\(\int \frac {d+e x+f x^2+g x^3+h x^4+i x^5+j x^8+k x^{11}}{(a+b x^2+c x^4)^3} \, dx\) [56]

Optimal result

Integrand size = 50, antiderivative size = 1177 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5+j x^8+k x^{11}}{\left (a+b x^2+c x^4\right )^3} \, dx =\text {Too large to display} \] Output:

-1/4*x*(c^2*(a*b*f-b^2*(d+a^2*j/c^2)+2*a*(c*d-a*h+a^2*j/c))+(2*a*c^3*f-a*b 
^3*j-b*c*(-3*a^2*j+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*c^3*(a*i+c*e)-a*b^4*k+4*a^2*b^2*c*k-2*a^3*c^2*k-2*a*c^4*g+(2*c^5*e 
+b^2*c^3*i-c^4*(2*a*i+b*g)-b^5*k+5*a*b^3*c*k-5*a^2*b*c^2*k)*x^2)/c^4/(-4*a 
*c+b^2)/(c*x^4+b*x^2+a)^2+1/8*x*(c*(a*b^3*f+8*a^2*b*c*f+4*a^2*(-9*a^2*j+a* 
c*h+7*c^2*d)+b^4*(3*d-2*a^2*j/c^2)-a*b^2*(25*c*d+7*a*h-11*a^2*j/c))+(a*b^2 
*c^2*f+20*a^2*c^3*f+b^3*(a^2*j+3*c^2*d)-4*a*b*c*(4*a^2*j+3*a*c*h+6*c^2*d)) 
*x^2)/a^2/c/(-4*a*c+b^2)^2/(c*x^4+b*x^2+a)+1/4*(b^3*c^2*i+2*b*c^3*(a*i+3*c 
*e)+11*a*b^4*k-b^6*k/c+32*a^3*c^2*k-3*b^2*(13*a^2*c*k+c^3*g)+2*(6*c^5*e+b^ 
2*c^3*i-c^4*(-2*a*i+3*b*g)+2*b^5*k-15*a*b^3*c*k+25*a^2*b*c^2*k)*x^2)/c^3/( 
-4*a*c+b^2)^2/(c*x^4+b*x^2+a)+1/16*(a*b^2*c^2*f+20*a^2*c^3*f+b^3*(a^2*j+3* 
c^2*d)-4*a*b*c*(4*a^2*j+3*a*c*h+6*c^2*d)+(a*b^3*c^2*f-52*a^2*b*c^3*f-6*a*b 
^2*c*(-3*a^2*j-3*a*c*h+5*c^2*d)+b^4*(-a^2*j+3*c^2*d)+8*a^2*c^2*(5*a^2*j+3* 
a*c*h+21*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^2/c^(3/2)/(-4*a*c+b^2)^2/(b-(-4*a*c+b^2)^(1/2) 
)^(1/2)+1/16*(a*b^2*c^2*f+20*a^2*c^3*f+b^3*(a^2*j+3*c^2*d)-4*a*b*c*(4*a^2* 
j+3*a*c*h+6*c^2*d)-(a*b^3*c^2*f-52*a^2*b*c^3*f-6*a*b^2*c*(-3*a^2*j-3*a*c*h 
+5*c^2*d)+b^4*(-a^2*j+3*c^2*d)+8*a^2*c^2*(5*a^2*j+3*a*c*h+21*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^2/c^(3/2)/(-4*a*c+b^2)^2/(b+(-4*a*c+b^2)^(1/2))^(1/2)-1/2*(12*c^5...
 

Mathematica [A] (verified)

Time = 7.33 (sec) , antiderivative size = 1649, normalized size of antiderivative = 1.40 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5+j x^8+k x^{11}}{\left (a+b x^2+c x^4\right )^3} \, dx =\text {Too large to display} \] Input:

Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5 + j*x^8 + k*x^11)/(a + 
b*x^2 + c*x^4)^3,x]
 

Output:

(a*b*c^4*e - 2*a^2*c^4*g + a^2*b*c^3*i - a^2*b^4*k + 4*a^3*b^2*c*k - 2*a^4 
*c^2*k - b^2*c^4*d*x + 2*a*c^5*d*x + a*b*c^4*f*x - 2*a^2*c^4*h*x - a^2*b^2 
*c^2*j*x + 2*a^3*c^3*j*x + 2*a*c^5*e*x^2 - a*b*c^4*g*x^2 + a*b^2*c^3*i*x^2 
 - 2*a^2*c^4*i*x^2 - a*b^5*k*x^2 + 5*a^2*b^3*c*k*x^2 - 5*a^3*b*c^2*k*x^2 - 
 b*c^5*d*x^3 + 2*a*c^5*f*x^3 - a*b*c^4*h*x^3 - a*b^3*c^2*j*x^3 + 3*a^2*b*c 
^3*j*x^3)/(4*a*c^4*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)^2) + (12*a^2*b*c^5*e 
 - 6*a^2*b^2*c^4*g + 2*a^2*b^3*c^3*i + 4*a^3*b*c^4*i - 2*a^2*b^6*k + 22*a^ 
3*b^4*c*k - 78*a^4*b^2*c^2*k + 64*a^5*c^3*k + 3*b^4*c^4*d*x - 25*a*b^2*c^5 
*d*x + 28*a^2*c^6*d*x + a*b^3*c^4*f*x + 8*a^2*b*c^5*f*x - 7*a^2*b^2*c^4*h* 
x + 4*a^3*c^5*h*x - 2*a^2*b^4*c^2*j*x + 11*a^3*b^2*c^3*j*x - 36*a^4*c^4*j* 
x + 24*a^2*c^6*e*x^2 - 12*a^2*b*c^5*g*x^2 + 4*a^2*b^2*c^4*i*x^2 + 8*a^3*c^ 
5*i*x^2 + 8*a^2*b^5*c*k*x^2 - 60*a^3*b^3*c^2*k*x^2 + 100*a^4*b*c^3*k*x^2 + 
 3*b^3*c^5*d*x^3 - 24*a*b*c^6*d*x^3 + a*b^2*c^5*f*x^3 + 20*a^2*c^6*f*x^3 - 
 12*a^2*b*c^5*h*x^3 + a^2*b^3*c^3*j*x^3 - 16*a^3*b*c^4*j*x^3)/(8*a^2*c^4*( 
-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 + 16 
8*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 - a^2*b^4*j + 18*a^3*b^2*c*j + 40*a^4*c^2*j + a^2*b^3* 
Sqrt[b^2 - 4*a*c]*j - 16*a^3*b*c*Sqrt[b^2 - 4*a*c]*j)*ArcTan[(Sqrt[2]*S...
 

Rubi [A] (verified)

Time = 3.04 (sec) , antiderivative size = 1236, normalized size of antiderivative = 1.05, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2202, 2194, 2191, 2191, 27, 1142, 1083, 219, 1103, 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.

\(\displaystyle \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5+j x^8+k x^{11}}{\left (a+b x^2+c x^4\right )^3} \, dx\)

\(\Big \downarrow \) 2202

\(\displaystyle \int \frac {j x^8+h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^3}dx+\int \frac {x \left (k x^{10}+i x^4+g x^2+e\right )}{\left (c x^4+b x^2+a\right )^3}dx\)

\(\Big \downarrow \) 2194

\(\displaystyle \int \frac {j x^8+h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^3}dx+\frac {1}{2} \int \frac {k x^{10}+i x^4+g x^2+e}{\left (c x^4+b x^2+a\right )^3}dx^2\)

\(\Big \downarrow \) 2191

\(\displaystyle \frac {1}{2} \left (\frac {c^4 \left (\frac {2 a^3 k}{c^2}-\frac {4 a^2 b^2 k}{c^3}+\frac {a b^4 k}{c^4}-b \left (\frac {a i}{c}+e\right )+2 a g\right )-x^2 \left (-5 a^2 b c^2 k+5 a b^3 c k-c^4 (2 a i+b g)+b^5 (-k)+b^2 c^3 i+2 c^5 e\right )}{2 c^4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\int \frac {2 \left (4 a-\frac {b^2}{c}\right ) k x^6+\frac {2 b \left (b^2-4 a c\right ) k x^4}{c^2}-\frac {2 \left (b^4-5 a c b^2+4 a^2 c^2\right ) k x^2}{c^3}+6 c e-3 b g+2 a i+\frac {b^2 i}{c}+\frac {a^2 b k}{c^2}+\frac {3 a b^3 k}{c^3}-\frac {b^5 k}{c^4}}{\left (c x^4+b x^2+a\right )^2}dx^2}{2 \left (b^2-4 a c\right )}\right )+\int \frac {j x^8+h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^3}dx\)

\(\Big \downarrow \) 2191

\(\displaystyle \frac {1}{2} \left (\frac {c^4 \left (\frac {2 a^3 k}{c^2}-\frac {4 a^2 b^2 k}{c^3}+\frac {a b^4 k}{c^4}-b \left (\frac {a i}{c}+e\right )+2 a g\right )-x^2 \left (-5 a^2 b c^2 k+5 a b^3 c k-c^4 (2 a i+b g)+b^5 (-k)+b^2 c^3 i+2 c^5 e\right )}{2 c^4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {-\frac {\int \frac {2 \left (\left (\frac {a k b^3}{c^2}+i b^2-3 c g b-\frac {7 a^2 k b}{c}+6 c^2 e+2 a c i\right ) c^2+\left (b^2-4 a c\right )^2 k x^2\right )}{c^2 \left (c x^4+b x^2+a\right )}dx^2}{b^2-4 a c}-\frac {2 x^2 \left (25 a^2 b c^2 k-15 a b^3 c k-c^4 (3 b g-2 a i)+2 b^5 k+b^2 c^3 i+6 c^5 e\right )+c^3 \left (\frac {32 a^3 k}{c}-b^2 \left (\frac {39 a^2 k}{c^2}+3 g\right )+\frac {11 a b^4 k}{c^3}+2 b (a i+3 c e)-\frac {b^6 k}{c^4}+\frac {b^3 i}{c}\right )}{c^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{2 \left (b^2-4 a c\right )}\right )+\int \frac {j x^8+h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^3}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {c^4 \left (\frac {2 a^3 k}{c^2}-\frac {4 a^2 b^2 k}{c^3}+\frac {a b^4 k}{c^4}-b \left (\frac {a i}{c}+e\right )+2 a g\right )-x^2 \left (-5 a^2 b c^2 k+5 a b^3 c k-c^4 (2 a i+b g)+b^5 (-k)+b^2 c^3 i+2 c^5 e\right )}{2 c^4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {-\frac {2 \int \frac {6 e c^4-(3 b g-2 a i) c^3+b^2 i c^2-7 a^2 b k c+\left (b^2-4 a c\right )^2 k x^2+a b^3 k}{c x^4+b x^2+a}dx^2}{c^2 \left (b^2-4 a c\right )}-\frac {2 x^2 \left (25 a^2 b c^2 k-15 a b^3 c k-c^4 (3 b g-2 a i)+2 b^5 k+b^2 c^3 i+6 c^5 e\right )+c^3 \left (\frac {32 a^3 k}{c}-b^2 \left (\frac {39 a^2 k}{c^2}+3 g\right )+\frac {11 a b^4 k}{c^3}+2 b (a i+3 c e)-\frac {b^6 k}{c^4}+\frac {b^3 i}{c}\right )}{c^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{2 \left (b^2-4 a c\right )}\right )+\int \frac {j x^8+h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^3}dx\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {1}{2} \left (\frac {c^4 \left (\frac {2 a^3 k}{c^2}-\frac {4 a^2 b^2 k}{c^3}+\frac {a b^4 k}{c^4}-b \left (\frac {a i}{c}+e\right )+2 a g\right )-x^2 \left (-5 a^2 b c^2 k+5 a b^3 c k-c^4 (2 a i+b g)+b^5 (-k)+b^2 c^3 i+2 c^5 e\right )}{2 c^4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {-\frac {2 \left (\frac {\left (-30 a^2 b c^2 k+10 a b^3 c k-c^4 (6 b g-4 a i)+b^5 (-k)+2 b^2 c^3 i+12 c^5 e\right ) \int \frac {1}{c x^4+b x^2+a}dx^2}{2 c}+\frac {k \left (b^2-4 a c\right )^2 \int \frac {2 c x^2+b}{c x^4+b x^2+a}dx^2}{2 c}\right )}{c^2 \left (b^2-4 a c\right )}-\frac {2 x^2 \left (25 a^2 b c^2 k-15 a b^3 c k-c^4 (3 b g-2 a i)+2 b^5 k+b^2 c^3 i+6 c^5 e\right )+c^3 \left (\frac {32 a^3 k}{c}-b^2 \left (\frac {39 a^2 k}{c^2}+3 g\right )+\frac {11 a b^4 k}{c^3}+2 b (a i+3 c e)-\frac {b^6 k}{c^4}+\frac {b^3 i}{c}\right )}{c^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{2 \left (b^2-4 a c\right )}\right )+\int \frac {j x^8+h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^3}dx\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {1}{2} \left (\frac {c^4 \left (\frac {2 a^3 k}{c^2}-\frac {4 a^2 b^2 k}{c^3}+\frac {a b^4 k}{c^4}-b \left (\frac {a i}{c}+e\right )+2 a g\right )-x^2 \left (-5 a^2 b c^2 k+5 a b^3 c k-c^4 (2 a i+b g)+b^5 (-k)+b^2 c^3 i+2 c^5 e\right )}{2 c^4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {-\frac {2 \left (\frac {k \left (b^2-4 a c\right )^2 \int \frac {2 c x^2+b}{c x^4+b x^2+a}dx^2}{2 c}-\frac {\left (-30 a^2 b c^2 k+10 a b^3 c k-c^4 (6 b g-4 a i)+b^5 (-k)+2 b^2 c^3 i+12 c^5 e\right ) \int \frac {1}{-x^4+b^2-4 a c}d\left (2 c x^2+b\right )}{c}\right )}{c^2 \left (b^2-4 a c\right )}-\frac {2 x^2 \left (25 a^2 b c^2 k-15 a b^3 c k-c^4 (3 b g-2 a i)+2 b^5 k+b^2 c^3 i+6 c^5 e\right )+c^3 \left (\frac {32 a^3 k}{c}-b^2 \left (\frac {39 a^2 k}{c^2}+3 g\right )+\frac {11 a b^4 k}{c^3}+2 b (a i+3 c e)-\frac {b^6 k}{c^4}+\frac {b^3 i}{c}\right )}{c^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{2 \left (b^2-4 a c\right )}\right )+\int \frac {j x^8+h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^3}dx\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{2} \left (\frac {c^4 \left (\frac {2 a^3 k}{c^2}-\frac {4 a^2 b^2 k}{c^3}+\frac {a b^4 k}{c^4}-b \left (\frac {a i}{c}+e\right )+2 a g\right )-x^2 \left (-5 a^2 b c^2 k+5 a b^3 c k-c^4 (2 a i+b g)+b^5 (-k)+b^2 c^3 i+2 c^5 e\right )}{2 c^4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {-\frac {2 \left (\frac {k \left (b^2-4 a c\right )^2 \int \frac {2 c x^2+b}{c x^4+b x^2+a}dx^2}{2 c}-\frac {\text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-30 a^2 b c^2 k+10 a b^3 c k-c^4 (6 b g-4 a i)+b^5 (-k)+2 b^2 c^3 i+12 c^5 e\right )}{c \sqrt {b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )}-\frac {2 x^2 \left (25 a^2 b c^2 k-15 a b^3 c k-c^4 (3 b g-2 a i)+2 b^5 k+b^2 c^3 i+6 c^5 e\right )+c^3 \left (\frac {32 a^3 k}{c}-b^2 \left (\frac {39 a^2 k}{c^2}+3 g\right )+\frac {11 a b^4 k}{c^3}+2 b (a i+3 c e)-\frac {b^6 k}{c^4}+\frac {b^3 i}{c}\right )}{c^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{2 \left (b^2-4 a c\right )}\right )+\int \frac {j x^8+h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^3}dx\)

\(\Big \downarrow \) 1103

\(\displaystyle \int \frac {j x^8+h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^3}dx+\frac {1}{2} \left (\frac {c^4 \left (\frac {2 a^3 k}{c^2}-\frac {4 a^2 b^2 k}{c^3}+\frac {a b^4 k}{c^4}-b \left (\frac {a i}{c}+e\right )+2 a g\right )-x^2 \left (-5 a^2 b c^2 k+5 a b^3 c k-c^4 (2 a i+b g)+b^5 (-k)+b^2 c^3 i+2 c^5 e\right )}{2 c^4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {-\frac {2 \left (\frac {k \left (b^2-4 a c\right )^2 \log \left (a+b x^2+c x^4\right )}{2 c}-\frac {\text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-30 a^2 b c^2 k+10 a b^3 c k-c^4 (6 b g-4 a i)+b^5 (-k)+2 b^2 c^3 i+12 c^5 e\right )}{c \sqrt {b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )}-\frac {2 x^2 \left (25 a^2 b c^2 k-15 a b^3 c k-c^4 (3 b g-2 a i)+2 b^5 k+b^2 c^3 i+6 c^5 e\right )+c^3 \left (\frac {32 a^3 k}{c}-b^2 \left (\frac {39 a^2 k}{c^2}+3 g\right )+\frac {11 a b^4 k}{c^3}+2 b (a i+3 c e)-\frac {b^6 k}{c^4}+\frac {b^3 i}{c}\right )}{c^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{2 \left (b^2-4 a c\right )}\right )\)

\(\Big \downarrow \) 2206

\(\displaystyle -\frac {\int -\frac {-4 a \left (4 a-\frac {b^2}{c}\right ) j x^4-\frac {\left (-a j b^3-c \left (j a^2+5 c h a+5 c^2 d\right ) b+10 a c^3 f\right ) x^2}{c^2}+a b f+b^2 \left (3 d-\frac {a^2 j}{c^2}\right )-2 a \left (-\frac {j a^2}{c}+h a+7 c d\right )}{\left (c x^4+b x^2+a\right )^2}dx}{4 a \left (b^2-4 a c\right )}-\frac {x \left (x^2 \left (-b c \left (-3 a^2 j+a c h+c^2 d\right )-a b^3 j+2 a c^3 f\right )+c^2 \left (-\left (b^2 \left (\frac {a^2 j}{c^2}+d\right )\right )+2 a \left (\frac {a^2 j}{c}-a h+c d\right )+a b f\right )\right )}{4 a c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {1}{2} \left (\frac {c^4 \left (\frac {2 a^3 k}{c^2}-\frac {4 a^2 b^2 k}{c^3}+\frac {a b^4 k}{c^4}-b \left (\frac {a i}{c}+e\right )+2 a g\right )-x^2 \left (-5 a^2 b c^2 k+5 a b^3 c k-c^4 (2 a i+b g)+b^5 (-k)+b^2 c^3 i+2 c^5 e\right )}{2 c^4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {-\frac {2 \left (\frac {k \left (b^2-4 a c\right )^2 \log \left (a+b x^2+c x^4\right )}{2 c}-\frac {\text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-30 a^2 b c^2 k+10 a b^3 c k-c^4 (6 b g-4 a i)+b^5 (-k)+2 b^2 c^3 i+12 c^5 e\right )}{c \sqrt {b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )}-\frac {2 x^2 \left (25 a^2 b c^2 k-15 a b^3 c k-c^4 (3 b g-2 a i)+2 b^5 k+b^2 c^3 i+6 c^5 e\right )+c^3 \left (\frac {32 a^3 k}{c}-b^2 \left (\frac {39 a^2 k}{c^2}+3 g\right )+\frac {11 a b^4 k}{c^3}+2 b (a i+3 c e)-\frac {b^6 k}{c^4}+\frac {b^3 i}{c}\right )}{c^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{2 \left (b^2-4 a c\right )}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {-4 a \left (4 a-\frac {b^2}{c}\right ) j x^4-\frac {\left (-a j b^3-c \left (j a^2+5 c h a+5 c^2 d\right ) b+10 a c^3 f\right ) x^2}{c^2}+a b f+b^2 \left (3 d-\frac {a^2 j}{c^2}\right )-2 a \left (-\frac {j a^2}{c}+h a+7 c d\right )}{\left (c x^4+b x^2+a\right )^2}dx}{4 a \left (b^2-4 a c\right )}-\frac {x \left (x^2 \left (-b c \left (-3 a^2 j+a c h+c^2 d\right )-a b^3 j+2 a c^3 f\right )+c^2 \left (-\left (b^2 \left (\frac {a^2 j}{c^2}+d\right )\right )+2 a \left (\frac {a^2 j}{c}-a h+c d\right )+a b f\right )\right )}{4 a c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {1}{2} \left (\frac {c^4 \left (\frac {2 a^3 k}{c^2}-\frac {4 a^2 b^2 k}{c^3}+\frac {a b^4 k}{c^4}-b \left (\frac {a i}{c}+e\right )+2 a g\right )-x^2 \left (-5 a^2 b c^2 k+5 a b^3 c k-c^4 (2 a i+b g)+b^5 (-k)+b^2 c^3 i+2 c^5 e\right )}{2 c^4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {-\frac {2 \left (\frac {k \left (b^2-4 a c\right )^2 \log \left (a+b x^2+c x^4\right )}{2 c}-\frac {\text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-30 a^2 b c^2 k+10 a b^3 c k-c^4 (6 b g-4 a i)+b^5 (-k)+2 b^2 c^3 i+12 c^5 e\right )}{c \sqrt {b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )}-\frac {2 x^2 \left (25 a^2 b c^2 k-15 a b^3 c k-c^4 (3 b g-2 a i)+2 b^5 k+b^2 c^3 i+6 c^5 e\right )+c^3 \left (\frac {32 a^3 k}{c}-b^2 \left (\frac {39 a^2 k}{c^2}+3 g\right )+\frac {11 a b^4 k}{c^3}+2 b (a i+3 c e)-\frac {b^6 k}{c^4}+\frac {b^3 i}{c}\right )}{c^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{2 \left (b^2-4 a c\right )}\right )\)

\(\Big \downarrow \) 2206

\(\displaystyle -\frac {x \left (\left (-\left (\left (\frac {j a^2}{c^2}+d\right ) b^2\right )+a f b+2 a \left (\frac {j a^2}{c}-h a+c d\right )\right ) c^2+\left (-a j b^3-c \left (-3 j a^2+c h a+c^2 d\right ) b+2 a c^3 f\right ) x^2\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 {c^4 \left (\frac {a k b^4}{c^4}-\frac {4 a^2 k b^2}{c^3}-\left (e+\frac {a i}{c}\right ) b+2 a g+\frac {2 a^3 k}{c^2}\right )-\left (-k b^5+5 a c k b^3+c^3 i b^2-5 a^2 c^2 k b+2 c^5 e-c^4 (b g+2 a i)\right ) x^2}{2 c^4 \left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )^2}-\frac {-\frac {\left (-\frac {k b^6}{c^4}+\frac {11 a k b^4}{c^3}+\frac {i b^3}{c}-\left (\frac {39 k a^2}{c^2}+3 g\right ) b^2+2 (3 c e+a i) b+\frac {32 a^3 k}{c}\right ) c^3+2 \left (2 k b^5-15 a c k b^3+c^3 i b^2+25 a^2 c^2 k b+6 c^5 e-c^4 (3 b g-2 a i)\right ) x^2}{c^3 \left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )}-\frac {2 \left (\frac {\left (b^2-4 a c\right )^2 k \log \left (c x^4+b x^2+a\right )}{2 c}-\frac {\left (-k b^5+10 a c k b^3+2 c^3 i b^2-30 a^2 c^2 k b+12 c^5 e-c^4 (6 b g-4 a i)\right ) \text {arctanh}\left (\frac {2 c x^2+b}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )}}{2 \left (b^2-4 a c\right )}\right )+\frac {\frac {x \left (\left (\left (j a^2+3 c^2 d\right ) b^3+a c^2 f b^2-4 a c \left (4 j a^2+3 c h a+6 c^2 d\right ) b+20 a^2 c^3 f\right ) x^2+c \left (\left (3 d-\frac {2 a^2 j}{c^2}\right ) b^4+a f b^3-a \left (-\frac {11 j a^2}{c}+7 h a+25 c d\right ) b^2+8 a^2 c f b+4 a^2 \left (-9 j 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 (j a^2+3 c^2 d\right ) b^3+a c^2 f b^2-4 a c \left (4 j a^2+3 c h a+6 c^2 d\right ) b+20 a^2 c^3 f\right ) x^2+c \left (3 d b^4+a f b^3-a \left (-\frac {j a^2}{c}-3 h a+27 c d\right ) b^2-16 a^2 c f b+4 a^2 \left (5 j 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 (\left (\frac {j a^2}{c^2}+d\right ) b^2\right )+a f b+2 a \left (\frac {j a^2}{c}-h a+c d\right )\right ) c^2+\left (-a j b^3-c \left (-3 j a^2+c h a+c^2 d\right ) b+2 a c^3 f\right ) x^2\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 {c^4 \left (\frac {a k b^4}{c^4}-\frac {4 a^2 k b^2}{c^3}-\left (e+\frac {a i}{c}\right ) b+2 a g+\frac {2 a^3 k}{c^2}\right )-\left (-k b^5+5 a c k b^3+c^3 i b^2-5 a^2 c^2 k b+2 c^5 e-c^4 (b g+2 a i)\right ) x^2}{2 c^4 \left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )^2}-\frac {-\frac {\left (-\frac {k b^6}{c^4}+\frac {11 a k b^4}{c^3}+\frac {i b^3}{c}-\left (\frac {39 k a^2}{c^2}+3 g\right ) b^2+2 (3 c e+a i) b+\frac {32 a^3 k}{c}\right ) c^3+2 \left (2 k b^5-15 a c k b^3+c^3 i b^2+25 a^2 c^2 k b+6 c^5 e-c^4 (3 b g-2 a i)\right ) x^2}{c^3 \left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )}-\frac {2 \left (\frac {\left (b^2-4 a c\right )^2 k \log \left (c x^4+b x^2+a\right )}{2 c}-\frac {\left (-k b^5+10 a c k b^3+2 c^3 i b^2-30 a^2 c^2 k b+12 c^5 e-c^4 (6 b g-4 a i)\right ) \text {arctanh}\left (\frac {2 c x^2+b}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )}}{2 \left (b^2-4 a c\right )}\right )+\frac {\frac {x \left (\left (\left (j a^2+3 c^2 d\right ) b^3+a c^2 f b^2-4 a c \left (4 j a^2+3 c h a+6 c^2 d\right ) b+20 a^2 c^3 f\right ) x^2+c \left (\left (3 d-\frac {2 a^2 j}{c^2}\right ) b^4+a f b^3-a \left (-\frac {11 j a^2}{c}+7 h a+25 c d\right ) b^2+8 a^2 c f b+4 a^2 \left (-9 j 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 (j a^2+3 c^2 d\right ) b^3+a c^2 f b^2-4 a c \left (4 j a^2+3 c h a+6 c^2 d\right ) b+20 a^2 c^3 f\right ) x^2+c \left (3 d b^4+a f b^3-a \left (-\frac {j a^2}{c}-3 h a+27 c d\right ) b^2-16 a^2 c f b+4 a^2 \left (5 j 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 (\left (\frac {j a^2}{c^2}+d\right ) b^2\right )+a f b+2 a \left (\frac {j a^2}{c}-h a+c d\right )\right ) c^2+\left (-a j b^3-c \left (-3 j a^2+c h a+c^2 d\right ) b+2 a c^3 f\right ) x^2\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 {c^4 \left (\frac {a k b^4}{c^4}-\frac {4 a^2 k b^2}{c^3}-\left (e+\frac {a i}{c}\right ) b+2 a g+\frac {2 a^3 k}{c^2}\right )-\left (-k b^5+5 a c k b^3+c^3 i b^2-5 a^2 c^2 k b+2 c^5 e-c^4 (b g+2 a i)\right ) x^2}{2 c^4 \left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )^2}-\frac {-\frac {\left (-\frac {k b^6}{c^4}+\frac {11 a k b^4}{c^3}+\frac {i b^3}{c}-\left (\frac {39 k a^2}{c^2}+3 g\right ) b^2+2 (3 c e+a i) b+\frac {32 a^3 k}{c}\right ) c^3+2 \left (2 k b^5-15 a c k b^3+c^3 i b^2+25 a^2 c^2 k b+6 c^5 e-c^4 (3 b g-2 a i)\right ) x^2}{c^3 \left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )}-\frac {2 \left (\frac {\left (b^2-4 a c\right )^2 k \log \left (c x^4+b x^2+a\right )}{2 c}-\frac {\left (-k b^5+10 a c k b^3+2 c^3 i b^2-30 a^2 c^2 k b+12 c^5 e-c^4 (6 b g-4 a i)\right ) \text {arctanh}\left (\frac {2 c x^2+b}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )}}{2 \left (b^2-4 a c\right )}\right )+\frac {\frac {x \left (\left (\left (j a^2+3 c^2 d\right ) b^3+a c^2 f b^2-4 a c \left (4 j a^2+3 c h a+6 c^2 d\right ) b+20 a^2 c^3 f\right ) x^2+c \left (\left (3 d-\frac {2 a^2 j}{c^2}\right ) b^4+a f b^3-a \left (-\frac {11 j a^2}{c}+7 h a+25 c d\right ) b^2+8 a^2 c f b+4 a^2 \left (-9 j 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 (j a^2+3 c^2 d\right ) b^3+a c^2 f b^2-4 a c \left (4 j a^2+3 c h a+6 c^2 d\right ) b+20 a^2 c^3 f\right ) x^2+c \left (3 d b^4+a f b^3-a \left (-\frac {j a^2}{c}-3 h a+27 c d\right ) b^2-16 a^2 c f b+4 a^2 \left (5 j 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 (\left (\frac {j a^2}{c^2}+d\right ) b^2\right )+a f b+2 a \left (\frac {j a^2}{c}-h a+c d\right )\right ) c^2+\left (-a j b^3-c \left (-3 j a^2+c h a+c^2 d\right ) b+2 a c^3 f\right ) x^2\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 {c^4 \left (\frac {a k b^4}{c^4}-\frac {4 a^2 k b^2}{c^3}-\left (e+\frac {a i}{c}\right ) b+2 a g+\frac {2 a^3 k}{c^2}\right )-\left (-k b^5+5 a c k b^3+c^3 i b^2-5 a^2 c^2 k b+2 c^5 e-c^4 (b g+2 a i)\right ) x^2}{2 c^4 \left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )^2}-\frac {-\frac {\left (-\frac {k b^6}{c^4}+\frac {11 a k b^4}{c^3}+\frac {i b^3}{c}-\left (\frac {39 k a^2}{c^2}+3 g\right ) b^2+2 (3 c e+a i) b+\frac {32 a^3 k}{c}\right ) c^3+2 \left (2 k b^5-15 a c k b^3+c^3 i b^2+25 a^2 c^2 k b+6 c^5 e-c^4 (3 b g-2 a i)\right ) x^2}{c^3 \left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )}-\frac {2 \left (\frac {\left (b^2-4 a c\right )^2 k \log \left (c x^4+b x^2+a\right )}{2 c}-\frac {\left (-k b^5+10 a c k b^3+2 c^3 i b^2-30 a^2 c^2 k b+12 c^5 e-c^4 (6 b g-4 a i)\right ) \text {arctanh}\left (\frac {2 c x^2+b}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )}}{2 \left (b^2-4 a c\right )}\right )+\frac {\frac {x \left (\left (\left (j a^2+3 c^2 d\right ) b^3+a c^2 f b^2-4 a c \left (4 j a^2+3 c h a+6 c^2 d\right ) b+20 a^2 c^3 f\right ) x^2+c \left (\left (3 d-\frac {2 a^2 j}{c^2}\right ) b^4+a f b^3-a \left (-\frac {11 j a^2}{c}+7 h a+25 c d\right ) b^2+8 a^2 c f b+4 a^2 \left (-9 j 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 (j a^2+3 c^2 d\right ) b^3+a c^2 f b^2-4 a c \left (4 j a^2+3 c h a+6 c^2 d\right ) b+20 a^2 c^3 f+\frac {\left (3 c^2 d-a^2 j\right ) b^4+a c^2 f b^3-6 a c \left (-3 j a^2-3 c h a+5 c^2 d\right ) b^2-52 a^2 c^3 f b+8 a^2 c^2 \left (5 j 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 (j a^2+3 c^2 d\right ) b^3+a c^2 f b^2-4 a c \left (4 j a^2+3 c h a+6 c^2 d\right ) b+20 a^2 c^3 f-\frac {\left (3 c^2 d-a^2 j\right ) b^4+a c^2 f b^3-6 a c \left (-3 j a^2-3 c h a+5 c^2 d\right ) b^2-52 a^2 c^3 f b+8 a^2 c^2 \left (5 j 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 (\left (\frac {j a^2}{c^2}+d\right ) b^2\right )+a f b+2 a \left (\frac {j a^2}{c}-h a+c d\right )\right ) c^2+\left (-a j b^3-c \left (-3 j a^2+c h a+c^2 d\right ) b+2 a c^3 f\right ) x^2\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 (j a^2+3 c^2 d\right ) b^3+a c^2 f b^2-4 a c \left (4 j a^2+3 c h a+6 c^2 d\right ) b+20 a^2 c^3 f\right ) x^2+c \left (\left (3 d-\frac {2 a^2 j}{c^2}\right ) b^4+a f b^3-a \left (-\frac {11 j a^2}{c}+7 h a+25 c d\right ) b^2+8 a^2 c f b+4 a^2 \left (-9 j 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 (j a^2+3 c^2 d\right ) b^3+a c^2 f b^2-4 a c \left (4 j a^2+3 c h a+6 c^2 d\right ) b+20 a^2 c^3 f+\frac {\left (3 c^2 d-a^2 j\right ) b^4+a c^2 f b^3-6 a c \left (-3 j a^2-3 c h a+5 c^2 d\right ) b^2-52 a^2 c^3 f b+8 a^2 c^2 \left (5 j 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 (j a^2+3 c^2 d\right ) b^3+a c^2 f b^2-4 a c \left (4 j a^2+3 c h a+6 c^2 d\right ) b+20 a^2 c^3 f-\frac {\left (3 c^2 d-a^2 j\right ) b^4+a c^2 f b^3-6 a c \left (-3 j a^2-3 c h a+5 c^2 d\right ) b^2-52 a^2 c^3 f b+8 a^2 c^2 \left (5 j 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 {c^4 \left (\frac {a k b^4}{c^4}-\frac {4 a^2 k b^2}{c^3}-\left (e+\frac {a i}{c}\right ) b+2 a g+\frac {2 a^3 k}{c^2}\right )-\left (-k b^5+5 a c k b^3+c^3 i b^2-5 a^2 c^2 k b+2 c^5 e-c^4 (b g+2 a i)\right ) x^2}{2 c^4 \left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )^2}-\frac {-\frac {\left (-\frac {k b^6}{c^4}+\frac {11 a k b^4}{c^3}+\frac {i b^3}{c}-\left (\frac {39 k a^2}{c^2}+3 g\right ) b^2+2 (3 c e+a i) b+\frac {32 a^3 k}{c}\right ) c^3+2 \left (2 k b^5-15 a c k b^3+c^3 i b^2+25 a^2 c^2 k b+6 c^5 e-c^4 (3 b g-2 a i)\right ) x^2}{c^3 \left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )}-\frac {2 \left (\frac {\left (b^2-4 a c\right )^2 k \log \left (c x^4+b x^2+a\right )}{2 c}-\frac {\left (-k b^5+10 a c k b^3+2 c^3 i b^2-30 a^2 c^2 k b+12 c^5 e-c^4 (6 b g-4 a i)\right ) \text {arctanh}\left (\frac {2 c x^2+b}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )}}{2 \left (b^2-4 a c\right )}\right )\)

Input:

Int[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5 + j*x^8 + k*x^11)/(a + b*x^2 
+ c*x^4)^3,x]
 

Output:

-1/4*(x*(c^2*(a*b*f - b^2*(d + (a^2*j)/c^2) + 2*a*(c*d - a*h + (a^2*j)/c)) 
 + (2*a*c^3*f - a*b^3*j - b*c*(c^2*d + a*c*h - 3*a^2*j))*x^2))/(a*c^2*(b^2 
 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + ((x*(c*(a*b^3*f + 8*a^2*b*c*f + 4*a^2*( 
7*c^2*d + a*c*h - 9*a^2*j) + b^4*(3*d - (2*a^2*j)/c^2) - a*b^2*(25*c*d + 7 
*a*h - (11*a^2*j)/c)) + (a*b^2*c^2*f + 20*a^2*c^3*f + b^3*(3*c^2*d + a^2*j 
) - 4*a*b*c*(6*c^2*d + 3*a*c*h + 4*a^2*j))*x^2))/(2*a*c*(b^2 - 4*a*c)*(a + 
 b*x^2 + c*x^4)) + (((a*b^2*c^2*f + 20*a^2*c^3*f + b^3*(3*c^2*d + a^2*j) - 
 4*a*b*c*(6*c^2*d + 3*a*c*h + 4*a^2*j) + (a*b^3*c^2*f - 52*a^2*b*c^3*f - 6 
*a*b^2*c*(5*c^2*d - 3*a*c*h - 3*a^2*j) + b^4*(3*c^2*d - a^2*j) + 8*a^2*c^2 
*(21*c^2*d + 3*a*c*h + 5*a^2*j))/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^2*f + 20*a^2*c^3*f + b^3*(3*c^2*d + a^2*j) - 4*a*b*c*(6 
*c^2*d + 3*a*c*h + 4*a^2*j) - (a*b^3*c^2*f - 52*a^2*b*c^3*f - 6*a*b^2*c*(5 
*c^2*d - 3*a*c*h - 3*a^2*j) + b^4*(3*c^2*d - a^2*j) + 8*a^2*c^2*(21*c^2*d 
+ 3*a*c*h + 5*a^2*j))/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]]))/(2* 
a*c*(b^2 - 4*a*c)))/(4*a*(b^2 - 4*a*c)) + ((c^4*(2*a*g - b*(e + (a*i)/c) + 
 (a*b^4*k)/c^4 - (4*a^2*b^2*k)/c^3 + (2*a^3*k)/c^2) - (2*c^5*e + b^2*c^3*i 
 - c^4*(b*g + 2*a*i) - b^5*k + 5*a*b^3*c*k - 5*a^2*b*c^2*k)*x^2)/(2*c^4*(b 
^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (-((c^3*((b^3*i)/c + 2*b*(3*c*e + ...
 

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 1103

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]
 

rule 1142

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]
 

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 = 0.82 (sec) , antiderivative size = 1182, normalized size of antiderivative = 1.00

method	result	size
risch	\(\text {Expression too large to display}\)	\(1182\)
default	\(\text {Expression too large to display}\)	\(2058\)

Input:

int((k*x^11+j*x^8+i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^3,x,metho 
d=_RETURNVERBOSE)
 

Output:

(-1/8*(16*a^3*b*c*j-a^2*b^3*j+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*(25*a^2*b*c^2 
*k-15*a*b^3*c*k+2*a*c^4*i+2*b^5*k+b^2*c^3*i-3*b*c^4*g+6*c^5*e)/c^2/(16*a^2 
*c^2-8*a*b^2*c+b^4)*x^6-1/8/a^2*(36*a^4*c^2*j+5*a^3*b^2*c*j-4*a^3*c^3*h+a^ 
2*b^4*j+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*(32*a^3*c^3*k+11 
*a^2*b^2*c^2*k-19*a*b^4*c*k+6*a*b*c^4*i+3*b^6*k+3*b^3*c^3*i-9*b^2*c^4*g+18 
*b*c^5*e)/(16*a^2*c^2-8*a*b^2*c+b^4)/c^3*x^4-1/8/c*(28*a^4*b*c*j+2*a^3*b^3 
*j+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^3*(31*a^3*b*c^2*k-22*a^2*b^3*c*k-2*a^2*c^4*i+3*a*b^5*k+5*a*b^2*c^3*i-5 
*a*b*c^4*g+10*a*c^5*e-b^3*c^3*g+2*b^2*c^4*e)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^ 
2-1/8*(20*a^4*c*j+a^3*b^2*j+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)/c/(16*a^2*c^2-8*a*b^2*c+b^4)/ 
a*x+1/4*(24*a^4*c^2*k-21*a^3*b^2*c*k+3*a^2*b^4*k+6*a^2*b*c^3*i-8*a^2*c^4*g 
-a*b^2*c^3*g+10*a*b*c^4*e-b^3*c^3*e)/(16*a^2*c^2-8*a*b^2*c+b^4)/c^3)/(c*x^ 
4+b*x^2+a)^2+1/16/c*sum((8/c*k*_R^3-1/a^2*(16*a^3*b*c*j-a^2*b^3*j+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)/(16*a^2*c^2-8*a* 
b^2*c+b^4)*_R^2-8/c*(7*a^2*b*c*k-a*b^3*k-2*a*c^3*i-b^2*c^2*i+3*b*c^3*g-6*c 
^4*e)/(16*a^2*c^2-8*a*b^2*c+b^4)*_R+1/a^2*(20*a^4*c*j+a^3*b^2*j+12*a^3*...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5+j x^8+k x^{11}}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Timed out} \] Input:

integrate((k*x^11+j*x^8+i*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")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5+j x^8+k x^{11}}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Timed out} \] Input:

integrate((k*x**11+j*x**8+i*x**5+h*x**4+g*x**3+f*x**2+e*x+d)/(c*x**4+b*x** 
2+a)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5+j x^8+k x^{11}}{\left (a+b x^2+c x^4\right )^3} \, dx=\int { \frac {k x^{11} + j x^{8} + i x^{5} + h x^{4} + g x^{3} + f x^{2} + e x + d}{{\left (c x^{4} + b x^{2} + a\right )}^{3}} \,d x } \] Input:

integrate((k*x^11+j*x^8+i*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")
 

Output:

1/8*(12*a^4*b*c^3*i - (12*a^2*b*c^5*h - 3*(b^3*c^5 - 8*a*b*c^6)*d - (a*b^2 
*c^5 + 20*a^2*c^6)*f - (a^2*b^3*c^3 - 16*a^3*b*c^4)*j)*x^7 + 4*(6*a^2*c^6* 
e - 3*a^2*b*c^5*g + (a^2*b^2*c^4 + 2*a^3*c^5)*i + (2*a^2*b^5*c - 15*a^3*b^ 
3*c^2 + 25*a^4*b*c^3)*k)*x^6 + ((6*b^4*c^4 - 49*a*b^2*c^5 + 28*a^2*c^6)*d 
+ 2*(a*b^3*c^4 + 14*a^2*b*c^5)*f - (19*a^2*b^2*c^4 - 4*a^3*c^5)*h - (a^2*b 
^4*c^2 + 5*a^3*b^2*c^3 + 36*a^4*c^4)*j)*x^5 + 2*(18*a^2*b*c^5*e - 9*a^2*b^ 
2*c^4*g + 3*(a^2*b^3*c^3 + 2*a^3*b*c^4)*i + (3*a^2*b^6 - 19*a^3*b^4*c + 11 
*a^4*b^2*c^2 + 32*a^5*c^3)*k)*x^4 + ((3*b^5*c^3 - 20*a*b^3*c^4 - 4*a^2*b*c 
^5)*d + (a*b^4*c^3 + 5*a^2*b^2*c^4 + 36*a^3*c^5)*f - (5*a^2*b^3*c^3 + 16*a 
^3*b*c^4)*h - 2*(a^3*b^3*c^2 + 14*a^4*b*c^3)*j)*x^3 + 4*(2*(a^2*b^2*c^4 + 
5*a^3*c^5)*e - (a^2*b^3*c^3 + 5*a^3*b*c^4)*g + (5*a^3*b^2*c^3 - 2*a^4*c^4) 
*i + (3*a^3*b^5 - 22*a^4*b^3*c + 31*a^5*b*c^2)*k)*x^2 - 2*(a^2*b^3*c^3 - 1 
0*a^3*b*c^4)*e - 2*(a^3*b^2*c^3 + 8*a^4*c^4)*g + 6*(a^4*b^4 - 7*a^5*b^2*c 
+ 8*a^6*c^2)*k + ((5*a*b^4*c^3 - 37*a^2*b^2*c^4 + 44*a^3*c^5)*d - (a^2*b^3 
*c^3 - 16*a^3*b*c^4)*f - 3*(a^3*b^2*c^3 + 4*a^4*c^4)*h - (a^4*b^2*c^2 + 20 
*a^5*c^3)*j)*x)/(a^4*b^4*c^3 - 8*a^5*b^2*c^4 + 16*a^6*c^5 + (a^2*b^4*c^5 - 
 8*a^3*b^2*c^6 + 16*a^4*c^7)*x^8 + 2*(a^2*b^5*c^4 - 8*a^3*b^3*c^5 + 16*a^4 
*b*c^6)*x^6 + (a^2*b^6*c^3 - 6*a^3*b^4*c^4 + 32*a^5*c^6)*x^4 + 2*(a^3*b^5* 
c^3 - 8*a^4*b^3*c^4 + 16*a^5*b*c^5)*x^2) + 1/8*integrate((8*(a^2*b^4 - 8*a 
^3*b^2*c + 16*a^4*c^2)*k*x^3 - (12*a^2*b*c^3*h - 3*(b^3*c^3 - 8*a*b*c^4...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 29140 vs. \(2 (1122) = 2244\).

Time = 3.92 (sec) , antiderivative size = 29140, normalized size of antiderivative = 24.76 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5+j x^8+k x^{11}}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((k*x^11+j*x^8+i*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")
 

Output:

-1/64*(3*(a^4*b^8*c^5 - 16*a^5*b^6*c^6 + 96*a^6*b^4*c^7 - 256*a^7*b^2*c^8 
+ 256*a^8*c^9)^2*(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)*sqrt(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)*s 
qrt(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 + sqrt(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)*d + (a^4*b^8*c^5 - 16*a^5*b^6*c^6 + 96*a^6*b^4*c^7 - 256*a^ 
7*b^2*c^8 + 256*a^8*c^9)^2*(2*a*b^4*c^4 + 32*a^2*b^2*c^5 - 160*a^3*c^6 - s 
qrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c^2 - 16*sq 
rt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^3 + 2*sq 
rt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^3 + 80*sqr 
t(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)*sq 
rt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^4 - 20*sqrt(2)*sqr 
t(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)*f - 12*(a^4*b^8*c^5 - 16*a^5*b^6*c^6 
+ 96*a^6*b^4*c^7 - 256*a^7*b^2*c^8 + 256*a^8*c^9)^2*(2*a^2*b^3*c^4 - 8*...
 

Mupad [B] (verification not implemented)

Time = 55.46 (sec) , antiderivative size = 97905, normalized size of antiderivative = 83.18 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5+j x^8+k x^{11}}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:

int((d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5 + j*x^8 + k*x^11)/(a + b*x^2 
+ c*x^4)^3,x)
 

Output:

((x^7*(3*b^3*c^2*d + 20*a^2*c^3*f + a^2*b^3*j - 24*a*b*c^3*d - 16*a^3*b*c* 
j + a*b^2*c^2*f - 12*a^2*b*c^2*h))/(8*a^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) 
- (b^3*c^3*e + 8*a^2*c^4*g - 3*a^2*b^4*k - 24*a^4*c^2*k - 10*a*b*c^4*e + a 
*b^2*c^3*g - 6*a^2*b*c^3*i + 21*a^3*b^2*c*k)/(4*c^3*(b^4 + 16*a^2*c^2 - 8* 
a*b^2*c)) + (x^4*(3*b^6*k - 9*b^2*c^4*g + 3*b^3*c^3*i + 32*a^3*c^3*k + 18* 
b*c^5*e + 11*a^2*b^2*c^2*k + 6*a*b*c^4*i - 19*a*b^4*c*k))/(4*c^3*(b^4 + 16 
*a^2*c^2 - 8*a*b^2*c)) + (x^2*(2*b^2*c^4*e - b^3*c^3*g - 2*a^2*c^4*i + 10* 
a*c^5*e + 3*a*b^5*k - 5*a*b*c^4*g + 5*a*b^2*c^3*i - 22*a^2*b^3*c*k + 31*a^ 
3*b*c^2*k))/(2*c^3*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x^6*(6*c^5*e + 2*b^5 
*k + b^2*c^3*i - 3*b*c^4*g + 2*a*c^4*i - 15*a*b^3*c*k + 25*a^2*b*c^2*k))/( 
2*c^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) - (x^3*(2*a^3*b^3*j - 36*a^3*c^3*f - 
 3*b^5*c*d - 5*a^2*b^2*c^2*f - a*b^4*c*f + 28*a^4*b*c*j + 20*a*b^3*c^2*d + 
 4*a^2*b*c^3*d + 5*a^2*b^3*c*h + 16*a^3*b*c^2*h))/(8*a^2*c*(b^4 + 16*a^2*c 
^2 - 8*a*b^2*c)) + (x^5*(28*a^2*c^4*d + 6*b^4*c^2*d + 4*a^3*c^3*h - a^2*b^ 
4*j - 36*a^4*c^2*j - 19*a^2*b^2*c^2*h - 49*a*b^2*c^3*d + 2*a*b^3*c^2*f + 2 
8*a^2*b*c^3*f - 5*a^3*b^2*c*j))/(8*a^2*c*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) - 
 (x*(12*a^3*c^2*h - 44*a^2*c^3*d + a^3*b^2*j - 5*b^4*c*d + 20*a^4*c*j + a* 
b^3*c*f + 37*a*b^2*c^2*d - 16*a^2*b*c^2*f + 3*a^2*b^2*c*h))/(8*a*c*(b^4 + 
16*a^2*c^2 - 8*a*b^2*c)))/(x^4*(2*a*c + b^2) + a^2 + c^2*x^8 + 2*a*b*x^2 + 
 2*b*c*x^6) + symsum(log((10368*a*b^5*c^10*d^3 - 8000*a^5*c^11*f^3 - 56...
 

Reduce [B] (verification not implemented)

Time = 20.05 (sec) , antiderivative size = 30922, normalized size of antiderivative = 26.27 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5+j x^8+k x^{11}}{\left (a+b x^2+c x^4\right )^3} \, dx =\text {Too large to display} \] Input:

int((k*x^11+j*x^8+i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^3,x)
 

Output:

(480*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**7*b**2 
*c**2*k - 160*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**6*b**4*c*k + 960*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**6*b**3*c**2*k*x**2 + 960*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*sq 
rt(c)*sqrt(a) + b))*a**6*b**2*c**3*k*x**4 - 64*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**6*b*c**4*i + 16*sqrt(2*sqrt(c)*sqrt(a) 
 + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sq 
rt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**5*b**6*k - 320*sqrt(2*sqrt(c)*sqr 
t(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**5*b**5*c*k*x**2 + 160*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**5*b**4*c**2*k*x**4 
 - 32*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**5*b** 
3*c**3*i + 960*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*...