\(\int \frac {A+B x^4}{x^4 (a+b x^4) (c+d x^4)} \, dx\) [17]

Optimal result

Integrand size = 29, antiderivative size = 389 \[ \int \frac {A+B x^4}{x^4 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=-\frac {A}{3 a c x^3}+\frac {b^{3/4} (A b-a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{7/4} (b c-a d)}-\frac {b^{3/4} (A b-a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{7/4} (b c-a d)}+\frac {d^{3/4} (B c-A d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt {2} c^{7/4} (b c-a d)}-\frac {d^{3/4} (B c-A d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt {2} c^{7/4} (b c-a d)}-\frac {b^{3/4} (A b-a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a}+\sqrt {b} x^2}\right )}{2 \sqrt {2} a^{7/4} (b c-a d)}-\frac {d^{3/4} (B c-A d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x}{\sqrt {c}+\sqrt {d} x^2}\right )}{2 \sqrt {2} c^{7/4} (b c-a d)} \] Output:

-1/3*A/a/c/x^3-1/4*b^(3/4)*(A*b-B*a)*arctan(-1+2^(1/2)*b^(1/4)*x/a^(1/4))* 
2^(1/2)/a^(7/4)/(-a*d+b*c)-1/4*b^(3/4)*(A*b-B*a)*arctan(1+2^(1/2)*b^(1/4)* 
x/a^(1/4))*2^(1/2)/a^(7/4)/(-a*d+b*c)-1/4*d^(3/4)*(-A*d+B*c)*arctan(-1+2^( 
1/2)*d^(1/4)*x/c^(1/4))*2^(1/2)/c^(7/4)/(-a*d+b*c)-1/4*d^(3/4)*(-A*d+B*c)* 
arctan(1+2^(1/2)*d^(1/4)*x/c^(1/4))*2^(1/2)/c^(7/4)/(-a*d+b*c)-1/4*b^(3/4) 
*(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 
)/a^(7/4)/(-a*d+b*c)-1/4*d^(3/4)*(-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)/c^(7/4)/(-a*d+b*c)
 

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.26 \[ \int \frac {A+B x^4}{x^4 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {8 a^{3/4} A c^{3/4} (-b c+a d)+6 \sqrt {2} b^{3/4} (A b-a B) c^{7/4} x^3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )-6 \sqrt {2} b^{3/4} (A b-a B) c^{7/4} x^3 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )-6 \sqrt {2} a^{7/4} d^{3/4} (-B c+A d) x^3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )+6 \sqrt {2} a^{7/4} d^{3/4} (-B c+A d) x^3 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )+3 \sqrt {2} b^{3/4} (A b-a B) c^{7/4} x^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )-3 \sqrt {2} b^{3/4} (A b-a B) c^{7/4} x^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )+3 \sqrt {2} a^{7/4} d^{3/4} (B c-A d) x^3 \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )+3 \sqrt {2} a^{7/4} d^{3/4} (-B c+A d) x^3 \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{24 a^{7/4} c^{7/4} (b c-a d) x^3} \] Input:

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

Output:

(8*a^(3/4)*A*c^(3/4)*(-(b*c) + a*d) + 6*Sqrt[2]*b^(3/4)*(A*b - a*B)*c^(7/4 
)*x^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] - 6*Sqrt[2]*b^(3/4)*(A*b - a 
*B)*c^(7/4)*x^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] - 6*Sqrt[2]*a^(7/4 
)*d^(3/4)*(-(B*c) + A*d)*x^3*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)] + 6*S 
qrt[2]*a^(7/4)*d^(3/4)*(-(B*c) + A*d)*x^3*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c 
^(1/4)] + 3*Sqrt[2]*b^(3/4)*(A*b - a*B)*c^(7/4)*x^3*Log[Sqrt[a] - Sqrt[2]* 
a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] - 3*Sqrt[2]*b^(3/4)*(A*b - a*B)*c^(7/4)*x 
^3*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + 3*Sqrt[2]*a^(7 
/4)*d^(3/4)*(B*c - A*d)*x^3*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt 
[d]*x^2] + 3*Sqrt[2]*a^(7/4)*d^(3/4)*(-(B*c) + A*d)*x^3*Log[Sqrt[c] + Sqrt 
[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(24*a^(7/4)*c^(7/4)*(b*c - a*d)*x^3)
 

Rubi [A] (verified)

Time = 1.24 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.20, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {1053, 27, 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.

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

\(\Big \downarrow \) 1053

\(\displaystyle -\frac {\int \frac {3 \left (A b d x^4+A b c-a B c+a A d\right )}{\left (b x^4+a\right ) \left (d x^4+c\right )}dx}{3 a c}-\frac {A}{3 a c x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {A b d x^4+A b c-a B c+a A d}{\left (b x^4+a\right ) \left (d x^4+c\right )}dx}{a c}-\frac {A}{3 a c x^3}\)

\(\Big \downarrow \) 1020

\(\displaystyle -\frac {\frac {b c (A b-a B) \int \frac {1}{b x^4+a}dx}{b c-a d}+\frac {a d (B c-A d) \int \frac {1}{d x^4+c}dx}{b c-a d}}{a c}-\frac {A}{3 a c x^3}\)

\(\Big \downarrow \) 755

\(\displaystyle -\frac {\frac {b c (A b-a B) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{b x^4+a}dx}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {b} x^2+\sqrt {a}}{b x^4+a}dx}{2 \sqrt {a}}\right )}{b c-a d}+\frac {a d (B c-A d) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x^2}{d x^4+c}dx}{2 \sqrt {c}}+\frac {\int \frac {\sqrt {d} x^2+\sqrt {c}}{d x^4+c}dx}{2 \sqrt {c}}\right )}{b c-a d}}{a c}-\frac {A}{3 a c x^3}\)

\(\Big \downarrow \) 1476

\(\displaystyle -\frac {\frac {b c (A b-a B) \left (\frac {\frac {\int \frac {1}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}dx}{2 \sqrt {b}}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}dx}{2 \sqrt {b}}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{b x^4+a}dx}{2 \sqrt {a}}\right )}{b c-a d}+\frac {a d (B c-A d) \left (\frac {\frac {\int \frac {1}{x^2-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}dx}{2 \sqrt {d}}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}dx}{2 \sqrt {d}}}{2 \sqrt {c}}+\frac {\int \frac {\sqrt {c}-\sqrt {d} x^2}{d x^4+c}dx}{2 \sqrt {c}}\right )}{b c-a d}}{a c}-\frac {A}{3 a c x^3}\)

\(\Big \downarrow \) 1082

\(\displaystyle -\frac {\frac {b c (A b-a B) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{b x^4+a}dx}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}+\frac {a d (B c-A d) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x^2}{d x^4+c}dx}{2 \sqrt {c}}+\frac {\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}}{a c}-\frac {A}{3 a c x^3}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {\frac {b c (A b-a B) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{b x^4+a}dx}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}+\frac {a d (B c-A d) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x^2}{d x^4+c}dx}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}}{a c}-\frac {A}{3 a c x^3}\)

\(\Big \downarrow \) 1479

\(\displaystyle -\frac {\frac {b c (A b-a B) \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} x}{\sqrt [4]{b} \left (x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} x+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}+\frac {a d (B c-A d) \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} x}{\sqrt [4]{d} \left (x^2-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}dx}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} x+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x^2+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}dx}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}}{a c}-\frac {A}{3 a c x^3}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {b c (A b-a B) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} x}{\sqrt [4]{b} \left (x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} x+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}+\frac {a d (B c-A d) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} x}{\sqrt [4]{d} \left (x^2-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}dx}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} x+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x^2+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}dx}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}}{a c}-\frac {A}{3 a c x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {b c (A b-a B) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} x}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} x+\sqrt [4]{a}}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}dx}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}+\frac {a d (B c-A d) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} x}{x^2-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}dx}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{d} x+\sqrt [4]{c}}{x^2+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}dx}{2 \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}}{a c}-\frac {A}{3 a c x^3}\)

\(\Big \downarrow \) 1103

\(\displaystyle -\frac {\frac {b c (A b-a B) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}+\frac {a d (B c-A d) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}}{a c}-\frac {A}{3 a c x^3}\)

Input:

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

Output:

-1/3*A/(a*c*x^3) - ((b*(A*b - a*B)*c*((-(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*Sqrt[a]) + (-1/2*Log[Sqrt[a] - Sqrt[2]*a^( 
1/4)*b^(1/4)*x + Sqrt[b]*x^2]/(Sqrt[2]*a^(1/4)*b^(1/4)) + Log[Sqrt[a] + Sq 
rt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2]/(2*Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqr 
t[a])))/(b*c - a*d) + (a*d*(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 + Sqrt[d]*x^2]/(2*Sqrt[2]*c^(1/4)*d^(1/4)))/(2*S 
qrt[c])))/(b*c - a*d))/(a*c)
 

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 1053

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_ 
))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b 
*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^n*( 
m + 1))   Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) 
- e*(b*c + a*d)*(m + n + 1) - e*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2 
) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 
0] && LtQ[m, -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 = 0.21 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.65

method	result	size
default	\(\frac {\left (A b -B a \right ) b \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a^{2} \left (a d -c b \right )}-\frac {A}{3 a c \,x^{3}}-\frac {\left (A d -B c \right ) d \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {c}{d}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {c}{d}}}{x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c^{2} \left (a d -c b \right )}\)	\(254\)
risch	\(\text {Expression too large to display}\)	\(4622\)

Input:

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

Output:

1/8/a^2*(A*b-B*a)*b/(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*arctan(2^(1/2)/(a/b)^(1/4)*x-1))-1/3*A/a/c/x^3-1/8/ 
c^2*(A*d-B*c)*d/(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. \[ \int \frac {A+B x^4}{x^4 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.20 \[ \int \frac {A+B x^4}{x^4 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {\frac {2 \, \sqrt {2} {\left (B a b - A b^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (B a b - A b^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (B a b - A b^{2}\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (B a b - A b^{2}\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}}{8 \, {\left (a b c - a^{2} d\right )}} - \frac {\frac {2 \, \sqrt {2} {\left (B c d - A d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {d} x + \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (B c d - A d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {d} x - \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (B c d - A d^{2}\right )} \log \left (\sqrt {d} x^{2} + \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (B c d - A d^{2}\right )} \log \left (\sqrt {d} x^{2} - \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{8 \, {\left (b c^{2} - a c d\right )}} - \frac {A}{3 \, a c x^{3}} \] Input:

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

Output:

1/8*(2*sqrt(2)*(B*a*b - A*b^2)*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 
*sqrt(2)*(B*a*b - A*b^2)*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*b - A*b^2)*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*b - A*b^2)*log(sqrt(b)*x^2 - sqrt(2)*a^(1/4 
)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(1/4)))/(a*b*c - a^2*d) - 1/8*(2*sqrt(2) 
*(B*c*d - A*d^2)*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* 
d - A*d^2)*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*d - A*d^ 
2)*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*d - A*d^2)*log(sqrt(d)*x^2 - sqrt(2)*c^(1/4)*d^(1/4)*x + 
sqrt(c))/(c^(3/4)*d^(1/4)))/(b*c^2 - a*c*d) - 1/3*A/(a*c*x^3)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 585, normalized size of antiderivative = 1.50 \[ \int \frac {A+B x^4}{x^4 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 11.97 (sec) , antiderivative size = 23502, normalized size of antiderivative = 60.42 \[ \int \frac {A+B x^4}{x^4 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

2*atan((A^6*a^2*b^5*d^7*x + A^6*b^7*c^2*d^5*x - 2*A^5*B*b^7*c^3*d^4*x + A^ 
4*B^2*a^4*b^3*d^7*x + A^4*B^2*b^7*c^4*d^3*x + 2*B^6*a^4*b^3*c^4*d^3*x - 2* 
A^5*B*a^3*b^4*d^7*x - (256*A^2*a^2*b^9*c^11*x*(A^4*d^7 + B^4*c^4*d^3 + 6*A 
^2*B^2*c^2*d^5 - 4*A^3*B*c*d^6 - 4*A*B^3*c^3*d^4))/(256*b^4*c^11 + 256*a^4 
*c^7*d^4 - 1024*a^3*b*c^8*d^3 + 1536*a^2*b^2*c^9*d^2 - 1024*a*b^3*c^10*d) 
- (256*B^2*a^4*b^7*c^11*x*(A^4*d^7 + B^4*c^4*d^3 + 6*A^2*B^2*c^2*d^5 - 4*A 
^3*B*c*d^6 - 4*A*B^3*c^3*d^4))/(256*b^4*c^11 + 256*a^4*c^7*d^4 - 1024*a^3* 
b*c^8*d^3 + 1536*a^2*b^2*c^9*d^2 - 1024*a*b^3*c^10*d) - (256*A^2*a^11*c^2* 
d^9*x*(A^4*d^7 + B^4*c^4*d^3 + 6*A^2*B^2*c^2*d^5 - 4*A^3*B*c*d^6 - 4*A*B^3 
*c^3*d^4))/(256*b^4*c^11 + 256*a^4*c^7*d^4 - 1024*a^3*b*c^8*d^3 + 1536*a^2 
*b^2*c^9*d^2 - 1024*a*b^3*c^10*d) - (256*B^2*a^11*c^4*d^7*x*(A^4*d^7 + B^4 
*c^4*d^3 + 6*A^2*B^2*c^2*d^5 - 4*A^3*B*c*d^6 - 4*A*B^3*c^3*d^4))/(256*b^4* 
c^11 + 256*a^4*c^7*d^4 - 1024*a^3*b*c^8*d^3 + 1536*a^2*b^2*c^9*d^2 - 1024* 
a*b^3*c^10*d) + 7*A^2*B^4*a^2*b^5*c^4*d^3*x + 16*A^2*B^4*a^3*b^4*c^3*d^4*x 
 + 7*A^2*B^4*a^4*b^3*c^2*d^5*x - 16*A^3*B^3*a^2*b^5*c^3*d^4*x - 16*A^3*B^3 
*a^3*b^4*c^2*d^5*x + 12*A^4*B^2*a^2*b^5*c^2*d^5*x - 4*A^5*B*a*b^6*c^2*d^5* 
x - 4*A^5*B*a^2*b^5*c*d^6*x - 6*A*B^5*a^3*b^4*c^4*d^3*x - 6*A*B^5*a^4*b^3* 
c^3*d^4*x - 4*A^3*B^3*a*b^6*c^4*d^3*x - 4*A^3*B^3*a^4*b^3*c*d^6*x + 8*A^4* 
B^2*a*b^6*c^3*d^4*x + 8*A^4*B^2*a^3*b^4*c*d^6*x + (512*A*B*a^11*c^3*d^8*x* 
(A^4*d^7 + B^4*c^4*d^3 + 6*A^2*B^2*c^2*d^5 - 4*A^3*B*c*d^6 - 4*A*B^3*c^...
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.41 \[ \int \frac {A+B x^4}{x^4 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {6 d^{\frac {3}{4}} c^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}-2 \sqrt {d}\, x}{d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}}\right ) x^{3}-6 d^{\frac {3}{4}} c^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}+2 \sqrt {d}\, x}{d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}}\right ) x^{3}+3 d^{\frac {3}{4}} c^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (-d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {c}+\sqrt {d}\, x^{2}\right ) x^{3}-3 d^{\frac {3}{4}} c^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {c}+\sqrt {d}\, x^{2}\right ) x^{3}-8 c}{24 c^{2} x^{3}} \] Input:

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

Output:

(6*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)))*x**3 - 6*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)))*x**3 + 
 3*d**(3/4)*c**(1/4)*sqrt(2)*log( - d**(1/4)*c**(1/4)*sqrt(2)*x + sqrt(c) 
+ sqrt(d)*x**2)*x**3 - 3*d**(3/4)*c**(1/4)*sqrt(2)*log(d**(1/4)*c**(1/4)*s 
qrt(2)*x + sqrt(c) + sqrt(d)*x**2)*x**3 - 8*c)/(24*c**2*x**3)