Integrand size = 55, antiderivative size = 770 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{\left (a+b x^2+c x^4\right )^2} \, dx=\frac {m x}{c^2}-\frac {b c (c e+a j)-a b^2 l-2 a c (c g-a l)+\left (2 c^3 e-c^2 (b g+2 a j)-b^3 l+b c (b j+3 a l)\right ) x^2}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {x \left (a b c (c f+a k)-b^2 \left (c^2 d+a^2 m\right )+2 a c \left (c^2 d-a c h+a^2 m\right )+\left (a b^2 c k+2 a c^2 (c f-a k)-a b^3 m-b c \left (c^2 d+a c h-3 a^2 m\right )\right ) x^2\right )}{2 a c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\left (a b^2 c k-2 a c^2 (c f+3 a k)-3 a b^3 m+b c \left (c^2 d+a c h+13 a^2 m\right )-\frac {a b^3 c k-4 a b c^2 (c f+2 a k)-3 a b^4 m-b^2 c \left (c^2 d-a c h-19 a^2 m\right )+4 a c^2 \left (3 c^2 d+a c h-5 a^2 m\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a c^{5/2} \left (b^2-4 a c\right ) \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (a b^2 c k-2 a c^2 (c f+3 a k)-3 a b^3 m+b c \left (c^2 d+a c h+13 a^2 m\right )+\frac {a b^3 c k-4 a b c^2 (c f+2 a k)-3 a b^4 m-b^2 c \left (c^2 d-a c h-19 a^2 m\right )+4 a c^2 \left (3 c^2 d+a c h-5 a^2 m\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a c^{5/2} \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}}}+\frac {\left (4 c^3 e-c^2 (2 b g-4 a j)+b^3 l-6 a b c l\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^2 \left (b^2-4 a c\right )^{3/2}}+\frac {l \log \left (a+b x^2+c x^4\right )}{4 c^2} \] Output:
m*x/c^2-1/2*(b*c*(a*j+c*e)-a*b^2*l-2*a*c*(-a*l+c*g)+(2*c^3*e-c^2*(2*a*j+b* g)-b^3*l+b*c*(3*a*l+b*j))*x^2)/c^2/(-4*a*c+b^2)/(c*x^4+b*x^2+a)-1/2*x*(a*b *c*(a*k+c*f)-b^2*(a^2*m+c^2*d)+2*a*c*(a^2*m-a*c*h+c^2*d)+(a*b^2*c*k+2*a*c^ 2*(-a*k+c*f)-a*b^3*m-b*c*(-3*a^2*m+a*c*h+c^2*d))*x^2)/a/c^2/(-4*a*c+b^2)/( c*x^4+b*x^2+a)+1/4*(a*b^2*c*k-2*a*c^2*(3*a*k+c*f)-3*a*b^3*m+b*c*(13*a^2*m+ a*c*h+c^2*d)-(a*b^3*c*k-4*a*b*c^2*(2*a*k+c*f)-3*a*b^4*m-b^2*c*(-19*a^2*m-a *c*h+c^2*d)+4*a*c^2*(-5*a^2*m+a*c*h+3*c^2*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^(5/2)/(-4*a*c+b ^2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)+1/4*(a*b^2*c*k-2*a*c^2*(3*a*k+c*f)-3*a*b^ 3*m+b*c*(13*a^2*m+a*c*h+c^2*d)+(a*b^3*c*k-4*a*b*c^2*(2*a*k+c*f)-3*a*b^4*m- b^2*c*(-19*a^2*m-a*c*h+c^2*d)+4*a*c^2*(-5*a^2*m+a*c*h+3*c^2*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^(5/2)/(-4*a*c+b^2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)+1/2*(4*c^3*e-c^2*(-4*a* j+2*b*g)+b^3*l-6*a*b*c*l)*arctanh((2*c*x^2+b)/(-4*a*c+b^2)^(1/2))/c^2/(-4* a*c+b^2)^(3/2)+1/4*l*ln(c*x^4+b*x^2+a)/c^2
Time = 5.62 (sec) , antiderivative size = 935, normalized size of antiderivative = 1.21 \[ \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}{\left (a+b x^2+c x^4\right )^2} \, dx=\frac {4 \sqrt {c} m x+\frac {2 \sqrt {c} \left (2 a^3 c (l+m x)-b c^2 d x \left (b+c x^2\right )+a \left (b^2 c x^2 (j+k x)-b^3 x^2 (l+m x)+2 c^3 x (d+x (e+f x))+b c^2 (e+x (f-x (g+h x)))\right )-a^2 \left (b^2 (l+m x)+2 c^2 (g+x (h+x (j+k x)))-b c (j+x (k+3 x (l+m x)))\right )\right )}{a \left (-b^2+4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\sqrt {2} \left (-3 a b^4 m+2 a c^2 \left (6 c^2 d+c \sqrt {b^2-4 a c} f+2 a c h+3 a \sqrt {b^2-4 a c} k-10 a^2 m\right )+a b^3 \left (c k+3 \sqrt {b^2-4 a c} m\right )-b c \left (c^2 \left (\sqrt {b^2-4 a c} d+4 a f\right )+a c \left (\sqrt {b^2-4 a c} h+8 a k\right )+13 a^2 \sqrt {b^2-4 a c} m\right )+b^2 c \left (-c^2 d+a c h+a \left (-\sqrt {b^2-4 a c} k+19 a m\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{a \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \left (3 a b^4 m+2 a c^2 \left (-6 c^2 d+c \sqrt {b^2-4 a c} f-2 a c h+3 a \sqrt {b^2-4 a c} k+10 a^2 m\right )+a b^3 \left (-c k+3 \sqrt {b^2-4 a c} m\right )-b c \left (c^2 \left (\sqrt {b^2-4 a c} d-4 a f\right )+a c \left (\sqrt {b^2-4 a c} h-8 a k\right )+13 a^2 \sqrt {b^2-4 a c} m\right )+b^2 c \left (c^2 d-a c h-a \left (\sqrt {b^2-4 a c} k+19 a m\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{a \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (-4 c^3 e+2 c^2 (b g-2 a j)+b^2 \left (-b+\sqrt {b^2-4 a c}\right ) l+a c \left (6 b l-4 \sqrt {b^2-4 a c} l\right )\right ) \log \left (-b+\sqrt {b^2-4 a c}-2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac {\sqrt {c} \left (4 c^3 e+c^2 (-2 b g+4 a j)+b^2 \left (b+\sqrt {b^2-4 a c}\right ) l-2 a c \left (3 b+2 \sqrt {b^2-4 a c}\right ) l\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}}{4 c^{5/2}} \] Input:
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)^2,x]
Output:
(4*Sqrt[c]*m*x + (2*Sqrt[c]*(2*a^3*c*(l + m*x) - b*c^2*d*x*(b + c*x^2) + a *(b^2*c*x^2*(j + k*x) - b^3*x^2*(l + m*x) + 2*c^3*x*(d + x*(e + f*x)) + b* c^2*(e + x*(f - x*(g + h*x)))) - a^2*(b^2*(l + m*x) + 2*c^2*(g + x*(h + x* (j + k*x))) - b*c*(j + x*(k + 3*x*(l + m*x))))))/(a*(-b^2 + 4*a*c)*(a + b* x^2 + c*x^4)) - (Sqrt[2]*(-3*a*b^4*m + 2*a*c^2*(6*c^2*d + c*Sqrt[b^2 - 4*a *c]*f + 2*a*c*h + 3*a*Sqrt[b^2 - 4*a*c]*k - 10*a^2*m) + a*b^3*(c*k + 3*Sqr t[b^2 - 4*a*c]*m) - b*c*(c^2*(Sqrt[b^2 - 4*a*c]*d + 4*a*f) + a*c*(Sqrt[b^2 - 4*a*c]*h + 8*a*k) + 13*a^2*Sqrt[b^2 - 4*a*c]*m) + b^2*c*(-(c^2*d) + a*c *h + a*(-(Sqrt[b^2 - 4*a*c]*k) + 19*a*m)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt [b - Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c ]]) - (Sqrt[2]*(3*a*b^4*m + 2*a*c^2*(-6*c^2*d + c*Sqrt[b^2 - 4*a*c]*f - 2* a*c*h + 3*a*Sqrt[b^2 - 4*a*c]*k + 10*a^2*m) + a*b^3*(-(c*k) + 3*Sqrt[b^2 - 4*a*c]*m) - b*c*(c^2*(Sqrt[b^2 - 4*a*c]*d - 4*a*f) + a*c*(Sqrt[b^2 - 4*a* c]*h - 8*a*k) + 13*a^2*Sqrt[b^2 - 4*a*c]*m) + b^2*c*(c^2*d - a*c*h - a*(Sq rt[b^2 - 4*a*c]*k + 19*a*m)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (Sqrt[c ]*(-4*c^3*e + 2*c^2*(b*g - 2*a*j) + b^2*(-b + Sqrt[b^2 - 4*a*c])*l + a*c*( 6*b*l - 4*Sqrt[b^2 - 4*a*c]*l))*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^ 2 - 4*a*c)^(3/2) + (Sqrt[c]*(4*c^3*e + c^2*(-2*b*g + 4*a*j) + b^2*(b + Sqr t[b^2 - 4*a*c])*l - 2*a*c*(3*b + 2*Sqrt[b^2 - 4*a*c])*l)*Log[b + Sqrt[b...
Time = 4.43 (sec) , antiderivative size = 788, normalized size of antiderivative = 1.02, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2202, 2194, 2191, 1142, 1083, 219, 1103, 2206, 25, 2205, 2009}
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.
| \(\displaystyle \int \frac {d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{\left (a+b x^2+c x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 2202 |
| \(\displaystyle \int \frac {m x^8+k x^6+h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^2}dx+\int \frac {x \left (l x^6+j x^4+g x^2+e\right )}{\left (c x^4+b x^2+a\right )^2}dx\) |
| \(\Big \downarrow \) 2194 |
| \(\displaystyle \int \frac {m x^8+k x^6+h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^2}dx+\frac {1}{2} \int \frac {l x^6+j x^4+g x^2+e}{\left (c x^4+b x^2+a\right )^2}dx^2\) |
| \(\Big \downarrow \) 2191 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {\left (4 a-\frac {b^2}{c}\right ) l x^2+2 c e-b g+2 a j-\frac {a b l}{c}}{c x^4+b x^2+a}dx^2}{b^2-4 a c}-\frac {x^2 \left (-c^2 (2 a j+b g)+b c (3 a l+b j)+b^3 (-l)+2 c^3 e\right )-a b^2 l+b c (a j+c e)-2 a c (c g-a l)}{c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )+\int \frac {m x^8+k x^6+h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^2}dx\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\frac {\left (-c^2 (2 b g-4 a j)-6 a b c l+b^3 l+4 c^3 e\right ) \int \frac {1}{c x^4+b x^2+a}dx^2}{2 c^2}-\frac {l \left (b^2-4 a c\right ) \int \frac {2 c x^2+b}{c x^4+b x^2+a}dx^2}{2 c^2}}{b^2-4 a c}-\frac {x^2 \left (-c^2 (2 a j+b g)+b c (3 a l+b j)+b^3 (-l)+2 c^3 e\right )-a b^2 l+b c (a j+c e)-2 a c (c g-a l)}{c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )+\int \frac {m x^8+k x^6+h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^2}dx\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{2} \left (-\frac {-\frac {l \left (b^2-4 a c\right ) \int \frac {2 c x^2+b}{c x^4+b x^2+a}dx^2}{2 c^2}-\frac {\left (-c^2 (2 b g-4 a j)-6 a b c l+b^3 l+4 c^3 e\right ) \int \frac {1}{-x^4+b^2-4 a c}d\left (2 c x^2+b\right )}{c^2}}{b^2-4 a c}-\frac {x^2 \left (-c^2 (2 a j+b g)+b c (3 a l+b j)+b^3 (-l)+2 c^3 e\right )-a b^2 l+b c (a j+c e)-2 a c (c g-a l)}{c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )+\int \frac {m x^8+k x^6+h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^2}dx\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (-\frac {-\frac {l \left (b^2-4 a c\right ) \int \frac {2 c x^2+b}{c x^4+b x^2+a}dx^2}{2 c^2}-\frac {\text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-c^2 (2 b g-4 a j)-6 a b c l+b^3 l+4 c^3 e\right )}{c^2 \sqrt {b^2-4 a c}}}{b^2-4 a c}-\frac {x^2 \left (-c^2 (2 a j+b g)+b c (3 a l+b j)+b^3 (-l)+2 c^3 e\right )-a b^2 l+b c (a j+c e)-2 a c (c g-a l)}{c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )+\int \frac {m x^8+k x^6+h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^2}dx\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \int \frac {m x^8+k x^6+h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^2}dx+\frac {1}{2} \left (-\frac {-\frac {\text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-c^2 (2 b g-4 a j)-6 a b c l+b^3 l+4 c^3 e\right )}{c^2 \sqrt {b^2-4 a c}}-\frac {l \left (b^2-4 a c\right ) \log \left (a+b x^2+c x^4\right )}{2 c^2}}{b^2-4 a c}-\frac {x^2 \left (-c^2 (2 a j+b g)+b c (3 a l+b j)+b^3 (-l)+2 c^3 e\right )-a b^2 l+b c (a j+c e)-2 a c (c g-a l)}{c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )\) |
| \(\Big \downarrow \) 2206 |
| \(\displaystyle -\frac {\int -\frac {-2 a \left (4 a-\frac {b^2}{c}\right ) m x^4+\frac {\left (-a m b^3+a c k b^2+c \left (5 m a^2+c h a+c^2 d\right ) b-2 a c^2 (c f+3 a k)\right ) x^2}{c^2}+\frac {\left (c^2 d-a^2 m\right ) b^2+a c (c f+a k) b-2 a c \left (-m a^2+c h a+3 c^2 d\right )}{c^2}}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}-\frac {x \left (-\left (b^2 \left (a^2 m+c^2 d\right )\right )+x^2 \left (-b c \left (-3 a^2 m+a c h+c^2 d\right )-a b^3 m+a b^2 c k+2 a c^2 (c f-a k)\right )+2 a c \left (a^2 m-a c h+c^2 d\right )+a b c (a k+c f)\right )}{2 a c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {1}{2} \left (-\frac {-\frac {\text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-c^2 (2 b g-4 a j)-6 a b c l+b^3 l+4 c^3 e\right )}{c^2 \sqrt {b^2-4 a c}}-\frac {l \left (b^2-4 a c\right ) \log \left (a+b x^2+c x^4\right )}{2 c^2}}{b^2-4 a c}-\frac {x^2 \left (-c^2 (2 a j+b g)+b c (3 a l+b j)+b^3 (-l)+2 c^3 e\right )-a b^2 l+b c (a j+c e)-2 a c (c g-a l)}{c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {-2 a \left (4 a-\frac {b^2}{c}\right ) m x^4+\frac {\left (-a m b^3+a c k b^2+c \left (5 m a^2+c h a+c^2 d\right ) b-2 a c^2 (c f+3 a k)\right ) x^2}{c^2}+\frac {\left (c^2 d-a^2 m\right ) b^2+a c (c f+a k) b-2 a c \left (-m a^2+c h a+3 c^2 d\right )}{c^2}}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}-\frac {x \left (-\left (b^2 \left (a^2 m+c^2 d\right )\right )+x^2 \left (-b c \left (-3 a^2 m+a c h+c^2 d\right )-a b^3 m+a b^2 c k+2 a c^2 (c f-a k)\right )+2 a c \left (a^2 m-a c h+c^2 d\right )+a b c (a k+c f)\right )}{2 a c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {1}{2} \left (-\frac {-\frac {\text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-c^2 (2 b g-4 a j)-6 a b c l+b^3 l+4 c^3 e\right )}{c^2 \sqrt {b^2-4 a c}}-\frac {l \left (b^2-4 a c\right ) \log \left (a+b x^2+c x^4\right )}{2 c^2}}{b^2-4 a c}-\frac {x^2 \left (-c^2 (2 a j+b g)+b c (3 a l+b j)+b^3 (-l)+2 c^3 e\right )-a b^2 l+b c (a j+c e)-2 a c (c g-a l)}{c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )\) |
| \(\Big \downarrow \) 2205 |
| \(\displaystyle \frac {\int \left (\frac {2 a \left (b^2-4 a c\right ) m}{c^2}+\frac {\left (c^2 d-3 a^2 m\right ) b^2+a c (c f+a k) b+\left (-3 a m b^3+a c k b^2+c \left (13 m a^2+c h a+c^2 d\right ) b-2 a c^2 (c f+3 a k)\right ) x^2-2 a c \left (-5 m a^2+c h a+3 c^2 d\right )}{c^2 \left (c x^4+b x^2+a\right )}\right )dx}{2 a \left (b^2-4 a c\right )}-\frac {x \left (-\left (b^2 \left (a^2 m+c^2 d\right )\right )+x^2 \left (-b c \left (-3 a^2 m+a c h+c^2 d\right )-a b^3 m+a b^2 c k+2 a c^2 (c f-a k)\right )+2 a c \left (a^2 m-a c h+c^2 d\right )+a b c (a k+c f)\right )}{2 a c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {1}{2} \left (-\frac {-\frac {\text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-c^2 (2 b g-4 a j)-6 a b c l+b^3 l+4 c^3 e\right )}{c^2 \sqrt {b^2-4 a c}}-\frac {l \left (b^2-4 a c\right ) \log \left (a+b x^2+c x^4\right )}{2 c^2}}{b^2-4 a c}-\frac {x^2 \left (-c^2 (2 a j+b g)+b c (3 a l+b j)+b^3 (-l)+2 c^3 e\right )-a b^2 l+b c (a j+c e)-2 a c (c g-a l)}{c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (-\frac {-b^2 c \left (-19 a^2 m-a c h+c^2 d\right )+4 a c^2 \left (-5 a^2 m+a c h+3 c^2 d\right )-3 a b^4 m+a b^3 c k-4 a b c^2 (2 a k+c f)}{\sqrt {b^2-4 a c}}+b c \left (13 a^2 m+a c h+c^2 d\right )-3 a b^3 m+a b^2 c k-2 a c^2 (3 a k+c f)\right )}{\sqrt {2} c^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (\frac {-b^2 c \left (-19 a^2 m-a c h+c^2 d\right )+4 a c^2 \left (-5 a^2 m+a c h+3 c^2 d\right )-3 a b^4 m+a b^3 c k-4 a b c^2 (2 a k+c f)}{\sqrt {b^2-4 a c}}+b c \left (13 a^2 m+a c h+c^2 d\right )-3 a b^3 m+a b^2 c k-2 a c^2 (3 a k+c f)\right )}{\sqrt {2} c^{5/2} \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {2 a m x \left (b^2-4 a c\right )}{c^2}}{2 a \left (b^2-4 a c\right )}-\frac {x \left (-\left (b^2 \left (a^2 m+c^2 d\right )\right )+x^2 \left (-b c \left (-3 a^2 m+a c h+c^2 d\right )-a b^3 m+a b^2 c k+2 a c^2 (c f-a k)\right )+2 a c \left (a^2 m-a c h+c^2 d\right )+a b c (a k+c f)\right )}{2 a c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {1}{2} \left (-\frac {-\frac {\text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-c^2 (2 b g-4 a j)-6 a b c l+b^3 l+4 c^3 e\right )}{c^2 \sqrt {b^2-4 a c}}-\frac {l \left (b^2-4 a c\right ) \log \left (a+b x^2+c x^4\right )}{2 c^2}}{b^2-4 a c}-\frac {x^2 \left (-c^2 (2 a j+b g)+b c (3 a l+b j)+b^3 (-l)+2 c^3 e\right )-a b^2 l+b c (a j+c e)-2 a c (c g-a l)}{c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )\) |
Input:
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)^2,x]
Output:
-1/2*(x*(a*b*c*(c*f + a*k) - b^2*(c^2*d + a^2*m) + 2*a*c*(c^2*d - a*c*h + a^2*m) + (a*b^2*c*k + 2*a*c^2*(c*f - a*k) - a*b^3*m - b*c*(c^2*d + a*c*h - 3*a^2*m))*x^2))/(a*c^2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + ((2*a*(b^2 - 4*a*c)*m*x)/c^2 + ((a*b^2*c*k - 2*a*c^2*(c*f + 3*a*k) - 3*a*b^3*m + b*c*(c ^2*d + a*c*h + 13*a^2*m) - (a*b^3*c*k - 4*a*b*c^2*(c*f + 2*a*k) - 3*a*b^4* m - b^2*c*(c^2*d - a*c*h - 19*a^2*m) + 4*a*c^2*(3*c^2*d + a*c*h - 5*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^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((a*b^2*c*k - 2*a*c^2* (c*f + 3*a*k) - 3*a*b^3*m + b*c*(c^2*d + a*c*h + 13*a^2*m) + (a*b^3*c*k - 4*a*b*c^2*(c*f + 2*a*k) - 3*a*b^4*m - b^2*c*(c^2*d - a*c*h - 19*a^2*m) + 4 *a*c^2*(3*c^2*d + a*c*h - 5*a^2*m))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqr t[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(2*a*(b^2 - 4*a*c)) + (-((b*c*(c*e + a*j) - a*b^2*l - 2*a*c*(c* g - a*l) + (2*c^3*e - c^2*(b*g + 2*a*j) - b^3*l + b*c*(b*j + 3*a*l))*x^2)/ (c^2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4))) - (-(((4*c^3*e - c^2*(2*b*g - 4*a *j) + b^3*l - 6*a*b*c*l)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(c^2*Sq rt[b^2 - 4*a*c])) - ((b^2 - 4*a*c)*l*Log[a + b*x^2 + c*x^4])/(2*c^2))/(b^2 - 4*a*c))/2
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])
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]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[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]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
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]
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]
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]
Int[(Px_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandInte grand[Px/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x^ 2] && Expon[Px, x^2] > 1
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.80 (sec) , antiderivative size = 506, normalized size of antiderivative = 0.66
| method | result | size |
| risch | \(\frac {m x}{c^{2}}+\frac {\frac {\left (3 a^{2} b c m -2 a^{2} c^{2} k -a \,b^{3} m +a \,b^{2} c k -a b \,c^{2} h +2 a \,c^{3} f -b \,c^{3} d \right ) x^{3}}{2 a \left (4 a c -b^{2}\right )}+\frac {\left (3 a b c l -2 a \,c^{2} j -b^{3} l +b^{2} c j -b \,c^{2} g +2 c^{3} e \right ) x^{2}}{8 a c -2 b^{2}}+\frac {\left (2 a^{3} c m -a^{2} b^{2} m +a^{2} b c k -2 a^{2} c^{2} h +a b \,c^{2} f +2 a \,c^{3} d -b^{2} c^{2} d \right ) x}{2 a \left (4 a c -b^{2}\right )}+\frac {2 a^{2} c l -a \,b^{2} l +a b c j -2 a \,c^{2} g +b \,c^{2} e}{8 a c -2 b^{2}}}{c^{2} \left (c \,x^{4}+b \,x^{2}+a \right )}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (2 l c \,\textit {\_R}^{3}-\frac {\left (13 a^{2} b c m -6 a^{2} c^{2} k -3 a \,b^{3} m +a \,b^{2} c k +a b \,c^{2} h -2 a \,c^{3} f +b \,c^{3} d \right ) \textit {\_R}^{2}}{a \left (4 a c -b^{2}\right )}-\frac {2 c \left (a b l -2 a c j +b c g -2 e \,c^{2}\right ) \textit {\_R}}{4 a c -b^{2}}-\frac {10 a^{3} c m -3 a^{2} b^{2} m +a^{2} b c k -2 a^{2} c^{2} h +a b \,c^{2} f -6 a \,c^{3} d +b^{2} c^{2} d}{a \left (4 a c -b^{2}\right )}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3} c +\textit {\_R} b}}{4 c^{2}}\) | \(506\) |
| default | \(\text {Expression too large to display}\) | \(1317\) |
Input:
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)^2,x, method=_RETURNVERBOSE)
Output:
m*x/c^2+(1/2/a*(3*a^2*b*c*m-2*a^2*c^2*k-a*b^3*m+a*b^2*c*k-a*b*c^2*h+2*a*c^ 3*f-b*c^3*d)/(4*a*c-b^2)*x^3+1/2*(3*a*b*c*l-2*a*c^2*j-b^3*l+b^2*c*j-b*c^2* g+2*c^3*e)/(4*a*c-b^2)*x^2+1/2*(2*a^3*c*m-a^2*b^2*m+a^2*b*c*k-2*a^2*c^2*h+ a*b*c^2*f+2*a*c^3*d-b^2*c^2*d)/a/(4*a*c-b^2)*x+1/2*(2*a^2*c*l-a*b^2*l+a*b* c*j-2*a*c^2*g+b*c^2*e)/(4*a*c-b^2))/c^2/(c*x^4+b*x^2+a)+1/4/c^2*sum((2*l*c *_R^3-(13*a^2*b*c*m-6*a^2*c^2*k-3*a*b^3*m+a*b^2*c*k+a*b*c^2*h-2*a*c^3*f+b* c^3*d)/a/(4*a*c-b^2)*_R^2-2*c*(a*b*l-2*a*c*j+b*c*g-2*c^2*e)/(4*a*c-b^2)*_R -(10*a^3*c*m-3*a^2*b^2*m+a^2*b*c*k-2*a^2*c^2*h+a*b*c^2*f-6*a*c^3*d+b^2*c^2 *d)/a/(4*a*c-b^2))/(2*_R^3*c+_R*b)*ln(x-_R),_R=RootOf(_Z^4*c+_Z^2*b+a))
Timed out. \[ \int \frac {d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \] Input:
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 )^2,x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \] Input:
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)**2,x)
Output:
Timed out
\[ \int \frac {d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{\left (a+b x^2+c x^4\right )^2} \, dx=\int { \frac {m x^{8} + l x^{7} + k x^{6} + j x^{5} + h x^{4} + g x^{3} + f x^{2} + e x + d}{{\left (c x^{4} + b x^{2} + a\right )}^{2}} \,d x } \] Input:
integrate((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 )^2,x, algorithm="maxima")
Output:
-1/2*(a*b*c^2*e - 2*a^2*c^2*g + a^2*b*c*j - (b*c^3*d - 2*a*c^3*f + a*b*c^2 *h - (a*b^2*c - 2*a^2*c^2)*k + (a*b^3 - 3*a^2*b*c)*m)*x^3 + (2*a*c^3*e - a *b*c^2*g + (a*b^2*c - 2*a^2*c^2)*j - (a*b^3 - 3*a^2*b*c)*l)*x^2 - (a^2*b^2 - 2*a^3*c)*l + (a*b*c^2*f - 2*a^2*c^2*h + a^2*b*c*k - (b^2*c^2 - 2*a*c^3) *d - (a^2*b^2 - 2*a^3*c)*m)*x)/(a^2*b^2*c^2 - 4*a^3*c^3 + (a*b^2*c^3 - 4*a ^2*c^4)*x^4 + (a*b^3*c^2 - 4*a^2*b*c^3)*x^2) + m*x/c^2 - 1/2*integrate(-(a *b*c^2*f - 2*a^2*c^2*h + a^2*b*c*k + 2*(a*b^2*c - 4*a^2*c^2)*l*x^3 + (b*c^ 3*d - 2*a*c^3*f + a*b*c^2*h + (a*b^2*c - 6*a^2*c^2)*k - (3*a*b^3 - 13*a^2* b*c)*m)*x^2 + (b^2*c^2 - 6*a*c^3)*d - (3*a^2*b^2 - 10*a^3*c)*m - 2*(2*a*c^ 3*e - a*b*c^2*g + 2*a^2*c^2*j - a^2*b*c*l)*x)/(c*x^4 + b*x^2 + a), x)/(a*b ^2*c^2 - 4*a^2*c^3)
Leaf count of result is larger than twice the leaf count of optimal. 20158 vs. \(2 (718) = 1436\).
Time = 2.57 (sec) , antiderivative size = 20158, normalized size of antiderivative = 26.18 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \] Input:
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 )^2,x, algorithm="giac")
Output:
m*x/c^2 + 1/4*l*log(abs(c*x^4 + b*x^2 + a))/c^2 + 1/16*((a^2*b^4*c^5 - 8*a ^3*b^2*c^6 + 16*a^4*c^7)^2*(2*b^3*c^5 - 8*a*b*c^6 - sqrt(2)*sqrt(b^2 - 4*a *c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)* sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^4 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt (b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b*c^5 - 2*(b^2 - 4*a*c)*b*c^5)*d - 2*(a^2*b^4*c^5 - 8*a^3*b^2*c^6 + 16*a^4*c^7)^2*(2*a*b^2*c^5 - 8*a^2*c^6 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^4 + 2*sqrt(2)*sqrt(b^2 - 4*a* c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqr t(b*c + sqrt(b^2 - 4*a*c)*c)*a*c^5 - 2*(b^2 - 4*a*c)*a*c^5)*f + (a^2*b^4*c ^5 - 8*a^3*b^2*c^6 + 16*a^4*c^7)^2*(2*a*b^3*c^4 - 8*a^2*b*c^5 - sqrt(2)*sq rt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^2 + 4*sqrt(2)*sqrt (b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^3 + 2*sqrt(2)*sqrt(b ^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^4 - 2*(b^2 - 4*a*c)*a*b*c^4) *h + (a^2*b^4*c^5 - 8*a^3*b^2*c^6 + 16*a^4*c^7)^2*(2*a*b^4*c^3 - 20*a^2*b^ 2*c^4 + 48*a^3*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c )*c)*a*b^4*c + 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c )*a^2*b^2*c^2 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c...
Time = 49.92 (sec) , antiderivative size = 82785, normalized size of antiderivative = 107.51 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \] Input:
int((d + e*x + f*x^2 + g*x^3 + h*x^4 + j*x^5 + k*x^6 + l*x^7 + m*x^8)/(a + b*x^2 + c*x^4)^2,x)
Output:
symsum(log(root(1572864*a^8*b^2*c^10*z^4 - 983040*a^7*b^4*c^9*z^4 + 327680 *a^6*b^6*c^8*z^4 - 61440*a^5*b^8*c^7*z^4 + 6144*a^4*b^10*c^6*z^4 - 256*a^3 *b^12*c^5*z^4 - 1048576*a^9*c^11*z^4 - 1572864*a^8*b^2*c^8*l*z^3 + 983040* a^7*b^4*c^7*l*z^3 - 327680*a^6*b^6*c^6*l*z^3 + 61440*a^5*b^8*c^5*l*z^3 - 6 144*a^4*b^10*c^4*l*z^3 + 256*a^3*b^12*c^3*l*z^3 + 1048576*a^9*c^9*l*z^3 + 96*a^3*b^12*c*k*m*z^2 + 98304*a^8*b*c^7*j*l*z^2 + 24576*a^8*b*c^7*h*m*z^2 + 155648*a^7*b*c^8*d*m*z^2 + 98304*a^7*b*c^8*e*l*z^2 + 57344*a^7*b*c^8*f*k *z^2 + 32768*a^7*b*c^8*g*j*z^2 + 57344*a^6*b*c^9*d*h*z^2 + 32768*a^6*b*c^9 *e*g*z^2 - 32*a*b^10*c^5*d*f*z^2 - 491520*a^8*b^2*c^6*k*m*z^2 + 358400*a^7 *b^4*c^5*k*m*z^2 - 129024*a^6*b^6*c^4*k*m*z^2 + 24768*a^5*b^8*c^3*k*m*z^2 - 2432*a^4*b^10*c^2*k*m*z^2 - 90112*a^7*b^3*c^6*j*l*z^2 + 30720*a^6*b^5*c^ 5*j*l*z^2 - 4608*a^5*b^7*c^4*j*l*z^2 + 256*a^4*b^9*c^3*j*l*z^2 - 21504*a^6 *b^5*c^5*h*m*z^2 + 9216*a^5*b^7*c^4*h*m*z^2 + 8192*a^7*b^3*c^6*h*m*z^2 - 1 568*a^4*b^9*c^3*h*m*z^2 + 96*a^3*b^11*c^2*h*m*z^2 - 172032*a^7*b^2*c^7*f*m *z^2 + 116736*a^6*b^4*c^6*f*m*z^2 - 49152*a^7*b^2*c^7*g*l*z^2 + 45056*a^6* b^4*c^6*g*l*z^2 - 35840*a^5*b^6*c^5*f*m*z^2 + 24576*a^7*b^2*c^7*h*k*z^2 - 15360*a^5*b^6*c^5*g*l*z^2 + 5184*a^4*b^8*c^4*f*m*z^2 - 3072*a^5*b^6*c^5*h* k*z^2 + 2304*a^4*b^8*c^4*g*l*z^2 + 2048*a^6*b^4*c^6*h*k*z^2 + 576*a^4*b^8* c^4*h*k*z^2 - 288*a^3*b^10*c^3*f*m*z^2 - 128*a^3*b^10*c^3*g*l*z^2 - 32*a^3 *b^10*c^3*h*k*z^2 - 147456*a^6*b^3*c^7*d*m*z^2 - 90112*a^6*b^3*c^7*e*l*...
Time = 2.18 (sec) , antiderivative size = 15020, normalized size of antiderivative = 19.51 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{\left (a+b x^2+c x^4\right )^2} \, dx =\text {Too large to display} \] Input:
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)^2,x)
Output:
(24*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*s qrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**4*b**2* c**2*l - 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**3*j - 4*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*at an((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b) )*a**3*b**4*c*l + 24*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**3*c**2*l*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*sqr t(c)*sqrt(a) + b))*a**3*b**2*c**3*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)/sq rt(2*sqrt(c)*sqrt(a) + b))*a**3*b**2*c**3*j*x**2 + 24*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**3*l*x**4 - 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**4*e - 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**3*b*c**4*j*x* *4 - 4*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sq...