Integrand size = 55, antiderivative size = 1150 \[ \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 )^3} \, dx=-\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}{4 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\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 )}{4 a c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {\frac {b^3 j}{c}+2 b (3 c e+a j)-16 a^2 l-\frac {b^4 l}{c^2}-b^2 \left (3 g-\frac {5 a l}{c}\right )+2 \left (6 c^2 e-3 b c g+b^2 j+2 a c j-3 a b l\right ) x^2}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {x \left (4 a^2 b c^2 (2 c f+a k)+a b^3 c (c f+2 a k)-a b^2 c \left (25 c^2 d+7 a c h-11 a^2 m\right )+4 a^2 c^2 \left (7 c^2 d+a c h-9 a^2 m\right )+b^4 \left (3 c^2 d-2 a^2 m\right )+c \left (a b^2 c (c f+3 a k)+4 a^2 c^2 (5 c f+3 a k)+b^3 \left (3 c^2 d+a^2 m\right )-4 a b c \left (6 c^2 d+3 a c h+4 a^2 m\right )\right ) x^2\right )}{8 a^2 c^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\left (a b^2 c (c f+3 a k)+4 a^2 c^2 (5 c f+3 a k)+b^3 \left (3 c^2 d+a^2 m\right )-4 a b c \left (6 c^2 d+3 a c h+4 a^2 m\right )+\frac {a b^3 c (c f-3 a k)-4 a^2 b c^2 (13 c f+9 a k)-6 a b^2 c \left (5 c^2 d-3 a c h-3 a^2 m\right )+b^4 \left (3 c^2 d-a^2 m\right )+8 a^2 c^2 \left (21 c^2 d+3 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 )}{8 \sqrt {2} a^2 c^{3/2} \left (b^2-4 a c\right )^2 \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (a b^2 c (c f+3 a k)+4 a^2 c^2 (5 c f+3 a k)+b^3 \left (3 c^2 d+a^2 m\right )-4 a b c \left (6 c^2 d+3 a c h+4 a^2 m\right )-\frac {a b^3 c (c f-3 a k)-4 a^2 b c^2 (13 c f+9 a k)-6 a b^2 c \left (5 c^2 d-3 a c h-3 a^2 m\right )+b^4 \left (3 c^2 d-a^2 m\right )+8 a^2 c^2 \left (21 c^2 d+3 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 )}{8 \sqrt {2} a^2 c^{3/2} \left (b^2-4 a c\right )^2 \sqrt {b+\sqrt {b^2-4 a c}}}-\frac {\left (6 c^2 e-3 b c g+b^2 j+2 a c j-3 a b l\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \]
1/4*(-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)^2-1/4*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)^2+1/4*(b^3*j/c+2*b*(a*j+3*c*e)-16*a^2*l-b^4*l/c^2-b^2*(3*g-5*a*l /c)+2*(-3*a*b*l+2*a*c*j+b^2*j-3*b*c*g+6*c^2*e)*x^2)/(-4*a*c+b^2)^2/(c*x^4+ b*x^2+a)+1/8*x*(4*a^2*b*c^2*(a*k+2*c*f)+a*b^3*c*(2*a*k+c*f)-a*b^2*c*(-11*a ^2*m+7*a*c*h+25*c^2*d)+4*a^2*c^2*(-9*a^2*m+a*c*h+7*c^2*d)+b^4*(-2*a^2*m+3* c^2*d)+c*(a*b^2*c*(3*a*k+c*f)+4*a^2*c^2*(3*a*k+5*c*f)+b^3*(a^2*m+3*c^2*d)- 4*a*b*c*(4*a^2*m+3*a*c*h+6*c^2*d))*x^2)/a^2/c^2/(-4*a*c+b^2)^2/(c*x^4+b*x^ 2+a)-(-3*a*b*l+2*a*c*j+b^2*j-3*b*c*g+6*c^2*e)*arctanh((2*c*x^2+b)/(-4*a*c+ b^2)^(1/2))/(-4*a*c+b^2)^(5/2)+1/16*arctan(x*2^(1/2)*c^(1/2)/(b-(-4*a*c+b^ 2)^(1/2))^(1/2))*(a*b^2*c*(3*a*k+c*f)+4*a^2*c^2*(3*a*k+5*c*f)+b^3*(a^2*m+3 *c^2*d)-4*a*b*c*(4*a^2*m+3*a*c*h+6*c^2*d)+(a*b^3*c*(-3*a*k+c*f)-4*a^2*b*c^ 2*(9*a*k+13*c*f)-6*a*b^2*c*(-3*a^2*m-3*a*c*h+5*c^2*d)+b^4*(-a^2*m+3*c^2*d) +8*a^2*c^2*(5*a^2*m+3*a*c*h+21*c^2*d))/(-4*a*c+b^2)^(1/2))/a^2/c^(3/2)/(-4 *a*c+b^2)^2*2^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)+1/16*arctan(x*2^(1/2)*c^( 1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*(a*b^2*c*(3*a*k+c*f)+4*a^2*c^2*(3*a*k+5 *c*f)+b^3*(a^2*m+3*c^2*d)-4*a*b*c*(4*a^2*m+3*a*c*h+6*c^2*d)+(-a*b^3*c*(-3* a*k+c*f)+4*a^2*b*c^2*(9*a*k+13*c*f)+6*a*b^2*c*(-3*a^2*m-3*a*c*h+5*c^2*d...
Time = 6.85 (sec) , antiderivative size = 1590, normalized size of antiderivative = 1.38 \[ \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 )^3} \, dx=\frac {a b c^2 e-2 a^2 c^2 g+a^2 b c j-a^2 b^2 l+2 a^3 c l-b^2 c^2 d x+2 a c^3 d x+a b c^2 f x-2 a^2 c^2 h x+a^2 b c k x-a^2 b^2 m x+2 a^3 c m x+2 a c^3 e x^2-a b c^2 g x^2+a b^2 c j x^2-2 a^2 c^2 j x^2-a b^3 l x^2+3 a^2 b c l x^2-b c^3 d x^3+2 a c^3 f x^3-a b c^2 h x^3+a b^2 c k x^3-2 a^2 c^2 k x^3-a b^3 m x^3+3 a^2 b c m x^3}{4 a c^2 \left (-b^2+4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {12 a^2 b c^3 e-6 a^2 b^2 c^2 g+2 a^2 b^3 c j+4 a^3 b c^2 j-2 a^2 b^4 l+10 a^3 b^2 c l-32 a^4 c^2 l+3 b^4 c^2 d x-25 a b^2 c^3 d x+28 a^2 c^4 d x+a b^3 c^2 f x+8 a^2 b c^3 f x-7 a^2 b^2 c^2 h x+4 a^3 c^3 h x+2 a^2 b^3 c k x+4 a^3 b c^2 k x-2 a^2 b^4 m x+11 a^3 b^2 c m x-36 a^4 c^2 m x+24 a^2 c^4 e x^2-12 a^2 b c^3 g x^2+4 a^2 b^2 c^2 j x^2+8 a^3 c^3 j x^2-12 a^3 b c^2 l x^2+3 b^3 c^3 d x^3-24 a b c^4 d x^3+a b^2 c^3 f x^3+20 a^2 c^4 f x^3-12 a^2 b c^3 h x^3+3 a^2 b^2 c^2 k x^3+12 a^3 c^3 k x^3+a^2 b^3 c m x^3-16 a^3 b c^2 m x^3}{8 a^2 c^2 \left (-b^2+4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\left (3 b^4 c^2 d-30 a b^2 c^3 d+168 a^2 c^4 d+3 b^3 c^2 \sqrt {b^2-4 a c} d-24 a b c^3 \sqrt {b^2-4 a c} d+a b^3 c^2 f-52 a^2 b c^3 f+a b^2 c^2 \sqrt {b^2-4 a c} f+20 a^2 c^3 \sqrt {b^2-4 a c} f+18 a^2 b^2 c^2 h+24 a^3 c^3 h-12 a^2 b c^2 \sqrt {b^2-4 a c} h-3 a^2 b^3 c k-36 a^3 b c^2 k+3 a^2 b^2 c \sqrt {b^2-4 a c} k+12 a^3 c^2 \sqrt {b^2-4 a c} k-a^2 b^4 m+18 a^3 b^2 c m+40 a^4 c^2 m+a^2 b^3 \sqrt {b^2-4 a c} m-16 a^3 b c \sqrt {b^2-4 a c} m\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a^2 c^{3/2} \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (-3 b^4 c^2 d+30 a b^2 c^3 d-168 a^2 c^4 d+3 b^3 c^2 \sqrt {b^2-4 a c} d-24 a b c^3 \sqrt {b^2-4 a c} d-a b^3 c^2 f+52 a^2 b c^3 f+a b^2 c^2 \sqrt {b^2-4 a c} f+20 a^2 c^3 \sqrt {b^2-4 a c} f-18 a^2 b^2 c^2 h-24 a^3 c^3 h-12 a^2 b c^2 \sqrt {b^2-4 a c} h+3 a^2 b^3 c k+36 a^3 b c^2 k+3 a^2 b^2 c \sqrt {b^2-4 a c} k+12 a^3 c^2 \sqrt {b^2-4 a c} k+a^2 b^4 m-18 a^3 b^2 c m-40 a^4 c^2 m+a^2 b^3 \sqrt {b^2-4 a c} m-16 a^3 b c \sqrt {b^2-4 a c} m\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a^2 c^{3/2} \left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}}}+\frac {\left (6 c^2 e-3 b c g+b^2 j+2 a c j-3 a b l\right ) \log \left (-b+\sqrt {b^2-4 a c}-2 c x^2\right )}{2 \left (b^2-4 a c\right )^{5/2}}+\frac {\left (-6 c^2 e+3 b c g-b^2 j-2 a c j+3 a b l\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{2 \left (b^2-4 a c\right )^{5/2}} \]
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)^3,x]
(a*b*c^2*e - 2*a^2*c^2*g + a^2*b*c*j - a^2*b^2*l + 2*a^3*c*l - b^2*c^2*d*x + 2*a*c^3*d*x + a*b*c^2*f*x - 2*a^2*c^2*h*x + a^2*b*c*k*x - a^2*b^2*m*x + 2*a^3*c*m*x + 2*a*c^3*e*x^2 - a*b*c^2*g*x^2 + a*b^2*c*j*x^2 - 2*a^2*c^2*j *x^2 - a*b^3*l*x^2 + 3*a^2*b*c*l*x^2 - b*c^3*d*x^3 + 2*a*c^3*f*x^3 - a*b*c ^2*h*x^3 + a*b^2*c*k*x^3 - 2*a^2*c^2*k*x^3 - a*b^3*m*x^3 + 3*a^2*b*c*m*x^3 )/(4*a*c^2*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)^2) + (12*a^2*b*c^3*e - 6*a^2 *b^2*c^2*g + 2*a^2*b^3*c*j + 4*a^3*b*c^2*j - 2*a^2*b^4*l + 10*a^3*b^2*c*l - 32*a^4*c^2*l + 3*b^4*c^2*d*x - 25*a*b^2*c^3*d*x + 28*a^2*c^4*d*x + a*b^3 *c^2*f*x + 8*a^2*b*c^3*f*x - 7*a^2*b^2*c^2*h*x + 4*a^3*c^3*h*x + 2*a^2*b^3 *c*k*x + 4*a^3*b*c^2*k*x - 2*a^2*b^4*m*x + 11*a^3*b^2*c*m*x - 36*a^4*c^2*m *x + 24*a^2*c^4*e*x^2 - 12*a^2*b*c^3*g*x^2 + 4*a^2*b^2*c^2*j*x^2 + 8*a^3*c ^3*j*x^2 - 12*a^3*b*c^2*l*x^2 + 3*b^3*c^3*d*x^3 - 24*a*b*c^4*d*x^3 + a*b^2 *c^3*f*x^3 + 20*a^2*c^4*f*x^3 - 12*a^2*b*c^3*h*x^3 + 3*a^2*b^2*c^2*k*x^3 + 12*a^3*c^3*k*x^3 + a^2*b^3*c*m*x^3 - 16*a^3*b*c^2*m*x^3)/(8*a^2*c^2*(-b^2 + 4*a*c)^2*(a + b*x^2 + c*x^4)) + ((3*b^4*c^2*d - 30*a*b^2*c^3*d + 168*a^ 2*c^4*d + 3*b^3*c^2*Sqrt[b^2 - 4*a*c]*d - 24*a*b*c^3*Sqrt[b^2 - 4*a*c]*d + a*b^3*c^2*f - 52*a^2*b*c^3*f + a*b^2*c^2*Sqrt[b^2 - 4*a*c]*f + 20*a^2*c^3 *Sqrt[b^2 - 4*a*c]*f + 18*a^2*b^2*c^2*h + 24*a^3*c^3*h - 12*a^2*b*c^2*Sqrt [b^2 - 4*a*c]*h - 3*a^2*b^3*c*k - 36*a^3*b*c^2*k + 3*a^2*b^2*c*Sqrt[b^2 - 4*a*c]*k + 12*a^3*c^2*Sqrt[b^2 - 4*a*c]*k - a^2*b^4*m + 18*a^3*b^2*c*m ...
Time = 2.87 (sec) , antiderivative size = 1172, normalized size of antiderivative = 1.02, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.236, Rules used = {2202, 2194, 2191, 1159, 1083, 219, 2206, 25, 2206, 25, 27, 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.
| \(\displaystyle \int \frac {d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{\left (a+b x^2+c x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 2202 |
| \(\displaystyle \int \frac {m x^8+k x^6+h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^3}dx+\int \frac {x \left (l x^6+j x^4+g x^2+e\right )}{\left (c x^4+b x^2+a\right )^3}dx\) |
| \(\Big \downarrow \) 2194 |
| \(\displaystyle \int \frac {m x^8+k x^6+h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^3}dx+\frac {1}{2} \int \frac {l x^6+j x^4+g x^2+e}{\left (c x^4+b x^2+a\right )^3}dx^2\) |
| \(\Big \downarrow \) 2191 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {-\frac {l b^3}{c^2}-3 g b+\frac {(b j+a l) b}{c}+2 \left (4 a-\frac {b^2}{c}\right ) l x^2+6 c e+2 a j}{\left (c x^4+b x^2+a\right )^2}dx^2}{2 \left (b^2-4 a c\right )}-\frac {x^2 \left (-c^2 (2 a j+b g)+b c (3 a l+b j)+b^3 (-l)+2 c^3 e\right )-a b^2 l+b c (a j+c e)-2 a c (c g-a l)}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )+\int \frac {m x^8+k x^6+h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^3}dx\) |
| \(\Big \downarrow \) 1159 |
| \(\displaystyle \frac {1}{2} \left (-\frac {-\frac {2 \left (-3 a b l+2 a c j+b^2 j-3 b c g+6 c^2 e\right ) \int \frac {1}{c x^4+b x^2+a}dx^2}{b^2-4 a c}-\frac {-16 a^2 l+2 x^2 \left (-3 a b l+2 a c j+b^2 j-3 b c g+6 c^2 e\right )-b^2 \left (3 g-\frac {5 a l}{c}\right )+2 b (a j+3 c e)-\frac {b^4 l}{c^2}+\frac {b^3 j}{c}}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{2 \left (b^2-4 a c\right )}-\frac {x^2 \left (-c^2 (2 a j+b g)+b c (3 a l+b j)+b^3 (-l)+2 c^3 e\right )-a b^2 l+b c (a j+c e)-2 a c (c g-a l)}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )+\int \frac {m x^8+k x^6+h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^3}dx\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\frac {4 \left (-3 a b l+2 a c j+b^2 j-3 b c g+6 c^2 e\right ) \int \frac {1}{-x^4+b^2-4 a c}d\left (2 c x^2+b\right )}{b^2-4 a c}-\frac {-16 a^2 l+2 x^2 \left (-3 a b l+2 a c j+b^2 j-3 b c g+6 c^2 e\right )-b^2 \left (3 g-\frac {5 a l}{c}\right )+2 b (a j+3 c e)-\frac {b^4 l}{c^2}+\frac {b^3 j}{c}}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{2 \left (b^2-4 a c\right )}-\frac {x^2 \left (-c^2 (2 a j+b g)+b c (3 a l+b j)+b^3 (-l)+2 c^3 e\right )-a b^2 l+b c (a j+c e)-2 a c (c g-a l)}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )+\int \frac {m x^8+k x^6+h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^3}dx\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \int \frac {m x^8+k x^6+h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^3}dx+\frac {1}{2} \left (-\frac {\frac {4 \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-3 a b l+2 a c j+b^2 j-3 b c g+6 c^2 e\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {-16 a^2 l+2 x^2 \left (-3 a b l+2 a c j+b^2 j-3 b c g+6 c^2 e\right )-b^2 \left (3 g-\frac {5 a l}{c}\right )+2 b (a j+3 c e)-\frac {b^4 l}{c^2}+\frac {b^3 j}{c}}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{2 \left (b^2-4 a c\right )}-\frac {x^2 \left (-c^2 (2 a j+b g)+b c (3 a l+b j)+b^3 (-l)+2 c^3 e\right )-a b^2 l+b c (a j+c e)-2 a c (c g-a l)}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 2206 |
| \(\displaystyle -\frac {\int -\frac {-4 a \left (4 a-\frac {b^2}{c}\right ) m x^4-\frac {\left (-a m b^3+a c k b^2-c \left (m a^2+5 c h a+5 c^2 d\right ) b+2 a c^2 (5 c f+3 a k)\right ) x^2}{c^2}+\frac {\left (3 c^2 d-a^2 m\right ) b^2+a c (c f+a k) b-2 a c \left (-m a^2+c h a+7 c^2 d\right )}{c^2}}{\left (c x^4+b x^2+a\right )^2}dx}{4 a \left (b^2-4 a c\right )}+\frac {1}{2} \left (-\frac {\frac {4 \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-3 a b l+2 a c j+b^2 j-3 b c g+6 c^2 e\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {-16 a^2 l+2 x^2 \left (-3 a b l+2 a c j+b^2 j-3 b c g+6 c^2 e\right )-b^2 \left (3 g-\frac {5 a l}{c}\right )+2 b (a j+3 c e)-\frac {b^4 l}{c^2}+\frac {b^3 j}{c}}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{2 \left (b^2-4 a c\right )}-\frac {x^2 \left (-c^2 (2 a j+b g)+b c (3 a l+b j)+b^3 (-l)+2 c^3 e\right )-a b^2 l+b c (a j+c e)-2 a c (c g-a l)}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )-\frac {x \left (-\left (b^2 \left (a^2 m+c^2 d\right )\right )+x^2 \left (-b c \left (-3 a^2 m+a c h+c^2 d\right )-a b^3 m+a b^2 c k+2 a c^2 (c f-a k)\right )+2 a c \left (a^2 m-a c h+c^2 d\right )+a b c (a k+c f)\right )}{4 a c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {-4 a \left (4 a-\frac {b^2}{c}\right ) m x^4-\frac {\left (-a m b^3+a c k b^2-c \left (m a^2+5 c h a+5 c^2 d\right ) b+2 a c^2 (5 c f+3 a k)\right ) x^2}{c^2}+\frac {\left (3 c^2 d-a^2 m\right ) b^2+a c (c f+a k) b-2 a c \left (-m a^2+c h a+7 c^2 d\right )}{c^2}}{\left (c x^4+b x^2+a\right )^2}dx}{4 a \left (b^2-4 a c\right )}+\frac {1}{2} \left (-\frac {\frac {4 \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-3 a b l+2 a c j+b^2 j-3 b c g+6 c^2 e\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {-16 a^2 l+2 x^2 \left (-3 a b l+2 a c j+b^2 j-3 b c g+6 c^2 e\right )-b^2 \left (3 g-\frac {5 a l}{c}\right )+2 b (a j+3 c e)-\frac {b^4 l}{c^2}+\frac {b^3 j}{c}}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{2 \left (b^2-4 a c\right )}-\frac {x^2 \left (-c^2 (2 a j+b g)+b c (3 a l+b j)+b^3 (-l)+2 c^3 e\right )-a b^2 l+b c (a j+c e)-2 a c (c g-a l)}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )-\frac {x \left (-\left (b^2 \left (a^2 m+c^2 d\right )\right )+x^2 \left (-b c \left (-3 a^2 m+a c h+c^2 d\right )-a b^3 m+a b^2 c k+2 a c^2 (c f-a k)\right )+2 a c \left (a^2 m-a c h+c^2 d\right )+a b c (a k+c f)\right )}{4 a c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 2206 |
| \(\displaystyle -\frac {x \left (-\left (\left (m a^2+c^2 d\right ) b^2\right )+a c (c f+a k) b+\left (-a m b^3+a c k b^2-c \left (-3 m a^2+c h a+c^2 d\right ) b+2 a c^2 (c f-a k)\right ) x^2+2 a c \left (m a^2-c h a+c^2 d\right )\right )}{4 a c^2 \left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )^2}+\frac {1}{2} \left (-\frac {-a l b^2+c (c e+a j) b+\left (-l b^3+c (b j+3 a l) b+2 c^3 e-c^2 (b g+2 a j)\right ) x^2-2 a c (c g-a l)}{2 c^2 \left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )^2}-\frac {\frac {4 \left (j b^2-3 c g b-3 a l b+6 c^2 e+2 a c j\right ) \text {arctanh}\left (\frac {2 c x^2+b}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {-\frac {l b^4}{c^2}+\frac {j b^3}{c}-\left (3 g-\frac {5 a l}{c}\right ) b^2+2 (3 c e+a j) b+2 \left (j b^2-3 c g b-3 a l b+6 c^2 e+2 a c j\right ) x^2-16 a^2 l}{\left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )}}{2 \left (b^2-4 a c\right )}\right )+\frac {\frac {x \left (\left (\left (m a^2+3 c^2 d\right ) b^3+a c (c f+3 a k) b^2-4 a c \left (4 m a^2+3 c h a+6 c^2 d\right ) b+4 a^2 c^2 (5 c f+3 a k)\right ) x^2+c \left (\left (3 d-\frac {2 a^2 m}{c^2}\right ) b^4+\frac {a (c f+2 a k) b^3}{c}-a \left (-\frac {11 m a^2}{c}+7 h a+25 c d\right ) b^2+4 a^2 (2 c f+a k) b+4 a^2 \left (-9 m a^2+c h a+7 c^2 d\right )\right )\right )}{2 a c \left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )}-\frac {\int -\frac {\left (\left (m a^2+3 c^2 d\right ) b^3+a c (c f+3 a k) b^2-4 a c \left (4 m a^2+3 c h a+6 c^2 d\right ) b+4 a^2 c^2 (5 c f+3 a k)\right ) x^2+c \left (3 d b^4+a f b^3-a \left (-\frac {m a^2}{c}-3 h a+27 c d\right ) b^2-4 a^2 (4 c f+3 a k) b+4 a^2 \left (5 m a^2+3 c h a+21 c^2 d\right )\right )}{c \left (c x^4+b x^2+a\right )}dx}{2 a \left (b^2-4 a c\right )}}{4 a \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x \left (-\left (\left (m a^2+c^2 d\right ) b^2\right )+a c (c f+a k) b+\left (-a m b^3+a c k b^2-c \left (-3 m a^2+c h a+c^2 d\right ) b+2 a c^2 (c f-a k)\right ) x^2+2 a c \left (m a^2-c h a+c^2 d\right )\right )}{4 a c^2 \left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )^2}+\frac {1}{2} \left (-\frac {-a l b^2+c (c e+a j) b+\left (-l b^3+c (b j+3 a l) b+2 c^3 e-c^2 (b g+2 a j)\right ) x^2-2 a c (c g-a l)}{2 c^2 \left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )^2}-\frac {\frac {4 \left (j b^2-3 c g b-3 a l b+6 c^2 e+2 a c j\right ) \text {arctanh}\left (\frac {2 c x^2+b}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {-\frac {l b^4}{c^2}+\frac {j b^3}{c}-\left (3 g-\frac {5 a l}{c}\right ) b^2+2 (3 c e+a j) b+2 \left (j b^2-3 c g b-3 a l b+6 c^2 e+2 a c j\right ) x^2-16 a^2 l}{\left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )}}{2 \left (b^2-4 a c\right )}\right )+\frac {\frac {x \left (\left (\left (m a^2+3 c^2 d\right ) b^3+a c (c f+3 a k) b^2-4 a c \left (4 m a^2+3 c h a+6 c^2 d\right ) b+4 a^2 c^2 (5 c f+3 a k)\right ) x^2+c \left (\left (3 d-\frac {2 a^2 m}{c^2}\right ) b^4+\frac {a (c f+2 a k) b^3}{c}-a \left (-\frac {11 m a^2}{c}+7 h a+25 c d\right ) b^2+4 a^2 (2 c f+a k) b+4 a^2 \left (-9 m a^2+c h a+7 c^2 d\right )\right )\right )}{2 a c \left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )}+\frac {\int \frac {\left (\left (m a^2+3 c^2 d\right ) b^3+a c (c f+3 a k) b^2-4 a c \left (4 m a^2+3 c h a+6 c^2 d\right ) b+4 a^2 c^2 (5 c f+3 a k)\right ) x^2+c \left (3 d b^4+a f b^3-a \left (-\frac {m a^2}{c}-3 h a+27 c d\right ) b^2-4 a^2 (4 c f+3 a k) b+4 a^2 \left (5 m a^2+3 c h a+21 c^2 d\right )\right )}{c \left (c x^4+b x^2+a\right )}dx}{2 a \left (b^2-4 a c\right )}}{4 a \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {x \left (-\left (\left (m a^2+c^2 d\right ) b^2\right )+a c (c f+a k) b+\left (-a m b^3+a c k b^2-c \left (-3 m a^2+c h a+c^2 d\right ) b+2 a c^2 (c f-a k)\right ) x^2+2 a c \left (m a^2-c h a+c^2 d\right )\right )}{4 a c^2 \left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )^2}+\frac {1}{2} \left (-\frac {-a l b^2+c (c e+a j) b+\left (-l b^3+c (b j+3 a l) b+2 c^3 e-c^2 (b g+2 a j)\right ) x^2-2 a c (c g-a l)}{2 c^2 \left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )^2}-\frac {\frac {4 \left (j b^2-3 c g b-3 a l b+6 c^2 e+2 a c j\right ) \text {arctanh}\left (\frac {2 c x^2+b}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {-\frac {l b^4}{c^2}+\frac {j b^3}{c}-\left (3 g-\frac {5 a l}{c}\right ) b^2+2 (3 c e+a j) b+2 \left (j b^2-3 c g b-3 a l b+6 c^2 e+2 a c j\right ) x^2-16 a^2 l}{\left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )}}{2 \left (b^2-4 a c\right )}\right )+\frac {\frac {x \left (\left (\left (m a^2+3 c^2 d\right ) b^3+a c (c f+3 a k) b^2-4 a c \left (4 m a^2+3 c h a+6 c^2 d\right ) b+4 a^2 c^2 (5 c f+3 a k)\right ) x^2+c \left (\left (3 d-\frac {2 a^2 m}{c^2}\right ) b^4+\frac {a (c f+2 a k) b^3}{c}-a \left (-\frac {11 m a^2}{c}+7 h a+25 c d\right ) b^2+4 a^2 (2 c f+a k) b+4 a^2 \left (-9 m a^2+c h a+7 c^2 d\right )\right )\right )}{2 a c \left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )}+\frac {\int \frac {\left (\left (m a^2+3 c^2 d\right ) b^3+a c (c f+3 a k) b^2-4 a c \left (4 m a^2+3 c h a+6 c^2 d\right ) b+4 a^2 c^2 (5 c f+3 a k)\right ) x^2+c \left (3 d b^4+a f b^3-a \left (-\frac {m a^2}{c}-3 h a+27 c d\right ) b^2-4 a^2 (4 c f+3 a k) b+4 a^2 \left (5 m a^2+3 c h a+21 c^2 d\right )\right )}{c x^4+b x^2+a}dx}{2 a c \left (b^2-4 a c\right )}}{4 a \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle -\frac {x \left (-\left (\left (m a^2+c^2 d\right ) b^2\right )+a c (c f+a k) b+\left (-a m b^3+a c k b^2-c \left (-3 m a^2+c h a+c^2 d\right ) b+2 a c^2 (c f-a k)\right ) x^2+2 a c \left (m a^2-c h a+c^2 d\right )\right )}{4 a c^2 \left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )^2}+\frac {1}{2} \left (-\frac {-a l b^2+c (c e+a j) b+\left (-l b^3+c (b j+3 a l) b+2 c^3 e-c^2 (b g+2 a j)\right ) x^2-2 a c (c g-a l)}{2 c^2 \left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )^2}-\frac {\frac {4 \left (j b^2-3 c g b-3 a l b+6 c^2 e+2 a c j\right ) \text {arctanh}\left (\frac {2 c x^2+b}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {-\frac {l b^4}{c^2}+\frac {j b^3}{c}-\left (3 g-\frac {5 a l}{c}\right ) b^2+2 (3 c e+a j) b+2 \left (j b^2-3 c g b-3 a l b+6 c^2 e+2 a c j\right ) x^2-16 a^2 l}{\left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )}}{2 \left (b^2-4 a c\right )}\right )+\frac {\frac {x \left (\left (\left (m a^2+3 c^2 d\right ) b^3+a c (c f+3 a k) b^2-4 a c \left (4 m a^2+3 c h a+6 c^2 d\right ) b+4 a^2 c^2 (5 c f+3 a k)\right ) x^2+c \left (\left (3 d-\frac {2 a^2 m}{c^2}\right ) b^4+\frac {a (c f+2 a k) b^3}{c}-a \left (-\frac {11 m a^2}{c}+7 h a+25 c d\right ) b^2+4 a^2 (2 c f+a k) b+4 a^2 \left (-9 m a^2+c h a+7 c^2 d\right )\right )\right )}{2 a c \left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )}+\frac {\frac {1}{2} \left (\left (m a^2+3 c^2 d\right ) b^3+a c (c f+3 a k) b^2-4 a c \left (4 m a^2+3 c h a+6 c^2 d\right ) b+4 a^2 c^2 (5 c f+3 a k)+\frac {\left (3 c^2 d-a^2 m\right ) b^4+a c (c f-3 a k) b^3-6 a c \left (-3 m a^2-3 c h a+5 c^2 d\right ) b^2-4 a^2 c^2 (13 c f+9 a k) b+8 a^2 c^2 \left (5 m a^2+3 c h a+21 c^2 d\right )}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx+\frac {1}{2} \left (\left (m a^2+3 c^2 d\right ) b^3+a c (c f+3 a k) b^2-4 a c \left (4 m a^2+3 c h a+6 c^2 d\right ) b+4 a^2 c^2 (5 c f+3 a k)-\frac {\left (3 c^2 d-a^2 m\right ) b^4+a c (c f-3 a k) b^3-6 a c \left (-3 m a^2-3 c h a+5 c^2 d\right ) b^2-4 a^2 c^2 (13 c f+9 a k) b+8 a^2 c^2 \left (5 m a^2+3 c h a+21 c^2 d\right )}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx}{2 a c \left (b^2-4 a c\right )}}{4 a \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {x \left (-\left (\left (m a^2+c^2 d\right ) b^2\right )+a c (c f+a k) b+\left (-a m b^3+a c k b^2-c \left (-3 m a^2+c h a+c^2 d\right ) b+2 a c^2 (c f-a k)\right ) x^2+2 a c \left (m a^2-c h a+c^2 d\right )\right )}{4 a c^2 \left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )^2}+\frac {\frac {x \left (\left (\left (m a^2+3 c^2 d\right ) b^3+a c (c f+3 a k) b^2-4 a c \left (4 m a^2+3 c h a+6 c^2 d\right ) b+4 a^2 c^2 (5 c f+3 a k)\right ) x^2+c \left (\left (3 d-\frac {2 a^2 m}{c^2}\right ) b^4+\frac {a (c f+2 a k) b^3}{c}-a \left (-\frac {11 m a^2}{c}+7 h a+25 c d\right ) b^2+4 a^2 (2 c f+a k) b+4 a^2 \left (-9 m a^2+c h a+7 c^2 d\right )\right )\right )}{2 a c \left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )}+\frac {\frac {\left (\left (m a^2+3 c^2 d\right ) b^3+a c (c f+3 a k) b^2-4 a c \left (4 m a^2+3 c h a+6 c^2 d\right ) b+4 a^2 c^2 (5 c f+3 a k)+\frac {\left (3 c^2 d-a^2 m\right ) b^4+a c (c f-3 a k) b^3-6 a c \left (-3 m a^2-3 c h a+5 c^2 d\right ) b^2-4 a^2 c^2 (13 c f+9 a k) b+8 a^2 c^2 \left (5 m a^2+3 c h a+21 c^2 d\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 )}{\sqrt {2} \sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (\left (m a^2+3 c^2 d\right ) b^3+a c (c f+3 a k) b^2-4 a c \left (4 m a^2+3 c h a+6 c^2 d\right ) b+4 a^2 c^2 (5 c f+3 a k)-\frac {\left (3 c^2 d-a^2 m\right ) b^4+a c (c f-3 a k) b^3-6 a c \left (-3 m a^2-3 c h a+5 c^2 d\right ) b^2-4 a^2 c^2 (13 c f+9 a k) b+8 a^2 c^2 \left (5 m a^2+3 c h a+21 c^2 d\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 )}{\sqrt {2} \sqrt {c} \sqrt {b+\sqrt {b^2-4 a c}}}}{2 a c \left (b^2-4 a c\right )}}{4 a \left (b^2-4 a c\right )}+\frac {1}{2} \left (-\frac {-a l b^2+c (c e+a j) b+\left (-l b^3+c (b j+3 a l) b+2 c^3 e-c^2 (b g+2 a j)\right ) x^2-2 a c (c g-a l)}{2 c^2 \left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )^2}-\frac {\frac {4 \left (j b^2-3 c g b-3 a l b+6 c^2 e+2 a c j\right ) \text {arctanh}\left (\frac {2 c x^2+b}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {-\frac {l b^4}{c^2}+\frac {j b^3}{c}-\left (3 g-\frac {5 a l}{c}\right ) b^2+2 (3 c e+a j) b+2 \left (j b^2-3 c g b-3 a l b+6 c^2 e+2 a c j\right ) x^2-16 a^2 l}{\left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )}}{2 \left (b^2-4 a c\right )}\right )\) |
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)^3,x]
-1/4*(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) + ((x*(c*(4*a ^2*b*(2*c*f + a*k) + (a*b^3*(c*f + 2*a*k))/c + 4*a^2*(7*c^2*d + a*c*h - 9* a^2*m) + b^4*(3*d - (2*a^2*m)/c^2) - a*b^2*(25*c*d + 7*a*h - (11*a^2*m)/c) ) + (a*b^2*c*(c*f + 3*a*k) + 4*a^2*c^2*(5*c*f + 3*a*k) + b^3*(3*c^2*d + a^ 2*m) - 4*a*b*c*(6*c^2*d + 3*a*c*h + 4*a^2*m))*x^2))/(2*a*c*(b^2 - 4*a*c)*( a + b*x^2 + c*x^4)) + (((a*b^2*c*(c*f + 3*a*k) + 4*a^2*c^2*(5*c*f + 3*a*k) + b^3*(3*c^2*d + a^2*m) - 4*a*b*c*(6*c^2*d + 3*a*c*h + 4*a^2*m) + (a*b^3* c*(c*f - 3*a*k) - 4*a^2*b*c^2*(13*c*f + 9*a*k) - 6*a*b^2*c*(5*c^2*d - 3*a* c*h - 3*a^2*m) + b^4*(3*c^2*d - a^2*m) + 8*a^2*c^2*(21*c^2*d + 3*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]*Sqrt[c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((a*b^2*c*(c*f + 3*a*k) + 4*a^2*c^2*(5*c*f + 3*a*k) + b^3*(3*c^2*d + a^2*m) - 4*a*b*c*(6* c^2*d + 3*a*c*h + 4*a^2*m) - (a*b^3*c*(c*f - 3*a*k) - 4*a^2*b*c^2*(13*c*f + 9*a*k) - 6*a*b^2*c*(5*c^2*d - 3*a*c*h - 3*a^2*m) + b^4*(3*c^2*d - a^2*m) + 8*a^2*c^2*(21*c^2*d + 3*a*c*h + 5*a^2*m))/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]]))/(2*a*c*(b^2 - 4*a*c)))/(4*a*(b^2 - 4*a*c)) + (-1/2*(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...
3.1.57.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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)^(p_), x_Symbol ] :> Simp[((b*d - 2*a*e + (2*c*d - b*e)*x)/((p + 1)*(b^2 - 4*a*c)))*(a + b* x + c*x^2)^(p + 1), x] - Simp[(2*p + 3)*((2*c*d - b*e)/((p + 1)*(b^2 - 4*a* c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] & & LtQ[p, -1] && NeQ[p, -3/2]
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]
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)^(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 = 1.39 (sec) , antiderivative size = 1167, normalized size of antiderivative = 1.01
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1167\) |
| default | \(\text {Expression too large to display}\) | \(1987\) |
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)^3,x, method=_RETURNVERBOSE)
(-1/8*(16*a^3*b*c*m-12*a^3*c^2*k-a^2*b^3*m-3*a^2*b^2*c*k+12*a^2*b*c^2*h-20 *a^2*c^3*f-a*b^2*c^2*f+24*a*b*c^3*d-3*b^3*c^2*d)/a^2/(16*a^2*c^2-8*a*b^2*c +b^4)*x^7-1/2*c*(3*a*b*l-2*a*c*j-b^2*j+3*b*c*g-6*c^2*e)/(16*a^2*c^2-8*a*b^ 2*c+b^4)*x^6-1/8/a^2*(36*a^4*c^2*m+5*a^3*b^2*c*m-16*a^3*b*c^2*k-4*a^3*c^3* h+a^2*b^4*m-5*a^2*b^3*c*k+19*a^2*b^2*c^2*h-28*a^2*b*c^3*f-28*a^2*c^4*d-2*a *b^3*c^2*f+49*a*b^2*c^3*d-6*b^4*c^2*d)/(16*a^2*c^2-8*a*b^2*c+b^4)/c*x^5-1/ 4*(16*a^2*c^2*l+a*b^2*c*l-6*a*b*c^2*j+b^4*l-3*b^3*c*j+9*b^2*c^2*g-18*b*c^3 *e)/(16*a^2*c^2-8*a*b^2*c+b^4)/c*x^4-1/8/c*(28*a^4*b*c*m+4*a^4*c^2*k+2*a^3 *b^3*m-19*a^3*b^2*c*k+16*a^3*b*c^2*h-36*a^3*c^3*f+5*a^2*b^3*c*h-5*a^2*b^2* c^2*f+4*a^2*b*c^3*d-a*b^4*c*f+20*a*b^3*c^2*d-3*b^5*c*d)/a^2/(16*a^2*c^2-8* a*b^2*c+b^4)*x^3-1/2/c*(5*a^2*b*c*l+2*a^2*c^2*j+a*b^3*l-5*a*b^2*c*j+5*a*b* c^2*g-10*a*c^3*e+b^3*c*g-2*b^2*c^2*e)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2-1/8*( 20*a^4*c*m+a^3*b^2*m-12*a^3*b*c*k+12*a^3*c^2*h+3*a^2*b^2*c*h-16*a^2*b*c^2* f-44*a^2*c^3*d+a*b^3*c*f+37*a*b^2*c^2*d-5*b^4*c*d)/(16*a^2*c^2-8*a*b^2*c+b ^4)/c/a*x-1/4/c*(8*a^3*c*l+a^2*b^2*l-6*a^2*b*c*j+8*a^2*c^2*g+a*b^2*c*g-10* a*b*c^2*e+b^3*c*e)/(16*a^2*c^2-8*a*b^2*c+b^4))/(c*x^4+b*x^2+a)^2+1/16*sum( (-(16*a^3*b*c*m-12*a^3*c^2*k-a^2*b^3*m-3*a^2*b^2*c*k+12*a^2*b*c^2*h-20*a^2 *c^3*f-a*b^2*c^2*f+24*a*b*c^3*d-3*b^3*c^2*d)/a^2/(16*a^2*c^2-8*a*b^2*c+b^4 )/c*_R^2-8*(3*a*b*l-2*a*c*j-b^2*j+3*b*c*g-6*c^2*e)/(16*a^2*c^2-8*a*b^2*c+b ^4)*_R+1/c*(20*a^4*c*m+a^3*b^2*m-12*a^3*b*c*k+12*a^3*c^2*h+3*a^2*b^2*c*...
Timed out. \[ \int \frac {d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Timed out} \]
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 )^3,x, algorithm="fricas")
Timed out. \[ \int \frac {d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{\left (a+b x^2+c x^4\right )^3} \, dx=\int { \frac {m x^{8} + l x^{7} + k x^{6} + j x^{5} + h x^{4} + g x^{3} + f x^{2} + e x + d}{{\left (c x^{4} + b x^{2} + a\right )}^{3}} \,d x } \]
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 )^3,x, algorithm="maxima")
-1/8*((12*a^2*b*c^3*h - 3*(b^3*c^3 - 8*a*b*c^4)*d - (a*b^2*c^3 + 20*a^2*c^ 4)*f - 3*(a^2*b^2*c^2 + 4*a^3*c^3)*k - (a^2*b^3*c - 16*a^3*b*c^2)*m)*x^7 - 12*a^4*b*c*j - 4*(6*a^2*c^4*e - 3*a^2*b*c^3*g - 3*a^3*b*c^2*l + (a^2*b^2* c^2 + 2*a^3*c^3)*j)*x^6 - ((6*b^4*c^2 - 49*a*b^2*c^3 + 28*a^2*c^4)*d + 2*( a*b^3*c^2 + 14*a^2*b*c^3)*f - (19*a^2*b^2*c^2 - 4*a^3*c^3)*h + (5*a^2*b^3* c + 16*a^3*b*c^2)*k - (a^2*b^4 + 5*a^3*b^2*c + 36*a^4*c^2)*m)*x^5 - 2*(18* a^2*b*c^3*e - 9*a^2*b^2*c^2*g + 3*(a^2*b^3*c + 2*a^3*b*c^2)*j - (a^2*b^4 + a^3*b^2*c + 16*a^4*c^2)*l)*x^4 - ((3*b^5*c - 20*a*b^3*c^2 - 4*a^2*b*c^3)* d + (a*b^4*c + 5*a^2*b^2*c^2 + 36*a^3*c^3)*f - (5*a^2*b^3*c + 16*a^3*b*c^2 )*h + (19*a^3*b^2*c - 4*a^4*c^2)*k - 2*(a^3*b^3 + 14*a^4*b*c)*m)*x^3 - 4*( 2*(a^2*b^2*c^2 + 5*a^3*c^3)*e - (a^2*b^3*c + 5*a^3*b*c^2)*g + (5*a^3*b^2*c - 2*a^4*c^2)*j - (a^3*b^3 + 5*a^4*b*c)*l)*x^2 + 2*(a^2*b^3*c - 10*a^3*b*c ^2)*e + 2*(a^3*b^2*c + 8*a^4*c^2)*g + 2*(a^4*b^2 + 8*a^5*c)*l - (12*a^4*b* c*k + (5*a*b^4*c - 37*a^2*b^2*c^2 + 44*a^3*c^3)*d - (a^2*b^3*c - 16*a^3*b* c^2)*f - 3*(a^3*b^2*c + 4*a^4*c^2)*h - (a^4*b^2 + 20*a^5*c)*m)*x)/(a^4*b^4 *c - 8*a^5*b^2*c^2 + 16*a^6*c^3 + (a^2*b^4*c^3 - 8*a^3*b^2*c^4 + 16*a^4*c^ 5)*x^8 + 2*(a^2*b^5*c^2 - 8*a^3*b^3*c^3 + 16*a^4*b*c^4)*x^6 + (a^2*b^6*c - 6*a^3*b^4*c^2 + 32*a^5*c^4)*x^4 + 2*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b *c^3)*x^2) - 1/8*integrate((12*a^3*b*c*k + (12*a^2*b*c^2*h - 3*(b^3*c^2 - 8*a*b*c^3)*d - (a*b^2*c^2 + 20*a^2*c^3)*f - 3*(a^2*b^2*c + 4*a^3*c^2)*k...
Leaf count of result is larger than twice the leaf count of optimal. 22429 vs. \(2 (1098) = 2196\).
Time = 3.50 (sec) , antiderivative size = 22429, normalized size of antiderivative = 19.50 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \]
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 )^3,x, algorithm="giac")
1/64*(3*(2*b^5*c^4 - 24*a*b^3*c^5 + 64*a^2*b*c^6 - sqrt(2)*sqrt(b^2 - 4*a* c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5*c^2 + 12*sqrt(2)*sqrt(b^2 - 4*a*c)* sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^3 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sq rt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^3 - 32*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt( b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^4 - 16*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)*b^3*c^4 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqr t(b^2 - 4*a*c)*c)*a*b*c^5 - 2*(b^2 - 4*a*c)*b^3*c^4 + 16*(b^2 - 4*a*c)*a*b *c^5)*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)^2*d + (2*a*b^4*c^4 + 32*a^2 *b^2*c^5 - 160*a^3*c^6 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4 *a*c)*c)*a*b^4*c^2 - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4* a*c)*c)*a^2*b^2*c^3 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4* a*c)*c)*a*b^3*c^3 + 80*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a *c)*c)*a^3*c^4 + 40*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c) *c)*a^2*b*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 - 20*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a ^2*c^5 - 2*(b^2 - 4*a*c)*a*b^2*c^4 - 40*(b^2 - 4*a*c)*a^2*c^5)*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)^2*f - 12*(2*a^2*b^3*c^4 - 8*a^3*b*c^5 - sqrt (2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^2 + 4*sqrt (2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b*c^3 + 2*sqr...
Time = 32.04 (sec) , antiderivative size = 114377, normalized size of antiderivative = 99.46 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \]
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)^3,x)
symsum(log(root(56371445760*a^11*b^8*c^9*z^4 - 503316480*a^8*b^14*c^6*z^4 + 47185920*a^7*b^16*c^5*z^4 - 2621440*a^6*b^18*c^4*z^4 + 65536*a^5*b^20*c^ 3*z^4 - 171798691840*a^14*b^2*c^12*z^4 + 193273528320*a^13*b^4*c^11*z^4 - 128849018880*a^12*b^6*c^10*z^4 - 16911433728*a^10*b^10*c^8*z^4 + 352321536 0*a^9*b^12*c^7*z^4 + 68719476736*a^15*c^13*z^4 + 1536*a^5*b^16*c*k*m*z^2 + 1536*a*b^18*c^3*d*f*z^2 - 2571632640*a^9*b^5*c^8*d*m*z^2 + 2548039680*a^9 *b^3*c^10*d*h*z^2 + 1509949440*a^10*b^3*c^9*e*l*z^2 + 1509949440*a^9*b^3*c ^10*e*g*z^2 - 1401421824*a^8*b^5*c^9*d*h*z^2 - 1321205760*a^9*b^2*c^11*d*f *z^2 - 2793406464*a^11*b*c^10*d*m*z^2 + 890634240*a^8*b^7*c^7*d*m*z^2 - 75 4974720*a^10*b^4*c^8*g*l*z^2 - 754974720*a^9*b^5*c^8*e*l*z^2 + 719585280*a ^8*b^6*c^8*d*k*z^2 - 707788800*a^9*b^4*c^9*d*k*z^2 - 754974720*a^8*b^5*c^9 *e*g*z^2 + 603979776*a^11*b^2*c^9*g*l*z^2 - 581959680*a^10*b^4*c^8*f*m*z^2 + 732168192*a^7*b^6*c^9*d*f*z^2 + 534773760*a^11*b^3*c^8*h*m*z^2 - 456130 560*a^11*b^4*c^7*k*m*z^2 - 603979776*a^10*b^2*c^10*e*j*z^2 + 534773760*a^1 0*b^3*c^9*f*k*z^2 + 384040960*a^9*b^6*c^7*f*m*z^2 + 377487360*a^9*b^6*c^7* g*l*z^2 - 456130560*a^9*b^4*c^9*f*h*z^2 + 301989888*a^11*b^3*c^8*j*l*z^2 - 415236096*a^10*b^2*c^10*d*k*z^2 + 254017536*a^10*b^6*c^6*k*m*z^2 - 330301 440*a^10*b^4*c^8*h*k*z^2 + 390463488*a^7*b^7*c^8*d*h*z^2 + 188743680*a^12* b^2*c^8*k*m*z^2 + 301989888*a^10*b^3*c^9*g*j*z^2 - 297861120*a^7*b^8*c^7*d *k*z^2 - 366280704*a^6*b^8*c^8*d*f*z^2 + 188743680*a^11*b^2*c^9*h*k*z^2...