\(\int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{(a+b x^2+c x^4)^3} \, dx\) [51]

Optimal result

Integrand size = 40, antiderivative size = 732 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {x \left (b^2 d-a b f-2 a (c d-a h)+(b c d-2 a c f+a b h) x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {2 a c g-b (c e+a i)-\left (2 c^2 e-b c g+b^2 i-2 a c i\right ) x^2}{4 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {\left (6 c^2 e-3 b c g+b^2 i+2 a c i\right ) \left (b+2 c x^2\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {x \left (3 b^4 d+a b^3 f+8 a^2 b c f+4 a^2 c (7 c d+a h)-a b^2 (25 c d+7 a h)+c \left (3 b^3 d+a b^2 f+20 a^2 c f-12 a b (2 c d+a h)\right ) x^2\right )}{8 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\sqrt {c} \left (3 b^3 d+a b^2 f+20 a^2 c f-12 a b (2 c d+a h)+\frac {3 b^4 d+a b^3 f-52 a^2 b c f-6 a b^2 (5 c d-3 a h)+24 a^2 c (7 c d+a h)}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a^2 \left (b^2-4 a c\right )^2 \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (3 b^3 d+a b^2 f+20 a^2 c f-12 a b (2 c d+a h)-\frac {3 b^4 d+a b^3 f-52 a^2 b c f-6 a b^2 (5 c d-3 a h)+24 a^2 c (7 c d+a h)}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a^2 \left (b^2-4 a c\right )^2 \sqrt {b+\sqrt {b^2-4 a c}}}-\frac {\left (6 c^2 e-3 b c g+b^2 i+2 a c i\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \] Output:

1/4*x*(b^2*d-a*b*f-2*a*(-a*h+c*d)+(a*b*h-2*a*c*f+b*c*d)*x^2)/a/(-4*a*c+b^2 
)/(c*x^4+b*x^2+a)^2+1/4*(2*a*c*g-b*(a*i+c*e)-(-2*a*c*i+b^2*i-b*c*g+2*c^2*e 
)*x^2)/c/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^2+1/4*(2*a*c*i+b^2*i-3*b*c*g+6*c^2*e 
)*(2*c*x^2+b)/c/(-4*a*c+b^2)^2/(c*x^4+b*x^2+a)+1/8*x*(3*b^4*d+a*b^3*f+8*a^ 
2*b*c*f+4*a^2*c*(a*h+7*c*d)-a*b^2*(7*a*h+25*c*d)+c*(3*b^3*d+a*b^2*f+20*a^2 
*c*f-12*a*b*(a*h+2*c*d))*x^2)/a^2/(-4*a*c+b^2)^2/(c*x^4+b*x^2+a)+1/16*c^(1 
/2)*(3*b^3*d+a*b^2*f+20*a^2*c*f-12*a*b*(a*h+2*c*d)+(3*b^4*d+a*b^3*f-52*a^2 
*b*c*f-6*a*b^2*(-3*a*h+5*c*d)+24*a^2*c*(a*h+7*c*d))/(-4*a*c+b^2)^(1/2))*ar 
ctan(2^(1/2)*c^(1/2)*x/(b-(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)/a^2/(-4*a*c+b 
^2)^2/(b-(-4*a*c+b^2)^(1/2))^(1/2)+1/16*c^(1/2)*(3*b^3*d+a*b^2*f+20*a^2*c* 
f-12*a*b*(a*h+2*c*d)-(3*b^4*d+a*b^3*f-52*a^2*b*c*f-6*a*b^2*(-3*a*h+5*c*d)+ 
24*a^2*c*(a*h+7*c*d))/(-4*a*c+b^2)^(1/2))*arctan(2^(1/2)*c^(1/2)*x/(b+(-4* 
a*c+b^2)^(1/2))^(1/2))*2^(1/2)/a^2/(-4*a*c+b^2)^2/(b+(-4*a*c+b^2)^(1/2))^( 
1/2)-(2*a*c*i+b^2*i-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)
 

Mathematica [A] (verified)

Time = 6.66 (sec) , antiderivative size = 980, normalized size of antiderivative = 1.34 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {a b c e-2 a^2 c g+a^2 b i-b^2 c d x+2 a c^2 d x+a b c f x-2 a^2 c h x+2 a c^2 e x^2-a b c g x^2+a b^2 i x^2-2 a^2 c i x^2-b c^2 d x^3+2 a c^2 f x^3-a b c h x^3}{4 a c \left (-b^2+4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {12 a^2 b c^2 e-6 a^2 b^2 c g+2 a^2 b^3 i+4 a^3 b c i+3 b^4 c d x-25 a b^2 c^2 d x+28 a^2 c^3 d x+a b^3 c f x+8 a^2 b c^2 f x-7 a^2 b^2 c h x+4 a^3 c^2 h x+24 a^2 c^3 e x^2-12 a^2 b c^2 g x^2+4 a^2 b^2 c i x^2+8 a^3 c^2 i x^2+3 b^3 c^2 d x^3-24 a b c^3 d x^3+a b^2 c^2 f x^3+20 a^2 c^3 f x^3-12 a^2 b c^2 h x^3}{8 a^2 c \left (-b^2+4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\sqrt {c} \left (3 b^4 d-30 a b^2 c d+168 a^2 c^2 d+3 b^3 \sqrt {b^2-4 a c} d-24 a b c \sqrt {b^2-4 a c} d+a b^3 f-52 a^2 b c f+a b^2 \sqrt {b^2-4 a c} f+20 a^2 c \sqrt {b^2-4 a c} f+18 a^2 b^2 h+24 a^3 c h-12 a^2 b \sqrt {b^2-4 a c} h\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (-3 b^4 d+30 a b^2 c d-168 a^2 c^2 d+3 b^3 \sqrt {b^2-4 a c} d-24 a b c \sqrt {b^2-4 a c} d-a b^3 f+52 a^2 b c f+a b^2 \sqrt {b^2-4 a c} f+20 a^2 c \sqrt {b^2-4 a c} f-18 a^2 b^2 h-24 a^3 c h-12 a^2 b \sqrt {b^2-4 a c} h\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}}}+\frac {\left (6 c^2 e-3 b c g+b^2 i+2 a c i\right ) \log \left (-b+\sqrt {b^2-4 a c}-2 c x^2\right )}{2 \left (b^2-4 a c\right )^{5/2}}+\frac {\left (-6 c^2 e+3 b c g-b^2 i-2 a c i\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{2 \left (b^2-4 a c\right )^{5/2}} \] Input:

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

Output:

(a*b*c*e - 2*a^2*c*g + a^2*b*i - b^2*c*d*x + 2*a*c^2*d*x + a*b*c*f*x - 2*a 
^2*c*h*x + 2*a*c^2*e*x^2 - a*b*c*g*x^2 + a*b^2*i*x^2 - 2*a^2*c*i*x^2 - b*c 
^2*d*x^3 + 2*a*c^2*f*x^3 - a*b*c*h*x^3)/(4*a*c*(-b^2 + 4*a*c)*(a + b*x^2 + 
 c*x^4)^2) + (12*a^2*b*c^2*e - 6*a^2*b^2*c*g + 2*a^2*b^3*i + 4*a^3*b*c*i + 
 3*b^4*c*d*x - 25*a*b^2*c^2*d*x + 28*a^2*c^3*d*x + a*b^3*c*f*x + 8*a^2*b*c 
^2*f*x - 7*a^2*b^2*c*h*x + 4*a^3*c^2*h*x + 24*a^2*c^3*e*x^2 - 12*a^2*b*c^2 
*g*x^2 + 4*a^2*b^2*c*i*x^2 + 8*a^3*c^2*i*x^2 + 3*b^3*c^2*d*x^3 - 24*a*b*c^ 
3*d*x^3 + a*b^2*c^2*f*x^3 + 20*a^2*c^3*f*x^3 - 12*a^2*b*c^2*h*x^3)/(8*a^2* 
c*(-b^2 + 4*a*c)^2*(a + b*x^2 + c*x^4)) + (Sqrt[c]*(3*b^4*d - 30*a*b^2*c*d 
 + 168*a^2*c^2*d + 3*b^3*Sqrt[b^2 - 4*a*c]*d - 24*a*b*c*Sqrt[b^2 - 4*a*c]* 
d + a*b^3*f - 52*a^2*b*c*f + a*b^2*Sqrt[b^2 - 4*a*c]*f + 20*a^2*c*Sqrt[b^2 
 - 4*a*c]*f + 18*a^2*b^2*h + 24*a^3*c*h - 12*a^2*b*Sqrt[b^2 - 4*a*c]*h)*Ar 
cTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2*(b^2 
 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[c]*(-3*b^4*d + 30*a*b 
^2*c*d - 168*a^2*c^2*d + 3*b^3*Sqrt[b^2 - 4*a*c]*d - 24*a*b*c*Sqrt[b^2 - 4 
*a*c]*d - a*b^3*f + 52*a^2*b*c*f + a*b^2*Sqrt[b^2 - 4*a*c]*f + 20*a^2*c*Sq 
rt[b^2 - 4*a*c]*f - 18*a^2*b^2*h - 24*a^3*c*h - 12*a^2*b*Sqrt[b^2 - 4*a*c] 
*h)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^ 
2*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + ((6*c^2*e - 3*b*c*g + 
 b^2*i + 2*a*c*i)*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(2*(b^2 - 4*a*...
 

Rubi [A] (verified)

Time = 1.55 (sec) , antiderivative size = 732, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.325, Rules used = {2202, 2194, 2191, 27, 1086, 1083, 219, 2206, 25, 1492, 25, 1480, 218}

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

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

\(\Big \downarrow \) 2202

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

\(\Big \downarrow \) 2194

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

\(\Big \downarrow \) 2191

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1086

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

\(\Big \downarrow \) 1083

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

\(\Big \downarrow \) 219

\(\displaystyle \int \frac {h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^3}dx+\frac {1}{2} \left (\frac {c \left (2 a g-b \left (\frac {a i}{c}+e\right )\right )-x^2 \left (-2 a c i+b^2 i-b c g+2 c^2 e\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\left (\frac {4 c \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {b+2 c x^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right ) \left (2 a i+\frac {b^2 i}{c}-3 b g+6 c e\right )}{2 \left (b^2-4 a c\right )}\right )\)

\(\Big \downarrow \) 2206

\(\displaystyle -\frac {\int -\frac {3 d b^2+a f b+5 (b c d-2 a c f+a b h) x^2-2 a (7 c d+a h)}{\left (c x^4+b x^2+a\right )^2}dx}{4 a \left (b^2-4 a c\right )}+\frac {1}{2} \left (\frac {c \left (2 a g-b \left (\frac {a i}{c}+e\right )\right )-x^2 \left (-2 a c i+b^2 i-b c g+2 c^2 e\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\left (\frac {4 c \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {b+2 c x^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right ) \left (2 a i+\frac {b^2 i}{c}-3 b g+6 c e\right )}{2 \left (b^2-4 a c\right )}\right )+\frac {x \left (x^2 (a b h-2 a c f+b c d)-a b f-2 a (c d-a h)+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {3 d b^2+a f b+5 (b c d-2 a c f+a b h) x^2-2 a (7 c d+a h)}{\left (c x^4+b x^2+a\right )^2}dx}{4 a \left (b^2-4 a c\right )}+\frac {1}{2} \left (\frac {c \left (2 a g-b \left (\frac {a i}{c}+e\right )\right )-x^2 \left (-2 a c i+b^2 i-b c g+2 c^2 e\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\left (\frac {4 c \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {b+2 c x^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right ) \left (2 a i+\frac {b^2 i}{c}-3 b g+6 c e\right )}{2 \left (b^2-4 a c\right )}\right )+\frac {x \left (x^2 (a b h-2 a c f+b c d)-a b f-2 a (c d-a h)+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\)

\(\Big \downarrow \) 1492

\(\displaystyle \frac {\frac {x \left (c x^2 \left (20 a^2 c f+a b^2 f-12 a b (a h+2 c d)+3 b^3 d\right )+8 a^2 b c f+4 a^2 c (a h+7 c d)+a b^3 f-a b^2 (7 a h+25 c d)+3 b^4 d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {3 d b^4+a f b^3-3 a (9 c d-a h) b^2-16 a^2 c f b+c \left (3 d b^3+a f b^2-12 a (2 c d+a h) b+20 a^2 c f\right ) x^2+12 a^2 c (7 c d+a h)}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}}{4 a \left (b^2-4 a c\right )}+\frac {1}{2} \left (\frac {c \left (2 a g-b \left (\frac {a i}{c}+e\right )\right )-x^2 \left (-2 a c i+b^2 i-b c g+2 c^2 e\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\left (\frac {4 c \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {b+2 c x^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right ) \left (2 a i+\frac {b^2 i}{c}-3 b g+6 c e\right )}{2 \left (b^2-4 a c\right )}\right )+\frac {x \left (x^2 (a b h-2 a c f+b c d)-a b f-2 a (c d-a h)+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {3 d b^4+a f b^3-3 a (9 c d-a h) b^2-16 a^2 c f b+c \left (3 d b^3+a f b^2-12 a (2 c d+a h) b+20 a^2 c f\right ) x^2+12 a^2 c (7 c d+a h)}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}+\frac {x \left (c x^2 \left (20 a^2 c f+a b^2 f-12 a b (a h+2 c d)+3 b^3 d\right )+8 a^2 b c f+4 a^2 c (a h+7 c d)+a b^3 f-a b^2 (7 a h+25 c d)+3 b^4 d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 a \left (b^2-4 a c\right )}+\frac {1}{2} \left (\frac {c \left (2 a g-b \left (\frac {a i}{c}+e\right )\right )-x^2 \left (-2 a c i+b^2 i-b c g+2 c^2 e\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\left (\frac {4 c \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {b+2 c x^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right ) \left (2 a i+\frac {b^2 i}{c}-3 b g+6 c e\right )}{2 \left (b^2-4 a c\right )}\right )+\frac {x \left (x^2 (a b h-2 a c f+b c d)-a b f-2 a (c d-a h)+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {\frac {\frac {1}{2} c \left (\frac {-52 a^2 b c f+24 a^2 c (a h+7 c d)+a b^3 f-6 a b^2 (5 c d-3 a h)+3 b^4 d}{\sqrt {b^2-4 a c}}+20 a^2 c f+a b^2 f-12 a b (a h+2 c d)+3 b^3 d\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx+\frac {1}{2} c \left (-\frac {-52 a^2 b c f+24 a^2 c (a h+7 c d)+a b^3 f-6 a b^2 (5 c d-3 a h)+3 b^4 d}{\sqrt {b^2-4 a c}}+20 a^2 c f+a b^2 f-12 a b (a h+2 c d)+3 b^3 d\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx}{2 a \left (b^2-4 a c\right )}+\frac {x \left (c x^2 \left (20 a^2 c f+a b^2 f-12 a b (a h+2 c d)+3 b^3 d\right )+8 a^2 b c f+4 a^2 c (a h+7 c d)+a b^3 f-a b^2 (7 a h+25 c d)+3 b^4 d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 a \left (b^2-4 a c\right )}+\frac {1}{2} \left (\frac {c \left (2 a g-b \left (\frac {a i}{c}+e\right )\right )-x^2 \left (-2 a c i+b^2 i-b c g+2 c^2 e\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\left (\frac {4 c \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {b+2 c x^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right ) \left (2 a i+\frac {b^2 i}{c}-3 b g+6 c e\right )}{2 \left (b^2-4 a c\right )}\right )+\frac {x \left (x^2 (a b h-2 a c f+b c d)-a b f-2 a (c d-a h)+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (\frac {-52 a^2 b c f+24 a^2 c (a h+7 c d)+a b^3 f-6 a b^2 (5 c d-3 a h)+3 b^4 d}{\sqrt {b^2-4 a c}}+20 a^2 c f+a b^2 f-12 a b (a h+2 c d)+3 b^3 d\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (-\frac {-52 a^2 b c f+24 a^2 c (a h+7 c d)+a b^3 f-6 a b^2 (5 c d-3 a h)+3 b^4 d}{\sqrt {b^2-4 a c}}+20 a^2 c f+a b^2 f-12 a b (a h+2 c d)+3 b^3 d\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}}{2 a \left (b^2-4 a c\right )}+\frac {x \left (c x^2 \left (20 a^2 c f+a b^2 f-12 a b (a h+2 c d)+3 b^3 d\right )+8 a^2 b c f+4 a^2 c (a h+7 c d)+a b^3 f-a b^2 (7 a h+25 c d)+3 b^4 d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 a \left (b^2-4 a c\right )}+\frac {1}{2} \left (\frac {c \left (2 a g-b \left (\frac {a i}{c}+e\right )\right )-x^2 \left (-2 a c i+b^2 i-b c g+2 c^2 e\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\left (\frac {4 c \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {b+2 c x^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right ) \left (2 a i+\frac {b^2 i}{c}-3 b g+6 c e\right )}{2 \left (b^2-4 a c\right )}\right )+\frac {x \left (x^2 (a b h-2 a c f+b c d)-a b f-2 a (c d-a h)+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\)

Input:

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

Output:

(x*(b^2*d - a*b*f - 2*a*(c*d - a*h) + (b*c*d - 2*a*c*f + a*b*h)*x^2))/(4*a 
*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + ((x*(3*b^4*d + a*b^3*f + 8*a^2*b*c 
*f + 4*a^2*c*(7*c*d + a*h) - a*b^2*(25*c*d + 7*a*h) + c*(3*b^3*d + a*b^2*f 
 + 20*a^2*c*f - 12*a*b*(2*c*d + a*h))*x^2))/(2*a*(b^2 - 4*a*c)*(a + b*x^2 
+ c*x^4)) + ((Sqrt[c]*(3*b^3*d + a*b^2*f + 20*a^2*c*f - 12*a*b*(2*c*d + a* 
h) + (3*b^4*d + a*b^3*f - 52*a^2*b*c*f - 6*a*b^2*(5*c*d - 3*a*h) + 24*a^2* 
c*(7*c*d + a*h))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sq 
rt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[c]*(3*b^3 
*d + a*b^2*f + 20*a^2*c*f - 12*a*b*(2*c*d + a*h) - (3*b^4*d + a*b^3*f - 52 
*a^2*b*c*f - 6*a*b^2*(5*c*d - 3*a*h) + 24*a^2*c*(7*c*d + a*h))/Sqrt[b^2 - 
4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]* 
Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(2*a*(b^2 - 4*a*c)))/(4*a*(b^2 - 4*a*c)) + ( 
(c*(2*a*g - b*(e + (a*i)/c)) - (2*c^2*e - b*c*g + b^2*i - 2*a*c*i)*x^2)/(2 
*c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - ((6*c*e - 3*b*g + 2*a*i + (b^2*i 
)/c)*(-((b + 2*c*x^2)/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4))) + (4*c*ArcTanh[ 
(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(3/2)))/(2*(b^2 - 4*a*c))) 
/2
 

Defintions of rubi rules used

rule 25

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

rule 27

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

rule 218

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

rule 219

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

rule 1083

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

rule 1086

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*c*((2*p + 
 3)/((p + 1)*(b^2 - 4*a*c)))   Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fre 
eQ[{a, b, c}, x] && ILtQ[p, -1]
 

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 1492

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + 
 c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 
 - 4*a*c))   Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 
7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, 
 b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && 
 LtQ[p, -1] && IntegerQ[2*p]
 

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 = 0.27 (sec) , antiderivative size = 795, normalized size of antiderivative = 1.09

method	result	size
risch	\(\frac {-\frac {c^{2} \left (12 a^{2} b h -20 a^{2} c f -a \,b^{2} f +24 a b c d -3 b^{3} d \right ) x^{7}}{8 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {c \left (2 a c i +b^{2} i -3 b c g +6 e \,c^{2}\right ) x^{6}}{32 a^{2} c^{2}-16 a \,b^{2} c +2 b^{4}}+\frac {c \left (4 a^{3} c h -19 a^{2} b^{2} h +28 a^{2} b c f +28 a^{2} c^{2} d +2 a \,b^{3} f -49 a \,b^{2} c d +6 b^{4} d \right ) x^{5}}{8 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {3 b \left (2 a c i +b^{2} i -3 b c g +6 e \,c^{2}\right ) x^{4}}{4 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {\left (16 a^{3} b c h -36 a^{3} c^{2} f +5 a^{2} b^{3} h -5 a^{2} b^{2} c f +4 a^{2} c^{2} b d -a \,b^{4} f +20 a \,b^{3} c d -3 b^{5} d \right ) x^{3}}{8 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {\left (2 a^{2} c i -5 a \,b^{2} i +5 a b c g -10 a \,c^{2} e +b^{3} g -2 b^{2} c e \right ) x^{2}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {\left (12 a^{3} c h +3 a^{2} b^{2} h -16 a^{2} b c f -44 a^{2} c^{2} d +a \,b^{3} f +37 a \,b^{2} c d -5 b^{4} d \right ) x}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {6 a^{2} b i -8 a^{2} c g -a \,b^{2} g +10 a b c e -b^{3} e}{64 a^{2} c^{2}-32 a \,b^{2} c +4 b^{4}}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (-\frac {c \left (12 a^{2} b h -20 a^{2} c f -a \,b^{2} f +24 a b c d -3 b^{3} d \right ) \textit {\_R}^{2}}{a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {8 \left (2 a c i +b^{2} i -3 b c g +6 e \,c^{2}\right ) \textit {\_R}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {12 a^{3} c h +3 a^{2} b^{2} h -16 a^{2} b c f +84 a^{2} c^{2} d +a \,b^{3} f -27 a \,b^{2} c d +3 b^{4} d}{a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3} c +\textit {\_R} b}\right )}{16}\)	\(795\)
default	\(\text {Expression too large to display}\)	\(1295\)

Input:

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

Output:

(-1/8*c^2*(12*a^2*b*h-20*a^2*c*f-a*b^2*f+24*a*b*c*d-3*b^3*d)/a^2/(16*a^2*c 
^2-8*a*b^2*c+b^4)*x^7+1/2*c*(2*a*c*i+b^2*i-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*c*(4*a^3*c*h-19*a^2*b^2*h+28*a^2*b*c*f+28*a^2*c^2 
*d+2*a*b^3*f-49*a*b^2*c*d+6*b^4*d)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^5+3/4*b*(2 
*a*c*i+b^2*i-3*b*c*g+6*c^2*e)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4-1/8*(16*a^3*b 
*c*h-36*a^3*c^2*f+5*a^2*b^3*h-5*a^2*b^2*c*f+4*a^2*b*c^2*d-a*b^4*f+20*a*b^3 
*c*d-3*b^5*d)/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3-1/2*(2*a^2*c*i-5*a*b^2*i+ 
5*a*b*c*g-10*a*c^2*e+b^3*g-2*b^2*c*e)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2-1/8*( 
12*a^3*c*h+3*a^2*b^2*h-16*a^2*b*c*f-44*a^2*c^2*d+a*b^3*f+37*a*b^2*c*d-5*b^ 
4*d)/(16*a^2*c^2-8*a*b^2*c+b^4)/a*x+1/4*(6*a^2*b*i-8*a^2*c*g-a*b^2*g+10*a* 
b*c*e-b^3*e)/(16*a^2*c^2-8*a*b^2*c+b^4))/(c*x^4+b*x^2+a)^2+1/16*sum((-c*(1 
2*a^2*b*h-20*a^2*c*f-a*b^2*f+24*a*b*c*d-3*b^3*d)/a^2/(16*a^2*c^2-8*a*b^2*c 
+b^4)*_R^2+8*(2*a*c*i+b^2*i-3*b*c*g+6*c^2*e)/(16*a^2*c^2-8*a*b^2*c+b^4)*_R 
+(12*a^3*c*h+3*a^2*b^2*h-16*a^2*b*c*f+84*a^2*c^2*d+a*b^3*f-27*a*b^2*c*d+3* 
b^4*d)/a^2/(16*a^2*c^2-8*a*b^2*c+b^4))/(2*_R^3*c+_R*b)*ln(x-_R),_R=RootOf( 
_Z^4*c+_Z^2*b+a))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-1/8*((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)*x^7 - 4*(6*a^2*c^3*e - 3*a^2*b*c^2*g + (a^2*b^2*c + 2*a^3*c^2)*i)*x^ 
6 - 12*a^4*b*i - ((6*b^4*c - 49*a*b^2*c^2 + 28*a^2*c^3)*d + 2*(a*b^3*c + 1 
4*a^2*b*c^2)*f - (19*a^2*b^2*c - 4*a^3*c^2)*h)*x^5 - 6*(6*a^2*b*c^2*e - 3* 
a^2*b^2*c*g + (a^2*b^3 + 2*a^3*b*c)*i)*x^4 - ((3*b^5 - 20*a*b^3*c - 4*a^2* 
b*c^2)*d + (a*b^4 + 5*a^2*b^2*c + 36*a^3*c^2)*f - (5*a^2*b^3 + 16*a^3*b*c) 
*h)*x^3 - 4*(2*(a^2*b^2*c + 5*a^3*c^2)*e - (a^2*b^3 + 5*a^3*b*c)*g + (5*a^ 
3*b^2 - 2*a^4*c)*i)*x^2 + 2*(a^2*b^3 - 10*a^3*b*c)*e + 2*(a^3*b^2 + 8*a^4* 
c)*g - ((5*a*b^4 - 37*a^2*b^2*c + 44*a^3*c^2)*d - (a^2*b^3 - 16*a^3*b*c)*f 
 - 3*(a^3*b^2 + 4*a^4*c)*h)*x)/((a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4) 
*x^8 + a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2 + 2*(a^2*b^5*c - 8*a^3*b^3*c^2 + 
 16*a^4*b*c^3)*x^6 + (a^2*b^6 - 6*a^3*b^4*c + 32*a^5*c^3)*x^4 + 2*(a^3*b^5 
 - 8*a^4*b^3*c + 16*a^5*b*c^2)*x^2) - 1/8*integrate(((12*a^2*b*c*h - 3*(b^ 
3*c - 8*a*b*c^2)*d - (a*b^2*c + 20*a^2*c^2)*f)*x^2 - 3*(b^4 - 9*a*b^2*c + 
28*a^2*c^2)*d - (a*b^3 - 16*a^2*b*c)*f - 3*(a^2*b^2 + 4*a^3*c)*h - 8*(6*a^ 
2*c^2*e - 3*a^2*b*c*g + (a^2*b^2 + 2*a^3*c)*i)*x)/(c*x^4 + b*x^2 + a), x)/ 
(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 7058 vs. \(2 (680) = 1360\).

Time = 2.62 (sec) , antiderivative size = 7058, normalized size of antiderivative = 9.64 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

1/32*(3*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^8 - 17*sqrt(2)*sqrt(b*c 
 + sqrt(b^2 - 4*a*c)*c)*a*b^6*c - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c 
)*b^7*c - 2*b^8*c + 116*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^ 
2 + 26*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5*c^2 + sqrt(2)*sqrt(b* 
c + sqrt(b^2 - 4*a*c)*c)*b^6*c^2 + 34*a*b^6*c^2 - 2*b^7*c^2 - 368*sqrt(2)* 
sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^3 - 128*sqrt(2)*sqrt(b*c + sqrt( 
b^2 - 4*a*c)*c)*a^2*b^3*c^3 - 13*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a 
*b^4*c^3 - 232*a^2*b^4*c^3 + 30*a*b^5*c^3 + 448*sqrt(2)*sqrt(b*c + sqrt(b^ 
2 - 4*a*c)*c)*a^4*c^4 + 224*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b* 
c^4 + 64*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^4 + 736*a^3*b^2 
*c^4 - 176*a^2*b^3*c^4 - 112*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c 
^5 - 896*a^4*c^5 + 352*a^3*b*c^5 + sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sq 
rt(b^2 - 4*a*c)*c)*b^7 - 15*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 
- 4*a*c)*c)*a*b^5*c - 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4* 
a*c)*c)*b^6*c + 88*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)* 
c)*a^2*b^3*c^2 + 22*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c) 
*c)*a*b^4*c^2 + sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)* 
b^5*c^2 - 176*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^ 
3*b*c^3 - 88*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2 
*b^2*c^3 - 11*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)...
 

Mupad [B] (verification not implemented)

Time = 25.90 (sec) , antiderivative size = 36653, normalized size of antiderivative = 50.07 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

((x^5*(28*a^2*c^3*d + 4*a^3*c^2*h + 6*b^4*c*d + 2*a*b^3*c*f - 49*a*b^2*c^2 
*d + 28*a^2*b*c^2*f - 19*a^2*b^2*c*h))/(8*a^2*(b^4 + 16*a^2*c^2 - 8*a*b^2* 
c)) - (x^2*(b^3*g - 10*a*c^2*e - 2*b^2*c*e - 5*a*b^2*i + 2*a^2*c*i + 5*a*b 
*c*g))/(2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) - (b^3*e + a*b^2*g + 8*a^2*c*g - 
 6*a^2*b*i - 10*a*b*c*e)/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (3*b*x^4*(6* 
c^2*e + b^2*i - 3*b*c*g + 2*a*c*i))/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + ( 
c*x^6*(6*c^2*e + b^2*i - 3*b*c*g + 2*a*c*i))/(2*(b^4 + 16*a^2*c^2 - 8*a*b^ 
2*c)) + (x^3*(3*b^5*d + 36*a^3*c^2*f - 5*a^2*b^3*h + a*b^4*f - 20*a*b^3*c* 
d - 16*a^3*b*c*h - 4*a^2*b*c^2*d + 5*a^2*b^2*c*f))/(8*a^2*(b^4 + 16*a^2*c^ 
2 - 8*a*b^2*c)) - (x*(3*a^2*b^2*h - 44*a^2*c^2*d - 5*b^4*d + a*b^3*f + 12* 
a^3*c*h + 37*a*b^2*c*d - 16*a^2*b*c*f))/(8*a*(b^4 + 16*a^2*c^2 - 8*a*b^2*c 
)) + (c*x^7*(20*a^2*c^2*f + 3*b^3*c*d - 24*a*b*c^2*d + a*b^2*c*f - 12*a^2* 
b*c*h))/(8*a^2*(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^6*d^3 - 8000 
*a^5*c^7*f^3 - 567*b^7*c^5*d^3 + 169344*a^3*b*c^8*d^3 + 193536*a^4*c^8*d*e 
^2 - 141120*a^4*c^8*d^2*f + 1728*a^6*b*c^5*h^3 + 315*b^8*c^4*d^2*f + 27648 
*a^5*c^7*e^2*h + 21504*a^6*c^6*d*i^2 - 135*b^9*c^3*d^2*h - 2880*a^6*c^6*f* 
h^2 + 3072*a^7*c^5*h*i^2 - 67824*a^2*b^3*c^7*d^3 + 35*a^2*b^6*c^4*f^3 + 84 
*a^3*b^4*c^5*f^3 - 12720*a^4*b^2*c^6*f^3 + 540*a^4*b^5*c^3*h^3 + 4320*a^5* 
b^3*c^4*h^3 + 129024*a^5*c^7*d*e*i - 40320*a^5*c^7*d*f*h + 18432*a^6*c^...
 

Reduce [B] (verification not implemented)

Time = 14.41 (sec) , antiderivative size = 20724, normalized size of antiderivative = 28.31 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (a+b x^2+c x^4\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 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*i - 32*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqr 
t(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**5* 
b**3*i + 96*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**2*c*g - 128*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**2*c*i*x**2 - 192*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sq 
rt(a) - b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c) 
*sqrt(a) + b))*a**5*b*c**2*e - 128*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*s 
qrt(c)*sqrt(a) + b))*a**5*b*c**2*i*x**4 - 64*sqrt(2*sqrt(c)*sqrt(a) + b)*s 
qrt(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**4*i*x**2 + 192*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**3*c*g*x**2 - 128*sqrt(2*sq 
rt(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**3*c*i*x**4 - 38 
4*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*...