Integrand size = 34, antiderivative size = 1379 \[ \int \frac {1}{\left (a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4\right )^2} \, dx =\text {Too large to display} \] Output:
(-2*b^2*c^2+6*b^7*d-2*a*b^3*c*d-12*a^2*b^4*d^2-2*(8*a^2*b*d+b^4-4*a*c)^(1/ 2)*(-3*b^5*d+2*a*b*c*d+c^2)-4*a*d*(-3*b^5*d+2*a*b*c*d+c^2)*x)/a/b^2/d/(8*a ^2*b*d+b^4-4*a*c)/(4*a^2*b^2*d^2-4*b^5*d+4*a*b*c*d+c^2)/(2*b/d+(b^2+(8*a^2 *b*d+b^4-4*a*c)^(1/2))*x/a/d+2*x^2)+4*(b^4-2*a*c-b^2*(8*a^2*b*d+b^4-4*a*c) ^(1/2)+a*d*(b^2-(8*a^2*b*d+b^4-4*a*c)^(1/2))*x)/a/b/d^2/(8*a^2*b*d+b^4-4*a *c)^(1/2)/(b^4-2*a*c-4*a^2*b*d-b^2*(8*a^2*b*d+b^4-4*a*c)^(1/2))/(2*b/d+(b^ 2-(8*a^2*b*d+b^4-4*a*c)^(1/2))*x/a/d+2*x^2)/(2*b/d+(b^2+(8*a^2*b*d+b^4-4*a *c)^(1/2))*x/a/d+2*x^2)+4*2^(1/2)*a^2*(4*b^7*c*d-18*a*b^8*d^2+8*a^2*b^4*c* d^2+c^3*(8*a^2*b*d+b^4-4*a*c)^(1/2)+3*a*b*c^2*d*(8*a^2*b*d+b^4-4*a*c)^(1/2 )+6*a*b^6*d^2*(8*a^2*b*d+b^4-4*a*c)^(1/2)+4*b^5*(9*a^3*d^3-c*d*(8*a^2*b*d+ b^4-4*a*c)^(1/2))-b^2*(c^3+4*a^2*c*d^2*(8*a^2*b*d+b^4-4*a*c)^(1/2))+b^3*(a *c^2*d-12*a^3*d^3*(8*a^2*b*d+b^4-4*a*c)^(1/2)))*arctanh(1/2*(b^2-(8*a^2*b* d+b^4-4*a*c)^(1/2)+4*a*d*x)*2^(1/2)/(b^4-2*a*c-4*a^2*b*d-b^2*(8*a^2*b*d+b^ 4-4*a*c)^(1/2))^(1/2))/b^3/(8*a^2*b*d+b^4-4*a*c)^(3/2)/(b^4-2*a*c-4*a^2*b* d-b^2*(8*a^2*b*d+b^4-4*a*c)^(1/2))^(3/2)/(b^4-2*a*c-4*a^2*b*d+b^2*(8*a^2*b *d+b^4-4*a*c)^(1/2))-4*2^(1/2)*a^2*(4*b^7*c*d-18*a*b^8*d^2+8*a^2*b^4*c*d^2 -c^3*(8*a^2*b*d+b^4-4*a*c)^(1/2)-3*a*b*c^2*d*(8*a^2*b*d+b^4-4*a*c)^(1/2)-6 *a*b^6*d^2*(8*a^2*b*d+b^4-4*a*c)^(1/2)+4*b^5*(9*a^3*d^3+c*d*(8*a^2*b*d+b^4 -4*a*c)^(1/2))+a*b^3*d*(c^2+12*a^2*d^2*(8*a^2*b*d+b^4-4*a*c)^(1/2))-b^2*(c ^3-4*a^2*c*d^2*(8*a^2*b*d+b^4-4*a*c)^(1/2)))*arctanh(1/2*(b^2+(8*a^2*b*...
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 1.04 (sec) , antiderivative size = 610, normalized size of antiderivative = 0.44 \[ \int \frac {1}{\left (a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4\right )^2} \, dx=\frac {-\frac {-4 b^8 d+8 a b^6 d^2 x-6 b^7 d^2 x^2+2 b^2 c d x \left (-2 a^2 d+c x\right )+2 c^2 x \left (c+a d^2 x^2\right )+4 a b c d x \left (c+a d^2 x^2\right )+a b^4 d \left (5 c+6 a d^2 x^2\right )+b^3 \left (c^2-8 a^3 d^3 x+3 a c d^2 x^2\right )-b^5 d \left (2 a^2 d+7 c x+6 a d^2 x^3\right )}{\left (b^4-4 a c+8 a^2 b d\right ) \left (c^2+4 a b c d-4 b^2 d \left (b^3-a^2 d\right )\right ) \left (x \left (b^3+c x+b^2 d x^2\right )+a \left (b^2+d^2 x^4\right )\right )}+\frac {2 \text {RootSum}\left [a b^2+b^3 \text {$\#$1}+c \text {$\#$1}^2+b^2 d \text {$\#$1}^3+a d^2 \text {$\#$1}^4\&,\frac {c^3 \log (x-\text {$\#$1})-4 b^5 c d \log (x-\text {$\#$1})+2 a b c^2 d \log (x-\text {$\#$1})+9 a b^6 d^2 \log (x-\text {$\#$1})-6 a^2 b^2 c d^2 \log (x-\text {$\#$1})-12 a^3 b^3 d^3 \log (x-\text {$\#$1})+b^2 c^2 d \log (x-\text {$\#$1}) \text {$\#$1}-3 b^7 d^2 \log (x-\text {$\#$1}) \text {$\#$1}+a b^3 c d^2 \log (x-\text {$\#$1}) \text {$\#$1}+6 a^2 b^4 d^3 \log (x-\text {$\#$1}) \text {$\#$1}+a c^2 d^2 \log (x-\text {$\#$1}) \text {$\#$1}^2-3 a b^5 d^3 \log (x-\text {$\#$1}) \text {$\#$1}^2+2 a^2 b c d^3 \log (x-\text {$\#$1}) \text {$\#$1}^2}{b^3+2 c \text {$\#$1}+3 b^2 d \text {$\#$1}^2+4 a d^2 \text {$\#$1}^3}\&\right ]}{-b^4 c^2+4 a c^3+4 b^9 d-20 a b^5 c d+8 a^2 b c^2 d+28 a^2 b^6 d^2-16 a^3 b^2 c d^2-32 a^4 b^3 d^3}}{b^2} \] Input:
Integrate[(a*b^2 + b^3*x + c*x^2 + b^2*d*x^3 + a*d^2*x^4)^(-2),x]
Output:
(-((-4*b^8*d + 8*a*b^6*d^2*x - 6*b^7*d^2*x^2 + 2*b^2*c*d*x*(-2*a^2*d + c*x ) + 2*c^2*x*(c + a*d^2*x^2) + 4*a*b*c*d*x*(c + a*d^2*x^2) + a*b^4*d*(5*c + 6*a*d^2*x^2) + b^3*(c^2 - 8*a^3*d^3*x + 3*a*c*d^2*x^2) - b^5*d*(2*a^2*d + 7*c*x + 6*a*d^2*x^3))/((b^4 - 4*a*c + 8*a^2*b*d)*(c^2 + 4*a*b*c*d - 4*b^2 *d*(b^3 - a^2*d))*(x*(b^3 + c*x + b^2*d*x^2) + a*(b^2 + d^2*x^4)))) + (2*R ootSum[a*b^2 + b^3*#1 + c*#1^2 + b^2*d*#1^3 + a*d^2*#1^4 & , (c^3*Log[x - #1] - 4*b^5*c*d*Log[x - #1] + 2*a*b*c^2*d*Log[x - #1] + 9*a*b^6*d^2*Log[x - #1] - 6*a^2*b^2*c*d^2*Log[x - #1] - 12*a^3*b^3*d^3*Log[x - #1] + b^2*c^2 *d*Log[x - #1]*#1 - 3*b^7*d^2*Log[x - #1]*#1 + a*b^3*c*d^2*Log[x - #1]*#1 + 6*a^2*b^4*d^3*Log[x - #1]*#1 + a*c^2*d^2*Log[x - #1]*#1^2 - 3*a*b^5*d^3* Log[x - #1]*#1^2 + 2*a^2*b*c*d^3*Log[x - #1]*#1^2)/(b^3 + 2*c*#1 + 3*b^2*d *#1^2 + 4*a*d^2*#1^3) & ])/(-(b^4*c^2) + 4*a*c^3 + 4*b^9*d - 20*a*b^5*c*d + 8*a^2*b*c^2*d + 28*a^2*b^6*d^2 - 16*a^3*b^2*c*d^2 - 32*a^4*b^3*d^3))/b^2
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.
| \(\displaystyle \int \frac {1}{\left (a b^2+a d^2 x^4+b^3 x+b^2 d x^3+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 2492 |
| \(\displaystyle \frac {\int \left (\frac {2 a^2 \left (-4 a d^2 b^3+c d b^2+3 a d \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b+2 a d^2 (c-3 a b d) x-c \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) d^7}{b^3 \left (d^2 \left (b^4+8 a^2 d b-4 a c\right )\right )^{3/2} \left (2 a d^2 x^2+\left (b^2 d-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )}-\frac {2 a^2 \left (-4 a d^2 b^3+c d b^2-3 a d \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b+2 a d^2 (c-3 a b d) x+c \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) d^7}{b^3 \left (d^2 \left (b^4+8 a^2 d b-4 a c\right )\right )^{3/2} \left (2 a d^2 x^2+\left (d b^2+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )}+\frac {2 a^2 \left (d b^4-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b^2+2 a^2 d^2 b-2 a c d+a d \left (b^2 d-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x\right ) d^5}{b^2 \left (b^4+8 a^2 d b-4 a c\right ) \left (2 a d^2 x^2+\left (b^2 d-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )^2}+\frac {2 a^2 \left (d b^4+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b^2+2 a^2 d^2 b-2 a c d+a d \left (d b^2+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x\right ) d^5}{b^2 \left (b^4+8 a^2 d b-4 a c\right ) \left (2 a d^2 x^2+\left (d b^2+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )^2}\right )dx}{a^2 d^4}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {a^2 d^4}{\left (a d^2 x^4+b^2 d x^3+c x^2+b^3 x+a b^2\right )^2}dx}{a^2 d^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {1}{\left (a b^2+a d^2 x^4+b^3 x+b^2 d x^3+c x^2\right )^2}dx\) |
| \(\Big \downarrow \) 2492 |
| \(\displaystyle \frac {\int \left (\frac {2 a^2 \left (-4 a d^2 b^3+c d b^2+3 a d \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b+2 a d^2 (c-3 a b d) x-c \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) d^7}{b^3 \left (d^2 \left (b^4+8 a^2 d b-4 a c\right )\right )^{3/2} \left (2 a d^2 x^2+\left (b^2 d-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )}-\frac {2 a^2 \left (-4 a d^2 b^3+c d b^2-3 a d \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b+2 a d^2 (c-3 a b d) x+c \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) d^7}{b^3 \left (d^2 \left (b^4+8 a^2 d b-4 a c\right )\right )^{3/2} \left (2 a d^2 x^2+\left (d b^2+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )}+\frac {2 a^2 \left (d b^4-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b^2+2 a^2 d^2 b-2 a c d+a d \left (b^2 d-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x\right ) d^5}{b^2 \left (b^4+8 a^2 d b-4 a c\right ) \left (2 a d^2 x^2+\left (b^2 d-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )^2}+\frac {2 a^2 \left (d b^4+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b^2+2 a^2 d^2 b-2 a c d+a d \left (d b^2+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x\right ) d^5}{b^2 \left (b^4+8 a^2 d b-4 a c\right ) \left (2 a d^2 x^2+\left (d b^2+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )^2}\right )dx}{a^2 d^4}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {a^2 d^4}{\left (a d^2 x^4+b^2 d x^3+c x^2+b^3 x+a b^2\right )^2}dx}{a^2 d^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {1}{\left (a b^2+a d^2 x^4+b^3 x+b^2 d x^3+c x^2\right )^2}dx\) |
| \(\Big \downarrow \) 2492 |
| \(\displaystyle \frac {\int \left (\frac {2 a^2 \left (-4 a d^2 b^3+c d b^2+3 a d \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b+2 a d^2 (c-3 a b d) x-c \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) d^7}{b^3 \left (d^2 \left (b^4+8 a^2 d b-4 a c\right )\right )^{3/2} \left (2 a d^2 x^2+\left (b^2 d-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )}-\frac {2 a^2 \left (-4 a d^2 b^3+c d b^2-3 a d \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b+2 a d^2 (c-3 a b d) x+c \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) d^7}{b^3 \left (d^2 \left (b^4+8 a^2 d b-4 a c\right )\right )^{3/2} \left (2 a d^2 x^2+\left (d b^2+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )}+\frac {2 a^2 \left (d b^4-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b^2+2 a^2 d^2 b-2 a c d+a d \left (b^2 d-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x\right ) d^5}{b^2 \left (b^4+8 a^2 d b-4 a c\right ) \left (2 a d^2 x^2+\left (b^2 d-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )^2}+\frac {2 a^2 \left (d b^4+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b^2+2 a^2 d^2 b-2 a c d+a d \left (d b^2+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x\right ) d^5}{b^2 \left (b^4+8 a^2 d b-4 a c\right ) \left (2 a d^2 x^2+\left (d b^2+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )^2}\right )dx}{a^2 d^4}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {a^2 d^4}{\left (a d^2 x^4+b^2 d x^3+c x^2+b^3 x+a b^2\right )^2}dx}{a^2 d^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {1}{\left (a b^2+a d^2 x^4+b^3 x+b^2 d x^3+c x^2\right )^2}dx\) |
| \(\Big \downarrow \) 2492 |
| \(\displaystyle \frac {\int \left (\frac {2 a^2 \left (-4 a d^2 b^3+c d b^2+3 a d \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b+2 a d^2 (c-3 a b d) x-c \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) d^7}{b^3 \left (d^2 \left (b^4+8 a^2 d b-4 a c\right )\right )^{3/2} \left (2 a d^2 x^2+\left (b^2 d-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )}-\frac {2 a^2 \left (-4 a d^2 b^3+c d b^2-3 a d \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b+2 a d^2 (c-3 a b d) x+c \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) d^7}{b^3 \left (d^2 \left (b^4+8 a^2 d b-4 a c\right )\right )^{3/2} \left (2 a d^2 x^2+\left (d b^2+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )}+\frac {2 a^2 \left (d b^4-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b^2+2 a^2 d^2 b-2 a c d+a d \left (b^2 d-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x\right ) d^5}{b^2 \left (b^4+8 a^2 d b-4 a c\right ) \left (2 a d^2 x^2+\left (b^2 d-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )^2}+\frac {2 a^2 \left (d b^4+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b^2+2 a^2 d^2 b-2 a c d+a d \left (d b^2+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x\right ) d^5}{b^2 \left (b^4+8 a^2 d b-4 a c\right ) \left (2 a d^2 x^2+\left (d b^2+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )^2}\right )dx}{a^2 d^4}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {a^2 d^4}{\left (a d^2 x^4+b^2 d x^3+c x^2+b^3 x+a b^2\right )^2}dx}{a^2 d^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {1}{\left (a b^2+a d^2 x^4+b^3 x+b^2 d x^3+c x^2\right )^2}dx\) |
| \(\Big \downarrow \) 2492 |
| \(\displaystyle \frac {\int \left (\frac {2 a^2 \left (-4 a d^2 b^3+c d b^2+3 a d \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b+2 a d^2 (c-3 a b d) x-c \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) d^7}{b^3 \left (d^2 \left (b^4+8 a^2 d b-4 a c\right )\right )^{3/2} \left (2 a d^2 x^2+\left (b^2 d-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )}-\frac {2 a^2 \left (-4 a d^2 b^3+c d b^2-3 a d \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b+2 a d^2 (c-3 a b d) x+c \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) d^7}{b^3 \left (d^2 \left (b^4+8 a^2 d b-4 a c\right )\right )^{3/2} \left (2 a d^2 x^2+\left (d b^2+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )}+\frac {2 a^2 \left (d b^4-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b^2+2 a^2 d^2 b-2 a c d+a d \left (b^2 d-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x\right ) d^5}{b^2 \left (b^4+8 a^2 d b-4 a c\right ) \left (2 a d^2 x^2+\left (b^2 d-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )^2}+\frac {2 a^2 \left (d b^4+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b^2+2 a^2 d^2 b-2 a c d+a d \left (d b^2+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x\right ) d^5}{b^2 \left (b^4+8 a^2 d b-4 a c\right ) \left (2 a d^2 x^2+\left (d b^2+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )^2}\right )dx}{a^2 d^4}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {a^2 d^4}{\left (a d^2 x^4+b^2 d x^3+c x^2+b^3 x+a b^2\right )^2}dx}{a^2 d^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {1}{\left (a b^2+a d^2 x^4+b^3 x+b^2 d x^3+c x^2\right )^2}dx\) |
| \(\Big \downarrow \) 2492 |
| \(\displaystyle \frac {\int \left (\frac {2 a^2 \left (-4 a d^2 b^3+c d b^2+3 a d \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b+2 a d^2 (c-3 a b d) x-c \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) d^7}{b^3 \left (d^2 \left (b^4+8 a^2 d b-4 a c\right )\right )^{3/2} \left (2 a d^2 x^2+\left (b^2 d-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )}-\frac {2 a^2 \left (-4 a d^2 b^3+c d b^2-3 a d \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b+2 a d^2 (c-3 a b d) x+c \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) d^7}{b^3 \left (d^2 \left (b^4+8 a^2 d b-4 a c\right )\right )^{3/2} \left (2 a d^2 x^2+\left (d b^2+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )}+\frac {2 a^2 \left (d b^4-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b^2+2 a^2 d^2 b-2 a c d+a d \left (b^2 d-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x\right ) d^5}{b^2 \left (b^4+8 a^2 d b-4 a c\right ) \left (2 a d^2 x^2+\left (b^2 d-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )^2}+\frac {2 a^2 \left (d b^4+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b^2+2 a^2 d^2 b-2 a c d+a d \left (d b^2+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x\right ) d^5}{b^2 \left (b^4+8 a^2 d b-4 a c\right ) \left (2 a d^2 x^2+\left (d b^2+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )^2}\right )dx}{a^2 d^4}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {a^2 d^4}{\left (a d^2 x^4+b^2 d x^3+c x^2+b^3 x+a b^2\right )^2}dx}{a^2 d^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {1}{\left (a b^2+a d^2 x^4+b^3 x+b^2 d x^3+c x^2\right )^2}dx\) |
| \(\Big \downarrow \) 2492 |
| \(\displaystyle \frac {\int \left (\frac {2 a^2 \left (-4 a d^2 b^3+c d b^2+3 a d \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b+2 a d^2 (c-3 a b d) x-c \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) d^7}{b^3 \left (d^2 \left (b^4+8 a^2 d b-4 a c\right )\right )^{3/2} \left (2 a d^2 x^2+\left (b^2 d-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )}-\frac {2 a^2 \left (-4 a d^2 b^3+c d b^2-3 a d \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b+2 a d^2 (c-3 a b d) x+c \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) d^7}{b^3 \left (d^2 \left (b^4+8 a^2 d b-4 a c\right )\right )^{3/2} \left (2 a d^2 x^2+\left (d b^2+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )}+\frac {2 a^2 \left (d b^4-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b^2+2 a^2 d^2 b-2 a c d+a d \left (b^2 d-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x\right ) d^5}{b^2 \left (b^4+8 a^2 d b-4 a c\right ) \left (2 a d^2 x^2+\left (b^2 d-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )^2}+\frac {2 a^2 \left (d b^4+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b^2+2 a^2 d^2 b-2 a c d+a d \left (d b^2+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x\right ) d^5}{b^2 \left (b^4+8 a^2 d b-4 a c\right ) \left (2 a d^2 x^2+\left (d b^2+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )^2}\right )dx}{a^2 d^4}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {a^2 d^4}{\left (a d^2 x^4+b^2 d x^3+c x^2+b^3 x+a b^2\right )^2}dx}{a^2 d^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {1}{\left (a b^2+a d^2 x^4+b^3 x+b^2 d x^3+c x^2\right )^2}dx\) |
| \(\Big \downarrow \) 2492 |
| \(\displaystyle \frac {\int \left (\frac {2 a^2 \left (-4 a d^2 b^3+c d b^2+3 a d \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b+2 a d^2 (c-3 a b d) x-c \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) d^7}{b^3 \left (d^2 \left (b^4+8 a^2 d b-4 a c\right )\right )^{3/2} \left (2 a d^2 x^2+\left (b^2 d-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )}-\frac {2 a^2 \left (-4 a d^2 b^3+c d b^2-3 a d \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b+2 a d^2 (c-3 a b d) x+c \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) d^7}{b^3 \left (d^2 \left (b^4+8 a^2 d b-4 a c\right )\right )^{3/2} \left (2 a d^2 x^2+\left (d b^2+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )}+\frac {2 a^2 \left (d b^4-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b^2+2 a^2 d^2 b-2 a c d+a d \left (b^2 d-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x\right ) d^5}{b^2 \left (b^4+8 a^2 d b-4 a c\right ) \left (2 a d^2 x^2+\left (b^2 d-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )^2}+\frac {2 a^2 \left (d b^4+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b^2+2 a^2 d^2 b-2 a c d+a d \left (d b^2+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x\right ) d^5}{b^2 \left (b^4+8 a^2 d b-4 a c\right ) \left (2 a d^2 x^2+\left (d b^2+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )^2}\right )dx}{a^2 d^4}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {a^2 d^4}{\left (a d^2 x^4+b^2 d x^3+c x^2+b^3 x+a b^2\right )^2}dx}{a^2 d^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {1}{\left (a b^2+a d^2 x^4+b^3 x+b^2 d x^3+c x^2\right )^2}dx\) |
| \(\Big \downarrow \) 2492 |
| \(\displaystyle \frac {\int \left (\frac {2 a^2 \left (-4 a d^2 b^3+c d b^2+3 a d \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b+2 a d^2 (c-3 a b d) x-c \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) d^7}{b^3 \left (d^2 \left (b^4+8 a^2 d b-4 a c\right )\right )^{3/2} \left (2 a d^2 x^2+\left (b^2 d-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )}-\frac {2 a^2 \left (-4 a d^2 b^3+c d b^2-3 a d \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b+2 a d^2 (c-3 a b d) x+c \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) d^7}{b^3 \left (d^2 \left (b^4+8 a^2 d b-4 a c\right )\right )^{3/2} \left (2 a d^2 x^2+\left (d b^2+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )}+\frac {2 a^2 \left (d b^4-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b^2+2 a^2 d^2 b-2 a c d+a d \left (b^2 d-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x\right ) d^5}{b^2 \left (b^4+8 a^2 d b-4 a c\right ) \left (2 a d^2 x^2+\left (b^2 d-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )^2}+\frac {2 a^2 \left (d b^4+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b^2+2 a^2 d^2 b-2 a c d+a d \left (d b^2+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x\right ) d^5}{b^2 \left (b^4+8 a^2 d b-4 a c\right ) \left (2 a d^2 x^2+\left (d b^2+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )^2}\right )dx}{a^2 d^4}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {a^2 d^4}{\left (a d^2 x^4+b^2 d x^3+c x^2+b^3 x+a b^2\right )^2}dx}{a^2 d^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {1}{\left (a b^2+a d^2 x^4+b^3 x+b^2 d x^3+c x^2\right )^2}dx\) |
| \(\Big \downarrow \) 2492 |
| \(\displaystyle \frac {\int \left (\frac {2 a^2 \left (-4 a d^2 b^3+c d b^2+3 a d \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b+2 a d^2 (c-3 a b d) x-c \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) d^7}{b^3 \left (d^2 \left (b^4+8 a^2 d b-4 a c\right )\right )^{3/2} \left (2 a d^2 x^2+\left (b^2 d-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )}-\frac {2 a^2 \left (-4 a d^2 b^3+c d b^2-3 a d \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b+2 a d^2 (c-3 a b d) x+c \sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) d^7}{b^3 \left (d^2 \left (b^4+8 a^2 d b-4 a c\right )\right )^{3/2} \left (2 a d^2 x^2+\left (d b^2+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )}+\frac {2 a^2 \left (d b^4-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b^2+2 a^2 d^2 b-2 a c d+a d \left (b^2 d-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x\right ) d^5}{b^2 \left (b^4+8 a^2 d b-4 a c\right ) \left (2 a d^2 x^2+\left (b^2 d-\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )^2}+\frac {2 a^2 \left (d b^4+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )} b^2+2 a^2 d^2 b-2 a c d+a d \left (d b^2+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x\right ) d^5}{b^2 \left (b^4+8 a^2 d b-4 a c\right ) \left (2 a d^2 x^2+\left (d b^2+\sqrt {d^2 \left (b^4+8 a^2 d b-4 a c\right )}\right ) x+2 a b d\right )^2}\right )dx}{a^2 d^4}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {a^2 d^4}{\left (a d^2 x^4+b^2 d x^3+c x^2+b^3 x+a b^2\right )^2}dx}{a^2 d^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {1}{\left (a b^2+a d^2 x^4+b^3 x+b^2 d x^3+c x^2\right )^2}dx\) |
Input:
Int[(a*b^2 + b^3*x + c*x^2 + b^2*d*x^3 + a*d^2*x^4)^(-2),x]
Output:
$Aborted
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Px_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2 + (d_.)*(x_)^3 + (e_.)*(x_)^4) ^(p_), x_Symbol] :> Simp[e^p Int[ExpandIntegrand[Px*(b/d + ((d + Sqrt[e*( (b^2 - 4*a*c)/a) + 8*a*d*(e/b)])/(2*e))*x + x^2)^p*(b/d + ((d - Sqrt[e*((b^ 2 - 4*a*c)/a) + 8*a*d*(e/b)])/(2*e))*x + x^2)^p, x], x], x] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x] && ILtQ[p, 0] && EqQ[a*d^2 - b^2*e, 0]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.22 (sec) , antiderivative size = 722, normalized size of antiderivative = 0.52
| method | result | size |
| default | \(\frac {-\frac {2 a \,d^{2} \left (-3 b^{5} d +2 a b c d +c^{2}\right ) x^{3}}{b^{2} \left (32 a^{4} b^{3} d^{3}-28 a^{2} b^{6} d^{2}-4 b^{9} d +16 a^{3} b^{2} c \,d^{2}+20 a \,b^{5} c d -8 a^{2} b \,c^{2} d +b^{4} c^{2}-4 a \,c^{3}\right )}-\frac {\left (6 a^{2} b^{2} d^{2}-6 b^{5} d +3 a b c d +2 c^{2}\right ) d \,x^{2}}{32 a^{4} b^{3} d^{3}-28 a^{2} b^{6} d^{2}-4 b^{9} d +16 a^{3} b^{2} c \,d^{2}+20 a \,b^{5} c d -8 a^{2} b \,c^{2} d +b^{4} c^{2}-4 a \,c^{3}}+\frac {\left (8 a^{3} b^{3} d^{3}-8 a \,b^{6} d^{2}+4 a^{2} b^{2} c \,d^{2}+7 b^{5} c d -4 d \,c^{2} b a -2 c^{3}\right ) x}{b^{2} \left (32 a^{4} b^{3} d^{3}-28 a^{2} b^{6} d^{2}-4 b^{9} d +16 a^{3} b^{2} c \,d^{2}+20 a \,b^{5} c d -8 a^{2} b \,c^{2} d +b^{4} c^{2}-4 a \,c^{3}\right )}+\frac {b \left (2 a^{2} b^{2} d^{2}+4 b^{5} d -5 a b c d -c^{2}\right )}{\left (8 a^{2} b d +b^{4}-4 a c \right ) \left (4 a^{2} b^{2} d^{2}-4 b^{5} d +4 a b c d +c^{2}\right )}}{a \,d^{2} x^{4}+b^{2} d \,x^{3}+b^{3} x +b^{2} a +c \,x^{2}}+\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,d^{2} \textit {\_Z}^{4}+d \,b^{2} \textit {\_Z}^{3}+b^{3} \textit {\_Z} +\textit {\_Z}^{2} c +b^{2} a \right )}{\sum }\frac {\left (a \,d^{2} \left (3 b^{5} d -2 a b c d -c^{2}\right ) \textit {\_R}^{2}+d \,b^{2} \left (-6 a^{2} b^{2} d^{2}+3 b^{5} d -a b c d -c^{2}\right ) \textit {\_R} +12 a^{3} b^{3} d^{3}-9 a \,b^{6} d^{2}+6 a^{2} b^{2} c \,d^{2}+4 b^{5} c d -2 d \,c^{2} b a -c^{3}\right ) \ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3} a \,d^{2}+3 \textit {\_R}^{2} b^{2} d +b^{3}+2 \textit {\_R} c}\right )}{b^{2} \left (32 a^{4} b^{3} d^{3}-28 a^{2} b^{6} d^{2}-4 b^{9} d +16 a^{3} b^{2} c \,d^{2}+20 a \,b^{5} c d -8 a^{2} b \,c^{2} d +b^{4} c^{2}-4 a \,c^{3}\right )}\) | \(722\) |
| risch | \(\frac {-\frac {2 a \,d^{2} \left (-3 b^{5} d +2 a b c d +c^{2}\right ) x^{3}}{b^{2} \left (32 a^{4} b^{3} d^{3}-28 a^{2} b^{6} d^{2}-4 b^{9} d +16 a^{3} b^{2} c \,d^{2}+20 a \,b^{5} c d -8 a^{2} b \,c^{2} d +b^{4} c^{2}-4 a \,c^{3}\right )}-\frac {\left (6 a^{2} b^{2} d^{2}-6 b^{5} d +3 a b c d +2 c^{2}\right ) d \,x^{2}}{32 a^{4} b^{3} d^{3}-28 a^{2} b^{6} d^{2}-4 b^{9} d +16 a^{3} b^{2} c \,d^{2}+20 a \,b^{5} c d -8 a^{2} b \,c^{2} d +b^{4} c^{2}-4 a \,c^{3}}+\frac {\left (8 a^{3} b^{3} d^{3}-8 a \,b^{6} d^{2}+4 a^{2} b^{2} c \,d^{2}+7 b^{5} c d -4 d \,c^{2} b a -2 c^{3}\right ) x}{b^{2} \left (32 a^{4} b^{3} d^{3}-28 a^{2} b^{6} d^{2}-4 b^{9} d +16 a^{3} b^{2} c \,d^{2}+20 a \,b^{5} c d -8 a^{2} b \,c^{2} d +b^{4} c^{2}-4 a \,c^{3}\right )}+\frac {b \left (2 a^{2} b^{2} d^{2}+4 b^{5} d -5 a b c d -c^{2}\right )}{\left (8 a^{2} b d +b^{4}-4 a c \right ) \left (4 a^{2} b^{2} d^{2}-4 b^{5} d +4 a b c d +c^{2}\right )}}{a \,d^{2} x^{4}+b^{2} d \,x^{3}+b^{3} x +b^{2} a +c \,x^{2}}+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,d^{2} \textit {\_Z}^{4}+d \,b^{2} \textit {\_Z}^{3}+b^{3} \textit {\_Z} +\textit {\_Z}^{2} c +b^{2} a \right )}{\sum }\frac {\left (-\frac {a \,d^{2} \left (-3 b^{5} d +2 a b c d +c^{2}\right ) \textit {\_R}^{2}}{b^{2} \left (32 a^{4} b^{3} d^{3}-28 a^{2} b^{6} d^{2}-4 b^{9} d +16 a^{3} b^{2} c \,d^{2}+20 a \,b^{5} c d -8 a^{2} b \,c^{2} d +b^{4} c^{2}-4 a \,c^{3}\right )}-\frac {d \left (6 a^{2} b^{2} d^{2}-3 b^{5} d +a b c d +c^{2}\right ) \textit {\_R}}{32 a^{4} b^{3} d^{3}-28 a^{2} b^{6} d^{2}-4 b^{9} d +16 a^{3} b^{2} c \,d^{2}+20 a \,b^{5} c d -8 a^{2} b \,c^{2} d +b^{4} c^{2}-4 a \,c^{3}}+\frac {12 a^{3} b^{3} d^{3}-9 a \,b^{6} d^{2}+6 a^{2} b^{2} c \,d^{2}+4 b^{5} c d -2 d \,c^{2} b a -c^{3}}{b^{2} \left (32 a^{4} b^{3} d^{3}-28 a^{2} b^{6} d^{2}-4 b^{9} d +16 a^{3} b^{2} c \,d^{2}+20 a \,b^{5} c d -8 a^{2} b \,c^{2} d +b^{4} c^{2}-4 a \,c^{3}\right )}\right ) \ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3} a \,d^{2}+3 \textit {\_R}^{2} b^{2} d +b^{3}+2 \textit {\_R} c}\right )\) | \(869\) |
Input:
int(1/(a*d^2*x^4+b^2*d*x^3+b^3*x+a*b^2+c*x^2)^2,x,method=_RETURNVERBOSE)
Output:
(-2*a*d^2*(-3*b^5*d+2*a*b*c*d+c^2)/b^2/(32*a^4*b^3*d^3-28*a^2*b^6*d^2-4*b^ 9*d+16*a^3*b^2*c*d^2+20*a*b^5*c*d-8*a^2*b*c^2*d+b^4*c^2-4*a*c^3)*x^3-(6*a^ 2*b^2*d^2-6*b^5*d+3*a*b*c*d+2*c^2)*d/(32*a^4*b^3*d^3-28*a^2*b^6*d^2-4*b^9* d+16*a^3*b^2*c*d^2+20*a*b^5*c*d-8*a^2*b*c^2*d+b^4*c^2-4*a*c^3)*x^2+(8*a^3* b^3*d^3-8*a*b^6*d^2+4*a^2*b^2*c*d^2+7*b^5*c*d-4*a*b*c^2*d-2*c^3)/b^2/(32*a ^4*b^3*d^3-28*a^2*b^6*d^2-4*b^9*d+16*a^3*b^2*c*d^2+20*a*b^5*c*d-8*a^2*b*c^ 2*d+b^4*c^2-4*a*c^3)*x+b*(2*a^2*b^2*d^2+4*b^5*d-5*a*b*c*d-c^2)/(8*a^2*b*d+ b^4-4*a*c)/(4*a^2*b^2*d^2-4*b^5*d+4*a*b*c*d+c^2))/(a*d^2*x^4+b^2*d*x^3+b^3 *x+a*b^2+c*x^2)+2/b^2/(32*a^4*b^3*d^3-28*a^2*b^6*d^2-4*b^9*d+16*a^3*b^2*c* d^2+20*a*b^5*c*d-8*a^2*b*c^2*d+b^4*c^2-4*a*c^3)*sum((a*d^2*(3*b^5*d-2*a*b* c*d-c^2)*_R^2+d*b^2*(-6*a^2*b^2*d^2+3*b^5*d-a*b*c*d-c^2)*_R+12*a^3*b^3*d^3 -9*a*b^6*d^2+6*a^2*b^2*c*d^2+4*b^5*c*d-2*d*c^2*b*a-c^3)/(4*_R^3*a*d^2+3*_R ^2*b^2*d+b^3+2*_R*c)*ln(x-_R),_R=RootOf(_Z^4*a*d^2+_Z^3*b^2*d+_Z*b^3+_Z^2* c+a*b^2))
Timed out. \[ \int \frac {1}{\left (a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate(1/(a*d^2*x^4+b^2*d*x^3+b^3*x+a*b^2+c*x^2)^2,x, algorithm="fricas ")
Output:
Timed out
Timed out. \[ \int \frac {1}{\left (a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate(1/(a*d**2*x**4+b**2*d*x**3+b**3*x+a*b**2+c*x**2)**2,x)
Output:
Timed out
\[ \int \frac {1}{\left (a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4\right )^2} \, dx=\int { \frac {1}{{\left (a d^{2} x^{4} + b^{2} d x^{3} + b^{3} x + a b^{2} + c x^{2}\right )}^{2}} \,d x } \] Input:
integrate(1/(a*d^2*x^4+b^2*d*x^3+b^3*x+a*b^2+c*x^2)^2,x, algorithm="maxima ")
Output:
(2*a^2*b^5*d^2 - b^3*c^2 - 2*(a*c^2*d^2 - (3*a*b^5 - 2*a^2*b*c)*d^3)*x^3 - (6*a^2*b^4*d^3 + 2*b^2*c^2*d - 3*(2*b^7 - a*b^3*c)*d^2)*x^2 + (4*b^8 - 5* a*b^4*c)*d + (8*a^3*b^3*d^3 - 2*c^3 - 4*(2*a*b^6 - a^2*b^2*c)*d^2 + (7*b^5 *c - 4*a*b*c^2)*d)*x)/(32*a^5*b^7*d^3 + a*b^8*c^2 - 4*a^2*b^4*c^3 + (32*a^ 5*b^5*d^5 - 4*(7*a^3*b^8 - 4*a^4*b^4*c)*d^4 - 4*(a*b^11 - 5*a^2*b^7*c + 2* a^3*b^3*c^2)*d^3 + (a*b^6*c^2 - 4*a^2*b^2*c^3)*d^2)*x^4 + (32*a^4*b^7*d^4 - 4*(7*a^2*b^10 - 4*a^3*b^6*c)*d^3 - 4*(b^13 - 5*a*b^9*c + 2*a^2*b^5*c^2)* d^2 + (b^8*c^2 - 4*a*b^4*c^3)*d)*x^3 - 4*(7*a^3*b^10 - 4*a^4*b^6*c)*d^2 + (32*a^4*b^5*c*d^3 + b^6*c^3 - 4*a*b^2*c^4 - 4*(7*a^2*b^8*c - 4*a^3*b^4*c^2 )*d^2 - 4*(b^11*c - 5*a*b^7*c^2 + 2*a^2*b^3*c^3)*d)*x^2 - 4*(a*b^13 - 5*a^ 2*b^9*c + 2*a^3*b^5*c^2)*d + (32*a^4*b^8*d^3 + b^9*c^2 - 4*a*b^5*c^3 - 4*( 7*a^2*b^11 - 4*a^3*b^7*c)*d^2 - 4*(b^14 - 5*a*b^10*c + 2*a^2*b^6*c^2)*d)*x ) - 2*integrate(-(12*a^3*b^3*d^3 - c^3 - 3*(3*a*b^6 - 2*a^2*b^2*c)*d^2 - ( a*c^2*d^2 - (3*a*b^5 - 2*a^2*b*c)*d^3)*x^2 + 2*(2*b^5*c - a*b*c^2)*d - (6* a^2*b^4*d^3 + b^2*c^2*d - (3*b^7 - a*b^3*c)*d^2)*x)/(a*d^2*x^4 + b^2*d*x^3 + b^3*x + a*b^2 + c*x^2), x)/(32*a^4*b^5*d^3 + b^6*c^2 - 4*a*b^2*c^3 - 4* (7*a^2*b^8 - 4*a^3*b^4*c)*d^2 - 4*(b^11 - 5*a*b^7*c + 2*a^2*b^3*c^2)*d)
\[ \int \frac {1}{\left (a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4\right )^2} \, dx=\int { \frac {1}{{\left (a d^{2} x^{4} + b^{2} d x^{3} + b^{3} x + a b^{2} + c x^{2}\right )}^{2}} \,d x } \] Input:
integrate(1/(a*d^2*x^4+b^2*d*x^3+b^3*x+a*b^2+c*x^2)^2,x, algorithm="giac")
Output:
integrate((a*d^2*x^4 + b^2*d*x^3 + b^3*x + a*b^2 + c*x^2)^(-2), x)
Time = 25.60 (sec) , antiderivative size = 16419, normalized size of antiderivative = 11.91 \[ \int \frac {1}{\left (a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4\right )^2} \, dx=\text {Too large to display} \] Input:
int(1/(a*b^2 + b^3*x + c*x^2 + a*d^2*x^4 + b^2*d*x^3)^2,x)
Output:
symsum(log((216*a^5*b^10*d^10 - 40*a^5*c^4*d^8 - 80*a^6*b*c^3*d^9 - 72*a^4 *b^9*c*d^9 - 720*a^6*b^6*c*d^10 + 576*a^8*b^3*c*d^11 + 16*a^4*b^4*c^3*d^8 + 216*a^5*b^5*c^2*d^9 + 288*a^7*b^2*c^2*d^10)/(b^16*c^4 + 16*b^26*d^2 - 12 *a*b^12*c^5 - 8*b^21*c^2*d - 64*a^3*b^4*c^7 + 48*a^2*b^8*c^6 + 352*a^2*b^2 3*d^3 + 2320*a^4*b^20*d^4 + 2432*a^6*b^17*d^5 - 13312*a^8*b^14*d^6 + 8192* a^10*b^11*d^7 - 128*a^4*b^5*c^6*d + 704*a^3*b^9*c^5*d - 456*a^2*b^13*c^4*d - 3424*a^3*b^19*c*d^3 - 11712*a^5*b^16*c*d^4 + 11264*a^7*b^13*c*d^5 + 409 6*a^9*b^10*c*d^6 + 768*a^5*b^6*c^5*d^2 - 1216*a^4*b^10*c^4*d^2 + 1536*a^6* b^7*c^4*d^3 - 1120*a^3*b^14*c^3*d^2 - 5632*a^5*b^11*c^3*d^3 - 3072*a^7*b^8 *c^3*d^4 + 984*a^2*b^18*c^2*d^2 + 9408*a^4*b^15*c^2*d^3 + 7424*a^6*b^12*c^ 2*d^4 - 6144*a^8*b^9*c^2*d^5 + 104*a*b^17*c^3*d - 224*a*b^22*c*d^2) - root (77856768*a^13*b^23*c*d^9*z^4 - 56623104*a^15*b^20*c*d^10*z^4 - 18800640*a ^9*b^29*c*d^7*z^4 - 6704640*a^7*b^32*c*d^6*z^4 + 5308416*a^11*b^26*c*d^8*z ^4 - 969408*a^5*b^35*c*d^5*z^4 - 89088*a^5*b^15*c^9*d*z^4 - 65664*a^3*b^38 *c*d^4*z^4 + 61440*a^6*b^11*c^10*d*z^4 + 53760*a^4*b^19*c^8*d*z^4 - 17280* a^3*b^23*c^7*d*z^4 + 3120*a^2*b^27*c^6*d*z^4 + 1248*a*b^36*c^3*d^2*z^4 - 1 728*a*b^41*c*d^3*z^4 - 300*a*b^31*c^5*d*z^4 - 85131264*a^12*b^22*c^2*d^8*z ^4 - 64880640*a^9*b^24*c^3*d^6*z^4 + 58195968*a^13*b^18*c^3*d^8*z^4 + 4512 1536*a^10*b^20*c^4*d^6*z^4 + 41680896*a^10*b^25*c^2*d^7*z^4 + 32870400*a^8 *b^28*c^2*d^6*z^4 + 32194560*a^8*b^23*c^4*d^5*z^4 + 25165824*a^16*b^16*...
\[ \int \frac {1}{\left (a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4\right )^2} \, dx=\text {too large to display} \] Input:
int(1/(a*d^2*x^4+b^2*d*x^3+b^3*x+a*b^2+c*x^2)^2,x)
Output:
( - 4*int(x**3/(a**2*b**4 + 2*a**2*b**2*d**2*x**4 + a**2*d**4*x**8 + 2*a*b **5*x + 2*a*b**4*d*x**3 + 2*a*b**3*d**2*x**5 + 2*a*b**2*c*x**2 + 2*a*b**2* d**3*x**7 + 2*a*c*d**2*x**6 + b**6*x**2 + 2*b**5*d*x**4 + b**4*d**2*x**6 + 2*b**3*c*x**3 + 2*b**2*c*d*x**5 + c**2*x**4),x)*a**2*b**2*d**2 - 4*int(x* *3/(a**2*b**4 + 2*a**2*b**2*d**2*x**4 + a**2*d**4*x**8 + 2*a*b**5*x + 2*a* b**4*d*x**3 + 2*a*b**3*d**2*x**5 + 2*a*b**2*c*x**2 + 2*a*b**2*d**3*x**7 + 2*a*c*d**2*x**6 + b**6*x**2 + 2*b**5*d*x**4 + b**4*d**2*x**6 + 2*b**3*c*x* *3 + 2*b**2*c*d*x**5 + c**2*x**4),x)*a**2*d**4*x**4 - 4*int(x**3/(a**2*b** 4 + 2*a**2*b**2*d**2*x**4 + a**2*d**4*x**8 + 2*a*b**5*x + 2*a*b**4*d*x**3 + 2*a*b**3*d**2*x**5 + 2*a*b**2*c*x**2 + 2*a*b**2*d**3*x**7 + 2*a*c*d**2*x **6 + b**6*x**2 + 2*b**5*d*x**4 + b**4*d**2*x**6 + 2*b**3*c*x**3 + 2*b**2* c*d*x**5 + c**2*x**4),x)*a*b**3*d**2*x - 4*int(x**3/(a**2*b**4 + 2*a**2*b* *2*d**2*x**4 + a**2*d**4*x**8 + 2*a*b**5*x + 2*a*b**4*d*x**3 + 2*a*b**3*d* *2*x**5 + 2*a*b**2*c*x**2 + 2*a*b**2*d**3*x**7 + 2*a*c*d**2*x**6 + b**6*x* *2 + 2*b**5*d*x**4 + b**4*d**2*x**6 + 2*b**3*c*x**3 + 2*b**2*c*d*x**5 + c* *2*x**4),x)*a*b**2*d**3*x**3 - 4*int(x**3/(a**2*b**4 + 2*a**2*b**2*d**2*x* *4 + a**2*d**4*x**8 + 2*a*b**5*x + 2*a*b**4*d*x**3 + 2*a*b**3*d**2*x**5 + 2*a*b**2*c*x**2 + 2*a*b**2*d**3*x**7 + 2*a*c*d**2*x**6 + b**6*x**2 + 2*b** 5*d*x**4 + b**4*d**2*x**6 + 2*b**3*c*x**3 + 2*b**2*c*d*x**5 + c**2*x**4),x )*a*c*d**2*x**2 - 3*int(x**2/(a**2*b**4 + 2*a**2*b**2*d**2*x**4 + a**2*...