### 3.54 $$\int \frac{d+e x^2+f x^4}{x^7 (a+b x^2+c x^4)} \, dx$$

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

[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)

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Rubi [A]  time = 0.572902, antiderivative size = 244, 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} $-\frac{-a b e-a (c d-a f)+b^2 d}{2 a^3 x^2}+\frac{\log \left (a+b x^2+c x^4\right ) \left (a^2 c e-a b^2 e-a b (2 c d-a f)+b^3 d\right )}{4 a^4}-\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (3 a^2 b c e+2 a^2 c (c d-a f)-a b^2 (4 c d-a f)-a b^3 e+b^4 d\right )}{2 a^4 \sqrt{b^2-4 a c}}-\frac{\log (x) \left (a^2 c e-a b^2 e-a b (2 c d-a f)+b^3 d\right )}{a^4}+\frac{b d-a e}{4 a^2 x^4}-\frac{d}{6 a x^6}$

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 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{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*}

Mathematica [A]  time = 0.351751, size = 416, normalized size = 1.7 $\frac{\frac{3 \log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (a^2 c \left (e \sqrt{b^2-4 a c}-2 a f+2 c d\right )+a b^2 \left (-e \sqrt{b^2-4 a c}+a f-4 c d\right )+a b \left (-2 c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}+3 a c e\right )+b^3 \left (d \sqrt{b^2-4 a c}-a e\right )+b^4 d\right )}{\sqrt{b^2-4 a c}}+\frac{3 \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (a^2 c \left (e \sqrt{b^2-4 a c}+2 a f-2 c d\right )-a b^2 \left (e \sqrt{b^2-4 a c}+a f-4 c d\right )+a b \left (-2 c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}-3 a c e\right )+b^3 \left (d \sqrt{b^2-4 a c}+a e\right )+b^4 (-d)\right )}{\sqrt{b^2-4 a c}}-12 \log (x) \left (a^2 c e-a b^2 e+a b (a f-2 c d)+b^3 d\right )+\frac{3 a^2 (b d-a e)}{x^4}-\frac{2 a^3 d}{x^6}+\frac{6 a \left (a b e+a (c d-a f)+b^2 (-d)\right )}{x^2}}{12 a^4}$

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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Maple [B]  time = 0.014, size = 523, normalized size = 2.1 \begin{align*} -{\frac{d}{6\,a{x}^{6}}}-{\frac{e}{4\,a{x}^{4}}}+{\frac{bd}{4\,{a}^{2}{x}^{4}}}-{\frac{f}{2\,a{x}^{2}}}+{\frac{be}{2\,{a}^{2}{x}^{2}}}+{\frac{cd}{2\,{a}^{2}{x}^{2}}}-{\frac{{b}^{2}d}{2\,{a}^{3}{x}^{2}}}-{\frac{\ln \left ( x \right ) bf}{{a}^{2}}}-{\frac{\ln \left ( x \right ) ce}{{a}^{2}}}+{\frac{\ln \left ( x \right ){b}^{2}e}{{a}^{3}}}+2\,{\frac{\ln \left ( x \right ) bcd}{{a}^{3}}}-{\frac{\ln \left ( x \right ){b}^{3}d}{{a}^{4}}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) bf}{4\,{a}^{2}}}+{\frac{c\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) e}{4\,{a}^{2}}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{2}e}{4\,{a}^{3}}}-{\frac{c\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) bd}{2\,{a}^{3}}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{3}d}{4\,{a}^{4}}}-{\frac{fc}{a}\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}f}{2\,{a}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{3\,bce}{2\,{a}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{c}^{2}d}{{a}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{e{b}^{3}}{2\,{a}^{3}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-2\,{\frac{{b}^{2}cd}{{a}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{4}d}{2\,{a}^{4}}\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((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., 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((f*x^4+e*x^2+d)/x^7/(c*x^4+b*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 13.7013, size = 1747, normalized size = 7.16 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [A]  time = 1.14577, size = 423, normalized size = 1.73 \begin{align*} \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{align*}

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)