3.25 \(\int \frac{d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{a+b x^2+c x^4} \, dx\)

Optimal. Leaf size=545 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (\frac{c^2 \left (2 a^2 m+3 a b k+b^2 h\right )-b^2 c (4 a m+b k)-c^3 (2 a h+b f)+b^4 m+2 c^4 d}{\sqrt{b^2-4 a c}}-c^2 (a k+b h)+b c (2 a m+b k)+b^3 (-m)+c^3 f\right )}{\sqrt{2} c^{7/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (-\frac{c^2 \left (2 a^2 m+3 a b k+b^2 h\right )-b^2 c (4 a m+b k)-c^3 (2 a h+b f)+b^4 m+2 c^4 d}{\sqrt{b^2-4 a c}}-c^2 (a k+b h)+b c (2 a m+b k)+b^3 (-m)+c^3 f\right )}{\sqrt{2} c^{7/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{\log \left (a+b x^2+c x^4\right ) \left (-c (a l+b j)+b^2 l+c^2 g\right )}{4 c^3}+\frac{x \left (-c (a m+b k)+b^2 m+c^2 h\right )}{c^3}-\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (-c^2 (2 a j+b g)+b c (3 a l+b j)+b^3 (-l)+2 c^3 e\right )}{2 c^3 \sqrt{b^2-4 a c}}+\frac{x^2 (c j-b l)}{2 c^2}+\frac{x^3 (c k-b m)}{3 c^2}+\frac{l x^4}{4 c}+\frac{m x^5}{5 c} \]

[Out]

((c^2*h + b^2*m - c*(b*k + a*m))*x)/c^3 + ((c*j - b*l)*x^2)/(2*c^2) + ((c*k - b*
m)*x^3)/(3*c^2) + (l*x^4)/(4*c) + (m*x^5)/(5*c) + ((c^3*f - c^2*(b*h + a*k) - b^
3*m + b*c*(b*k + 2*a*m) + (2*c^4*d - c^3*(b*f + 2*a*h) + b^4*m - b^2*c*(b*k + 4*
a*m) + c^2*(b^2*h + 3*a*b*k + 2*a^2*m))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[
c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(7/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]
) + ((c^3*f - c^2*(b*h + a*k) - b^3*m + b*c*(b*k + 2*a*m) - (2*c^4*d - c^3*(b*f
+ 2*a*h) + b^4*m - b^2*c*(b*k + 4*a*m) + c^2*(b^2*h + 3*a*b*k + 2*a^2*m))/Sqrt[b
^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c
^(7/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) - ((2*c^3*e - c^2*(b*g + 2*a*j) - b^3*l + b*
c*(b*j + 3*a*l))*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^3*Sqrt[b^2 - 4*a
*c]) + ((c^2*g + b^2*l - c*(b*j + a*l))*Log[a + b*x^2 + c*x^4])/(4*c^3)

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Rubi [A]  time = 10.7722, antiderivative size = 545, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 55, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (\frac{c^2 \left (2 a^2 m+3 a b k+b^2 h\right )-b^2 c (4 a m+b k)-c^3 (2 a h+b f)+b^4 m+2 c^4 d}{\sqrt{b^2-4 a c}}-c^2 (a k+b h)+b c (2 a m+b k)+b^3 (-m)+c^3 f\right )}{\sqrt{2} c^{7/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (-\frac{c^2 \left (2 a^2 m+3 a b k+b^2 h\right )-b^2 c (4 a m+b k)-c^3 (2 a h+b f)+b^4 m+2 c^4 d}{\sqrt{b^2-4 a c}}-c^2 (a k+b h)+b c (2 a m+b k)+b^3 (-m)+c^3 f\right )}{\sqrt{2} c^{7/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{\log \left (a+b x^2+c x^4\right ) \left (-c (a l+b j)+b^2 l+c^2 g\right )}{4 c^3}+\frac{x \left (-c (a m+b k)+b^2 m+c^2 h\right )}{c^3}-\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (-c^2 (2 a j+b g)+b c (3 a l+b j)+b^3 (-l)+2 c^3 e\right )}{2 c^3 \sqrt{b^2-4 a c}}+\frac{x^2 (c j-b l)}{2 c^2}+\frac{x^3 (c k-b m)}{3 c^2}+\frac{l x^4}{4 c}+\frac{m x^5}{5 c} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x + f*x^2 + g*x^3 + h*x^4 + j*x^5 + k*x^6 + l*x^7 + m*x^8)/(a + b*x^2 + c*x^4),x]

[Out]

((c^2*h + b^2*m - c*(b*k + a*m))*x)/c^3 + ((c*j - b*l)*x^2)/(2*c^2) + ((c*k - b*
m)*x^3)/(3*c^2) + (l*x^4)/(4*c) + (m*x^5)/(5*c) + ((c^3*f - c^2*(b*h + a*k) - b^
3*m + b*c*(b*k + 2*a*m) + (2*c^4*d - c^3*(b*f + 2*a*h) + b^4*m - b^2*c*(b*k + 4*
a*m) + c^2*(b^2*h + 3*a*b*k + 2*a^2*m))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[
c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(7/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]
) + ((c^3*f - c^2*(b*h + a*k) - b^3*m + b*c*(b*k + 2*a*m) - (2*c^4*d - c^3*(b*f
+ 2*a*h) + b^4*m - b^2*c*(b*k + 4*a*m) + c^2*(b^2*h + 3*a*b*k + 2*a^2*m))/Sqrt[b
^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c
^(7/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) - ((2*c^3*e - c^2*(b*g + 2*a*j) - b^3*l + b*
c*(b*j + 3*a*l))*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^3*Sqrt[b^2 - 4*a
*c]) + ((c^2*g + b^2*l - c*(b*j + a*l))*Log[a + b*x^2 + c*x^4])/(4*c^3)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((m*x**8+l*x**7+k*x**6+j*x**5+h*x**4+g*x**3+f*x**2+e*x+d)/(c*x**4+b*x**2+a),x)

[Out]

Timed out

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Mathematica [A]  time = 3.52472, size = 816, normalized size = 1.5 \[ \frac{m x^5}{5 c}+\frac{l x^4}{4 c}+\frac{(c k-b m) x^3}{3 c^2}+\frac{(c j-b l) x^2}{2 c^2}+\frac{\left (m b^2+c^2 h-c (b k+a m)\right ) x}{c^3}+\frac{\left (2 d c^4+\left (-b f+\sqrt{b^2-4 a c} f-2 a h\right ) c^3+\left (2 m a^2+3 b k a-\sqrt{b^2-4 a c} k a+b^2 h-b \sqrt{b^2-4 a c} h\right ) c^2+b \left (-k b^2+\sqrt{b^2-4 a c} k b-4 a m b+2 a \sqrt{b^2-4 a c} m\right ) c+b^3 \left (b-\sqrt{b^2-4 a c}\right ) m\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} c^{7/2} \sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left (2 d c^4-\left (b f+\sqrt{b^2-4 a c} f+2 a h\right ) c^3+\left (2 m a^2+3 b k a+\sqrt{b^2-4 a c} k a+b^2 h+b \sqrt{b^2-4 a c} h\right ) c^2-b \left (k b^2+\sqrt{b^2-4 a c} k b+4 a m b+2 a \sqrt{b^2-4 a c} m\right ) c+b^3 \left (b+\sqrt{b^2-4 a c}\right ) m\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} c^{7/2} \sqrt{b^2-4 a c} \sqrt{b+\sqrt{b^2-4 a c}}}+\frac{\left (2 e c^3+\left (-b g+\sqrt{b^2-4 a c} g-2 a j\right ) c^2+\left (j b^2-\sqrt{b^2-4 a c} j b+3 a l b-a \sqrt{b^2-4 a c} l\right ) c+b^2 \left (\sqrt{b^2-4 a c}-b\right ) l\right ) \log \left (-2 c x^2-b+\sqrt{b^2-4 a c}\right )}{4 c^3 \sqrt{b^2-4 a c}}+\frac{\left (-2 e c^3+\left (b g+\sqrt{b^2-4 a c} g+2 a j\right ) c^2-\left (j b^2+\sqrt{b^2-4 a c} j b+3 a l b+a \sqrt{b^2-4 a c} l\right ) c+b^2 \left (b+\sqrt{b^2-4 a c}\right ) l\right ) \log \left (2 c x^2+b+\sqrt{b^2-4 a c}\right )}{4 c^3 \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + j*x^5 + k*x^6 + l*x^7 + m*x^8)/(a + b*x^2 + c*x^4),x]

[Out]

((c^2*h + b^2*m - c*(b*k + a*m))*x)/c^3 + ((c*j - b*l)*x^2)/(2*c^2) + ((c*k - b*
m)*x^3)/(3*c^2) + (l*x^4)/(4*c) + (m*x^5)/(5*c) + ((2*c^4*d + c^3*(-(b*f) + Sqrt
[b^2 - 4*a*c]*f - 2*a*h) + b^3*(b - Sqrt[b^2 - 4*a*c])*m + c^2*(b^2*h - b*Sqrt[b
^2 - 4*a*c]*h + 3*a*b*k - a*Sqrt[b^2 - 4*a*c]*k + 2*a^2*m) + b*c*(-(b^2*k) + b*S
qrt[b^2 - 4*a*c]*k - 4*a*b*m + 2*a*Sqrt[b^2 - 4*a*c]*m))*ArcTan[(Sqrt[2]*Sqrt[c]
*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(7/2)*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sq
rt[b^2 - 4*a*c]]) - ((2*c^4*d - c^3*(b*f + Sqrt[b^2 - 4*a*c]*f + 2*a*h) + b^3*(b
+ Sqrt[b^2 - 4*a*c])*m + c^2*(b^2*h + b*Sqrt[b^2 - 4*a*c]*h + 3*a*b*k + a*Sqrt[
b^2 - 4*a*c]*k + 2*a^2*m) - b*c*(b^2*k + b*Sqrt[b^2 - 4*a*c]*k + 4*a*b*m + 2*a*S
qrt[b^2 - 4*a*c]*m))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(S
qrt[2]*c^(7/2)*Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + ((2*c^3*e + c^2*
(-(b*g) + Sqrt[b^2 - 4*a*c]*g - 2*a*j) + b^2*(-b + Sqrt[b^2 - 4*a*c])*l + c*(b^2
*j - b*Sqrt[b^2 - 4*a*c]*j + 3*a*b*l - a*Sqrt[b^2 - 4*a*c]*l))*Log[-b + Sqrt[b^2
- 4*a*c] - 2*c*x^2])/(4*c^3*Sqrt[b^2 - 4*a*c]) + ((-2*c^3*e + c^2*(b*g + Sqrt[b
^2 - 4*a*c]*g + 2*a*j) + b^2*(b + Sqrt[b^2 - 4*a*c])*l - c*(b^2*j + b*Sqrt[b^2 -
4*a*c]*j + 3*a*b*l + a*Sqrt[b^2 - 4*a*c]*l))*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^
2])/(4*c^3*Sqrt[b^2 - 4*a*c])

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Maple [B]  time = 0.067, size = 3835, normalized size = 7. \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((m*x^8+l*x^7+k*x^6+j*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a),x)

[Out]

1/4/c^2/(4*a*c-b^2)*ln(2*c*x^2+(-4*a*c+b^2)^(1/2)+b)*b^3*j+1/4/c^2/(4*a*c-b^2)*l
n(-2*c*x^2+(-4*a*c+b^2)^(1/2)-b)*b^3*j-1/4/c^3/(4*a*c-b^2)*ln(-2*c*x^2+(-4*a*c+b
^2)^(1/2)-b)*b^4*l-1/4/c^3/(4*a*c-b^2)*ln(2*c*x^2+(-4*a*c+b^2)^(1/2)+b)*b^4*l-(-
4*a*c+b^2)^(1/2)/(4*a*c-b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c
*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a*h-(-4*a*c+b^2)^(1/2)/(4*a*c-b^2)
*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1
/2))*c)^(1/2))*a*h+1/4*l*x^4/c+1/5*m*x^5/c+h*x/c-1/2/(4*a*c-b^2)*2^(1/2)/((b+(-4
*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*f
*b^2+1/2/(4*a*c-b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x*2^(1/
2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*f*b^2+2*c/(4*a*c-b^2)*2^(1/2)/((b+(-4*a*c+
b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*f*a-1/
2*(-4*a*c+b^2)^(1/2)/(4*a*c-b^2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan
(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b*f+c*(-4*a*c+b^2)^(1/2)/(4*a*c-b
^2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)
^(1/2))*c)^(1/2))*d-2*c/(4*a*c-b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*ar
ctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*f*a-1/2*(-4*a*c+b^2)^(1/2)/
(4*a*c-b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(
-4*a*c+b^2)^(1/2))*c)^(1/2))*b*f+1/2*(-4*a*c+b^2)^(1/2)/(4*a*c-b^2)/c*2^(1/2)/((
-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^
(1/2))*b^2*h+1/(4*a*c-b^2)*ln(2*c*x^2+(-4*a*c+b^2)^(1/2)+b)*g*a+1/(4*a*c-b^2)*ln
(-2*c*x^2+(-4*a*c+b^2)^(1/2)-b)*g*a+1/2*(-4*a*c+b^2)^(1/2)/(4*a*c-b^2)*e*ln(2*c*
x^2+(-4*a*c+b^2)^(1/2)+b)-1/2*(-4*a*c+b^2)^(1/2)/(4*a*c-b^2)*e*ln(-2*c*x^2+(-4*a
*c+b^2)^(1/2)-b)-2/c^2*(-4*a*c+b^2)^(1/2)/(4*a*c-b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^
(1/2))*c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a*b^2*m+3
/2/c*(-4*a*c+b^2)^(1/2)/(4*a*c-b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*ar
ctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a*b*k-2/c^2*(-4*a*c+b^2)^(1
/2)/(4*a*c-b^2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+
(-4*a*c+b^2)^(1/2))*c)^(1/2))*a*b^2*m+3/2/c*(-4*a*c+b^2)^(1/2)/(4*a*c-b^2)*2^(1/
2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c
)^(1/2))*a*b*k+3/4/c^2*(-4*a*c+b^2)^(1/2)/(4*a*c-b^2)*ln(2*c*x^2+(-4*a*c+b^2)^(1
/2)+b)*a*b*l+1/2*(-4*a*c+b^2)^(1/2)/(4*a*c-b^2)/c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2)
)*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^2*h-1/2/c^2*(-
4*a*c+b^2)^(1/2)/(4*a*c-b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c
*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^3*k-4/c/(4*a*c-b^2)*2^(1/2)/((-b
+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1
/2))*a^2*b*m-3/c^2/(4*a*c-b^2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c
*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a*b^3*m-5/2/c/(4*a*c-b^2)*2^(1/2)/(
(-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)
^(1/2))*b^2*k*a+3/c^2/(4*a*c-b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arct
anh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^3*m*a+4/c/(4*a*c-b^2)*2^(1/
2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c
)^(1/2))*a^2*b*m+1/2/c^3*(-4*a*c+b^2)^(1/2)/(4*a*c-b^2)*2^(1/2)/((b+(-4*a*c+b^2)
^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^4*m-1/2/
c^2*(-4*a*c+b^2)^(1/2)/(4*a*c-b^2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arct
an(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^3*k-1/2/c^2/(4*a*c-b^2)*2^(1/
2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c
)^(1/2))*b^4*k-3/4/c^2*(-4*a*c+b^2)^(1/2)/(4*a*c-b^2)*ln(-2*c*x^2+(-4*a*c+b^2)^(
1/2)-b)*a*b*l-1/2/c^3/(4*a*c-b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arct
anh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^5*m+1/2/c/(4*a*c-b^2)*2^(1/
2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c
)^(1/2))*b^3*h+1/c*(-4*a*c+b^2)^(1/2)/(4*a*c-b^2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2)
)*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a^2*m+1/2/c^3*(-
4*a*c+b^2)^(1/2)/(4*a*c-b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c
*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^4*m+5/2/c/(4*a*c-b^2)*2^(1/2)/((
b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/
2))*a*k*b^2+1/c*(-4*a*c+b^2)^(1/2)/(4*a*c-b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*
c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a^2*m+1/2/c^3/(4
*a*c-b^2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*
c+b^2)^(1/2))*c)^(1/2))*b^5*m+1/2/c^2/(4*a*c-b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2
))*c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^4*k-1/c/(4*
a*c-b^2)*ln(2*c*x^2+(-4*a*c+b^2)^(1/2)+b)*a^2*l-1/c/(4*a*c-b^2)*ln(-2*c*x^2+(-4*
a*c+b^2)^(1/2)-b)*a^2*l+1/4*(-4*a*c+b^2)^(1/2)/(4*a*c-b^2)/c*ln(-2*c*x^2+(-4*a*c
+b^2)^(1/2)-b)*b*g-1/4/(4*a*c-b^2)/c*ln(2*c*x^2+(-4*a*c+b^2)^(1/2)+b)*g*b^2-1/3/
c^2*x^3*b*m-1/2/c^2*x^2*b*l-1/c^2*a*m*x+1/c^3*b^2*m*x-1/c^2*b*k*x-2/(4*a*c-b^2)*
2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/
2))*c)^(1/2))*b*h*a+2/(4*a*c-b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arct
anh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b*h*a-1/2/c/(4*a*c-b^2)*2^(1/
2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2)
)*c)^(1/2))*b^3*h-1/4*(-4*a*c+b^2)^(1/2)/(4*a*c-b^2)/c*ln(2*c*x^2+(-4*a*c+b^2)^(
1/2)+b)*b*g-1/4/(4*a*c-b^2)/c*ln(-2*c*x^2+(-4*a*c+b^2)^(1/2)-b)*g*b^2+1/3/c*x^3*
k+1/2/c*x^2*j+c*(-4*a*c+b^2)^(1/2)/(4*a*c-b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*
c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*d-1/c/(4*a*c-b^2
)*ln(2*c*x^2+(-4*a*c+b^2)^(1/2)+b)*b*j*a-1/c/(4*a*c-b^2)*ln(-2*c*x^2+(-4*a*c+b^2
)^(1/2)-b)*b*j*a+5/4/c^2/(4*a*c-b^2)*ln(2*c*x^2+(-4*a*c+b^2)^(1/2)+b)*a*l*b^2+5/
4/c^2/(4*a*c-b^2)*ln(-2*c*x^2+(-4*a*c+b^2)^(1/2)-b)*a*l*b^2+1/2/c*(-4*a*c+b^2)^(
1/2)/(4*a*c-b^2)*ln(-2*c*x^2+(-4*a*c+b^2)^(1/2)-b)*a*j+1/4/c^3*(-4*a*c+b^2)^(1/2
)/(4*a*c-b^2)*ln(-2*c*x^2+(-4*a*c+b^2)^(1/2)-b)*b^3*l-1/4/c^2*(-4*a*c+b^2)^(1/2)
/(4*a*c-b^2)*ln(-2*c*x^2+(-4*a*c+b^2)^(1/2)-b)*b^2*j-1/2/c*(-4*a*c+b^2)^(1/2)/(4
*a*c-b^2)*ln(2*c*x^2+(-4*a*c+b^2)^(1/2)+b)*a*j-1/4/c^3*(-4*a*c+b^2)^(1/2)/(4*a*c
-b^2)*ln(2*c*x^2+(-4*a*c+b^2)^(1/2)+b)*b^3*l+1/4/c^2*(-4*a*c+b^2)^(1/2)/(4*a*c-b
^2)*ln(2*c*x^2+(-4*a*c+b^2)^(1/2)+b)*b^2*j-2/(4*a*c-b^2)*2^(1/2)/((b+(-4*a*c+b^2
)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a^2*k+2/(
4*a*c-b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(-
4*a*c+b^2)^(1/2))*c)^(1/2))*a^2*k

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{12 \, c^{2} m x^{5} + 15 \, c^{2} l x^{4} + 20 \,{\left (c^{2} k - b c m\right )} x^{3} + 30 \,{\left (c^{2} j - b c l\right )} x^{2} + 60 \,{\left (c^{2} h - b c k +{\left (b^{2} - a c\right )} m\right )} x}{60 \, c^{3}} - \frac{-\int \frac{c^{3} d - a c^{2} h + a b c k +{\left (c^{3} g - b c^{2} j +{\left (b^{2} c - a c^{2}\right )} l\right )} x^{3} +{\left (c^{3} f - b c^{2} h +{\left (b^{2} c - a c^{2}\right )} k -{\left (b^{3} - 2 \, a b c\right )} m\right )} x^{2} -{\left (a b^{2} - a^{2} c\right )} m +{\left (c^{3} e - a c^{2} j + a b c l\right )} x}{c x^{4} + b x^{2} + a}\,{d x}}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((m*x^8 + l*x^7 + k*x^6 + j*x^5 + h*x^4 + g*x^3 + f*x^2 + e*x + d)/(c*x^4 + b*x^2 + a),x, algorithm="maxima")

[Out]

1/60*(12*c^2*m*x^5 + 15*c^2*l*x^4 + 20*(c^2*k - b*c*m)*x^3 + 30*(c^2*j - b*c*l)*
x^2 + 60*(c^2*h - b*c*k + (b^2 - a*c)*m)*x)/c^3 - integrate(-(c^3*d - a*c^2*h +
a*b*c*k + (c^3*g - b*c^2*j + (b^2*c - a*c^2)*l)*x^3 + (c^3*f - b*c^2*h + (b^2*c
- a*c^2)*k - (b^3 - 2*a*b*c)*m)*x^2 - (a*b^2 - a^2*c)*m + (c^3*e - a*c^2*j + a*b
*c*l)*x)/(c*x^4 + b*x^2 + a), x)/c^3

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((m*x^8 + l*x^7 + k*x^6 + j*x^5 + h*x^4 + g*x^3 + f*x^2 + e*x + d)/(c*x^4 + b*x^2 + a),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((m*x**8+l*x**7+k*x**6+j*x**5+h*x**4+g*x**3+f*x**2+e*x+d)/(c*x**4+b*x**2+a),x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((m*x^8 + l*x^7 + k*x^6 + j*x^5 + h*x^4 + g*x^3 + f*x^2 + e*x + d)/(c*x^4 + b*x^2 + a),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError