Integrand size = 15, antiderivative size = 1693 \[ \int \frac {1}{a+b \left (c+d x^2\right )^4} \, dx =\text {Too large to display} \] Output:
-1/8*(2^(1/2)+(2*a^(1/4)+2^(1/2)*b^(1/4)*c)/(a^(1/2)+2^(1/2)*a^(1/4)*b^(1/ 4)*c+b^(1/2)*c^2)^(1/2))*arctan(((-2^(1/2)*a^(1/4)-2*b^(1/4)*c+2*(a^(1/2)+ 2^(1/2)*a^(1/4)*b^(1/4)*c+b^(1/2)*c^2)^(1/2))^(1/2)-2*b^(1/8)*d^(1/2)*x)/( 2^(1/2)*a^(1/4)+2*b^(1/4)*c+2*(a^(1/2)+2^(1/2)*a^(1/4)*b^(1/4)*c+b^(1/2)*c ^2)^(1/2))^(1/2))/a^(3/4)/b^(1/8)/(2^(1/2)*a^(1/4)+2*b^(1/4)*c+2*(a^(1/2)+ 2^(1/2)*a^(1/4)*b^(1/4)*c+b^(1/2)*c^2)^(1/2))^(1/2)/d^(1/2)+1/8*(2^(1/2)+( 2*a^(1/4)+2^(1/2)*b^(1/4)*c)/(a^(1/2)+2^(1/2)*a^(1/4)*b^(1/4)*c+b^(1/2)*c^ 2)^(1/2))*arctan(((-2^(1/2)*a^(1/4)-2*b^(1/4)*c+2*(a^(1/2)+2^(1/2)*a^(1/4) *b^(1/4)*c+b^(1/2)*c^2)^(1/2))^(1/2)+2*b^(1/8)*d^(1/2)*x)/(2^(1/2)*a^(1/4) +2*b^(1/4)*c+2*(a^(1/2)+2^(1/2)*a^(1/4)*b^(1/4)*c+b^(1/2)*c^2)^(1/2))^(1/2 ))/a^(3/4)/b^(1/8)/(2^(1/2)*a^(1/4)+2*b^(1/4)*c+2*(a^(1/2)+2^(1/2)*a^(1/4) *b^(1/4)*c+b^(1/2)*c^2)^(1/2))^(1/2)/d^(1/2)-1/8*(2^(1/2)-(2*a^(1/4)+2^(1/ 2)*b^(1/4)*c)/(a^(1/2)+2^(1/2)*a^(1/4)*b^(1/4)*c+b^(1/2)*c^2)^(1/2))*arcta n(b^(1/8)*(2^(1/2)*a^(1/4)+2*b^(1/4)*c-2*(a^(1/2)+2^(1/2)*a^(1/4)*b^(1/4)* c+b^(1/2)*c^2)^(1/2))^(1/2)*d^(1/2)*x/((a^(1/2)+2^(1/2)*a^(1/4)*b^(1/4)*c+ b^(1/2)*c^2)^(1/2)+b^(1/4)*d*x^2))/a^(3/4)/b^(1/8)/(2^(1/2)*a^(1/4)+2*b^(1 /4)*c-2*(a^(1/2)+2^(1/2)*a^(1/4)*b^(1/4)*c+b^(1/2)*c^2)^(1/2))^(1/2)/d^(1/ 2)-1/8*(2^(1/2)-(2*a^(1/4)-2^(1/2)*b^(1/4)*c)/(a^(1/2)-2^(1/2)*a^(1/4)*b^( 1/4)*c+b^(1/2)*c^2)^(1/2))*arctanh(((2^(1/2)*a^(1/4)-2*b^(1/4)*c+2*(a^(1/2 )-2^(1/2)*a^(1/4)*b^(1/4)*c+b^(1/2)*c^2)^(1/2))^(1/2)-2*b^(1/8)*d^(1/2)...
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.06 \[ \int \frac {1}{a+b \left (c+d x^2\right )^4} \, dx=\frac {\text {RootSum}\left [a+b c^4+4 b c^3 d \text {$\#$1}^2+6 b c^2 d^2 \text {$\#$1}^4+4 b c d^3 \text {$\#$1}^6+b d^4 \text {$\#$1}^8\&,\frac {\log (x-\text {$\#$1})}{c^3 \text {$\#$1}+3 c^2 d \text {$\#$1}^3+3 c d^2 \text {$\#$1}^5+d^3 \text {$\#$1}^7}\&\right ]}{8 b d} \] Input:
Integrate[(a + b*(c + d*x^2)^4)^(-1),x]
Output:
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 ) & ]/(8*b*d)
Time = 2.35 (sec) , antiderivative size = 885, normalized size of antiderivative = 0.52, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.867, 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 |
\(\displaystyle \frac {\sqrt {-a} \int \frac {1}{-\sqrt {b} d^2 x^4-2 \sqrt {b} c d x^2-\sqrt {b} c^2+\sqrt {-a}}dx}{2 a}+\frac {\sqrt {-a} \int \frac {1}{\sqrt {b} d^2 x^4+2 \sqrt {b} c d x^2+\sqrt {b} c^2+\sqrt {-a}}dx}{2 a}\) |
\(\Big \downarrow \) 1406 |
\(\displaystyle \frac {\sqrt {-a} \int \frac {1}{\sqrt {b} d^2 x^4+2 \sqrt {b} c d x^2+\sqrt {b} c^2+\sqrt {-a}}dx}{2 a}+\frac {\sqrt {-a} \left (\frac {\sqrt [4]{b} d \int \frac {1}{\sqrt [4]{b} \left (\sqrt [4]{-a}-\sqrt [4]{b} c\right ) d-\sqrt {b} d^2 x^2}dx}{2 \sqrt [4]{-a}}-\frac {\sqrt [4]{b} d \int \frac {1}{-\sqrt {b} d^2 x^2-\sqrt [4]{b} \left (\sqrt [4]{b} c+\sqrt [4]{-a}\right ) d}dx}{2 \sqrt [4]{-a}}\right )}{2 a}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {\sqrt {-a} \left (\frac {\sqrt [4]{b} d \int \frac {1}{\sqrt [4]{b} \left (\sqrt [4]{-a}-\sqrt [4]{b} c\right ) d-\sqrt {b} d^2 x^2}dx}{2 \sqrt [4]{-a}}+\frac {\arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{-a}+\sqrt [4]{b} c}}\right )}{2 \sqrt [4]{-a} \sqrt [8]{b} \sqrt {d} \sqrt {\sqrt [4]{-a}+\sqrt [4]{b} c}}\right )}{2 a}+\frac {\sqrt {-a} \int \frac {1}{\sqrt {b} d^2 x^4+2 \sqrt {b} c d x^2+\sqrt {b} c^2+\sqrt {-a}}dx}{2 a}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\sqrt {-a} \int \frac {1}{\sqrt {b} d^2 x^4+2 \sqrt {b} c d x^2+\sqrt {b} c^2+\sqrt {-a}}dx}{2 a}+\frac {\sqrt {-a} \left (\frac {\arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{-a}+\sqrt [4]{b} c}}\right )}{2 \sqrt [4]{-a} \sqrt [8]{b} \sqrt {d} \sqrt {\sqrt [4]{-a}+\sqrt [4]{b} c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{-a}-\sqrt [4]{b} c}}\right )}{2 \sqrt [4]{-a} \sqrt [8]{b} \sqrt {d} \sqrt {\sqrt [4]{-a}-\sqrt [4]{b} c}}\right )}{2 a}\) |
\(\Big \downarrow \) 1407 |
\(\displaystyle \frac {\sqrt {-a} \left (\frac {\int \frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c}-\sqrt [8]{b} \sqrt {d} x}{\sqrt [8]{b} \sqrt {d} \left (x^2-\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {-a}}}{\sqrt [4]{b} d}\right )}dx}{2 \sqrt {2} \sqrt [8]{b} \sqrt {d} \sqrt {\sqrt {-a}+\sqrt {b} c^2} \sqrt {\sqrt {\sqrt {-a}+\sqrt {b} c^2}-\sqrt [4]{b} c}}+\frac {\int \frac {\sqrt [8]{b} \sqrt {d} x+\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c}}{\sqrt [8]{b} \sqrt {d} \left (x^2+\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {-a}}}{\sqrt [4]{b} d}\right )}dx}{2 \sqrt {2} \sqrt [8]{b} \sqrt {d} \sqrt {\sqrt {-a}+\sqrt {b} c^2} \sqrt {\sqrt {\sqrt {-a}+\sqrt {b} c^2}-\sqrt [4]{b} c}}\right )}{2 a}+\frac {\sqrt {-a} \left (\frac {\arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{-a}+\sqrt [4]{b} c}}\right )}{2 \sqrt [4]{-a} \sqrt [8]{b} \sqrt {d} \sqrt {\sqrt [4]{-a}+\sqrt [4]{b} c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{-a}-\sqrt [4]{b} c}}\right )}{2 \sqrt [4]{-a} \sqrt [8]{b} \sqrt {d} \sqrt {\sqrt [4]{-a}-\sqrt [4]{b} c}}\right )}{2 a}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {-a} \left (\frac {\int \frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c}-\sqrt [8]{b} \sqrt {d} x}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {-a}}}{\sqrt [4]{b} d}}dx}{2 \sqrt {2} \sqrt [4]{b} d \sqrt {\sqrt {-a}+\sqrt {b} c^2} \sqrt {\sqrt {\sqrt {-a}+\sqrt {b} c^2}-\sqrt [4]{b} c}}+\frac {\int \frac {\sqrt [8]{b} \sqrt {d} x+\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c}}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {-a}}}{\sqrt [4]{b} d}}dx}{2 \sqrt {2} \sqrt [4]{b} d \sqrt {\sqrt {-a}+\sqrt {b} c^2} \sqrt {\sqrt {\sqrt {-a}+\sqrt {b} c^2}-\sqrt [4]{b} c}}\right )}{2 a}+\frac {\sqrt {-a} \left (\frac {\arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{-a}+\sqrt [4]{b} c}}\right )}{2 \sqrt [4]{-a} \sqrt [8]{b} \sqrt {d} \sqrt {\sqrt [4]{-a}+\sqrt [4]{b} c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{-a}-\sqrt [4]{b} c}}\right )}{2 \sqrt [4]{-a} \sqrt [8]{b} \sqrt {d} \sqrt {\sqrt [4]{-a}-\sqrt [4]{b} c}}\right )}{2 a}\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {\sqrt {-a} \left (\frac {\arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt [4]{-a}}}\right )}{2 \sqrt [4]{-a} \sqrt [8]{b} \sqrt {\sqrt [4]{b} c+\sqrt [4]{-a}} \sqrt {d}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{-a}-\sqrt [4]{b} c}}\right )}{2 \sqrt [4]{-a} \sqrt [8]{b} \sqrt {\sqrt [4]{-a}-\sqrt [4]{b} c} \sqrt {d}}\right )}{2 a}+\frac {\sqrt {-a} \left (\frac {\frac {\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} \int \frac {1}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {-a}}}{\sqrt [4]{b} d}}dx}{\sqrt {2}}-\frac {1}{2} \sqrt [8]{b} \sqrt {d} \int -\frac {\sqrt {2} \left (\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c}-\sqrt {2} \sqrt [8]{b} \sqrt {d} x\right )}{\sqrt [8]{b} \sqrt {d} \left (x^2-\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {-a}}}{\sqrt [4]{b} d}\right )}dx}{2 \sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c^2+\sqrt {-a}} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} d}+\frac {\frac {\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} \int \frac {1}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {-a}}}{\sqrt [4]{b} d}}dx}{\sqrt {2}}+\frac {1}{2} \sqrt [8]{b} \sqrt {d} \int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{b} \sqrt {d} x+\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c}\right )}{\sqrt [8]{b} \sqrt {d} \left (x^2+\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {-a}}}{\sqrt [4]{b} d}\right )}dx}{2 \sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c^2+\sqrt {-a}} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} d}\right )}{2 a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\sqrt {-a} \left (\frac {\arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt [4]{-a}}}\right )}{2 \sqrt [4]{-a} \sqrt [8]{b} \sqrt {\sqrt [4]{b} c+\sqrt [4]{-a}} \sqrt {d}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{-a}-\sqrt [4]{b} c}}\right )}{2 \sqrt [4]{-a} \sqrt [8]{b} \sqrt {\sqrt [4]{-a}-\sqrt [4]{b} c} \sqrt {d}}\right )}{2 a}+\frac {\sqrt {-a} \left (\frac {\frac {\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} \int \frac {1}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {-a}}}{\sqrt [4]{b} d}}dx}{\sqrt {2}}+\frac {1}{2} \sqrt [8]{b} \sqrt {d} \int \frac {\sqrt {2} \left (\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c}-\sqrt {2} \sqrt [8]{b} \sqrt {d} x\right )}{\sqrt [8]{b} \sqrt {d} \left (x^2-\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {-a}}}{\sqrt [4]{b} d}\right )}dx}{2 \sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c^2+\sqrt {-a}} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} d}+\frac {\frac {\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} \int \frac {1}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {-a}}}{\sqrt [4]{b} d}}dx}{\sqrt {2}}+\frac {1}{2} \sqrt [8]{b} \sqrt {d} \int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{b} \sqrt {d} x+\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c}\right )}{\sqrt [8]{b} \sqrt {d} \left (x^2+\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {-a}}}{\sqrt [4]{b} d}\right )}dx}{2 \sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c^2+\sqrt {-a}} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} d}\right )}{2 a}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {-a} \left (\frac {\arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt [4]{-a}}}\right )}{2 \sqrt [4]{-a} \sqrt [8]{b} \sqrt {\sqrt [4]{b} c+\sqrt [4]{-a}} \sqrt {d}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{-a}-\sqrt [4]{b} c}}\right )}{2 \sqrt [4]{-a} \sqrt [8]{b} \sqrt {\sqrt [4]{-a}-\sqrt [4]{b} c} \sqrt {d}}\right )}{2 a}+\frac {\sqrt {-a} \left (\frac {\frac {\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} \int \frac {1}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {-a}}}{\sqrt [4]{b} d}}dx}{\sqrt {2}}+\frac {\int \frac {\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c}-\sqrt {2} \sqrt [8]{b} \sqrt {d} x}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {-a}}}{\sqrt [4]{b} d}}dx}{\sqrt {2}}}{2 \sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c^2+\sqrt {-a}} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} d}+\frac {\frac {\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} \int \frac {1}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {-a}}}{\sqrt [4]{b} d}}dx}{\sqrt {2}}+\frac {\int \frac {\sqrt {2} \sqrt [8]{b} \sqrt {d} x+\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c}}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} x}{\sqrt [8]{b} \sqrt {d}}+\frac {\sqrt {\sqrt {b} c^2+\sqrt {-a}}}{\sqrt [4]{b} d}}dx}{\sqrt {2}}}{2 \sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c^2+\sqrt {-a}} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} d}\right )}{2 a}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {\sqrt {-a} \left (\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}}\right )}{2 a}+\frac {\sqrt {-a} \left (\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}\right )}{2 a}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {\sqrt {-a} \left (\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}}\right )}{2 a}+\frac {\sqrt {-a} \left (\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}\right )}{2 a}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {\sqrt {-a} \left (\frac {\arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt [4]{-a}}}\right )}{2 \sqrt [4]{-a} \sqrt [8]{b} \sqrt {\sqrt [4]{b} c+\sqrt [4]{-a}} \sqrt {d}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{-a}-\sqrt [4]{b} c}}\right )}{2 \sqrt [4]{-a} \sqrt [8]{b} \sqrt {\sqrt [4]{-a}-\sqrt [4]{b} c} \sqrt {d}}\right )}{2 a}+\frac {\sqrt {-a} \left (\frac {\frac {\sqrt [8]{b} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} \sqrt {d} \arctan \left (\frac {\sqrt [8]{b} \sqrt {d} \left (2 x-\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c}}{\sqrt [8]{b} \sqrt {d}}\right )}{\sqrt {2} \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {-a}}}}\right )}{\sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {-a}}}}-\frac {1}{2} \sqrt [8]{b} \sqrt {d} \log \left (\sqrt [4]{b} d x^2-\sqrt {2} \sqrt [8]{b} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} \sqrt {d} x+\sqrt {\sqrt {b} c^2+\sqrt {-a}}\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c^2+\sqrt {-a}} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} d}+\frac {\frac {\sqrt [8]{b} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} \sqrt {d} \arctan \left (\frac {\sqrt [8]{b} \sqrt {d} \left (2 x+\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c}}{\sqrt [8]{b} \sqrt {d}}\right )}{\sqrt {2} \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {-a}}}}\right )}{\sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {-a}}}}+\frac {1}{2} \sqrt [8]{b} \sqrt {d} \log \left (\sqrt [4]{b} d x^2+\sqrt {2} \sqrt [8]{b} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} \sqrt {d} x+\sqrt {\sqrt {b} c^2+\sqrt {-a}}\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c^2+\sqrt {-a}} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} d}\right )}{2 a}\) |
Input:
Int[(a + b*(c + d*x^2)^4)^(-1),x]
Output:
(Sqrt[-a]*(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)*S qrt[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*a) + (Sqrt[-a]*(((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] + Sqr t[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)*Sqr t[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] + S qrt[b]*c^2]]*d)))/(2*a)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
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]
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]
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]
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]
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]
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]
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.09 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.06
method | result | size |
default | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{4} \textit {\_Z}^{8}+4 b c \,d^{3} \textit {\_Z}^{6}+6 b \,c^{2} d^{2} \textit {\_Z}^{4}+4 b \,c^{3} d \,\textit {\_Z}^{2}+b \,c^{4}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{d^{3} \textit {\_R}^{7}+3 c \,d^{2} \textit {\_R}^{5}+3 c^{2} d \,\textit {\_R}^{3}+c^{3} \textit {\_R}}}{8 d b}\) | \(100\) |
risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{4} \textit {\_Z}^{8}+4 b c \,d^{3} \textit {\_Z}^{6}+6 b \,c^{2} d^{2} \textit {\_Z}^{4}+4 b \,c^{3} d \,\textit {\_Z}^{2}+b \,c^{4}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{d^{3} \textit {\_R}^{7}+3 c \,d^{2} \textit {\_R}^{5}+3 c^{2} d \,\textit {\_R}^{3}+c^{3} \textit {\_R}}}{8 d b}\) | \(100\) |
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=Root Of(_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))
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, algorithm="fricas")
Output:
Timed out
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
\[ \int \frac {1}{a+b \left (c+d x^2\right )^4} \, dx=\int { \frac {1}{{\left (d x^{2} + c\right )}^{4} b + a} \,d x } \] Input:
integrate(1/(a+b*(d*x^2+c)^4),x, algorithm="maxima")
Output:
integrate(1/((d*x^2 + c)^4*b + a), x)
\[ \int \frac {1}{a+b \left (c+d x^2\right )^4} \, dx=\int { \frac {1}{{\left (d x^{2} + c\right )}^{4} b + a} \,d x } \] Input:
integrate(1/(a+b*(d*x^2+c)^4),x, algorithm="giac")
Output:
integrate(1/((d*x^2 + c)^4*b + a), x)
Time = 9.88 (sec) , antiderivative size = 433, normalized size of antiderivative = 0.26 \[ \int \frac {1}{a+b \left (c+d x^2\right )^4} \, dx=\sum _{k=1}^8\ln \left (-\mathrm {root}\left (16777216\,a^6\,b^2\,c^4\,d^4\,z^8+16777216\,a^7\,b\,d^4\,z^8-1048576\,a^5\,b\,c\,d^3\,z^6+8192\,a^3\,b\,c^2\,d^2\,z^4+1,z,k\right )\,b^7\,d^{28}\,\left (x+\mathrm {root}\left (16777216\,a^6\,b^2\,c^4\,d^4\,z^8+16777216\,a^7\,b\,d^4\,z^8-1048576\,a^5\,b\,c\,d^3\,z^6+8192\,a^3\,b\,c^2\,d^2\,z^4+1,z,k\right )\,a\,8-{\mathrm {root}\left (16777216\,a^6\,b^2\,c^4\,d^4\,z^8+16777216\,a^7\,b\,d^4\,z^8-1048576\,a^5\,b\,c\,d^3\,z^6+8192\,a^3\,b\,c^2\,d^2\,z^4+1,z,k\right )}^5\,a^4\,b\,c^2\,d^2\,32768+{\mathrm {root}\left (16777216\,a^6\,b^2\,c^4\,d^4\,z^8+16777216\,a^7\,b\,d^4\,z^8-1048576\,a^5\,b\,c\,d^3\,z^6+8192\,a^3\,b\,c^2\,d^2\,z^4+1,z,k\right )}^4\,a^3\,b\,c^2\,d^2\,x\,4096-{\mathrm {root}\left (16777216\,a^6\,b^2\,c^4\,d^4\,z^8+16777216\,a^7\,b\,d^4\,z^8-1048576\,a^5\,b\,c\,d^3\,z^6+8192\,a^3\,b\,c^2\,d^2\,z^4+1,z,k\right )}^6\,a^5\,b\,c\,d^3\,x\,262144\right )\,8\right )\,\mathrm {root}\left (16777216\,a^6\,b^2\,c^4\,d^4\,z^8+16777216\,a^7\,b\,d^4\,z^8-1048576\,a^5\,b\,c\,d^3\,z^6+8192\,a^3\,b\,c^2\,d^2\,z^4+1,z,k\right ) \] 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)
\[ \int \frac {1}{a+b \left (c+d x^2\right )^4} \, dx=\int \frac {1}{b \,d^{4} x^{8}+4 b c \,d^{3} x^{6}+6 b \,c^{2} d^{2} x^{4}+4 b \,c^{3} d \,x^{2}+b \,c^{4}+a}d x \] Input:
int(1/(a+b*(d*x^2+c)^4),x)
Output:
int(1/(a + b*c**4 + 4*b*c**3*d*x**2 + 6*b*c**2*d**2*x**4 + 4*b*c*d**3*x**6 + b*d**4*x**8),x)