\(\int \frac {1}{a-b (c+d x^2)^4} \, dx\) [158]

Optimal result

Integrand size = 16, antiderivative size = 607 \[ \int \frac {1}{a-b \left (c+d x^2\right )^4} \, dx=\frac {\arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{a}+\sqrt [4]{b} c}}\right )}{4 a^{3/4} \sqrt [8]{b} \sqrt {\sqrt [4]{a}+\sqrt [4]{b} c} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt {-\sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}}-\sqrt {2} \sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}}}\right )}{4 \sqrt {2} \sqrt {a} \sqrt [8]{b} \sqrt {\sqrt {a}+\sqrt {b} c^2} \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}} \sqrt {d}}+\frac {\arctan \left (\frac {\sqrt {-\sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}}+\sqrt {2} \sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}}}\right )}{4 \sqrt {2} \sqrt {a} \sqrt [8]{b} \sqrt {\sqrt {a}+\sqrt {b} c^2} \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}} \sqrt {d}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{a}-\sqrt [4]{b} c}}\right )}{4 a^{3/4} \sqrt [8]{b} \sqrt {\sqrt [4]{a}-\sqrt [4]{b} c} \sqrt {d}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [8]{b} \sqrt {-\sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}} \sqrt {d} x}{\sqrt {\sqrt {a}+\sqrt {b} c^2}+\sqrt [4]{b} d x^2}\right )}{4 \sqrt {2} \sqrt {a} \sqrt [8]{b} \sqrt {\sqrt {a}+\sqrt {b} c^2} \sqrt {-\sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}} \sqrt {d}} \] Output:

1/4*arctan(b^(1/8)*d^(1/2)*x/(a^(1/4)+b^(1/4)*c)^(1/2))/a^(3/4)/b^(1/8)/(a 
^(1/4)+b^(1/4)*c)^(1/2)/d^(1/2)-1/8*arctan(((-b^(1/4)*c+(a^(1/2)+b^(1/2)*c 
^2)^(1/2))^(1/2)-2^(1/2)*b^(1/8)*d^(1/2)*x)/(b^(1/4)*c+(a^(1/2)+b^(1/2)*c^ 
2)^(1/2))^(1/2))*2^(1/2)/a^(1/2)/b^(1/8)/(a^(1/2)+b^(1/2)*c^2)^(1/2)/(b^(1 
/4)*c+(a^(1/2)+b^(1/2)*c^2)^(1/2))^(1/2)/d^(1/2)+1/8*arctan(((-b^(1/4)*c+( 
a^(1/2)+b^(1/2)*c^2)^(1/2))^(1/2)+2^(1/2)*b^(1/8)*d^(1/2)*x)/(b^(1/4)*c+(a 
^(1/2)+b^(1/2)*c^2)^(1/2))^(1/2))*2^(1/2)/a^(1/2)/b^(1/8)/(a^(1/2)+b^(1/2) 
*c^2)^(1/2)/(b^(1/4)*c+(a^(1/2)+b^(1/2)*c^2)^(1/2))^(1/2)/d^(1/2)+1/4*arct 
anh(b^(1/8)*d^(1/2)*x/(a^(1/4)-b^(1/4)*c)^(1/2))/a^(3/4)/b^(1/8)/(a^(1/4)- 
b^(1/4)*c)^(1/2)/d^(1/2)+1/8*arctanh(2^(1/2)*b^(1/8)*(-b^(1/4)*c+(a^(1/2)+ 
b^(1/2)*c^2)^(1/2))^(1/2)*d^(1/2)*x/((a^(1/2)+b^(1/2)*c^2)^(1/2)+b^(1/4)*d 
*x^2))*2^(1/2)/a^(1/2)/b^(1/8)/(a^(1/2)+b^(1/2)*c^2)^(1/2)/(-b^(1/4)*c+(a^ 
(1/2)+b^(1/2)*c^2)^(1/2))^(1/2)/d^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.05 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.18 \[ \int \frac {1}{a-b \left (c+d x^2\right )^4} \, dx=-\frac {\text {RootSum}\left [a-b c^4-4 b c^3 d \text {$\#$1}^2-6 b c^2 d^2 \text {$\#$1}^4-4 b c d^3 \text {$\#$1}^6-b d^4 \text {$\#$1}^8\&,\frac {\log (x-\text {$\#$1})}{c^3 \text {$\#$1}+3 c^2 d \text {$\#$1}^3+3 c d^2 \text {$\#$1}^5+d^3 \text {$\#$1}^7}\&\right ]}{8 b d} \] Input:

Integrate[(a - b*(c + d*x^2)^4)^(-1),x]
 

Output:

-1/8*RootSum[a - b*c^4 - 4*b*c^3*d*#1^2 - 6*b*c^2*d^2*#1^4 - 4*b*c*d^3*#1^ 
6 - b*d^4*#1^8 & , Log[x - #1]/(c^3*#1 + 3*c^2*d*#1^3 + 3*c*d^2*#1^5 + d^3 
*#1^7) & ]/(b*d)
 

Rubi [A] (verified)

Time = 2.37 (sec) , antiderivative size = 831, normalized size of antiderivative = 1.37, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.812, Rules used = {7289, 27, 1406, 218, 221, 1407, 27, 1142, 25, 27, 1083, 217, 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.

\(\displaystyle \int \frac {1}{a-b \left (c+d x^2\right )^4} \, dx\)

\(\Big \downarrow \) 7289

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1406

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

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\sqrt [4]{b} d \int \frac {1}{\sqrt [4]{b} \left (\sqrt [4]{a}-\sqrt [4]{b} c\right ) d-\sqrt {b} d^2 x^2}dx}{2 \sqrt [4]{a}}+\frac {\arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{a}+\sqrt [4]{b} c}}\right )}{2 \sqrt [4]{a} \sqrt [8]{b} \sqrt {d} \sqrt {\sqrt [4]{a}+\sqrt [4]{b} c}}}{2 \sqrt {a}}+\frac {\int \frac {1}{\sqrt {b} d^2 x^4+2 \sqrt {b} c d x^2+\sqrt {b} c^2+\sqrt {a}}dx}{2 \sqrt {a}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\int \frac {1}{\sqrt {b} d^2 x^4+2 \sqrt {b} c d x^2+\sqrt {b} c^2+\sqrt {a}}dx}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{a}+\sqrt [4]{b} c}}\right )}{2 \sqrt [4]{a} \sqrt [8]{b} \sqrt {d} \sqrt {\sqrt [4]{a}+\sqrt [4]{b} c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{a}-\sqrt [4]{b} c}}\right )}{2 \sqrt [4]{a} \sqrt [8]{b} \sqrt {d} \sqrt {\sqrt [4]{a}-\sqrt [4]{b} c}}}{2 \sqrt {a}}\)

\(\Big \downarrow \) 1407

\(\displaystyle \frac {\frac {\int \frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}-\sqrt [8]{b} \sqrt {d} x}{\sqrt [8]{b} \sqrt {d} \left (x^2-\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {a}}}{\sqrt [4]{b} d}\right )}dx}{2 \sqrt {2} \sqrt [8]{b} \sqrt {d} \sqrt {\sqrt {a}+\sqrt {b} c^2} \sqrt {\sqrt {\sqrt {a}+\sqrt {b} c^2}-\sqrt [4]{b} c}}+\frac {\int \frac {\sqrt [8]{b} \sqrt {d} x+\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}}{\sqrt [8]{b} \sqrt {d} \left (x^2+\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {a}}}{\sqrt [4]{b} d}\right )}dx}{2 \sqrt {2} \sqrt [8]{b} \sqrt {d} \sqrt {\sqrt {a}+\sqrt {b} c^2} \sqrt {\sqrt {\sqrt {a}+\sqrt {b} c^2}-\sqrt [4]{b} c}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{a}+\sqrt [4]{b} c}}\right )}{2 \sqrt [4]{a} \sqrt [8]{b} \sqrt {d} \sqrt {\sqrt [4]{a}+\sqrt [4]{b} c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{a}-\sqrt [4]{b} c}}\right )}{2 \sqrt [4]{a} \sqrt [8]{b} \sqrt {d} \sqrt {\sqrt [4]{a}-\sqrt [4]{b} c}}}{2 \sqrt {a}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}-\sqrt [8]{b} \sqrt {d} x}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {a}}}{\sqrt [4]{b} d}}dx}{2 \sqrt {2} \sqrt [4]{b} d \sqrt {\sqrt {a}+\sqrt {b} c^2} \sqrt {\sqrt {\sqrt {a}+\sqrt {b} c^2}-\sqrt [4]{b} c}}+\frac {\int \frac {\sqrt [8]{b} \sqrt {d} x+\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {a}}}{\sqrt [4]{b} d}}dx}{2 \sqrt {2} \sqrt [4]{b} d \sqrt {\sqrt {a}+\sqrt {b} c^2} \sqrt {\sqrt {\sqrt {a}+\sqrt {b} c^2}-\sqrt [4]{b} c}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{a}+\sqrt [4]{b} c}}\right )}{2 \sqrt [4]{a} \sqrt [8]{b} \sqrt {d} \sqrt {\sqrt [4]{a}+\sqrt [4]{b} c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{a}-\sqrt [4]{b} c}}\right )}{2 \sqrt [4]{a} \sqrt [8]{b} \sqrt {d} \sqrt {\sqrt [4]{a}-\sqrt [4]{b} c}}}{2 \sqrt {a}}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt [4]{a}}}\right )}{2 \sqrt [4]{a} \sqrt [8]{b} \sqrt {\sqrt [4]{b} c+\sqrt [4]{a}} \sqrt {d}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{a}-\sqrt [4]{b} c}}\right )}{2 \sqrt [4]{a} \sqrt [8]{b} \sqrt {\sqrt [4]{a}-\sqrt [4]{b} c} \sqrt {d}}}{2 \sqrt {a}}+\frac {\frac {\frac {\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \int \frac {1}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {a}}}{\sqrt [4]{b} d}}dx}{\sqrt {2}}-\frac {1}{2} \sqrt [8]{b} \sqrt {d} \int -\frac {\sqrt {2} \left (\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}-\sqrt {2} \sqrt [8]{b} \sqrt {d} x\right )}{\sqrt [8]{b} \sqrt {d} \left (x^2-\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {a}}}{\sqrt [4]{b} d}\right )}dx}{2 \sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c^2+\sqrt {a}} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} d}+\frac {\frac {\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \int \frac {1}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {a}}}{\sqrt [4]{b} d}}dx}{\sqrt {2}}+\frac {1}{2} \sqrt [8]{b} \sqrt {d} \int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{b} \sqrt {d} x+\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}\right )}{\sqrt [8]{b} \sqrt {d} \left (x^2+\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {a}}}{\sqrt [4]{b} d}\right )}dx}{2 \sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c^2+\sqrt {a}} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} d}}{2 \sqrt {a}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt [4]{a}}}\right )}{2 \sqrt [4]{a} \sqrt [8]{b} \sqrt {\sqrt [4]{b} c+\sqrt [4]{a}} \sqrt {d}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{a}-\sqrt [4]{b} c}}\right )}{2 \sqrt [4]{a} \sqrt [8]{b} \sqrt {\sqrt [4]{a}-\sqrt [4]{b} c} \sqrt {d}}}{2 \sqrt {a}}+\frac {\frac {\frac {\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \int \frac {1}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {a}}}{\sqrt [4]{b} d}}dx}{\sqrt {2}}+\frac {1}{2} \sqrt [8]{b} \sqrt {d} \int \frac {\sqrt {2} \left (\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}-\sqrt {2} \sqrt [8]{b} \sqrt {d} x\right )}{\sqrt [8]{b} \sqrt {d} \left (x^2-\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {a}}}{\sqrt [4]{b} d}\right )}dx}{2 \sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c^2+\sqrt {a}} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} d}+\frac {\frac {\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \int \frac {1}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {a}}}{\sqrt [4]{b} d}}dx}{\sqrt {2}}+\frac {1}{2} \sqrt [8]{b} \sqrt {d} \int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{b} \sqrt {d} x+\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}\right )}{\sqrt [8]{b} \sqrt {d} \left (x^2+\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {a}}}{\sqrt [4]{b} d}\right )}dx}{2 \sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c^2+\sqrt {a}} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} d}}{2 \sqrt {a}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt [4]{a}}}\right )}{2 \sqrt [4]{a} \sqrt [8]{b} \sqrt {\sqrt [4]{b} c+\sqrt [4]{a}} \sqrt {d}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{a}-\sqrt [4]{b} c}}\right )}{2 \sqrt [4]{a} \sqrt [8]{b} \sqrt {\sqrt [4]{a}-\sqrt [4]{b} c} \sqrt {d}}}{2 \sqrt {a}}+\frac {\frac {\frac {\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \int \frac {1}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {a}}}{\sqrt [4]{b} d}}dx}{\sqrt {2}}+\frac {\int \frac {\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}-\sqrt {2} \sqrt [8]{b} \sqrt {d} x}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {a}}}{\sqrt [4]{b} d}}dx}{\sqrt {2}}}{2 \sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c^2+\sqrt {a}} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} d}+\frac {\frac {\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \int \frac {1}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {a}}}{\sqrt [4]{b} d}}dx}{\sqrt {2}}+\frac {\int \frac {\sqrt {2} \sqrt [8]{b} \sqrt {d} x+\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {a}}}{\sqrt [4]{b} d}}dx}{\sqrt {2}}}{2 \sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c^2+\sqrt {a}} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} d}}{2 \sqrt {a}}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt [4]{a}}}\right )}{2 \sqrt [4]{a} \sqrt [8]{b} \sqrt {\sqrt [4]{b} c+\sqrt [4]{a}} \sqrt {d}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{a}-\sqrt [4]{b} c}}\right )}{2 \sqrt [4]{a} \sqrt [8]{b} \sqrt {\sqrt [4]{a}-\sqrt [4]{b} c} \sqrt {d}}}{2 \sqrt {a}}+\frac {\frac {\frac {\int \frac {\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}-\sqrt {2} \sqrt [8]{b} \sqrt {d} x}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {a}}}{\sqrt [4]{b} d}}dx}{\sqrt {2}}-\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \int \frac {1}{-\left (2 x-\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}}{\sqrt [8]{b} \sqrt {d}}\right )^2-\frac {2 \left (c+\frac {\sqrt {\sqrt {b} c^2+\sqrt {a}}}{\sqrt [4]{b}}\right )}{d}}d\left (2 x-\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}}{\sqrt [8]{b} \sqrt {d}}\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c^2+\sqrt {a}} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} d}+\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{b} \sqrt {d} x+\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {a}}}{\sqrt [4]{b} d}}dx}{\sqrt {2}}-\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \int \frac {1}{-\left (2 x+\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}}{\sqrt [8]{b} \sqrt {d}}\right )^2-\frac {2 \left (c+\frac {\sqrt {\sqrt {b} c^2+\sqrt {a}}}{\sqrt [4]{b}}\right )}{d}}d\left (2 x+\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}}{\sqrt [8]{b} \sqrt {d}}\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c^2+\sqrt {a}} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} d}}{2 \sqrt {a}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt [4]{a}}}\right )}{2 \sqrt [4]{a} \sqrt [8]{b} \sqrt {\sqrt [4]{b} c+\sqrt [4]{a}} \sqrt {d}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{a}-\sqrt [4]{b} c}}\right )}{2 \sqrt [4]{a} \sqrt [8]{b} \sqrt {\sqrt [4]{a}-\sqrt [4]{b} c} \sqrt {d}}}{2 \sqrt {a}}+\frac {\frac {\frac {\sqrt [8]{b} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \sqrt {d} \arctan \left (\frac {\sqrt [8]{b} \sqrt {d} \left (2 x-\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}}{\sqrt [8]{b} \sqrt {d}}\right )}{\sqrt {2} \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}}}\right )}{\sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}}}+\frac {\int \frac {\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}-\sqrt {2} \sqrt [8]{b} \sqrt {d} x}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {a}}}{\sqrt [4]{b} d}}dx}{\sqrt {2}}}{2 \sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c^2+\sqrt {a}} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} d}+\frac {\frac {\sqrt [8]{b} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \sqrt {d} \arctan \left (\frac {\sqrt [8]{b} \sqrt {d} \left (2 x+\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}}{\sqrt [8]{b} \sqrt {d}}\right )}{\sqrt {2} \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}}}\right )}{\sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}}}+\frac {\int \frac {\sqrt {2} \sqrt [8]{b} \sqrt {d} x+\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {a}}}{\sqrt [4]{b} d}}dx}{\sqrt {2}}}{2 \sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c^2+\sqrt {a}} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} d}}{2 \sqrt {a}}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt [4]{a}}}\right )}{2 \sqrt [4]{a} \sqrt [8]{b} \sqrt {\sqrt [4]{b} c+\sqrt [4]{a}} \sqrt {d}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{a}-\sqrt [4]{b} c}}\right )}{2 \sqrt [4]{a} \sqrt [8]{b} \sqrt {\sqrt [4]{a}-\sqrt [4]{b} c} \sqrt {d}}}{2 \sqrt {a}}+\frac {\frac {\frac {\sqrt [8]{b} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \sqrt {d} \arctan \left (\frac {\sqrt [8]{b} \sqrt {d} \left (2 x-\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}}{\sqrt [8]{b} \sqrt {d}}\right )}{\sqrt {2} \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}}}\right )}{\sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}}}-\frac {1}{2} \sqrt [8]{b} \sqrt {d} \log \left (\sqrt [4]{b} d x^2-\sqrt {2} \sqrt [8]{b} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \sqrt {d} x+\sqrt {\sqrt {b} c^2+\sqrt {a}}\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c^2+\sqrt {a}} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} d}+\frac {\frac {\sqrt [8]{b} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \sqrt {d} \arctan \left (\frac {\sqrt [8]{b} \sqrt {d} \left (2 x+\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}}{\sqrt [8]{b} \sqrt {d}}\right )}{\sqrt {2} \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}}}\right )}{\sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}}}+\frac {1}{2} \sqrt [8]{b} \sqrt {d} \log \left (\sqrt [4]{b} d x^2+\sqrt {2} \sqrt [8]{b} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \sqrt {d} x+\sqrt {\sqrt {b} c^2+\sqrt {a}}\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c^2+\sqrt {a}} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} d}}{2 \sqrt {a}}\)

Input:

Int[(a - b*(c + d*x^2)^4)^(-1),x]
 

Output:

(ArcTan[(b^(1/8)*Sqrt[d]*x)/Sqrt[a^(1/4) + b^(1/4)*c]]/(2*a^(1/4)*b^(1/8)* 
Sqrt[a^(1/4) + b^(1/4)*c]*Sqrt[d]) + ArcTanh[(b^(1/8)*Sqrt[d]*x)/Sqrt[a^(1 
/4) - b^(1/4)*c]]/(2*a^(1/4)*b^(1/8)*Sqrt[a^(1/4) - b^(1/4)*c]*Sqrt[d]))/( 
2*Sqrt[a]) + (((b^(1/8)*Sqrt[-(b^(1/4)*c) + Sqrt[Sqrt[a] + Sqrt[b]*c^2]]*S 
qrt[d]*ArcTan[(b^(1/8)*Sqrt[d]*(-((Sqrt[2]*Sqrt[-(b^(1/4)*c) + Sqrt[Sqrt[a 
] + Sqrt[b]*c^2]])/(b^(1/8)*Sqrt[d])) + 2*x))/(Sqrt[2]*Sqrt[b^(1/4)*c + Sq 
rt[Sqrt[a] + Sqrt[b]*c^2]])])/Sqrt[b^(1/4)*c + Sqrt[Sqrt[a] + Sqrt[b]*c^2] 
] - (b^(1/8)*Sqrt[d]*Log[Sqrt[Sqrt[a] + Sqrt[b]*c^2] - Sqrt[2]*b^(1/8)*Sqr 
t[-(b^(1/4)*c) + Sqrt[Sqrt[a] + Sqrt[b]*c^2]]*Sqrt[d]*x + b^(1/4)*d*x^2])/ 
2)/(2*Sqrt[2]*b^(1/4)*Sqrt[Sqrt[a] + Sqrt[b]*c^2]*Sqrt[-(b^(1/4)*c) + Sqrt 
[Sqrt[a] + Sqrt[b]*c^2]]*d) + ((b^(1/8)*Sqrt[-(b^(1/4)*c) + Sqrt[Sqrt[a] + 
 Sqrt[b]*c^2]]*Sqrt[d]*ArcTan[(b^(1/8)*Sqrt[d]*((Sqrt[2]*Sqrt[-(b^(1/4)*c) 
 + Sqrt[Sqrt[a] + Sqrt[b]*c^2]])/(b^(1/8)*Sqrt[d]) + 2*x))/(Sqrt[2]*Sqrt[b 
^(1/4)*c + Sqrt[Sqrt[a] + Sqrt[b]*c^2]])])/Sqrt[b^(1/4)*c + Sqrt[Sqrt[a] + 
 Sqrt[b]*c^2]] + (b^(1/8)*Sqrt[d]*Log[Sqrt[Sqrt[a] + Sqrt[b]*c^2] + Sqrt[2 
]*b^(1/8)*Sqrt[-(b^(1/4)*c) + Sqrt[Sqrt[a] + Sqrt[b]*c^2]]*Sqrt[d]*x + b^( 
1/4)*d*x^2])/2)/(2*Sqrt[2]*b^(1/4)*Sqrt[Sqrt[a] + Sqrt[b]*c^2]*Sqrt[-(b^(1 
/4)*c) + Sqrt[Sqrt[a] + Sqrt[b]*c^2]]*d))/(2*Sqrt[a])
 

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 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 221

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

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 1406

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^ 
2 - 4*a*c, 2]}, Simp[c/q   Int[1/(b/2 - q/2 + c*x^2), x], x] - Simp[c/q   I 
nt[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c 
, 0] && PosQ[b^2 - 4*a*c]
 

rule 1407

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[a/ 
c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r)   Int[(r - x)/(q - r* 
x + x^2), x], x] + Simp[1/(2*c*q*r)   Int[(r + x)/(q + r*x + x^2), x], x]]] 
 /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]
 

rule 7289

Int[((a_) + (b_.)*(v_)^(n_))^(-1), x_Symbol] :> Simp[2/(a*n)   Sum[Int[Toge 
ther[1/(1 - v^2/((-1)^(4*(k/n))*Rt[-a/b, n/2]))], x], {k, 1, n/2}], x] /; F 
reeQ[{a, b}, x] && IGtQ[n/2, 1]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.10 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.17

method	result	size
default	\(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{4} \textit {\_Z}^{8}+4 b c \,d^{3} \textit {\_Z}^{6}+6 b \,c^{2} d^{2} \textit {\_Z}^{4}+4 b \,c^{3} d \,\textit {\_Z}^{2}+b \,c^{4}-a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{-d^{3} \textit {\_R}^{7}-3 c \,d^{2} \textit {\_R}^{5}-3 c^{2} d \,\textit {\_R}^{3}-c^{3} \textit {\_R}}}{8 d b}\)	\(104\)
risch	\(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{4} \textit {\_Z}^{8}+4 b c \,d^{3} \textit {\_Z}^{6}+6 b \,c^{2} d^{2} \textit {\_Z}^{4}+4 b \,c^{3} d \,\textit {\_Z}^{2}+b \,c^{4}-a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{-d^{3} \textit {\_R}^{7}-3 c \,d^{2} \textit {\_R}^{5}-3 c^{2} d \,\textit {\_R}^{3}-c^{3} \textit {\_R}}}{8 d b}\)	\(104\)

Input:

int(1/(a-b*(d*x^2+c)^4),x,method=_RETURNVERBOSE)
 

Output:

1/8/d/b*sum(1/(-_R^7*d^3-3*_R^5*c*d^2-3*_R^3*c^2*d-_R*c^3)*ln(x-_R),_R=Roo 
tOf(_Z^8*b*d^4+4*_Z^6*b*c*d^3+6*_Z^4*b*c^2*d^2+4*_Z^2*b*c^3*d+b*c^4-a))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 6.14 (sec) , antiderivative size = 475271, normalized size of antiderivative = 782.98 \[ \int \frac {1}{a-b \left (c+d x^2\right )^4} \, dx=\text {Too large to display} \] Input:

integrate(1/(a-b*(d*x^2+c)^4),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

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

integrate(1/(a-b*(d*x**2+c)**4),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{a-b \left (c+d x^2\right )^4} \, dx=\int { -\frac {1}{{\left (d x^{2} + c\right )}^{4} b - a} \,d x } \] Input:

integrate(1/(a-b*(d*x^2+c)^4),x, algorithm="maxima")
 

Output:

-integrate(1/((d*x^2 + c)^4*b - a), x)
 

Giac [F]

\[ \int \frac {1}{a-b \left (c+d x^2\right )^4} \, dx=\int { -\frac {1}{{\left (d x^{2} + c\right )}^{4} b - a} \,d x } \] Input:

integrate(1/(a-b*(d*x^2+c)^4),x, algorithm="giac")
 

Output:

integrate(-1/((d*x^2 + c)^4*b - a), x)
 

Mupad [B] (verification not implemented)

Time = 10.00 (sec) , antiderivative size = 433, normalized size of antiderivative = 0.71 \[ \int \frac {1}{a-b \left (c+d x^2\right )^4} \, dx=\sum _{k=1}^8\ln \left (-\mathrm {root}\left (16777216\,a^6\,b^2\,c^4\,d^4\,z^8-16777216\,a^7\,b\,d^4\,z^8+1048576\,a^5\,b\,c\,d^3\,z^6-8192\,a^3\,b\,c^2\,d^2\,z^4+1,z,k\right )\,b^7\,d^{28}\,\left (x+\mathrm {root}\left (16777216\,a^6\,b^2\,c^4\,d^4\,z^8-16777216\,a^7\,b\,d^4\,z^8+1048576\,a^5\,b\,c\,d^3\,z^6-8192\,a^3\,b\,c^2\,d^2\,z^4+1,z,k\right )\,a\,8+{\mathrm {root}\left (16777216\,a^6\,b^2\,c^4\,d^4\,z^8-16777216\,a^7\,b\,d^4\,z^8+1048576\,a^5\,b\,c\,d^3\,z^6-8192\,a^3\,b\,c^2\,d^2\,z^4+1,z,k\right )}^5\,a^4\,b\,c^2\,d^2\,32768-{\mathrm {root}\left (16777216\,a^6\,b^2\,c^4\,d^4\,z^8-16777216\,a^7\,b\,d^4\,z^8+1048576\,a^5\,b\,c\,d^3\,z^6-8192\,a^3\,b\,c^2\,d^2\,z^4+1,z,k\right )}^4\,a^3\,b\,c^2\,d^2\,x\,4096+{\mathrm {root}\left (16777216\,a^6\,b^2\,c^4\,d^4\,z^8-16777216\,a^7\,b\,d^4\,z^8+1048576\,a^5\,b\,c\,d^3\,z^6-8192\,a^3\,b\,c^2\,d^2\,z^4+1,z,k\right )}^6\,a^5\,b\,c\,d^3\,x\,262144\right )\,8\right )\,\mathrm {root}\left (16777216\,a^6\,b^2\,c^4\,d^4\,z^8-16777216\,a^7\,b\,d^4\,z^8+1048576\,a^5\,b\,c\,d^3\,z^6-8192\,a^3\,b\,c^2\,d^2\,z^4+1,z,k\right ) \] Input:

int(1/(a - b*(c + d*x^2)^4),x)
 

Output:

symsum(log(-8*root(16777216*a^6*b^2*c^4*d^4*z^8 - 16777216*a^7*b*d^4*z^8 + 
 1048576*a^5*b*c*d^3*z^6 - 8192*a^3*b*c^2*d^2*z^4 + 1, z, k)*b^7*d^28*(x + 
 8*root(16777216*a^6*b^2*c^4*d^4*z^8 - 16777216*a^7*b*d^4*z^8 + 1048576*a^ 
5*b*c*d^3*z^6 - 8192*a^3*b*c^2*d^2*z^4 + 1, z, k)*a + 32768*root(16777216* 
a^6*b^2*c^4*d^4*z^8 - 16777216*a^7*b*d^4*z^8 + 1048576*a^5*b*c*d^3*z^6 - 8 
192*a^3*b*c^2*d^2*z^4 + 1, z, k)^5*a^4*b*c^2*d^2 - 4096*root(16777216*a^6* 
b^2*c^4*d^4*z^8 - 16777216*a^7*b*d^4*z^8 + 1048576*a^5*b*c*d^3*z^6 - 8192* 
a^3*b*c^2*d^2*z^4 + 1, z, k)^4*a^3*b*c^2*d^2*x + 262144*root(16777216*a^6* 
b^2*c^4*d^4*z^8 - 16777216*a^7*b*d^4*z^8 + 1048576*a^5*b*c*d^3*z^6 - 8192* 
a^3*b*c^2*d^2*z^4 + 1, z, k)^6*a^5*b*c*d^3*x))*root(16777216*a^6*b^2*c^4*d 
^4*z^8 - 16777216*a^7*b*d^4*z^8 + 1048576*a^5*b*c*d^3*z^6 - 8192*a^3*b*c^2 
*d^2*z^4 + 1, z, k), k, 1, 8)
 

Reduce [F]

\[ \int \frac {1}{a-b \left (c+d x^2\right )^4} \, dx=\int \frac {1}{a -b \left (d \,x^{2}+c \right )^{4}}d x \] Input:

int(1/(a-b*(d*x^2+c)^4),x)
 

Output:

int(1/(a-b*(d*x^2+c)^4),x)