### 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 $\frac{x^2 \left (-c (a f+b e)+b^2 f+c^2 d\right )}{2 c^3}+\frac{\log \left (a+b x^2+c x^4\right ) \left (-b c (c d-2 a f)-a c^2 e+b^2 c e+b^3 (-f)\right )}{4 c^4}+\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (-b^2 c (c d-4 a f)-3 a b c^2 e+2 a c^2 (c d-a f)+b^3 c e+b^4 (-f)\right )}{2 c^4 \sqrt{b^2-4 a c}}+\frac{x^4 (c e-b f)}{4 c^2}+\frac{f x^6}{6 c}$

[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)

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Rubi [A]  time = 0.423681, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 30, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.2, 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 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]

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 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 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 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 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]

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*}

Mathematica [A]  time = 0.143298, size = 193, normalized size = 0.95 $\frac{6 c x^2 \left (-c (a f+b e)+b^2 f+c^2 d\right )-3 \log \left (a+b x^2+c x^4\right ) \left (b c (c d-2 a f)+a c^2 e-b^2 c e+b^3 f\right )+\frac{6 \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right ) \left (b^2 c (c d-4 a f)+3 a b c^2 e+2 a c^2 (a f-c d)-b^3 c e+b^4 f\right )}{\sqrt{4 a c-b^2}}+3 c^2 x^4 (c e-b f)+2 c^3 f x^6}{12 c^4}$

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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Maple [B]  time = 0.006, size = 474, normalized size = 2.3 \begin{align*}{\frac{f{x}^{6}}{6\,c}}-{\frac{{x}^{4}bf}{4\,{c}^{2}}}+{\frac{{x}^{4}e}{4\,c}}-{\frac{{x}^{2}af}{2\,{c}^{2}}}+{\frac{{b}^{2}f{x}^{2}}{2\,{c}^{3}}}-{\frac{be{x}^{2}}{2\,{c}^{2}}}+{\frac{{x}^{2}d}{2\,c}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) abf}{2\,{c}^{3}}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) ae}{4\,{c}^{2}}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{3}f}{4\,{c}^{4}}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{2}e}{4\,{c}^{3}}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) bd}{4\,{c}^{2}}}+{\frac{{a}^{2}f}{{c}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-2\,{\frac{a{b}^{2}f}{{c}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{3\,abe}{2\,{c}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{ad}{c}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{4}f}{2\,{c}^{4}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{{b}^{3}e}{2\,{c}^{3}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{2}d}{2\,{c}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \end{align*}

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*b^2*f*x^2-1/2/c^2*b*e*x^2+1/2/c*d*x^2+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., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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

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Fricas [A]  time = 2.82308, size = 1404, normalized size = 6.92 \begin{align*} \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 ] \end{align*}

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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Sympy [B]  time = 35.7045, size = 1044, normalized size = 5.14 \begin{align*} \text{result too large to display} \end{align*}

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]

(-sqrt(-4*a*c + b**2)*(2*a**2*c**2*f - 4*a*b**2*c*f + 3*a*b*c**2*e - 2*a*c**3*d + b**4*f - b**3*c*e + b**2*c**
2*d)/(4*c**4*(4*a*c - b**2)) + (2*a*b*c*f - a*c**2*e - b**3*f + b**2*c*e - b*c**2*d)/(4*c**4))*log(x**2 + (-3*
a**2*b*c*f + 2*a**2*c**2*e + a*b**3*f - a*b**2*c*e + a*b*c**2*d + 8*a*c**4*(-sqrt(-4*a*c + b**2)*(2*a**2*c**2*
f - 4*a*b**2*c*f + 3*a*b*c**2*e - 2*a*c**3*d + b**4*f - b**3*c*e + b**2*c**2*d)/(4*c**4*(4*a*c - b**2)) + (2*a
*b*c*f - a*c**2*e - b**3*f + b**2*c*e - b*c**2*d)/(4*c**4)) - 2*b**2*c**3*(-sqrt(-4*a*c + b**2)*(2*a**2*c**2*f
- 4*a*b**2*c*f + 3*a*b*c**2*e - 2*a*c**3*d + b**4*f - b**3*c*e + b**2*c**2*d)/(4*c**4*(4*a*c - b**2)) + (2*a*
b*c*f - a*c**2*e - b**3*f + b**2*c*e - b*c**2*d)/(4*c**4)))/(2*a**2*c**2*f - 4*a*b**2*c*f + 3*a*b*c**2*e - 2*a
*c**3*d + b**4*f - b**3*c*e + b**2*c**2*d)) + (sqrt(-4*a*c + b**2)*(2*a**2*c**2*f - 4*a*b**2*c*f + 3*a*b*c**2*
e - 2*a*c**3*d + b**4*f - b**3*c*e + b**2*c**2*d)/(4*c**4*(4*a*c - b**2)) + (2*a*b*c*f - a*c**2*e - b**3*f + b
**2*c*e - b*c**2*d)/(4*c**4))*log(x**2 + (-3*a**2*b*c*f + 2*a**2*c**2*e + a*b**3*f - a*b**2*c*e + a*b*c**2*d +
8*a*c**4*(sqrt(-4*a*c + b**2)*(2*a**2*c**2*f - 4*a*b**2*c*f + 3*a*b*c**2*e - 2*a*c**3*d + b**4*f - b**3*c*e +
b**2*c**2*d)/(4*c**4*(4*a*c - b**2)) + (2*a*b*c*f - a*c**2*e - b**3*f + b**2*c*e - b*c**2*d)/(4*c**4)) - 2*b*
*2*c**3*(sqrt(-4*a*c + b**2)*(2*a**2*c**2*f - 4*a*b**2*c*f + 3*a*b*c**2*e - 2*a*c**3*d + b**4*f - b**3*c*e + b
**2*c**2*d)/(4*c**4*(4*a*c - b**2)) + (2*a*b*c*f - a*c**2*e - b**3*f + b**2*c*e - b*c**2*d)/(4*c**4)))/(2*a**2
*c**2*f - 4*a*b**2*c*f + 3*a*b*c**2*e - 2*a*c**3*d + b**4*f - b**3*c*e + b**2*c**2*d)) + f*x**6/(6*c) - x**4*(
b*f - c*e)/(4*c**2) - x**2*(a*c*f - b**2*f + b*c*e - c**2*d)/(2*c**3)

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Giac [A]  time = 1.1619, size = 289, normalized size = 1.42 \begin{align*} \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}} \end{align*}

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)