\(\int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{(a+b x^2+c x^4)^2} \, dx\) [50]

Optimal result

Integrand size = 40, antiderivative size = 468 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (a+b x^2+c x^4\right )^2} \, dx=\frac {x \left (b^2 d-a b f-2 a (c d-a h)+(b c d-2 a c f+a b h) x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {2 a c g-b (c e+a i)-\left (2 c^2 e-b c g+b^2 i-2 a c i\right ) x^2}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\left (b c d-2 a c f+a b h+\frac {4 a b c f+b^2 (c d-a h)-4 a c (3 c d+a h)}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a \sqrt {c} \left (b^2-4 a c\right ) \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (b c d-2 a c f+a b h-\frac {4 a b c f+b^2 (c d-a h)-4 a c (3 c d+a h)}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a \sqrt {c} \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}}}+\frac {(2 c e-b g+2 a i) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \] Output:

1/2*x*(b^2*d-a*b*f-2*a*(-a*h+c*d)+(a*b*h-2*a*c*f+b*c*d)*x^2)/a/(-4*a*c+b^2 
)/(c*x^4+b*x^2+a)+1/2*(2*a*c*g-b*(a*i+c*e)-(-2*a*c*i+b^2*i-b*c*g+2*c^2*e)* 
x^2)/c/(-4*a*c+b^2)/(c*x^4+b*x^2+a)+1/4*(b*c*d-2*a*c*f+a*b*h+(4*a*b*c*f+b^ 
2*(-a*h+c*d)-4*a*c*(a*h+3*c*d))/(-4*a*c+b^2)^(1/2))*arctan(2^(1/2)*c^(1/2) 
*x/(b-(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)/a/c^(1/2)/(-4*a*c+b^2)/(b-(-4*a*c 
+b^2)^(1/2))^(1/2)+1/4*(b*c*d-2*a*c*f+a*b*h-(4*a*b*c*f+b^2*(-a*h+c*d)-4*a* 
c*(a*h+3*c*d))/(-4*a*c+b^2)^(1/2))*arctan(2^(1/2)*c^(1/2)*x/(b+(-4*a*c+b^2 
)^(1/2))^(1/2))*2^(1/2)/a/c^(1/2)/(-4*a*c+b^2)/(b+(-4*a*c+b^2)^(1/2))^(1/2 
)+(2*a*i-b*g+2*c*e)*arctanh((2*c*x^2+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^( 
3/2)
 

Mathematica [A] (verified)

Time = 2.15 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.12 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (a+b x^2+c x^4\right )^2} \, dx=\frac {1}{4} \left (\frac {2 \left (-b c d x \left (b+c x^2\right )+a^2 (b i-2 c (g+x (h+i x)))+a \left (b^2 i x^2+2 c^2 x (d+x (e+f x))+b c (e+x (f-x (g+h x)))\right )\right )}{a c \left (-b^2+4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\sqrt {2} \left (b^2 (c d-a h)-2 a c \left (6 c d+\sqrt {b^2-4 a c} f+2 a h\right )+b \left (c \sqrt {b^2-4 a c} d+4 a c f+a \sqrt {b^2-4 a c} h\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{a \sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \left (b^2 (-c d+a h)+2 a c \left (6 c d-\sqrt {b^2-4 a c} f+2 a h\right )+b \left (c \sqrt {b^2-4 a c} d-4 a c f+a \sqrt {b^2-4 a c} h\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{a \sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}}+\frac {2 (-2 c e+b g-2 a i) \log \left (-b+\sqrt {b^2-4 a c}-2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac {2 (2 c e-b g+2 a i) \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}\right ) \] Input:

Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(a + b*x^2 + c*x^4)^2, 
x]
 

Output:

((2*(-(b*c*d*x*(b + c*x^2)) + a^2*(b*i - 2*c*(g + x*(h + i*x))) + a*(b^2*i 
*x^2 + 2*c^2*x*(d + x*(e + f*x)) + b*c*(e + x*(f - x*(g + h*x))))))/(a*c*( 
-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*(b^2*(c*d - a*h) - 2*a*c*(6* 
c*d + Sqrt[b^2 - 4*a*c]*f + 2*a*h) + b*(c*Sqrt[b^2 - 4*a*c]*d + 4*a*c*f + 
a*Sqrt[b^2 - 4*a*c]*h))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a 
*c]]])/(a*Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt 
[2]*(b^2*(-(c*d) + a*h) + 2*a*c*(6*c*d - Sqrt[b^2 - 4*a*c]*f + 2*a*h) + b* 
(c*Sqrt[b^2 - 4*a*c]*d - 4*a*c*f + a*Sqrt[b^2 - 4*a*c]*h))*ArcTan[(Sqrt[2] 
*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a*Sqrt[c]*(b^2 - 4*a*c)^(3/2)*S 
qrt[b + Sqrt[b^2 - 4*a*c]]) + (2*(-2*c*e + b*g - 2*a*i)*Log[-b + Sqrt[b^2 
- 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(3/2) + (2*(2*c*e - b*g + 2*a*i)*Log[b 
+ Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/4
 

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 460, normalized size of antiderivative = 0.98, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2202, 2194, 2191, 27, 1083, 219, 2206, 25, 1480, 218}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (a+b x^2+c x^4\right )^2} \, dx\)

\(\Big \downarrow \) 2202

\(\displaystyle \int \frac {h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^2}dx+\int \frac {x \left (i x^4+g x^2+e\right )}{\left (c x^4+b x^2+a\right )^2}dx\)

\(\Big \downarrow \) 2194

\(\displaystyle \int \frac {h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^2}dx+\frac {1}{2} \int \frac {i x^4+g x^2+e}{\left (c x^4+b x^2+a\right )^2}dx^2\)

\(\Big \downarrow \) 2191

\(\displaystyle \frac {1}{2} \left (\frac {c \left (2 a g-b \left (\frac {a i}{c}+e\right )\right )-x^2 \left (-2 a c i+b^2 i-b c g+2 c^2 e\right )}{c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int \frac {2 c e-b g+2 a i}{c x^4+b x^2+a}dx^2}{b^2-4 a c}\right )+\int \frac {h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {c \left (2 a g-b \left (\frac {a i}{c}+e\right )\right )-x^2 \left (-2 a c i+b^2 i-b c g+2 c^2 e\right )}{c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {(2 a i-b g+2 c e) \int \frac {1}{c x^4+b x^2+a}dx^2}{b^2-4 a c}\right )+\int \frac {h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^2}dx\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {1}{2} \left (\frac {2 (2 a i-b g+2 c e) \int \frac {1}{-x^4+b^2-4 a c}d\left (2 c x^2+b\right )}{b^2-4 a c}+\frac {c \left (2 a g-b \left (\frac {a i}{c}+e\right )\right )-x^2 \left (-2 a c i+b^2 i-b c g+2 c^2 e\right )}{c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )+\int \frac {h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^2}dx\)

\(\Big \downarrow \) 219

\(\displaystyle \int \frac {h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^2}dx+\frac {1}{2} \left (\frac {2 \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) (2 a i-b g+2 c e)}{\left (b^2-4 a c\right )^{3/2}}+\frac {c \left (2 a g-b \left (\frac {a i}{c}+e\right )\right )-x^2 \left (-2 a c i+b^2 i-b c g+2 c^2 e\right )}{c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )\)

\(\Big \downarrow \) 2206

\(\displaystyle -\frac {\int -\frac {d b^2+a f b+(b c d-2 a c f+a b h) x^2-2 a (3 c d+a h)}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}+\frac {1}{2} \left (\frac {2 \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) (2 a i-b g+2 c e)}{\left (b^2-4 a c\right )^{3/2}}+\frac {c \left (2 a g-b \left (\frac {a i}{c}+e\right )\right )-x^2 \left (-2 a c i+b^2 i-b c g+2 c^2 e\right )}{c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )+\frac {x \left (x^2 (a b h-2 a c f+b c d)-a b f-2 a (c d-a h)+b^2 d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {d b^2+a f b+(b c d-2 a c f+a b h) x^2-2 a (3 c d+a h)}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}+\frac {1}{2} \left (\frac {2 \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) (2 a i-b g+2 c e)}{\left (b^2-4 a c\right )^{3/2}}+\frac {c \left (2 a g-b \left (\frac {a i}{c}+e\right )\right )-x^2 \left (-2 a c i+b^2 i-b c g+2 c^2 e\right )}{c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )+\frac {x \left (x^2 (a b h-2 a c f+b c d)-a b f-2 a (c d-a h)+b^2 d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {\frac {1}{2} \left (\frac {b^2 (c d-a h)+4 a b c f-4 a c (a h+3 c d)}{\sqrt {b^2-4 a c}}+a b h-2 a c f+b c d\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx+\frac {1}{2} \left (-\frac {b^2 (c d-a h)+4 a b c f-4 a c (a h+3 c d)}{\sqrt {b^2-4 a c}}+a b h-2 a c f+b c d\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx}{2 a \left (b^2-4 a c\right )}+\frac {1}{2} \left (\frac {2 \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) (2 a i-b g+2 c e)}{\left (b^2-4 a c\right )^{3/2}}+\frac {c \left (2 a g-b \left (\frac {a i}{c}+e\right )\right )-x^2 \left (-2 a c i+b^2 i-b c g+2 c^2 e\right )}{c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )+\frac {x \left (x^2 (a b h-2 a c f+b c d)-a b f-2 a (c d-a h)+b^2 d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (\frac {b^2 (c d-a h)+4 a b c f-4 a c (a h+3 c d)}{\sqrt {b^2-4 a c}}+a b h-2 a c f+b c d\right )}{\sqrt {2} \sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (-\frac {b^2 (c d-a h)+4 a b c f-4 a c (a h+3 c d)}{\sqrt {b^2-4 a c}}+a b h-2 a c f+b c d\right )}{\sqrt {2} \sqrt {c} \sqrt {\sqrt {b^2-4 a c}+b}}}{2 a \left (b^2-4 a c\right )}+\frac {1}{2} \left (\frac {2 \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) (2 a i-b g+2 c e)}{\left (b^2-4 a c\right )^{3/2}}+\frac {c \left (2 a g-b \left (\frac {a i}{c}+e\right )\right )-x^2 \left (-2 a c i+b^2 i-b c g+2 c^2 e\right )}{c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )+\frac {x \left (x^2 (a b h-2 a c f+b c d)-a b f-2 a (c d-a h)+b^2 d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\)

Input:

Int[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(a + b*x^2 + c*x^4)^2,x]
 

Output:

(x*(b^2*d - a*b*f - 2*a*(c*d - a*h) + (b*c*d - 2*a*c*f + a*b*h)*x^2))/(2*a 
*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (((b*c*d - 2*a*c*f + a*b*h + (4*a*b* 
c*f + b^2*(c*d - a*h) - 4*a*c*(3*c*d + a*h))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sq 
rt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[c]*Sqrt[b - S 
qrt[b^2 - 4*a*c]]) + ((b*c*d - 2*a*c*f + a*b*h - (4*a*b*c*f + b^2*(c*d - a 
*h) - 4*a*c*(3*c*d + a*h))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/S 
qrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]) 
)/(2*a*(b^2 - 4*a*c)) + ((c*(2*a*g - b*(e + (a*i)/c)) - (2*c^2*e - b*c*g + 
 b^2*i - 2*a*c*i)*x^2)/(c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (2*(2*c*e - 
 b*g + 2*a*i)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(3/2 
))/2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

rule 2191

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = 
PolynomialQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[P 
q, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x + 
c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*c*f - b*g)*x)*((a + b*x + c*x^2)^ 
(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c))   Int 
[(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (2*p + 3)* 
(2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^ 
2 - 4*a*c, 0] && LtQ[p, -1]
 

rule 2194

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] : 
> Simp[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] && IntegerQ 
[(m - 1)/2]
 

rule 2202

Int[(Pn_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n 
 = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b 
*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 
1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] 
 &&  !PolyQ[Pn, x^2]
 

rule 2206

Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = 
 Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly 
nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 
4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b 
^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c))   Int[(a + b*x^2 + c 
*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Px, 
a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* 
p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x 
^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.17 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.65

method	result	size
risch	\(\frac {-\frac {\left (a b h -2 a c f +b c d \right ) x^{3}}{2 a \left (4 a c -b^{2}\right )}-\frac {\left (2 a c i -b^{2} i +b c g -2 e \,c^{2}\right ) x^{2}}{2 c \left (4 a c -b^{2}\right )}-\frac {\left (2 a^{2} h -a b f -2 d a c +b^{2} d \right ) x}{2 a \left (4 a c -b^{2}\right )}+\frac {a b i -2 a c g +c e b}{2 c \left (4 a c -b^{2}\right )}}{c \,x^{4}+b \,x^{2}+a}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (-\frac {\left (a b h -2 a c f +b c d \right ) \textit {\_R}^{2}}{a \left (4 a c -b^{2}\right )}+\frac {2 \left (2 a i -b g +2 c e \right ) \textit {\_R}}{4 a c -b^{2}}+\frac {2 a^{2} h -a b f +6 d a c -b^{2} d}{a \left (4 a c -b^{2}\right )}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3} c +\textit {\_R} b}\right )}{4}\)	\(304\)
default	\(\frac {-\frac {\left (a b h -2 a c f +b c d \right ) x^{3}}{2 a \left (4 a c -b^{2}\right )}-\frac {\left (2 a c i -b^{2} i +b c g -2 e \,c^{2}\right ) x^{2}}{2 c \left (4 a c -b^{2}\right )}-\frac {\left (2 a^{2} h -a b f -2 d a c +b^{2} d \right ) x}{2 a \left (4 a c -b^{2}\right )}+\frac {a b i -2 a c g +c e b}{2 c \left (4 a c -b^{2}\right )}}{c \,x^{4}+b \,x^{2}+a}+\frac {2 c \left (\frac {\frac {\left (8 \sqrt {-4 a c +b^{2}}\, a^{2} c i -4 \sqrt {-4 a c +b^{2}}\, a b c g +8 \sqrt {-4 a c +b^{2}}\, a \,c^{2} e \right ) \ln \left (2 c \,x^{2}+\sqrt {-4 a c +b^{2}}+b \right )}{4 c}+\frac {\left (4 \sqrt {-4 a c +b^{2}}\, a^{2} c h +\sqrt {-4 a c +b^{2}}\, a \,b^{2} h -4 \sqrt {-4 a c +b^{2}}\, a b c f +12 \sqrt {-4 a c +b^{2}}\, a \,c^{2} d -\sqrt {-4 a c +b^{2}}\, b^{2} c d -4 a^{2} b c h +8 a^{2} c^{2} f +a \,b^{3} h -2 a \,b^{2} c f -4 a b \,c^{2} d +b^{3} c d \right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}}{4 c \left (4 a c -b^{2}\right )}+\frac {-\frac {\left (8 \sqrt {-4 a c +b^{2}}\, a^{2} c i -4 \sqrt {-4 a c +b^{2}}\, a b c g +8 \sqrt {-4 a c +b^{2}}\, a \,c^{2} e \right ) \ln \left (-2 c \,x^{2}+\sqrt {-4 a c +b^{2}}-b \right )}{4 c}+\frac {\left (4 \sqrt {-4 a c +b^{2}}\, a^{2} c h +\sqrt {-4 a c +b^{2}}\, a \,b^{2} h -4 \sqrt {-4 a c +b^{2}}\, a b c f +12 \sqrt {-4 a c +b^{2}}\, a \,c^{2} d -\sqrt {-4 a c +b^{2}}\, b^{2} c d +4 a^{2} b c h -8 a^{2} c^{2} f -a \,b^{3} h +2 a \,b^{2} c f +4 a b \,c^{2} d -b^{3} c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}}{4 c \left (4 a c -b^{2}\right )}\right )}{a \left (4 a c -b^{2}\right )}\)	\(724\)

Input:

int((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^2,x,method=_RETURNVERB 
OSE)
 

Output:

(-1/2/a*(a*b*h-2*a*c*f+b*c*d)/(4*a*c-b^2)*x^3-1/2*(2*a*c*i-b^2*i+b*c*g-2*c 
^2*e)/c/(4*a*c-b^2)*x^2-1/2*(2*a^2*h-a*b*f-2*a*c*d+b^2*d)/a/(4*a*c-b^2)*x+ 
1/2/c*(a*b*i-2*a*c*g+b*c*e)/(4*a*c-b^2))/(c*x^4+b*x^2+a)+1/4*sum((-1/a*(a* 
b*h-2*a*c*f+b*c*d)/(4*a*c-b^2)*_R^2+2*(2*a*i-b*g+2*c*e)/(4*a*c-b^2)*_R+(2* 
a^2*h-a*b*f+6*a*c*d-b^2*d)/a/(4*a*c-b^2))/(2*_R^3*c+_R*b)*ln(x-_R),_R=Root 
Of(_Z^4*c+_Z^2*b+a))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \] Input:

integrate((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^2,x, algorithm=" 
fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \] Input:

integrate((i*x**5+h*x**4+g*x**3+f*x**2+e*x+d)/(c*x**4+b*x**2+a)**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (a+b x^2+c x^4\right )^2} \, dx=\int { \frac {i x^{5} + h x^{4} + g x^{3} + f x^{2} + e x + d}{{\left (c x^{4} + b x^{2} + a\right )}^{2}} \,d x } \] Input:

integrate((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^2,x, algorithm=" 
maxima")
 

Output:

-1/2*(a*b*c*e - 2*a^2*c*g + a^2*b*i - (b*c^2*d - 2*a*c^2*f + a*b*c*h)*x^3 
+ (2*a*c^2*e - a*b*c*g + (a*b^2 - 2*a^2*c)*i)*x^2 + (a*b*c*f - 2*a^2*c*h - 
 (b^2*c - 2*a*c^2)*d)*x)/(a^2*b^2*c - 4*a^3*c^2 + (a*b^2*c^2 - 4*a^2*c^3)* 
x^4 + (a*b^3*c - 4*a^2*b*c^2)*x^2) + 1/2*integrate((a*b*f - 2*a^2*h + (b*c 
*d - 2*a*c*f + a*b*h)*x^2 + (b^2 - 6*a*c)*d - 2*(2*a*c*e - a*b*g + 2*a^2*i 
)*x)/(c*x^4 + b*x^2 + a), x)/(a*b^2 - 4*a^2*c)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 7965 vs. \(2 (420) = 840\).

Time = 1.34 (sec) , antiderivative size = 7965, normalized size of antiderivative = 17.02 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \] Input:

integrate((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^2,x, algorithm=" 
giac")
 

Output:

1/2*(b*c^2*d*x^3 - 2*a*c^2*f*x^3 + a*b*c*h*x^3 - 2*a*c^2*e*x^2 + a*b*c*g*x 
^2 - a*b^2*i*x^2 + 2*a^2*c*i*x^2 + b^2*c*d*x - 2*a*c^2*d*x - a*b*c*f*x + 2 
*a^2*c*h*x - a*b*c*e + 2*a^2*c*g - a^2*b*i)/((c*x^4 + b*x^2 + a)*(a*b^2*c 
- 4*a^2*c^2)) + 1/16*((2*b^3*c^3 - 8*a*b*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*s 
qrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b* 
c + sqrt(b^2 - 4*a*c)*c)*a*b*c^2 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + 
sqrt(b^2 - 4*a*c)*c)*b^2*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b 
^2 - 4*a*c)*c)*b*c^3 - 2*(b^2 - 4*a*c)*b*c^3)*(a*b^2 - 4*a^2*c)^2*d - 2*(2 
*a*b^2*c^3 - 8*a^2*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4 
*a*c)*c)*a*b^2*c + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c 
)*c)*a^2*c^2 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c) 
*a*b*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*c^3 
 - 2*(b^2 - 4*a*c)*a*c^3)*(a*b^2 - 4*a^2*c)^2*f + (2*a*b^3*c^2 - 8*a^2*b*c 
^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3 + 4*s 
qrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c + 2*sqrt( 
2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c - sqrt(2)*sqr 
t(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^2 - 2*(b^2 - 4*a*c)*a 
*b*c^2)*(a*b^2 - 4*a^2*c)^2*h + 2*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c) 
*a*b^6*c - 14*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^2 - 2*sqrt 
(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5*c^2 - 2*a*b^6*c^2 + 64*sqrt(2...
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 20.88 (sec) , antiderivative size = 18449, normalized size of antiderivative = 39.42 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \] Input:

int((d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(a + b*x^2 + c*x^4)^2,x)
 

Output:

((b*c*e - 2*a*c*g + a*b*i)/(2*c*(4*a*c - b^2)) - (x*(b^2*d + 2*a^2*h - 2*a 
*c*d - a*b*f))/(2*a*(4*a*c - b^2)) + (x^2*(2*c^2*e + b^2*i - b*c*g - 2*a*c 
*i))/(2*c*(4*a*c - b^2)) - (x^3*(b*c*d - 2*a*c*f + a*b*h))/(2*a*(4*a*c - b 
^2)))/(a + b*x^2 + c*x^4) + symsum(log((5*b^3*c^4*d^3 + 8*a^3*c^4*f^3 - 96 
*a^2*c^5*d*e^2 + 72*a^2*c^5*d^2*f - 3*a^3*b^3*c*h^3 - 4*a^4*b*c^2*h^3 - 3* 
b^4*c^3*d^2*f - 32*a^3*c^4*e^2*h - 96*a^4*c^3*d*i^2 + b^5*c^2*d^2*h + 8*a^ 
4*c^3*f*h^2 - 32*a^5*c^2*h*i^2 + 6*a^2*b^2*c^3*f^3 - 36*a*b*c^5*d^3 + a*b^ 
5*c*d*h^2 - 192*a^3*c^4*d*e*i + 48*a^3*c^4*d*f*h - 64*a^4*c^3*e*h*i + 16*a 
*b^2*c^4*d*e^2 + 18*a*b^2*c^4*d^2*f + 3*a*b^3*c^3*d*f^2 - 60*a^2*b*c^4*d*f 
^2 + 4*a*b^4*c^2*d*g^2 + 16*a^2*b*c^4*e^2*f - a*b^3*c^3*d^2*h - 60*a^2*b*c 
^4*d^2*h - 28*a^3*b*c^3*d*h^2 + a^2*b^4*c*f*h^2 - 28*a^3*b*c^3*f^2*h + 16* 
a^4*b*c^2*f*i^2 - 24*a^2*b^2*c^3*d*g^2 - 9*a^2*b^3*c^2*d*h^2 + 4*a^2*b^3*c 
^2*f*g^2 + 16*a^3*b^2*c^2*d*i^2 - 5*a^2*b^3*c^2*f^2*h + 18*a^3*b^2*c^2*f*h 
^2 - 8*a^3*b^2*c^2*g^2*h - 16*a*b^3*c^3*d*e*g + 96*a^2*b*c^4*d*e*g - 4*a*b 
^4*c^2*d*f*h + 96*a^3*b*c^3*d*g*i + 32*a^3*b*c^3*e*f*i + 32*a^3*b*c^3*e*g* 
h + 32*a^4*b*c^2*g*h*i + 32*a^2*b^2*c^3*d*e*i + 52*a^2*b^2*c^3*d*f*h - 16* 
a^2*b^2*c^3*e*f*g - 16*a^2*b^3*c^2*d*g*i - 16*a^3*b^2*c^2*f*g*i)/(8*(a^2*b 
^6 - 64*a^5*c^3 - 12*a^3*b^4*c + 48*a^4*b^2*c^2)) - root(1572864*a^8*b^2*c 
^6*z^4 - 983040*a^7*b^4*c^5*z^4 + 327680*a^6*b^6*c^4*z^4 - 61440*a^5*b^8*c 
^3*z^4 + 6144*a^4*b^10*c^2*z^4 - 256*a^3*b^12*c*z^4 - 1048576*a^9*c^7*z...
 

Reduce [B] (verification not implemented)

Time = 0.80 (sec) , antiderivative size = 7607, normalized size of antiderivative = 16.25 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (a+b x^2+c x^4\right )^2} \, dx =\text {Too large to display} \] Input:

int((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^2,x)
 

Output:

( - 16*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt( 
2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**4*b* 
c*i + 8*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt 
(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**3*b 
**2*c*g - 16*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan( 
(sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a 
**3*b**2*c*i*x**2 - 16*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) 
- b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt( 
a) + b))*a**3*b*c**2*e - 16*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqr 
t(a) - b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)* 
sqrt(a) + b))*a**3*b*c**2*i*x**4 + 8*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sq 
rt(c)*sqrt(a) - b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2 
*sqrt(c)*sqrt(a) + b))*a**2*b**3*c*g*x**2 - 16*sqrt(2*sqrt(c)*sqrt(a) + b) 
*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c) 
*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**2*b**2*c**2*e*x**2 + 8*sqrt(2*sqrt(c)* 
sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) 
 - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**2*b**2*c**2*g*x**4 - 16*sq 
rt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*sqrt(c) 
*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**2*b*c**3*e*x* 
*4 + 8*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a)...