### 3.47 $$\int \frac{x^7 (d+e x^2+f x^4)}{a+b x^2+c x^4} \, dx$$

Optimal. Leaf size=273 $-\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (2 a^2 c^3 e-4 a b^2 c^2 e-b^3 c (c d-5 a f)+a b c^2 (3 c d-5 a f)+b^4 c e+b^5 (-f)\right )}{2 c^5 \sqrt{b^2-4 a c}}+\frac{x^4 \left (-c (a f+b e)+b^2 f+c^2 d\right )}{4 c^3}+\frac{x^2 \left (-b c (c d-2 a f)-a c^2 e+b^2 c e+b^3 (-f)\right )}{2 c^4}-\frac{\log \left (a+b x^2+c x^4\right ) \left (-b^2 c (c d-3 a f)-2 a b c^2 e+a c^2 (c d-a f)+b^3 c e+b^4 (-f)\right )}{4 c^5}+\frac{x^6 (c e-b f)}{6 c^2}+\frac{f x^8}{8 c}$

[Out]

((b^2*c*e - a*c^2*e - b^3*f - b*c*(c*d - 2*a*f))*x^2)/(2*c^4) + ((c^2*d + b^2*f - c*(b*e + a*f))*x^4)/(4*c^3)
+ ((c*e - b*f)*x^6)/(6*c^2) + (f*x^8)/(8*c) - ((b^4*c*e - 4*a*b^2*c^2*e + 2*a^2*c^3*e - b^5*f - b^3*c*(c*d - 5
*a*f) + a*b*c^2*(3*c*d - 5*a*f))*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^5*Sqrt[b^2 - 4*a*c]) - ((b^3*c
*e - 2*a*b*c^2*e - b^4*f - b^2*c*(c*d - 3*a*f) + a*c^2*(c*d - a*f))*Log[a + b*x^2 + c*x^4])/(4*c^5)

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Rubi [A]  time = 0.8541, antiderivative size = 273, 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} $-\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (2 a^2 c^3 e-4 a b^2 c^2 e-b^3 c (c d-5 a f)+a b c^2 (3 c d-5 a f)+b^4 c e+b^5 (-f)\right )}{2 c^5 \sqrt{b^2-4 a c}}+\frac{x^4 \left (-c (a f+b e)+b^2 f+c^2 d\right )}{4 c^3}+\frac{x^2 \left (-b c (c d-2 a f)-a c^2 e+b^2 c e+b^3 (-f)\right )}{2 c^4}-\frac{\log \left (a+b x^2+c x^4\right ) \left (-b^2 c (c d-3 a f)-2 a b c^2 e+a c^2 (c d-a f)+b^3 c e+b^4 (-f)\right )}{4 c^5}+\frac{x^6 (c e-b f)}{6 c^2}+\frac{f x^8}{8 c}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((b^2*c*e - a*c^2*e - b^3*f - b*c*(c*d - 2*a*f))*x^2)/(2*c^4) + ((c^2*d + b^2*f - c*(b*e + a*f))*x^4)/(4*c^3)
+ ((c*e - b*f)*x^6)/(6*c^2) + (f*x^8)/(8*c) - ((b^4*c*e - 4*a*b^2*c^2*e + 2*a^2*c^3*e - b^5*f - b^3*c*(c*d - 5
*a*f) + a*b*c^2*(3*c*d - 5*a*f))*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^5*Sqrt[b^2 - 4*a*c]) - ((b^3*c
*e - 2*a*b*c^2*e - b^4*f - b^2*c*(c*d - 3*a*f) + a*c^2*(c*d - a*f))*Log[a + b*x^2 + c*x^4])/(4*c^5)

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^7 \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^3 \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{b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)}{c^4}+\frac{\left (c^2 d+b^2 f-c (b e+a f)\right ) x}{c^3}+\frac{(c e-b f) x^2}{c^2}+\frac{f x^3}{c}+\frac{-a \left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right )-\left (b^3 c e-2 a b c^2 e-b^4 f-b^2 c (c d-3 a f)+a c^2 (c d-a f)\right ) x}{c^4 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=\frac{\left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right ) x^2}{2 c^4}+\frac{\left (c^2 d+b^2 f-c (b e+a f)\right ) x^4}{4 c^3}+\frac{(c e-b f) x^6}{6 c^2}+\frac{f x^8}{8 c}+\frac{\operatorname{Subst}\left (\int \frac{-a \left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right )-\left (b^3 c e-2 a b c^2 e-b^4 f-b^2 c (c d-3 a f)+a c^2 (c d-a f)\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 c^4}\\ &=\frac{\left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right ) x^2}{2 c^4}+\frac{\left (c^2 d+b^2 f-c (b e+a f)\right ) x^4}{4 c^3}+\frac{(c e-b f) x^6}{6 c^2}+\frac{f x^8}{8 c}+\frac{\left (b^4 c e-4 a b^2 c^2 e+2 a^2 c^3 e-b^5 f-b^3 c (c d-5 a f)+a b c^2 (3 c d-5 a f)\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^5}-\frac{\left (b^3 c e-2 a b c^2 e-b^4 f-b^2 c (c d-3 a f)+a c^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 c^5}\\ &=\frac{\left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right ) x^2}{2 c^4}+\frac{\left (c^2 d+b^2 f-c (b e+a f)\right ) x^4}{4 c^3}+\frac{(c e-b f) x^6}{6 c^2}+\frac{f x^8}{8 c}-\frac{\left (b^3 c e-2 a b c^2 e-b^4 f-b^2 c (c d-3 a f)+a c^2 (c d-a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 c^5}-\frac{\left (b^4 c e-4 a b^2 c^2 e+2 a^2 c^3 e-b^5 f-b^3 c (c d-5 a f)+a b c^2 (3 c d-5 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^5}\\ &=\frac{\left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right ) x^2}{2 c^4}+\frac{\left (c^2 d+b^2 f-c (b e+a f)\right ) x^4}{4 c^3}+\frac{(c e-b f) x^6}{6 c^2}+\frac{f x^8}{8 c}-\frac{\left (b^4 c e-4 a b^2 c^2 e+2 a^2 c^3 e-b^5 f-b^3 c (c d-5 a f)+a b c^2 (3 c d-5 a f)\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^5 \sqrt{b^2-4 a c}}-\frac{\left (b^3 c e-2 a b c^2 e-b^4 f-b^2 c (c d-3 a f)+a c^2 (c d-a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 c^5}\\ \end{align*}

Mathematica [A]  time = 0.208072, size = 260, normalized size = 0.95 $\frac{-\frac{12 \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right ) \left (-2 a^2 c^3 e+4 a b^2 c^2 e+b^3 c (c d-5 a f)+a b c^2 (5 a f-3 c d)-b^4 c e+b^5 f\right )}{\sqrt{4 a c-b^2}}+6 c^2 x^4 \left (-c (a f+b e)+b^2 f+c^2 d\right )-12 c x^2 \left (b c (c d-2 a f)+a c^2 e-b^2 c e+b^3 f\right )+6 \log \left (a+b x^2+c x^4\right ) \left (b^2 c (c d-3 a f)+2 a b c^2 e+a c^2 (a f-c d)-b^3 c e+b^4 f\right )+4 c^3 x^6 (c e-b f)+3 c^4 f x^8}{24 c^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-12*c*(-(b^2*c*e) + a*c^2*e + b^3*f + b*c*(c*d - 2*a*f))*x^2 + 6*c^2*(c^2*d + b^2*f - c*(b*e + a*f))*x^4 + 4*
c^3*(c*e - b*f)*x^6 + 3*c^4*f*x^8 - (12*(-(b^4*c*e) + 4*a*b^2*c^2*e - 2*a^2*c^3*e + b^5*f + b^3*c*(c*d - 5*a*f
) + a*b*c^2*(-3*c*d + 5*a*f))*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + 6*(-(b^3*c*e) + 2
*a*b*c^2*e + b^4*f + b^2*c*(c*d - 3*a*f) + a*c^2*(-(c*d) + a*f))*Log[a + b*x^2 + c*x^4])/(24*c^5)

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Maple [B]  time = 0.007, size = 622, normalized size = 2.3 \begin{align*} -{\frac{5\,{a}^{2}bf}{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{5\,a{b}^{3}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}}}}}-2\,{\frac{a{b}^{2}e}{{c}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{3\,bad}{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{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{2}d}{4\,{c}^{3}}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){a}^{2}f}{4\,{c}^{3}}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) ad}{4\,{c}^{2}}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{4}f}{4\,{c}^{5}}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{3}e}{4\,{c}^{4}}}+{\frac{{b}^{2}e{x}^{2}}{2\,{c}^{3}}}-{\frac{bd{x}^{2}}{2\,{c}^{2}}}-{\frac{{x}^{6}bf}{6\,{c}^{2}}}-{\frac{{x}^{4}af}{4\,{c}^{2}}}+{\frac{{x}^{4}{b}^{2}f}{4\,{c}^{3}}}-{\frac{be{x}^{4}}{4\,{c}^{2}}}-{\frac{ae{x}^{2}}{2\,{c}^{2}}}-{\frac{{b}^{3}f{x}^{2}}{2\,{c}^{4}}}-{\frac{3\,\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) a{b}^{2}f}{4\,{c}^{4}}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) abe}{2\,{c}^{3}}}+{\frac{{a}^{2}e}{{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{{b}^{5}f}{2\,{c}^{5}}\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}e}{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}d}{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{abf{x}^{2}}{{c}^{3}}}+{\frac{{x}^{4}d}{4\,c}}+{\frac{{x}^{6}e}{6\,c}}+{\frac{f{x}^{8}}{8\,c}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^7*(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 = 4.16819, size = 1854, normalized size = 6.79 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/24*(3*(b^2*c^4 - 4*a*c^5)*f*x^8 + 4*((b^2*c^4 - 4*a*c^5)*e - (b^3*c^3 - 4*a*b*c^4)*f)*x^6 + 6*((b^2*c^4 - 4
*a*c^5)*d - (b^3*c^3 - 4*a*b*c^4)*e + (b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*f)*x^4 - 12*((b^3*c^3 - 4*a*b*c^4)*d
- (b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*e + (b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*f)*x^2 + 6*sqrt(b^2 - 4*a*c)*((
b^3*c^2 - 3*a*b*c^3)*d - (b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*e + (b^5 - 5*a*b^3*c + 5*a^2*b*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)) + 6*((b^4*c^2 - 5*a*b^2*c
^3 + 4*a^2*c^4)*d - (b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*e + (b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*f)*
log(c*x^4 + b*x^2 + a))/(b^2*c^5 - 4*a*c^6), 1/24*(3*(b^2*c^4 - 4*a*c^5)*f*x^8 + 4*((b^2*c^4 - 4*a*c^5)*e - (b
^3*c^3 - 4*a*b*c^4)*f)*x^6 + 6*((b^2*c^4 - 4*a*c^5)*d - (b^3*c^3 - 4*a*b*c^4)*e + (b^4*c^2 - 5*a*b^2*c^3 + 4*a
^2*c^4)*f)*x^4 - 12*((b^3*c^3 - 4*a*b*c^4)*d - (b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*e + (b^5*c - 6*a*b^3*c^2 +
8*a^2*b*c^3)*f)*x^2 + 12*sqrt(-b^2 + 4*a*c)*((b^3*c^2 - 3*a*b*c^3)*d - (b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*e + (
b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*f)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) + 6*((b^4*c^2 - 5*a*
b^2*c^3 + 4*a^2*c^4)*d - (b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*e + (b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3
)*f)*log(c*x^4 + b*x^2 + a))/(b^2*c^5 - 4*a*c^6)]

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Sympy [B]  time = 49.7414, size = 1392, normalized size = 5.1 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-sqrt(-4*a*c + b**2)*(5*a**2*b*c**2*f - 2*a**2*c**3*e - 5*a*b**3*c*f + 4*a*b**2*c**2*e - 3*a*b*c**3*d + b**5*
f - b**4*c*e + b**3*c**2*d)/(4*c**5*(4*a*c - b**2)) + (a**2*c**2*f - 3*a*b**2*c*f + 2*a*b*c**2*e - a*c**3*d +
b**4*f - b**3*c*e + b**2*c**2*d)/(4*c**5))*log(x**2 + (2*a**3*c**2*f - 4*a**2*b**2*c*f + 3*a**2*b*c**2*e - 2*a
**2*c**3*d + a*b**4*f - a*b**3*c*e + a*b**2*c**2*d - 8*a*c**5*(-sqrt(-4*a*c + b**2)*(5*a**2*b*c**2*f - 2*a**2*
c**3*e - 5*a*b**3*c*f + 4*a*b**2*c**2*e - 3*a*b*c**3*d + b**5*f - b**4*c*e + b**3*c**2*d)/(4*c**5*(4*a*c - b**
2)) + (a**2*c**2*f - 3*a*b**2*c*f + 2*a*b*c**2*e - a*c**3*d + b**4*f - b**3*c*e + b**2*c**2*d)/(4*c**5)) + 2*b
**2*c**4*(-sqrt(-4*a*c + b**2)*(5*a**2*b*c**2*f - 2*a**2*c**3*e - 5*a*b**3*c*f + 4*a*b**2*c**2*e - 3*a*b*c**3*
d + b**5*f - b**4*c*e + b**3*c**2*d)/(4*c**5*(4*a*c - b**2)) + (a**2*c**2*f - 3*a*b**2*c*f + 2*a*b*c**2*e - a*
c**3*d + b**4*f - b**3*c*e + b**2*c**2*d)/(4*c**5)))/(5*a**2*b*c**2*f - 2*a**2*c**3*e - 5*a*b**3*c*f + 4*a*b**
2*c**2*e - 3*a*b*c**3*d + b**5*f - b**4*c*e + b**3*c**2*d)) + (sqrt(-4*a*c + b**2)*(5*a**2*b*c**2*f - 2*a**2*c
**3*e - 5*a*b**3*c*f + 4*a*b**2*c**2*e - 3*a*b*c**3*d + b**5*f - b**4*c*e + b**3*c**2*d)/(4*c**5*(4*a*c - b**2
)) + (a**2*c**2*f - 3*a*b**2*c*f + 2*a*b*c**2*e - a*c**3*d + b**4*f - b**3*c*e + b**2*c**2*d)/(4*c**5))*log(x*
*2 + (2*a**3*c**2*f - 4*a**2*b**2*c*f + 3*a**2*b*c**2*e - 2*a**2*c**3*d + a*b**4*f - a*b**3*c*e + a*b**2*c**2*
d - 8*a*c**5*(sqrt(-4*a*c + b**2)*(5*a**2*b*c**2*f - 2*a**2*c**3*e - 5*a*b**3*c*f + 4*a*b**2*c**2*e - 3*a*b*c*
*3*d + b**5*f - b**4*c*e + b**3*c**2*d)/(4*c**5*(4*a*c - b**2)) + (a**2*c**2*f - 3*a*b**2*c*f + 2*a*b*c**2*e -
a*c**3*d + b**4*f - b**3*c*e + b**2*c**2*d)/(4*c**5)) + 2*b**2*c**4*(sqrt(-4*a*c + b**2)*(5*a**2*b*c**2*f - 2
*a**2*c**3*e - 5*a*b**3*c*f + 4*a*b**2*c**2*e - 3*a*b*c**3*d + b**5*f - b**4*c*e + b**3*c**2*d)/(4*c**5*(4*a*c
- b**2)) + (a**2*c**2*f - 3*a*b**2*c*f + 2*a*b*c**2*e - a*c**3*d + b**4*f - b**3*c*e + b**2*c**2*d)/(4*c**5))
)/(5*a**2*b*c**2*f - 2*a**2*c**3*e - 5*a*b**3*c*f + 4*a*b**2*c**2*e - 3*a*b*c**3*d + b**5*f - b**4*c*e + b**3*
c**2*d)) + f*x**8/(8*c) - x**6*(b*f - c*e)/(6*c**2) - x**4*(a*c*f - b**2*f + b*c*e - c**2*d)/(4*c**3) + x**2*(
2*a*b*c*f - a*c**2*e - b**3*f + b**2*c*e - b*c**2*d)/(2*c**4)

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Giac [A]  time = 1.15015, size = 413, normalized size = 1.51 \begin{align*} \frac{3 \, c^{3} f x^{8} - 4 \, b c^{2} f x^{6} + 4 \, c^{3} x^{6} e + 6 \, c^{3} d x^{4} + 6 \, b^{2} c f x^{4} - 6 \, a c^{2} f x^{4} - 6 \, b c^{2} x^{4} e - 12 \, b c^{2} d x^{2} - 12 \, b^{3} f x^{2} + 24 \, a b c f x^{2} + 12 \, b^{2} c x^{2} e - 12 \, a c^{2} x^{2} e}{24 \, c^{4}} + \frac{{\left (b^{2} c^{2} d - a c^{3} d + b^{4} f - 3 \, a b^{2} c f + a^{2} c^{2} f - b^{3} c e + 2 \, a b c^{2} e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, c^{5}} - \frac{{\left (b^{3} c^{2} d - 3 \, a b c^{3} d + b^{5} f - 5 \, a b^{3} c f + 5 \, a^{2} b c^{2} f - b^{4} c e + 4 \, a b^{2} c^{2} e - 2 \, a^{2} c^{3} 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^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(3*c^3*f*x^8 - 4*b*c^2*f*x^6 + 4*c^3*x^6*e + 6*c^3*d*x^4 + 6*b^2*c*f*x^4 - 6*a*c^2*f*x^4 - 6*b*c^2*x^4*e
- 12*b*c^2*d*x^2 - 12*b^3*f*x^2 + 24*a*b*c*f*x^2 + 12*b^2*c*x^2*e - 12*a*c^2*x^2*e)/c^4 + 1/4*(b^2*c^2*d - a*c
^3*d + b^4*f - 3*a*b^2*c*f + a^2*c^2*f - b^3*c*e + 2*a*b*c^2*e)*log(c*x^4 + b*x^2 + a)/c^5 - 1/2*(b^3*c^2*d -
3*a*b*c^3*d + b^5*f - 5*a*b^3*c*f + 5*a^2*b*c^2*f - b^4*c*e + 4*a*b^2*c^2*e - 2*a^2*c^3*e)*arctan((2*c*x^2 + b
)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^5)