Integrand size = 79, antiderivative size = 203 \[ \int \frac {\left (-2 x+(1+k) x^2\right ) \left (a-a (1+k) x+(1+a k) x^2\right )}{(-1+x) ((1-x) x (1-k x))^{2/3} (-1+k x) \left (b-b (1+k) x+(-1+b k) x^2\right )} \, dx=\frac {3 x^2}{2 \left (x-x^2-k x^2+k x^3\right )^{2/3}}+\frac {\left (\sqrt {3} a+\sqrt {3} b\right ) \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{\sqrt [3]{b}}+\frac {(a+b) \log \left (x-\sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{\sqrt [3]{b}}+\frac {(-a-b) \log \left (x^2+\sqrt [3]{b} x \sqrt [3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left (x+(-1-k) x^2+k x^3\right )^{2/3}\right )}{2 \sqrt [3]{b}} \]
3/2*x^2/(k*x^3-k*x^2-x^2+x)^(2/3)+(3^(1/2)*a+3^(1/2)*b)*arctan(3^(1/2)*x/( x+2*b^(1/3)*(x+(-1-k)*x^2+k*x^3)^(1/3)))/b^(1/3)+(a+b)*ln(x-b^(1/3)*(x+(-1 -k)*x^2+k*x^3)^(1/3))/b^(1/3)+1/2*(-a-b)*ln(x^2+b^(1/3)*x*(x+(-1-k)*x^2+k* x^3)^(1/3)+b^(2/3)*(x+(-1-k)*x^2+k*x^3)^(2/3))/b^(1/3)
Time = 34.46 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.73 \[ \int \frac {\left (-2 x+(1+k) x^2\right ) \left (a-a (1+k) x+(1+a k) x^2\right )}{(-1+x) ((1-x) x (1-k x))^{2/3} (-1+k x) \left (b-b (1+k) x+(-1+b k) x^2\right )} \, dx=\frac {3 x^2}{2 ((-1+x) x (-1+k x))^{2/3}}+\frac {(a+b) \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{b} \sqrt [3]{(-1+x) x (-1+k x)}}\right )+2 \log \left (x-\sqrt [3]{b} \sqrt [3]{(-1+x) x (-1+k x)}\right )-\log \left (x^2+\sqrt [3]{b} x \sqrt [3]{(-1+x) x (-1+k x)}+b^{2/3} ((-1+x) x (-1+k x))^{2/3}\right )\right )}{2 \sqrt [3]{b}} \]
Integrate[((-2*x + (1 + k)*x^2)*(a - a*(1 + k)*x + (1 + a*k)*x^2))/((-1 + x)*((1 - x)*x*(1 - k*x))^(2/3)*(-1 + k*x)*(b - b*(1 + k)*x + (-1 + b*k)*x^ 2)),x]
(3*x^2)/(2*((-1 + x)*x*(-1 + k*x))^(2/3)) + ((a + b)*(2*Sqrt[3]*ArcTan[(Sq rt[3]*x)/(x + 2*b^(1/3)*((-1 + x)*x*(-1 + k*x))^(1/3))] + 2*Log[x - b^(1/3 )*((-1 + x)*x*(-1 + k*x))^(1/3)] - Log[x^2 + b^(1/3)*x*((-1 + x)*x*(-1 + k *x))^(1/3) + b^(2/3)*((-1 + x)*x*(-1 + k*x))^(2/3)]))/(2*b^(1/3))
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.
| \(\displaystyle \int \frac {\left ((k+1) x^2-2 x\right ) \left (x^2 (a k+1)-a (k+1) x+a\right )}{(x-1) ((1-x) x (1-k x))^{2/3} (k x-1) \left (x^2 (b k-1)-b (k+1) x+b\right )} \, dx\) |
| \(\Big \downarrow \) 2027 |
| \(\displaystyle \int \frac {x ((k+1) x-2) \left (x^2 (a k+1)-a (k+1) x+a\right )}{(x-1) ((1-x) x (1-k x))^{2/3} (k x-1) \left (x^2 (b k-1)-b (k+1) x+b\right )}dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \left (k x^2-(k+1) x+1\right )^{2/3} \int -\frac {\sqrt [3]{x} (2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x) (1-k x) \left (k x^2-(k+1) x+1\right )^{2/3} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}dx}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^{2/3} \left (k x^2-(k+1) x+1\right )^{2/3} \int \frac {\sqrt [3]{x} (2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x) (1-k x) \left (k x^2-(k+1) x+1\right )^{2/3} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}dx}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 x^{2/3} \left (k x^2-(k+1) x+1\right )^{2/3} \int \frac {x (2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x) (1-k x) \left (k x^2-(k+1) x+1\right )^{2/3} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 1395 |
| \(\displaystyle -\frac {3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3} \int \frac {x (2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x)^{5/3} (1-k x)^{5/3} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3} \int \left (\frac {(k+1) (a k+1) x^2}{(1-b k) (1-x)^{5/3} (1-k x)^{5/3}}-\frac {\left (b \left (k^2+1\right )+a \left ((1-2 b) k^2+4 k+1\right )+2\right ) x}{(1-b k)^2 (1-x)^{5/3} (1-k x)^{5/3}}+\frac {b (a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right )-(a+b) \left (\left (k^4+1\right ) b^2+4 \left (k^2+k+1\right ) b+2\right ) x}{(b k-1)^3 (1-x)^{5/3} (1-k x)^{5/3} \left ((b k-1) x^2-b (k+1) x+b\right )}+\frac {(a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right )}{(1-b k)^3 (1-x)^{5/3} (1-k x)^{5/3}}\right )d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3} \int \frac {x (2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x)^{5/3} (1-k x)^{5/3} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3} \int \left (\frac {(k+1) (a k+1) x^2}{(1-b k) (1-x)^{5/3} (1-k x)^{5/3}}-\frac {\left (b \left (k^2+1\right )+a \left ((1-2 b) k^2+4 k+1\right )+2\right ) x}{(1-b k)^2 (1-x)^{5/3} (1-k x)^{5/3}}+\frac {b (a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right )-(a+b) \left (\left (k^4+1\right ) b^2+4 \left (k^2+k+1\right ) b+2\right ) x}{(b k-1)^3 (1-x)^{5/3} (1-k x)^{5/3} \left ((b k-1) x^2-b (k+1) x+b\right )}+\frac {(a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right )}{(1-b k)^3 (1-x)^{5/3} (1-k x)^{5/3}}\right )d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3} \int \frac {x (2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x)^{5/3} (1-k x)^{5/3} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3} \int \left (\frac {(k+1) (a k+1) x^2}{(1-b k) (1-x)^{5/3} (1-k x)^{5/3}}-\frac {\left (b \left (k^2+1\right )+a \left ((1-2 b) k^2+4 k+1\right )+2\right ) x}{(1-b k)^2 (1-x)^{5/3} (1-k x)^{5/3}}+\frac {b (a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right )-(a+b) \left (\left (k^4+1\right ) b^2+4 \left (k^2+k+1\right ) b+2\right ) x}{(b k-1)^3 (1-x)^{5/3} (1-k x)^{5/3} \left ((b k-1) x^2-b (k+1) x+b\right )}+\frac {(a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right )}{(1-b k)^3 (1-x)^{5/3} (1-k x)^{5/3}}\right )d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3} \int \frac {x (2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x)^{5/3} (1-k x)^{5/3} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3} \int \left (\frac {(k+1) (a k+1) x^2}{(1-b k) (1-x)^{5/3} (1-k x)^{5/3}}-\frac {\left (b \left (k^2+1\right )+a \left ((1-2 b) k^2+4 k+1\right )+2\right ) x}{(1-b k)^2 (1-x)^{5/3} (1-k x)^{5/3}}+\frac {b (a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right )-(a+b) \left (\left (k^4+1\right ) b^2+4 \left (k^2+k+1\right ) b+2\right ) x}{(b k-1)^3 (1-x)^{5/3} (1-k x)^{5/3} \left ((b k-1) x^2-b (k+1) x+b\right )}+\frac {(a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right )}{(1-b k)^3 (1-x)^{5/3} (1-k x)^{5/3}}\right )d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3} \int \frac {x (2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x)^{5/3} (1-k x)^{5/3} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3} \int \left (\frac {(k+1) (a k+1) x^2}{(1-b k) (1-x)^{5/3} (1-k x)^{5/3}}-\frac {\left (b \left (k^2+1\right )+a \left ((1-2 b) k^2+4 k+1\right )+2\right ) x}{(1-b k)^2 (1-x)^{5/3} (1-k x)^{5/3}}+\frac {b (a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right )-(a+b) \left (\left (k^4+1\right ) b^2+4 \left (k^2+k+1\right ) b+2\right ) x}{(b k-1)^3 (1-x)^{5/3} (1-k x)^{5/3} \left ((b k-1) x^2-b (k+1) x+b\right )}+\frac {(a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right )}{(1-b k)^3 (1-x)^{5/3} (1-k x)^{5/3}}\right )d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3} \int \frac {x (2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x)^{5/3} (1-k x)^{5/3} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3} \int \left (\frac {(k+1) (a k+1) x^2}{(1-b k) (1-x)^{5/3} (1-k x)^{5/3}}-\frac {\left (b \left (k^2+1\right )+a \left ((1-2 b) k^2+4 k+1\right )+2\right ) x}{(1-b k)^2 (1-x)^{5/3} (1-k x)^{5/3}}+\frac {b (a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right )-(a+b) \left (\left (k^4+1\right ) b^2+4 \left (k^2+k+1\right ) b+2\right ) x}{(b k-1)^3 (1-x)^{5/3} (1-k x)^{5/3} \left ((b k-1) x^2-b (k+1) x+b\right )}+\frac {(a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right )}{(1-b k)^3 (1-x)^{5/3} (1-k x)^{5/3}}\right )d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3} \int \frac {x (2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x)^{5/3} (1-k x)^{5/3} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3} \int \left (\frac {(k+1) (a k+1) x^2}{(1-b k) (1-x)^{5/3} (1-k x)^{5/3}}-\frac {\left (b \left (k^2+1\right )+a \left ((1-2 b) k^2+4 k+1\right )+2\right ) x}{(1-b k)^2 (1-x)^{5/3} (1-k x)^{5/3}}+\frac {b (a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right )-(a+b) \left (\left (k^4+1\right ) b^2+4 \left (k^2+k+1\right ) b+2\right ) x}{(b k-1)^3 (1-x)^{5/3} (1-k x)^{5/3} \left ((b k-1) x^2-b (k+1) x+b\right )}+\frac {(a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right )}{(1-b k)^3 (1-x)^{5/3} (1-k x)^{5/3}}\right )d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3} \int \frac {x (2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x)^{5/3} (1-k x)^{5/3} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3} \int \left (\frac {(k+1) (a k+1) x^2}{(1-b k) (1-x)^{5/3} (1-k x)^{5/3}}-\frac {\left (b \left (k^2+1\right )+a \left ((1-2 b) k^2+4 k+1\right )+2\right ) x}{(1-b k)^2 (1-x)^{5/3} (1-k x)^{5/3}}+\frac {b (a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right )-(a+b) \left (\left (k^4+1\right ) b^2+4 \left (k^2+k+1\right ) b+2\right ) x}{(b k-1)^3 (1-x)^{5/3} (1-k x)^{5/3} \left ((b k-1) x^2-b (k+1) x+b\right )}+\frac {(a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right )}{(1-b k)^3 (1-x)^{5/3} (1-k x)^{5/3}}\right )d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3} \int \frac {x (2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x)^{5/3} (1-k x)^{5/3} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3} \int \left (\frac {(k+1) (a k+1) x^2}{(1-b k) (1-x)^{5/3} (1-k x)^{5/3}}-\frac {\left (b \left (k^2+1\right )+a \left ((1-2 b) k^2+4 k+1\right )+2\right ) x}{(1-b k)^2 (1-x)^{5/3} (1-k x)^{5/3}}+\frac {b (a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right )-(a+b) \left (\left (k^4+1\right ) b^2+4 \left (k^2+k+1\right ) b+2\right ) x}{(b k-1)^3 (1-x)^{5/3} (1-k x)^{5/3} \left ((b k-1) x^2-b (k+1) x+b\right )}+\frac {(a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right )}{(1-b k)^3 (1-x)^{5/3} (1-k x)^{5/3}}\right )d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3} \int \frac {x (2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x)^{5/3} (1-k x)^{5/3} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3} \int \left (\frac {(k+1) (a k+1) x^2}{(1-b k) (1-x)^{5/3} (1-k x)^{5/3}}-\frac {\left (b \left (k^2+1\right )+a \left ((1-2 b) k^2+4 k+1\right )+2\right ) x}{(1-b k)^2 (1-x)^{5/3} (1-k x)^{5/3}}+\frac {b (a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right )-(a+b) \left (\left (k^4+1\right ) b^2+4 \left (k^2+k+1\right ) b+2\right ) x}{(b k-1)^3 (1-x)^{5/3} (1-k x)^{5/3} \left ((b k-1) x^2-b (k+1) x+b\right )}+\frac {(a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right )}{(1-b k)^3 (1-x)^{5/3} (1-k x)^{5/3}}\right )d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3} \int \frac {x (2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x)^{5/3} (1-k x)^{5/3} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3} \int \left (\frac {(k+1) (a k+1) x^2}{(1-b k) (1-x)^{5/3} (1-k x)^{5/3}}-\frac {\left (b \left (k^2+1\right )+a \left ((1-2 b) k^2+4 k+1\right )+2\right ) x}{(1-b k)^2 (1-x)^{5/3} (1-k x)^{5/3}}+\frac {b (a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right )-(a+b) \left (\left (k^4+1\right ) b^2+4 \left (k^2+k+1\right ) b+2\right ) x}{(b k-1)^3 (1-x)^{5/3} (1-k x)^{5/3} \left ((b k-1) x^2-b (k+1) x+b\right )}+\frac {(a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right )}{(1-b k)^3 (1-x)^{5/3} (1-k x)^{5/3}}\right )d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3} \int \frac {x (2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x)^{5/3} (1-k x)^{5/3} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3} \int \left (\frac {(k+1) (a k+1) x^2}{(1-b k) (1-x)^{5/3} (1-k x)^{5/3}}-\frac {\left (b \left (k^2+1\right )+a \left ((1-2 b) k^2+4 k+1\right )+2\right ) x}{(1-b k)^2 (1-x)^{5/3} (1-k x)^{5/3}}+\frac {b (a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right )-(a+b) \left (\left (k^4+1\right ) b^2+4 \left (k^2+k+1\right ) b+2\right ) x}{(b k-1)^3 (1-x)^{5/3} (1-k x)^{5/3} \left ((b k-1) x^2-b (k+1) x+b\right )}+\frac {(a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right )}{(1-b k)^3 (1-x)^{5/3} (1-k x)^{5/3}}\right )d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3} \int \frac {x (2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x)^{5/3} (1-k x)^{5/3} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3} \int \left (\frac {(k+1) (a k+1) x^2}{(1-b k) (1-x)^{5/3} (1-k x)^{5/3}}-\frac {\left (b \left (k^2+1\right )+a \left ((1-2 b) k^2+4 k+1\right )+2\right ) x}{(1-b k)^2 (1-x)^{5/3} (1-k x)^{5/3}}+\frac {b (a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right )-(a+b) \left (\left (k^4+1\right ) b^2+4 \left (k^2+k+1\right ) b+2\right ) x}{(b k-1)^3 (1-x)^{5/3} (1-k x)^{5/3} \left ((b k-1) x^2-b (k+1) x+b\right )}+\frac {(a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right )}{(1-b k)^3 (1-x)^{5/3} (1-k x)^{5/3}}\right )d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\) |
Int[((-2*x + (1 + k)*x^2)*(a - a*(1 + k)*x + (1 + a*k)*x^2))/((-1 + x)*((1 - x)*x*(1 - k*x))^(2/3)*(-1 + k*x)*(b - b*(1 + k)*x + (-1 + b*k)*x^2)),x]
3.25.72.3.1 Defintions of rubi rules used
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_)*((d_) + (e_.)*( x_)^(n_))^(q_.), x_Symbol] :> Simp[(a + b*x^n + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + c*(x^n/e))^FracPart[p]) Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && E qQ[n2, 2*n] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && !(EqQ[q, 1] && EqQ[n, 2])
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
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]]
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 1.01 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.83
| method | result | size |
| pseudoelliptic | \(-\frac {-\frac {3 x^{2} b \left (\frac {1}{b}\right )^{\frac {2}{3}}}{2}+\left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {2}{3}} \left (a +b \right ) \left (\arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {1}{b}\right )^{\frac {1}{3}} x +2 \left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {1}{3}}\right )}{3 \left (\frac {1}{b}\right )^{\frac {1}{3}} x}\right ) \sqrt {3}-\ln \left (\frac {-\left (\frac {1}{b}\right )^{\frac {1}{3}} x +\left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {1}{3}}}{x}\right )+\frac {\ln \left (\frac {\left (\frac {1}{b}\right )^{\frac {2}{3}} x^{2}+\left (\frac {1}{b}\right )^{\frac {1}{3}} \left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {1}{3}} x +\left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{2}\right )}{\left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {2}{3}} \left (\frac {1}{b}\right )^{\frac {2}{3}} b}\) | \(168\) |
int((-2*x+(1+k)*x^2)*(a-a*(1+k)*x+(a*k+1)*x^2)/(-1+x)/((1-x)*x*(-k*x+1))^( 2/3)/(k*x-1)/(b-b*(1+k)*x+(b*k-1)*x^2),x,method=_RETURNVERBOSE)
-1/((-1+x)*x*(k*x-1))^(2/3)*(-3/2*x^2*b*(1/b)^(2/3)+((-1+x)*x*(k*x-1))^(2/ 3)*(a+b)*(arctan(1/3*3^(1/2)*((1/b)^(1/3)*x+2*((-1+x)*x*(k*x-1))^(1/3))/(1 /b)^(1/3)/x)*3^(1/2)-ln((-(1/b)^(1/3)*x+((-1+x)*x*(k*x-1))^(1/3))/x)+1/2*l n(((1/b)^(2/3)*x^2+(1/b)^(1/3)*((-1+x)*x*(k*x-1))^(1/3)*x+((-1+x)*x*(k*x-1 ))^(2/3))/x^2)))/(1/b)^(2/3)/b
Timed out. \[ \int \frac {\left (-2 x+(1+k) x^2\right ) \left (a-a (1+k) x+(1+a k) x^2\right )}{(-1+x) ((1-x) x (1-k x))^{2/3} (-1+k x) \left (b-b (1+k) x+(-1+b k) x^2\right )} \, dx=\text {Timed out} \]
integrate((-2*x+(1+k)*x^2)*(a-a*(1+k)*x+(a*k+1)*x^2)/(-1+x)/((1-x)*x*(-k*x +1))^(2/3)/(k*x-1)/(b-b*(1+k)*x+(b*k-1)*x^2),x, algorithm="fricas")
Timed out. \[ \int \frac {\left (-2 x+(1+k) x^2\right ) \left (a-a (1+k) x+(1+a k) x^2\right )}{(-1+x) ((1-x) x (1-k x))^{2/3} (-1+k x) \left (b-b (1+k) x+(-1+b k) x^2\right )} \, dx=\text {Timed out} \]
integrate((-2*x+(1+k)*x**2)*(a-a*(1+k)*x+(a*k+1)*x**2)/(-1+x)/((1-x)*x*(-k *x+1))**(2/3)/(k*x-1)/(b-b*(1+k)*x+(b*k-1)*x**2),x)
\[ \int \frac {\left (-2 x+(1+k) x^2\right ) \left (a-a (1+k) x+(1+a k) x^2\right )}{(-1+x) ((1-x) x (1-k x))^{2/3} (-1+k x) \left (b-b (1+k) x+(-1+b k) x^2\right )} \, dx=\int { \frac {{\left (a {\left (k + 1\right )} x - {\left (a k + 1\right )} x^{2} - a\right )} {\left ({\left (k + 1\right )} x^{2} - 2 \, x\right )}}{{\left (b {\left (k + 1\right )} x - {\left (b k - 1\right )} x^{2} - b\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {2}{3}} {\left (k x - 1\right )} {\left (x - 1\right )}} \,d x } \]
integrate((-2*x+(1+k)*x^2)*(a-a*(1+k)*x+(a*k+1)*x^2)/(-1+x)/((1-x)*x*(-k*x +1))^(2/3)/(k*x-1)/(b-b*(1+k)*x+(b*k-1)*x^2),x, algorithm="maxima")
integrate((a*(k + 1)*x - (a*k + 1)*x^2 - a)*((k + 1)*x^2 - 2*x)/((b*(k + 1 )*x - (b*k - 1)*x^2 - b)*((k*x - 1)*(x - 1)*x)^(2/3)*(k*x - 1)*(x - 1)), x )
\[ \int \frac {\left (-2 x+(1+k) x^2\right ) \left (a-a (1+k) x+(1+a k) x^2\right )}{(-1+x) ((1-x) x (1-k x))^{2/3} (-1+k x) \left (b-b (1+k) x+(-1+b k) x^2\right )} \, dx=\int { \frac {{\left (a {\left (k + 1\right )} x - {\left (a k + 1\right )} x^{2} - a\right )} {\left ({\left (k + 1\right )} x^{2} - 2 \, x\right )}}{{\left (b {\left (k + 1\right )} x - {\left (b k - 1\right )} x^{2} - b\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {2}{3}} {\left (k x - 1\right )} {\left (x - 1\right )}} \,d x } \]
integrate((-2*x+(1+k)*x^2)*(a-a*(1+k)*x+(a*k+1)*x^2)/(-1+x)/((1-x)*x*(-k*x +1))^(2/3)/(k*x-1)/(b-b*(1+k)*x+(b*k-1)*x^2),x, algorithm="giac")
integrate((a*(k + 1)*x - (a*k + 1)*x^2 - a)*((k + 1)*x^2 - 2*x)/((b*(k + 1 )*x - (b*k - 1)*x^2 - b)*((k*x - 1)*(x - 1)*x)^(2/3)*(k*x - 1)*(x - 1)), x )
Timed out. \[ \int \frac {\left (-2 x+(1+k) x^2\right ) \left (a-a (1+k) x+(1+a k) x^2\right )}{(-1+x) ((1-x) x (1-k x))^{2/3} (-1+k x) \left (b-b (1+k) x+(-1+b k) x^2\right )} \, dx=-\int \frac {\left (2\,x-x^2\,\left (k+1\right )\right )\,\left (\left (a\,k+1\right )\,x^2-a\,\left (k+1\right )\,x+a\right )}{\left (k\,x-1\right )\,\left (x-1\right )\,{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{2/3}\,\left (\left (b\,k-1\right )\,x^2-b\,\left (k+1\right )\,x+b\right )} \,d x \]
int(-((2*x - x^2*(k + 1))*(a + x^2*(a*k + 1) - a*x*(k + 1)))/((k*x - 1)*(x - 1)*(x*(k*x - 1)*(x - 1))^(2/3)*(b + x^2*(b*k - 1) - b*x*(k + 1))),x)