\(\int \frac {x^8 (A+B x^4)}{(a+b x^4) (c+d x^4)} \, dx\) [11]

Optimal result

Integrand size = 29, antiderivative size = 412 \[ \int \frac {x^8 \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=-\frac {(b B c-A b d+a B d) x}{b^2 d^2}+\frac {B x^5}{5 b d}-\frac {a^{5/4} (A b-a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} b^{9/4} (b c-a d)}+\frac {a^{5/4} (A b-a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} b^{9/4} (b c-a d)}-\frac {c^{5/4} (B c-A d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt {2} d^{9/4} (b c-a d)}+\frac {c^{5/4} (B c-A d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt {2} d^{9/4} (b c-a d)}+\frac {a^{5/4} (A b-a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a}+\sqrt {b} x^2}\right )}{2 \sqrt {2} b^{9/4} (b c-a d)}+\frac {c^{5/4} (B c-A d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x}{\sqrt {c}+\sqrt {d} x^2}\right )}{2 \sqrt {2} d^{9/4} (b c-a d)} \] Output:

-(-A*b*d+B*a*d+B*b*c)*x/b^2/d^2+1/5*B*x^5/b/d+1/4*a^(5/4)*(A*b-B*a)*arctan 
(-1+2^(1/2)*b^(1/4)*x/a^(1/4))*2^(1/2)/b^(9/4)/(-a*d+b*c)+1/4*a^(5/4)*(A*b 
-B*a)*arctan(1+2^(1/2)*b^(1/4)*x/a^(1/4))*2^(1/2)/b^(9/4)/(-a*d+b*c)+1/4*c 
^(5/4)*(-A*d+B*c)*arctan(-1+2^(1/2)*d^(1/4)*x/c^(1/4))*2^(1/2)/d^(9/4)/(-a 
*d+b*c)+1/4*c^(5/4)*(-A*d+B*c)*arctan(1+2^(1/2)*d^(1/4)*x/c^(1/4))*2^(1/2) 
/d^(9/4)/(-a*d+b*c)+1/4*a^(5/4)*(A*b-B*a)*arctanh(2^(1/2)*a^(1/4)*b^(1/4)* 
x/(a^(1/2)+b^(1/2)*x^2))*2^(1/2)/b^(9/4)/(-a*d+b*c)+1/4*c^(5/4)*(-A*d+B*c) 
*arctanh(2^(1/2)*c^(1/4)*d^(1/4)*x/(c^(1/2)+d^(1/2)*x^2))*2^(1/2)/d^(9/4)/ 
(-a*d+b*c)
 

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.21 \[ \int \frac {x^8 \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {-40 \sqrt [4]{b} \sqrt [4]{d} (b c-a d) (b B c-A b d+a B d) x+8 b^{5/4} B d^{5/4} (b c-a d) x^5+10 \sqrt {2} a^{5/4} (-A b+a B) d^{9/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+10 \sqrt {2} a^{5/4} (A b-a B) d^{9/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )-10 \sqrt {2} b^{9/4} c^{5/4} (B c-A d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )+10 \sqrt {2} b^{9/4} c^{5/4} (B c-A d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )+5 \sqrt {2} a^{5/4} (-A b+a B) d^{9/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )+5 \sqrt {2} a^{5/4} (A b-a B) d^{9/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )-5 \sqrt {2} b^{9/4} c^{5/4} (B c-A d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )+5 \sqrt {2} b^{9/4} c^{5/4} (B c-A d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{40 b^{9/4} d^{9/4} (b c-a d)} \] Input:

Integrate[(x^8*(A + B*x^4))/((a + b*x^4)*(c + d*x^4)),x]
 

Output:

(-40*b^(1/4)*d^(1/4)*(b*c - a*d)*(b*B*c - A*b*d + a*B*d)*x + 8*b^(5/4)*B*d 
^(5/4)*(b*c - a*d)*x^5 + 10*Sqrt[2]*a^(5/4)*(-(A*b) + a*B)*d^(9/4)*ArcTan[ 
1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 10*Sqrt[2]*a^(5/4)*(A*b - a*B)*d^(9/4)* 
ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] - 10*Sqrt[2]*b^(9/4)*c^(5/4)*(B*c 
- A*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)] + 10*Sqrt[2]*b^(9/4)*c^(5/4 
)*(B*c - A*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)] + 5*Sqrt[2]*a^(5/4)* 
(-(A*b) + a*B)*d^(9/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x 
^2] + 5*Sqrt[2]*a^(5/4)*(A*b - a*B)*d^(9/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)* 
b^(1/4)*x + Sqrt[b]*x^2] - 5*Sqrt[2]*b^(9/4)*c^(5/4)*(B*c - A*d)*Log[Sqrt[ 
c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2] + 5*Sqrt[2]*b^(9/4)*c^(5/4)* 
(B*c - A*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(40*b^ 
(9/4)*d^(9/4)*(b*c - a*d))
 

Rubi [A] (verified)

Time = 1.47 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.23, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {1052, 27, 1052, 1020, 755, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

\(\displaystyle \int \frac {x^8 \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx\)

\(\Big \downarrow \) 1052

\(\displaystyle \frac {B x^5}{5 b d}-\frac {\int \frac {5 x^4 \left ((b B c-A b d+a B d) x^4+a B c\right )}{\left (b x^4+a\right ) \left (d x^4+c\right )}dx}{5 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {B x^5}{5 b d}-\frac {\int \frac {x^4 \left ((b B c-A b d+a B d) x^4+a B c\right )}{\left (b x^4+a\right ) \left (d x^4+c\right )}dx}{b d}\)

\(\Big \downarrow \) 1052

\(\displaystyle \frac {B x^5}{5 b d}-\frac {\frac {x (a B d-A b d+b B c)}{b d}-\frac {\int \frac {\left (c (B c-A d) b^2+a d (B c-A d) b+a^2 B d^2\right ) x^4+a c (b B c-A b d+a B d)}{\left (b x^4+a\right ) \left (d x^4+c\right )}dx}{b d}}{b d}\)

\(\Big \downarrow \) 1020

\(\displaystyle \frac {B x^5}{5 b d}-\frac {\frac {x (a B d-A b d+b B c)}{b d}-\frac {\frac {a^2 d^2 (A b-a B) \int \frac {1}{b x^4+a}dx}{b c-a d}+\frac {b^2 c^2 (B c-A d) \int \frac {1}{d x^4+c}dx}{b c-a d}}{b d}}{b d}\)

\(\Big \downarrow \) 755

\(\displaystyle \frac {B x^5}{5 b d}-\frac {\frac {x (a B d-A b d+b B c)}{b d}-\frac {\frac {a^2 d^2 (A b-a B) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{b x^4+a}dx}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {b} x^2+\sqrt {a}}{b x^4+a}dx}{2 \sqrt {a}}\right )}{b c-a d}+\frac {b^2 c^2 (B c-A d) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x^2}{d x^4+c}dx}{2 \sqrt {c}}+\frac {\int \frac {\sqrt {d} x^2+\sqrt {c}}{d x^4+c}dx}{2 \sqrt {c}}\right )}{b c-a d}}{b d}}{b d}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {B x^5}{5 b d}-\frac {\frac {x (a B d-A b d+b B c)}{b d}-\frac {\frac {a^2 d^2 (A b-a B) \left (\frac {\frac {\int \frac {1}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}dx}{2 \sqrt {b}}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}dx}{2 \sqrt {b}}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{b x^4+a}dx}{2 \sqrt {a}}\right )}{b c-a d}+\frac {b^2 c^2 (B c-A d) \left (\frac {\frac {\int \frac {1}{x^2-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}dx}{2 \sqrt {d}}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}dx}{2 \sqrt {d}}}{2 \sqrt {c}}+\frac {\int \frac {\sqrt {c}-\sqrt {d} x^2}{d x^4+c}dx}{2 \sqrt {c}}\right )}{b c-a d}}{b d}}{b d}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {B x^5}{5 b d}-\frac {\frac {x (a B d-A b d+b B c)}{b d}-\frac {\frac {a^2 d^2 (A b-a B) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{b x^4+a}dx}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}+\frac {b^2 c^2 (B c-A d) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x^2}{d x^4+c}dx}{2 \sqrt {c}}+\frac {\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}}{b d}}{b d}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {B x^5}{5 b d}-\frac {\frac {x (a B d-A b d+b B c)}{b d}-\frac {\frac {a^2 d^2 (A b-a B) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{b x^4+a}dx}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}+\frac {b^2 c^2 (B c-A d) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x^2}{d x^4+c}dx}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}}{b d}}{b d}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {B x^5}{5 b d}-\frac {\frac {x (a B d-A b d+b B c)}{b d}-\frac {\frac {a^2 d^2 (A b-a B) \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} x}{\sqrt [4]{b} \left (x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} x+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}+\frac {b^2 c^2 (B c-A d) \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} x}{\sqrt [4]{d} \left (x^2-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}dx}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} x+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x^2+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}dx}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}}{b d}}{b d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {B x^5}{5 b d}-\frac {\frac {x (a B d-A b d+b B c)}{b d}-\frac {\frac {a^2 d^2 (A b-a B) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} x}{\sqrt [4]{b} \left (x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} x+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}+\frac {b^2 c^2 (B c-A d) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} x}{\sqrt [4]{d} \left (x^2-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}dx}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} x+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x^2+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}dx}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}}{b d}}{b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {B x^5}{5 b d}-\frac {\frac {x (a B d-A b d+b B c)}{b d}-\frac {\frac {a^2 d^2 (A b-a B) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} x}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} x+\sqrt [4]{a}}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}dx}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}+\frac {b^2 c^2 (B c-A d) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} x}{x^2-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}dx}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{d} x+\sqrt [4]{c}}{x^2+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}dx}{2 \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}}{b d}}{b d}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {B x^5}{5 b d}-\frac {\frac {x (a B d-A b d+b B c)}{b d}-\frac {\frac {a^2 d^2 (A b-a B) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}+\frac {b^2 c^2 (B c-A d) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}}{b d}}{b d}\)

Input:

Int[(x^8*(A + B*x^4))/((a + b*x^4)*(c + d*x^4)),x]
 

Output:

(B*x^5)/(5*b*d) - (((b*B*c - A*b*d + a*B*d)*x)/(b*d) - ((a^2*(A*b - a*B)*d 
^2*((-(ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4))) 
+ ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sq 
rt[a]) + (-1/2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2]/(Sqr 
t[2]*a^(1/4)*b^(1/4)) + Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]* 
x^2]/(2*Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[a])))/(b*c - a*d) + (b^2*c^2*(B* 
c - A*d)*((-(ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)]/(Sqrt[2]*c^(1/4)*d^(1 
/4))) + ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)]/(Sqrt[2]*c^(1/4)*d^(1/4))) 
/(2*Sqrt[c]) + (-1/2*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2 
]/(Sqrt[2]*c^(1/4)*d^(1/4)) + Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sq 
rt[d]*x^2]/(2*Sqrt[2]*c^(1/4)*d^(1/4)))/(2*Sqrt[c])))/(b*c - a*d))/(b*d))/ 
(b*d)
 

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 217

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

rule 755

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] 
], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r)   Int[(r - s*x^2)/(a + b*x^4) 
, x], x] + Simp[1/(2*r)   Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, 
 b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & 
& AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 1020

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^( 
n_))), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(a + b*x^n), x], x 
] - Simp[(d*e - c*f)/(b*c - a*d)   Int[1/(c + d*x^n), x], x] /; FreeQ[{a, b 
, c, d, e, f, n}, x]
 

rule 1052

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[f*g^(n - 1)*(g*x)^(m 
- n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*d*(m + n*(p + q + 1) + 
 1))), x] - Simp[g^n/(b*d*(m + n*(p + q + 1) + 1))   Int[(g*x)^(m - n)*(a + 
 b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m - n + 1) + (a*f*d*(m + n*q + 1) + b*( 
f*c*(m + n*p + 1) - e*d*(m + n*(p + q + 1) + 1)))*x^n, x], x], x] /; FreeQ[ 
{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && GtQ[m, n - 1]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{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 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.67

method	result	size
default	\(\frac {\frac {1}{5} B \,x^{5} b d +A b d x -B a d x -B b c x}{b^{2} d^{2}}-\frac {a \left (A b -B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b^{2} \left (a d -c b \right )}+\frac {c \left (A d -B c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {c}{d}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {c}{d}}}{x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{8 d^{2} \left (a d -c b \right )}\)	\(275\)
risch	\(\text {Expression too large to display}\)	\(3265\)

Input:

int(x^8*(B*x^4+A)/(b*x^4+a)/(d*x^4+c),x,method=_RETURNVERBOSE)
 

Output:

1/b^2/d^2*(1/5*B*x^5*b*d+A*b*d*x-B*a*d*x-B*b*c*x)-1/8/b^2*a*(A*b-B*a)/(a*d 
-b*c)*(a/b)^(1/4)*2^(1/2)*(ln((x^2+(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2))/(x^2 
-(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+2*a 
rctan(2^(1/2)/(a/b)^(1/4)*x-1))+1/8/d^2*c*(A*d-B*c)/(a*d-b*c)*(c/d)^(1/4)* 
2^(1/2)*(ln((x^2+(c/d)^(1/4)*x*2^(1/2)+(c/d)^(1/2))/(x^2-(c/d)^(1/4)*x*2^( 
1/2)+(c/d)^(1/2)))+2*arctan(2^(1/2)/(c/d)^(1/4)*x+1)+2*arctan(2^(1/2)/(c/d 
)^(1/4)*x-1))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {x^8 \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\text {Timed out} \] Input:

integrate(x^8*(B*x^4+A)/(b*x^4+a)/(d*x^4+c),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^8 \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\text {Timed out} \] Input:

integrate(x**8*(B*x**4+A)/(b*x**4+a)/(d*x**4+c),x)
 

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.15 \[ \int \frac {x^8 \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=-\frac {{\left (\frac {2 \, \sqrt {2} {\left (B a - A b\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (B a - A b\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (B a - A b\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (B a - A b\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}\right )} a^{2}}{8 \, {\left (b^{3} c - a b^{2} d\right )}} + \frac {{\left (\frac {2 \, \sqrt {2} {\left (B c - A d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {d} x + \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (B c - A d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {d} x - \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (B c - A d\right )} \log \left (\sqrt {d} x^{2} + \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (B c - A d\right )} \log \left (\sqrt {d} x^{2} - \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}\right )} c^{2}}{8 \, {\left (b c d^{2} - a d^{3}\right )}} + \frac {B b d x^{5} - 5 \, {\left (B b c + {\left (B a - A b\right )} d\right )} x}{5 \, b^{2} d^{2}} \] Input:

integrate(x^8*(B*x^4+A)/(b*x^4+a)/(d*x^4+c),x, algorithm="maxima")
 

Output:

-1/8*(2*sqrt(2)*(B*a - A*b)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x + sqrt(2)*a^(1 
/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sq 
rt(2)*(B*a - A*b)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x - sqrt(2)*a^(1/4)*b^(1/4 
))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*(B*a - 
 A*b)*log(sqrt(b)*x^2 + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(1 
/4)) - sqrt(2)*(B*a - A*b)*log(sqrt(b)*x^2 - sqrt(2)*a^(1/4)*b^(1/4)*x + s 
qrt(a))/(a^(3/4)*b^(1/4)))*a^2/(b^3*c - a*b^2*d) + 1/8*(2*sqrt(2)*(B*c - A 
*d)*arctan(1/2*sqrt(2)*(2*sqrt(d)*x + sqrt(2)*c^(1/4)*d^(1/4))/sqrt(sqrt(c 
)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + 2*sqrt(2)*(B*c - A*d)*arctan 
(1/2*sqrt(2)*(2*sqrt(d)*x - sqrt(2)*c^(1/4)*d^(1/4))/sqrt(sqrt(c)*sqrt(d)) 
)/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + sqrt(2)*(B*c - A*d)*log(sqrt(d)*x^2 + 
sqrt(2)*c^(1/4)*d^(1/4)*x + sqrt(c))/(c^(3/4)*d^(1/4)) - sqrt(2)*(B*c - A* 
d)*log(sqrt(d)*x^2 - sqrt(2)*c^(1/4)*d^(1/4)*x + sqrt(c))/(c^(3/4)*d^(1/4) 
))*c^2/(b*c*d^2 - a*d^3) + 1/5*(B*b*d*x^5 - 5*(B*b*c + (B*a - A*b)*d)*x)/( 
b^2*d^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 649 vs. \(2 (316) = 632\).

Time = 0.13 (sec) , antiderivative size = 649, normalized size of antiderivative = 1.58 \[ \int \frac {x^8 \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx =\text {Too large to display} \] Input:

integrate(x^8*(B*x^4+A)/(b*x^4+a)/(d*x^4+c),x, algorithm="giac")
 

Output:

-1/2*((a*b^3)^(1/4)*B*a^2 - (a*b^3)^(1/4)*A*a*b)*arctan(1/2*sqrt(2)*(2*x + 
 sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(sqrt(2)*b^4*c - sqrt(2)*a*b^3*d) - 1/2 
*((a*b^3)^(1/4)*B*a^2 - (a*b^3)^(1/4)*A*a*b)*arctan(1/2*sqrt(2)*(2*x - sqr 
t(2)*(a/b)^(1/4))/(a/b)^(1/4))/(sqrt(2)*b^4*c - sqrt(2)*a*b^3*d) + 1/2*((c 
*d^3)^(1/4)*B*c^2 - (c*d^3)^(1/4)*A*c*d)*arctan(1/2*sqrt(2)*(2*x + sqrt(2) 
*(c/d)^(1/4))/(c/d)^(1/4))/(sqrt(2)*b*c*d^3 - sqrt(2)*a*d^4) + 1/2*((c*d^3 
)^(1/4)*B*c^2 - (c*d^3)^(1/4)*A*c*d)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(c/ 
d)^(1/4))/(c/d)^(1/4))/(sqrt(2)*b*c*d^3 - sqrt(2)*a*d^4) - 1/4*((a*b^3)^(1 
/4)*B*a^2 - (a*b^3)^(1/4)*A*a*b)*log(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/ 
b))/(sqrt(2)*b^4*c - sqrt(2)*a*b^3*d) + 1/4*((a*b^3)^(1/4)*B*a^2 - (a*b^3) 
^(1/4)*A*a*b)*log(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(sqrt(2)*b^4*c 
- sqrt(2)*a*b^3*d) + 1/4*((c*d^3)^(1/4)*B*c^2 - (c*d^3)^(1/4)*A*c*d)*log(x 
^2 + sqrt(2)*x*(c/d)^(1/4) + sqrt(c/d))/(sqrt(2)*b*c*d^3 - sqrt(2)*a*d^4) 
- 1/4*((c*d^3)^(1/4)*B*c^2 - (c*d^3)^(1/4)*A*c*d)*log(x^2 - sqrt(2)*x*(c/d 
)^(1/4) + sqrt(c/d))/(sqrt(2)*b*c*d^3 - sqrt(2)*a*d^4) + 1/5*(B*b^4*d^4*x^ 
5 - 5*B*b^4*c*d^3*x - 5*B*a*b^3*d^4*x + 5*A*b^4*d^4*x)/(b^5*d^5)
 

Mupad [B] (verification not implemented)

Time = 14.10 (sec) , antiderivative size = 33507, normalized size of antiderivative = 81.33 \[ \int \frac {x^8 \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\text {Too large to display} \] Input:

int((x^8*(A + B*x^4))/((a + b*x^4)*(c + d*x^4)),x)
 

Output:

x*(A/(b*d) - (B*(a*d + b*c))/(b^2*d^2)) + (atan((((((((-(B^4*a^9 + A^4*a^5 
*b^4 + 6*A^2*B^2*a^7*b^2 - 4*A*B^3*a^8*b - 4*A^3*B*a^6*b^3)/(b^13*c^4 + a^ 
4*b^9*d^4 - 4*a^3*b^10*c*d^3 + 6*a^2*b^11*c^2*d^2 - 4*a*b^12*c^3*d))^(3/4) 
*((4*x*(256*A^2*a^3*b^13*c^8*d^8 - 768*A^2*a^4*b^12*c^7*d^9 + 512*A^2*a^5* 
b^11*c^6*d^10 + 512*A^2*a^6*b^10*c^5*d^11 - 768*A^2*a^7*b^9*c^4*d^12 + 256 
*A^2*a^8*b^8*c^3*d^13 + 256*B^2*a^3*b^13*c^10*d^6 - 1024*B^2*a^4*b^12*c^9* 
d^7 + 1536*B^2*a^5*b^11*c^8*d^8 - 768*B^2*a^6*b^10*c^7*d^9 - 768*B^2*a^7*b 
^9*c^6*d^10 + 1536*B^2*a^8*b^8*c^5*d^11 - 1024*B^2*a^9*b^7*c^4*d^12 + 256* 
B^2*a^10*b^6*c^3*d^13 - 512*A*B*a^3*b^13*c^9*d^7 + 2048*A*B*a^4*b^12*c^8*d 
^8 - 3584*A*B*a^5*b^11*c^7*d^9 + 4096*A*B*a^6*b^10*c^6*d^10 - 3584*A*B*a^7 
*b^9*c^5*d^11 + 2048*A*B*a^8*b^8*c^4*d^12 - 512*A*B*a^9*b^7*c^3*d^13))/(b^ 
5*d^5) - ((-(B^4*a^9 + A^4*a^5*b^4 + 6*A^2*B^2*a^7*b^2 - 4*A*B^3*a^8*b - 4 
*A^3*B*a^6*b^3)/(b^13*c^4 + a^4*b^9*d^4 - 4*a^3*b^10*c*d^3 + 6*a^2*b^11*c^ 
2*d^2 - 4*a*b^12*c^3*d))^(1/4)*(256*B*a^3*b^14*c^9*d^8 - 1536*B*a^4*b^13*c 
^8*d^9 + 3840*B*a^5*b^12*c^7*d^10 - 5120*B*a^6*b^11*c^6*d^11 + 3840*B*a^7* 
b^10*c^5*d^12 - 1536*B*a^8*b^9*c^4*d^13 + 256*B*a^9*b^8*c^3*d^14)*4i)/(b^5 
*d^5))*1i)/64 - (16*(B^5*a^4*b^9*c^13 + B^5*a^13*c^4*d^9 - A^5*a^3*b^10*c^ 
9*d^4 + A^5*a^4*b^9*c^8*d^5 + A^5*a^8*b^5*c^4*d^9 - A^5*a^9*b^4*c^3*d^10 - 
 A*B^4*a^3*b^10*c^13 - A*B^4*a^13*c^3*d^10 - B^5*a^5*b^8*c^12*d - B^5*a^12 
*b*c^5*d^8 + 4*A*B^4*a^5*b^8*c^11*d^2 + 4*A*B^4*a^11*b^2*c^5*d^8 + 4*A^...
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.37 \[ \int \frac {x^8 \left (A+B x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {-10 d^{\frac {3}{4}} c^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}-2 \sqrt {d}\, x}{d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}}\right )+10 d^{\frac {3}{4}} c^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}+2 \sqrt {d}\, x}{d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}}\right )-5 d^{\frac {3}{4}} c^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (-d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {c}+\sqrt {d}\, x^{2}\right )+5 d^{\frac {3}{4}} c^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {c}+\sqrt {d}\, x^{2}\right )-40 c d x +8 d^{2} x^{5}}{40 d^{3}} \] Input:

int(x^8*(B*x^4+A)/(b*x^4+a)/(d*x^4+c),x)
 

Output:

( - 10*d**(3/4)*c**(1/4)*sqrt(2)*atan((d**(1/4)*c**(1/4)*sqrt(2) - 2*sqrt( 
d)*x)/(d**(1/4)*c**(1/4)*sqrt(2)))*c + 10*d**(3/4)*c**(1/4)*sqrt(2)*atan(( 
d**(1/4)*c**(1/4)*sqrt(2) + 2*sqrt(d)*x)/(d**(1/4)*c**(1/4)*sqrt(2)))*c - 
5*d**(3/4)*c**(1/4)*sqrt(2)*log( - d**(1/4)*c**(1/4)*sqrt(2)*x + sqrt(c) + 
 sqrt(d)*x**2)*c + 5*d**(3/4)*c**(1/4)*sqrt(2)*log(d**(1/4)*c**(1/4)*sqrt( 
2)*x + sqrt(c) + sqrt(d)*x**2)*c - 40*c*d*x + 8*d**2*x**5)/(40*d**3)