Integrand size = 95, antiderivative size = 109 \[ \int \frac {(-2+(1+k) x) \left (1-(1+k) x+(a+k) x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (1-2 (1+k) x+\left (1+c+4 k+k^2\right ) x^2-\left (c+2 k+c k+2 k^2\right ) x^3+\left (-b+c k+k^2\right ) x^4\right )} \, dx=-\text {RootSum}\left [b-c \text {$\#$1}^3-\text {$\#$1}^6\&,\frac {a \log (x)-a \log \left (\sqrt [3]{x+(-1-k) x^2+k x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{x+(-1-k) x^2+k x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{c \text {$\#$1}+2 \text {$\#$1}^4}\&\right ] \]
Time = 10.63 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.89 \[ \int \frac {(-2+(1+k) x) \left (1-(1+k) x+(a+k) x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (1-2 (1+k) x+\left (1+c+4 k+k^2\right ) x^2-\left (c+2 k+c k+2 k^2\right ) x^3+\left (-b+c k+k^2\right ) x^4\right )} \, dx=-\text {RootSum}\left [b-c \text {$\#$1}^3-\text {$\#$1}^6\&,\frac {a \log (x)-a \log \left (\sqrt [3]{(-1+x) x (-1+k x)}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{(-1+x) x (-1+k x)}-x \text {$\#$1}\right ) \text {$\#$1}^3}{c \text {$\#$1}+2 \text {$\#$1}^4}\&\right ] \]
Integrate[((-2 + (1 + k)*x)*(1 - (1 + k)*x + (a + k)*x^2))/(((1 - x)*x*(1 - k*x))^(1/3)*(1 - 2*(1 + k)*x + (1 + c + 4*k + k^2)*x^2 - (c + 2*k + c*k + 2*k^2)*x^3 + (-b + c*k + k^2)*x^4)),x]
-RootSum[b - c*#1^3 - #1^6 & , (a*Log[x] - a*Log[((-1 + x)*x*(-1 + k*x))^( 1/3) - x*#1] + Log[x]*#1^3 - Log[((-1 + x)*x*(-1 + k*x))^(1/3) - x*#1]*#1^ 3)/(c*#1 + 2*#1^4) & ]
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.
| \(\displaystyle \int \frac {((k+1) x-2) \left (x^2 (a+k)-(k+1) x+1\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (x^4 \left (-b+c k+k^2\right )-x^3 \left (c k+c+2 k^2+2 k\right )+x^2 \left (c+k^2+4 k+1\right )-2 (k+1) x+1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int -\frac {(2-(k+1) x) \left ((a+k) x^2-(k+1) x+1\right )}{\sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \left (-\left ((b-k (c+k)) x^4\right )-(k+1) (c+2 k) x^3+\left (k^2+4 k+c+1\right ) x^2-2 (k+1) x+1\right )}dx}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \frac {(2-(k+1) x) \left ((a+k) x^2-(k+1) x+1\right )}{\sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \left (-\left ((b-k (c+k)) x^4\right )-(k+1) (c+2 k) x^3+\left (k^2+4 k+c+1\right ) x^2-2 (k+1) x+1\right )}dx}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \frac {\sqrt [3]{x} (2-(k+1) x) \left ((a+k) x^2-(k+1) x+1\right )}{\sqrt [3]{k x^2-(k+1) x+1} \left (-\left ((b-k (c+k)) x^4\right )-(k+1) (c+2 k) x^3+\left (k^2+4 k+c+1\right ) x^2-2 (k+1) x+1\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \left (\frac {(-a-k) (k+1) x^{10/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {(k+1)^2 \left (\frac {2 (a+k)}{(k+1)^2}+1\right ) x^{7/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {3 (-k-1) x^{4/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {2 \sqrt [3]{x}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \frac {\sqrt [3]{x} (2-(k+1) x) \left ((a+k) x^2-(k+1) x+1\right )}{\sqrt [3]{k x^2-(k+1) x+1} \left ((k (c+k)-b) x^4-(k+1) (c+2 k) x^3+\left (k^2+4 k+c+1\right ) x^2-2 (k+1) x+1\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \left (\frac {(-a-k) (k+1) x^{10/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {(k+1)^2 \left (\frac {2 (a+k)}{(k+1)^2}+1\right ) x^{7/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {3 (-k-1) x^{4/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {2 \sqrt [3]{x}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \frac {\sqrt [3]{x} (2-(k+1) x) \left ((a+k) x^2-(k+1) x+1\right )}{\sqrt [3]{k x^2-(k+1) x+1} \left ((k (c+k)-b) x^4-(k+1) (c+2 k) x^3+\left (k^2+4 k+c+1\right ) x^2-2 (k+1) x+1\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \left (\frac {(-a-k) (k+1) x^{10/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {(k+1)^2 \left (\frac {2 (a+k)}{(k+1)^2}+1\right ) x^{7/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {3 (-k-1) x^{4/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {2 \sqrt [3]{x}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \frac {\sqrt [3]{x} (2-(k+1) x) \left ((a+k) x^2-(k+1) x+1\right )}{\sqrt [3]{k x^2-(k+1) x+1} \left ((k (c+k)-b) x^4-(k+1) (c+2 k) x^3+\left (k^2+4 k+c+1\right ) x^2-2 (k+1) x+1\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \left (\frac {(-a-k) (k+1) x^{10/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {(k+1)^2 \left (\frac {2 (a+k)}{(k+1)^2}+1\right ) x^{7/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {3 (-k-1) x^{4/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {2 \sqrt [3]{x}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \frac {\sqrt [3]{x} (2-(k+1) x) \left ((a+k) x^2-(k+1) x+1\right )}{\sqrt [3]{k x^2-(k+1) x+1} \left ((k (c+k)-b) x^4-(k+1) (c+2 k) x^3+\left (k^2+4 k+c+1\right ) x^2-2 (k+1) x+1\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \left (\frac {(-a-k) (k+1) x^{10/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {(k+1)^2 \left (\frac {2 (a+k)}{(k+1)^2}+1\right ) x^{7/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {3 (-k-1) x^{4/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {2 \sqrt [3]{x}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \frac {\sqrt [3]{x} (2-(k+1) x) \left ((a+k) x^2-(k+1) x+1\right )}{\sqrt [3]{k x^2-(k+1) x+1} \left ((k (c+k)-b) x^4-(k+1) (c+2 k) x^3+\left (k^2+4 k+c+1\right ) x^2-2 (k+1) x+1\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \left (\frac {(-a-k) (k+1) x^{10/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {(k+1)^2 \left (\frac {2 (a+k)}{(k+1)^2}+1\right ) x^{7/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {3 (-k-1) x^{4/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {2 \sqrt [3]{x}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \frac {\sqrt [3]{x} (2-(k+1) x) \left ((a+k) x^2-(k+1) x+1\right )}{\sqrt [3]{k x^2-(k+1) x+1} \left ((k (c+k)-b) x^4-(k+1) (c+2 k) x^3+\left (k^2+4 k+c+1\right ) x^2-2 (k+1) x+1\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \left (\frac {(-a-k) (k+1) x^{10/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {(k+1)^2 \left (\frac {2 (a+k)}{(k+1)^2}+1\right ) x^{7/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {3 (-k-1) x^{4/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {2 \sqrt [3]{x}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \frac {\sqrt [3]{x} (2-(k+1) x) \left ((a+k) x^2-(k+1) x+1\right )}{\sqrt [3]{k x^2-(k+1) x+1} \left ((k (c+k)-b) x^4-(k+1) (c+2 k) x^3+\left (k^2+4 k+c+1\right ) x^2-2 (k+1) x+1\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \left (\frac {(-a-k) (k+1) x^{10/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {(k+1)^2 \left (\frac {2 (a+k)}{(k+1)^2}+1\right ) x^{7/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {3 (-k-1) x^{4/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {2 \sqrt [3]{x}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \frac {\sqrt [3]{x} (2-(k+1) x) \left ((a+k) x^2-(k+1) x+1\right )}{\sqrt [3]{k x^2-(k+1) x+1} \left ((k (c+k)-b) x^4-(k+1) (c+2 k) x^3+\left (k^2+4 k+c+1\right ) x^2-2 (k+1) x+1\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \left (\frac {(-a-k) (k+1) x^{10/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {(k+1)^2 \left (\frac {2 (a+k)}{(k+1)^2}+1\right ) x^{7/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {3 (-k-1) x^{4/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {2 \sqrt [3]{x}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \frac {\sqrt [3]{x} (2-(k+1) x) \left ((a+k) x^2-(k+1) x+1\right )}{\sqrt [3]{k x^2-(k+1) x+1} \left ((k (c+k)-b) x^4-(k+1) (c+2 k) x^3+\left (k^2+4 k+c+1\right ) x^2-2 (k+1) x+1\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \left (\frac {(-a-k) (k+1) x^{10/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {(k+1)^2 \left (\frac {2 (a+k)}{(k+1)^2}+1\right ) x^{7/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {3 (-k-1) x^{4/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {2 \sqrt [3]{x}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \frac {\sqrt [3]{x} (2-(k+1) x) \left ((a+k) x^2-(k+1) x+1\right )}{\sqrt [3]{k x^2-(k+1) x+1} \left ((k (c+k)-b) x^4-(k+1) (c+2 k) x^3+\left (k^2+4 k+c+1\right ) x^2-2 (k+1) x+1\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \left (\frac {(-a-k) (k+1) x^{10/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {(k+1)^2 \left (\frac {2 (a+k)}{(k+1)^2}+1\right ) x^{7/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {3 (-k-1) x^{4/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {2 \sqrt [3]{x}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \frac {\sqrt [3]{x} (2-(k+1) x) \left ((a+k) x^2-(k+1) x+1\right )}{\sqrt [3]{k x^2-(k+1) x+1} \left ((k (c+k)-b) x^4-(k+1) (c+2 k) x^3+\left (k^2+4 k+c+1\right ) x^2-2 (k+1) x+1\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \left (\frac {(-a-k) (k+1) x^{10/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {(k+1)^2 \left (\frac {2 (a+k)}{(k+1)^2}+1\right ) x^{7/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {3 (-k-1) x^{4/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {2 \sqrt [3]{x}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \frac {\sqrt [3]{x} (2-(k+1) x) \left ((a+k) x^2-(k+1) x+1\right )}{\sqrt [3]{k x^2-(k+1) x+1} \left ((k (c+k)-b) x^4-(k+1) (c+2 k) x^3+\left (k^2+4 k+c+1\right ) x^2-2 (k+1) x+1\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \left (\frac {(-a-k) (k+1) x^{10/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {(k+1)^2 \left (\frac {2 (a+k)}{(k+1)^2}+1\right ) x^{7/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {3 (-k-1) x^{4/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {2 \sqrt [3]{x}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \frac {\sqrt [3]{x} (2-(k+1) x) \left ((a+k) x^2-(k+1) x+1\right )}{\sqrt [3]{k x^2-(k+1) x+1} \left ((k (c+k)-b) x^4-(k+1) (c+2 k) x^3+\left (k^2+4 k+c+1\right ) x^2-2 (k+1) x+1\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \left (\frac {(-a-k) (k+1) x^{10/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {(k+1)^2 \left (\frac {2 (a+k)}{(k+1)^2}+1\right ) x^{7/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {3 (-k-1) x^{4/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}+\frac {2 \sqrt [3]{x}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b \left (1-\frac {k (c+k)}{b}\right ) x^4-c \left (\frac {k (c+2 k+2)}{c}+1\right ) x^3+(c+k (k+4)+1) x^2-2 (k+1) x+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
Int[((-2 + (1 + k)*x)*(1 - (1 + k)*x + (a + k)*x^2))/(((1 - x)*x*(1 - k*x) )^(1/3)*(1 - 2*(1 + k)*x + (1 + c + 4*k + k^2)*x^2 - (c + 2*k + c*k + 2*k^ 2)*x^3 + (-b + c*k + k^2)*x^4)),x]
3.16.93.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 1.42 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.52
| method | result | size |
| pseudoelliptic | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+c \,\textit {\_Z}^{3}-b \right )}{\sum }\frac {\left (\textit {\_R}^{3}+a \right ) \ln \left (\frac {-\textit {\_R} x +\left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R} \left (2 \textit {\_R}^{3}+c \right )}\) | \(57\) |
int((-2+(1+k)*x)*(1-(1+k)*x+(a+k)*x^2)/((1-x)*x*(-k*x+1))^(1/3)/(1-2*(1+k) *x+(k^2+c+4*k+1)*x^2-(c*k+2*k^2+c+2*k)*x^3+(c*k+k^2-b)*x^4),x,method=_RETU RNVERBOSE)
sum(1/_R*(_R^3+a)*ln((-_R*x+((-1+x)*x*(k*x-1))^(1/3))/x)/(2*_R^3+c),_R=Roo tOf(_Z^6+_Z^3*c-b))
Timed out. \[ \int \frac {(-2+(1+k) x) \left (1-(1+k) x+(a+k) x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (1-2 (1+k) x+\left (1+c+4 k+k^2\right ) x^2-\left (c+2 k+c k+2 k^2\right ) x^3+\left (-b+c k+k^2\right ) x^4\right )} \, dx=\text {Timed out} \]
integrate((-2+(1+k)*x)*(1-(1+k)*x+(a+k)*x^2)/((1-x)*x*(-k*x+1))^(1/3)/(1-2 *(1+k)*x+(k^2+c+4*k+1)*x^2-(c*k+2*k^2+c+2*k)*x^3+(c*k+k^2-b)*x^4),x, algor ithm="fricas")
Timed out. \[ \int \frac {(-2+(1+k) x) \left (1-(1+k) x+(a+k) x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (1-2 (1+k) x+\left (1+c+4 k+k^2\right ) x^2-\left (c+2 k+c k+2 k^2\right ) x^3+\left (-b+c k+k^2\right ) x^4\right )} \, dx=\text {Timed out} \]
integrate((-2+(1+k)*x)*(1-(1+k)*x+(a+k)*x**2)/((1-x)*x*(-k*x+1))**(1/3)/(1 -2*(1+k)*x+(k**2+c+4*k+1)*x**2-(c*k+2*k**2+c+2*k)*x**3+(c*k+k**2-b)*x**4), x)
Not integrable
Time = 0.35 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.84 \[ \int \frac {(-2+(1+k) x) \left (1-(1+k) x+(a+k) x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (1-2 (1+k) x+\left (1+c+4 k+k^2\right ) x^2-\left (c+2 k+c k+2 k^2\right ) x^3+\left (-b+c k+k^2\right ) x^4\right )} \, dx=\int { \frac {{\left ({\left (a + k\right )} x^{2} - {\left (k + 1\right )} x + 1\right )} {\left ({\left (k + 1\right )} x - 2\right )}}{{\left ({\left (c k + k^{2} - b\right )} x^{4} - {\left (c k + 2 \, k^{2} + c + 2 \, k\right )} x^{3} + {\left (k^{2} + c + 4 \, k + 1\right )} x^{2} - 2 \, {\left (k + 1\right )} x + 1\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}}} \,d x } \]
integrate((-2+(1+k)*x)*(1-(1+k)*x+(a+k)*x^2)/((1-x)*x*(-k*x+1))^(1/3)/(1-2 *(1+k)*x+(k^2+c+4*k+1)*x^2-(c*k+2*k^2+c+2*k)*x^3+(c*k+k^2-b)*x^4),x, algor ithm="maxima")
integrate(((a + k)*x^2 - (k + 1)*x + 1)*((k + 1)*x - 2)/(((c*k + k^2 - b)* x^4 - (c*k + 2*k^2 + c + 2*k)*x^3 + (k^2 + c + 4*k + 1)*x^2 - 2*(k + 1)*x + 1)*((k*x - 1)*(x - 1)*x)^(1/3)), x)
Not integrable
Time = 0.69 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.84 \[ \int \frac {(-2+(1+k) x) \left (1-(1+k) x+(a+k) x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (1-2 (1+k) x+\left (1+c+4 k+k^2\right ) x^2-\left (c+2 k+c k+2 k^2\right ) x^3+\left (-b+c k+k^2\right ) x^4\right )} \, dx=\int { \frac {{\left ({\left (a + k\right )} x^{2} - {\left (k + 1\right )} x + 1\right )} {\left ({\left (k + 1\right )} x - 2\right )}}{{\left ({\left (c k + k^{2} - b\right )} x^{4} - {\left (c k + 2 \, k^{2} + c + 2 \, k\right )} x^{3} + {\left (k^{2} + c + 4 \, k + 1\right )} x^{2} - 2 \, {\left (k + 1\right )} x + 1\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}}} \,d x } \]
integrate((-2+(1+k)*x)*(1-(1+k)*x+(a+k)*x^2)/((1-x)*x*(-k*x+1))^(1/3)/(1-2 *(1+k)*x+(k^2+c+4*k+1)*x^2-(c*k+2*k^2+c+2*k)*x^3+(c*k+k^2-b)*x^4),x, algor ithm="giac")
integrate(((a + k)*x^2 - (k + 1)*x + 1)*((k + 1)*x - 2)/(((c*k + k^2 - b)* x^4 - (c*k + 2*k^2 + c + 2*k)*x^3 + (k^2 + c + 4*k + 1)*x^2 - 2*(k + 1)*x + 1)*((k*x - 1)*(x - 1)*x)^(1/3)), x)
Not integrable
Time = 8.11 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.84 \[ \int \frac {(-2+(1+k) x) \left (1-(1+k) x+(a+k) x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (1-2 (1+k) x+\left (1+c+4 k+k^2\right ) x^2-\left (c+2 k+c k+2 k^2\right ) x^3+\left (-b+c k+k^2\right ) x^4\right )} \, dx=\int \frac {\left (x\,\left (k+1\right )-2\right )\,\left (\left (a+k\right )\,x^2+\left (-k-1\right )\,x+1\right )}{{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{1/3}\,\left (x^4\,\left (k^2+c\,k-b\right )-x^3\,\left (c+2\,k+c\,k+2\,k^2\right )+x^2\,\left (k^2+4\,k+c+1\right )-2\,x\,\left (k+1\right )+1\right )} \,d x \]
int(((x*(k + 1) - 2)*(x^2*(a + k) - x*(k + 1) + 1))/((x*(k*x - 1)*(x - 1)) ^(1/3)*(x^4*(c*k - b + k^2) - x^3*(c + 2*k + c*k + 2*k^2) + x^2*(c + 4*k + k^2 + 1) - 2*x*(k + 1) + 1)),x)