d + b^{3} f - 2 \, a b c f - b^{2} c e + a c^{2} e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, c^{4}} + \frac {{\left (b^{2} c^{2} d - 2 \, a c^{3} d + b^{4} f - 4 \, a b^{2} c f + 2 \, a^{2} c^{2} f - b^{3} c e + 3 \, a b c^{2} e\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt {-b^{2} + 4 \, a c} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.01, size = 474, normalized size = 2.33 \[ \frac {f \,x^{6}}{6 c}-\frac {b f \,x^{4}}{4 c^{2}}+\frac {e \,x^{4}}{4 c}+\frac {a^{2} f \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{2}}-\frac {2 a \,b^{2} f \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{3}}+\frac {3 a b e \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, c^{2}}-\frac {a d \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c}-\frac {a f \,x^{2}}{2 c^{2}}+\frac {b^{4} f \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, c^{4}}-\frac {b^{3} e \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, c^{3}}+\frac {b^{2} d \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, c^{2}}+\frac {b^{2} f \,x^{2}}{2 c^{3}}-\frac {b e \,x^{2}}{2 c^{2}}+\frac {d \,x^{2}}{2 c}+\frac {a b f \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c^{3}}-\frac {a e \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 c^{2}}-\frac {b^{3} f \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 c^{4}}+\frac {b^{2} e \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 c^{3}}-\frac {b d \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*f*x^6/c-1/4/c^2*x^4*b*f+1/4/c*x^4*e-1/2/c^2*x^2*a*f+1/2/c^3*x^2*b^2*f-1/2/c^2*x^2*b*e+1/2/c*x^2*d+1/2/c^3*
ln(c*x^4+b*x^2+a)*a*b*f-1/4/c^2*ln(c*x^4+b*x^2+a)*a*e-1/4/c^4*ln(c*x^4+b*x^2+a)*b^3*f+1/4/c^3*ln(c*x^4+b*x^2+a
)*b^2*e-1/4/c^2*ln(c*x^4+b*x^2+a)*b*d+1/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*a^2*f-2/c^
3/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*a*b^2*f+3/2/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)
/(4*a*c-b^2)^(1/2))*a*b*e-1/c/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*a*d+1/2/c^4/(4*a*c-b^2)^
(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b^4*f-1/2/c^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/
2))*b^3*e+1/2/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b^2*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(x^5*(f*x^4+e*x^2+d)/(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 = 1.63, size = 2295, normalized size = 11.31 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^4*(e/(4*c) - (b*f)/(4*c^2)) - x^2*((b*(e/c - (b*f)/c^2))/(2*c) - d/(2*c) + (a*f)/(2*c^2)) + (log(a + b*x^2 +
c*x^4)*(2*b^5*f - 8*a^2*c^3*e + 2*b^3*c^2*d - 2*b^4*c*e - 8*a*b*c^3*d - 12*a*b^3*c*f + 10*a*b^2*c^2*e + 16*a^
2*b*c^2*f))/(2*(16*a*c^5 - 4*b^2*c^4)) + (f*x^6)/(6*c) + (atan((2*c^6*(4*a*c - b^2)*(x^2*(((((6*b^2*c^6*d + 4*
a^2*c^6*f - 6*b^3*c^5*e + 6*b^4*c^4*f - 4*a*c^7*d + 10*a*b*c^6*e - 16*a*b^2*c^5*f)/c^6 + (4*b*c^2*(2*b^5*f - 8
*a^2*c^3*e + 2*b^3*c^2*d - 2*b^4*c*e - 8*a*b*c^3*d - 12*a*b^3*c*f + 10*a*b^2*c^2*e + 16*a^2*b*c^2*f))/(16*a*c^
5 - 4*b^2*c^4))*(b^4*f + b^2*c^2*d + 2*a^2*c^2*f - 2*a*c^3*d - b^3*c*e + 3*a*b*c^2*e - 4*a*b^2*c*f))/(8*c^4*(4
*a*c - b^2)^(1/2)) + (b*(b^4*f + b^2*c^2*d + 2*a^2*c^2*f - 2*a*c^3*d - b^3*c*e + 3*a*b*c^2*e - 4*a*b^2*c*f)*(2
*b^5*f - 8*a^2*c^3*e + 2*b^3*c^2*d - 2*b^4*c*e - 8*a*b*c^3*d - 12*a*b^3*c*f + 10*a*b^2*c^2*e + 16*a^2*b*c^2*f)
)/(2*c^2*(4*a*c - b^2)^(1/2)*(16*a*c^5 - 4*b^2*c^4)))/a - (b*((b^7*f^2 + b^3*c^4*d^2 + b^5*c^2*e^2 - 3*a*b^3*c
^3*e^2 + 2*a^2*b*c^4*e^2 - 2*a^3*b*c^3*f^2 - 2*b^6*c*e*f + 7*a^2*b^3*c^2*f^2 - a*b*c^5*d^2 - 5*a*b^5*c*f^2 - a
^2*c^5*d*e - 2*b^4*c^3*d*e + a^3*c^4*e*f + 2*b^5*c^2*d*f + 4*a*b^2*c^4*d*e - 6*a*b^3*c^3*d*f + 3*a^2*b*c^4*d*f
+ 8*a*b^4*c^2*e*f - 8*a^2*b^2*c^3*e*f)/c^6 + (((6*b^2*c^6*d + 4*a^2*c^6*f - 6*b^3*c^5*e + 6*b^4*c^4*f - 4*a*c
^7*d + 10*a*b*c^6*e - 16*a*b^2*c^5*f)/c^6 + (4*b*c^2*(2*b^5*f - 8*a^2*c^3*e + 2*b^3*c^2*d - 2*b^4*c*e - 8*a*b*
c^3*d - 12*a*b^3*c*f + 10*a*b^2*c^2*e + 16*a^2*b*c^2*f))/(16*a*c^5 - 4*b^2*c^4))*(2*b^5*f - 8*a^2*c^3*e + 2*b^
3*c^2*d - 2*b^4*c*e - 8*a*b*c^3*d - 12*a*b^3*c*f + 10*a*b^2*c^2*e + 16*a^2*b*c^2*f))/(2*(16*a*c^5 - 4*b^2*c^4)
) - (b*(b^4*f + b^2*c^2*d + 2*a^2*c^2*f - 2*a*c^3*d - b^3*c*e + 3*a*b*c^2*e - 4*a*b^2*c*f)^2)/(2*c^6*(4*a*c -
b^2))))/(2*a*(4*a*c - b^2)^(1/2))) + ((((8*a^2*c^6*e + 8*a*b*c^6*d - 8*a*b^2*c^5*e + 8*a*b^3*c^4*f - 16*a^2*b*
c^5*f)/c^6 + (8*a*c^2*(2*b^5*f - 8*a^2*c^3*e + 2*b^3*c^2*d - 2*b^4*c*e - 8*a*b*c^3*d - 12*a*b^3*c*f + 10*a*b^2
*c^2*e + 16*a^2*b*c^2*f))/(16*a*c^5 - 4*b^2*c^4))*(b^4*f + b^2*c^2*d + 2*a^2*c^2*f - 2*a*c^3*d - b^3*c*e + 3*a
*b*c^2*e - 4*a*b^2*c*f))/(8*c^4*(4*a*c - b^2)^(1/2)) + (a*(b^4*f + b^2*c^2*d + 2*a^2*c^2*f - 2*a*c^3*d - b^3*c
*e + 3*a*b*c^2*e - 4*a*b^2*c*f)*(2*b^5*f - 8*a^2*c^3*e + 2*b^3*c^2*d - 2*b^4*c*e - 8*a*b*c^3*d - 12*a*b^3*c*f
+ 10*a*b^2*c^2*e + 16*a^2*b*c^2*f))/(c^2*(4*a*c - b^2)^(1/2)*(16*a*c^5 - 4*b^2*c^4)))/a - (b*((a*b^6*f^2 + a^3
*c^4*e^2 + a*b^2*c^4*d^2 + a*b^4*c^2*e^2 - 4*a^2*b^4*c*f^2 - 2*a^2*b^2*c^3*e^2 + 4*a^3*b^2*c^2*f^2 - 2*a*b^3*c
^3*d*e + 2*a^2*b*c^4*d*e + 2*a*b^4*c^2*d*f - 4*a^3*b*c^3*e*f - 4*a^2*b^2*c^3*d*f + 6*a^2*b^3*c^2*e*f - 2*a*b^5
*c*e*f)/c^6 + (((8*a^2*c^6*e + 8*a*b*c^6*d - 8*a*b^2*c^5*e + 8*a*b^3*c^4*f - 16*a^2*b*c^5*f)/c^6 + (8*a*c^2*(2
*b^5*f - 8*a^2*c^3*e + 2*b^3*c^2*d - 2*b^4*c*e - 8*a*b*c^3*d - 12*a*b^3*c*f + 10*a*b^2*c^2*e + 16*a^2*b*c^2*f)
)/(16*a*c^5 - 4*b^2*c^4))*(2*b^5*f - 8*a^2*c^3*e + 2*b^3*c^2*d - 2*b^4*c*e - 8*a*b*c^3*d - 12*a*b^3*c*f + 10*a
*b^2*c^2*e + 16*a^2*b*c^2*f))/(2*(16*a*c^5 - 4*b^2*c^4)) - (a*(b^4*f + b^2*c^2*d + 2*a^2*c^2*f - 2*a*c^3*d - b
^3*c*e + 3*a*b*c^2*e - 4*a*b^2*c*f)^2)/(c^6*(4*a*c - b^2))))/(2*a*(4*a*c - b^2)^(1/2))))/(b^8*f^2 + 4*a^2*c^6*
d^2 + b^4*c^4*d^2 + 4*a^4*c^4*f^2 + b^6*c^2*e^2 - 4*a*b^2*c^5*d^2 - 6*a*b^4*c^3*e^2 - 2*b^7*c*e*f + 9*a^2*b^2*
c^4*e^2 + 20*a^2*b^4*c^2*f^2 - 16*a^3*b^2*c^3*f^2 - 8*a*b^6*c*f^2 - 8*a^3*c^5*d*f - 2*b^5*c^3*d*e + 2*b^6*c^2*
d*f + 10*a*b^3*c^4*d*e - 12*a^2*b*c^5*d*e - 12*a*b^4*c^3*d*f + 14*a*b^5*c^2*e*f + 12*a^3*b*c^4*e*f + 20*a^2*b^
2*c^4*d*f - 28*a^2*b^3*c^3*e*f))*(b^4*f + b^2*c^2*d + 2*a^2*c^2*f - 2*a*c^3*d - b^3*c*e + 3*a*b*c^2*e - 4*a*b^
2*c*f))/(2*c^4*(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(x**5*(f*x**4+e*x**2+d)/(c*x**4+b*x**2+a),x)

[Out]

Timed out

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