Integrand size = 75, antiderivative size = 212 \[ \int \frac {(-2+(1+k) x) \left (a-a (1+k) x+(1+a k) x^2\right )}{(-1+x) \sqrt [3]{(1-x) x (1-k x)} (-1+k x) \left (b-b (1+k) x+(-1+b k) x^2\right )} \, dx=\frac {3 \left (x-x^2-k x^2+k x^3\right )^{2/3}}{(-1+x) (-1+k x)}+\frac {\left (-\sqrt {3} a-\sqrt {3} b\right ) \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{b^{2/3}}+\frac {(a+b) \log \left (x-\sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{b^{2/3}}+\frac {(-a-b) \log \left (x^2+\sqrt [3]{b} x \sqrt [3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left (x+(-1-k) x^2+k x^3\right )^{2/3}\right )}{2 b^{2/3}} \]
3*(k*x^3-k*x^2-x^2+x)^(2/3)/(-1+x)/(k*x-1)+(-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^(2/3)+(a+b)*ln(x-b^(1 /3)*(x+(-1-k)*x^2+k*x^3)^(1/3))/b^(2/3)+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^(2/3)
Time = 29.38 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.14 \[ \int \frac {(-2+(1+k) x) \left (a-a (1+k) x+(1+a k) x^2\right )}{(-1+x) \sqrt [3]{(1-x) x (1-k x)} (-1+k x) \left (b-b (1+k) x+(-1+b k) x^2\right )} \, dx=\frac {(-1+x) \left (\frac {6 x}{-1+x}+\frac {(a+b) \sqrt [3]{\frac {x}{-1+x}} \sqrt [3]{\frac {-1+k x}{-1+x}} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} \sqrt [3]{\frac {-1+k x}{-1+x}}}{2 \left (\frac {x}{-1+x}\right )^{2/3}+\sqrt [3]{b} \sqrt [3]{\frac {-1+k x}{-1+x}}}\right )+2 \log \left (\left (\frac {x}{-1+x}\right )^{2/3}-\sqrt [3]{b} \sqrt [3]{\frac {-1+k x}{-1+x}}\right )-\log \left (\left (\frac {x}{-1+x}\right )^{4/3}+\sqrt [3]{b} \left (\frac {x}{-1+x}\right )^{2/3} \sqrt [3]{\frac {-1+k x}{-1+x}}+b^{2/3} \left (\frac {-1+k x}{-1+x}\right )^{2/3}\right )\right )}{b^{2/3}}\right )}{2 \sqrt [3]{(-1+x) x (-1+k x)}} \]
Integrate[((-2 + (1 + k)*x)*(a - a*(1 + k)*x + (1 + a*k)*x^2))/((-1 + x)*( (1 - x)*x*(1 - k*x))^(1/3)*(-1 + k*x)*(b - b*(1 + k)*x + (-1 + b*k)*x^2)), x]
((-1 + x)*((6*x)/(-1 + x) + ((a + b)*(x/(-1 + x))^(1/3)*((-1 + k*x)/(-1 + x))^(1/3)*(2*Sqrt[3]*ArcTan[(Sqrt[3]*b^(1/3)*((-1 + k*x)/(-1 + x))^(1/3))/ (2*(x/(-1 + x))^(2/3) + b^(1/3)*((-1 + k*x)/(-1 + x))^(1/3))] + 2*Log[(x/( -1 + x))^(2/3) - b^(1/3)*((-1 + k*x)/(-1 + x))^(1/3)] - Log[(x/(-1 + x))^( 4/3) + b^(1/3)*(x/(-1 + x))^(2/3)*((-1 + k*x)/(-1 + x))^(1/3) + b^(2/3)*(( -1 + k*x)/(-1 + x))^(2/3)]))/b^(2/3)))/(2*((-1 + x)*x*(-1 + k*x))^(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.
| \(\displaystyle \int \frac {((k+1) x-2) \left (x^2 (a k+1)-a (k+1) x+a\right )}{(x-1) \sqrt [3]{(1-x) x (1-k x)} (k x-1) \left (x^2 (b k-1)-b (k+1) x+b\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int -\frac {(2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x) \sqrt [3]{x} (1-k x) \sqrt [3]{k x^2-(k+1) x+1} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}dx}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \frac {(2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x) \sqrt [3]{x} (1-k x) \sqrt [3]{k x^2-(k+1) x+1} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}dx}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \frac {\sqrt [3]{x} (2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x) (1-k x) \sqrt [3]{k x^2-(k+1) x+1} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 1395 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \frac {\sqrt [3]{x} (2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x)^{4/3} (1-k x)^{4/3} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \left (\frac {(k+1) (a k+1) x^{4/3}}{(1-b k) (1-x)^{4/3} (1-k x)^{4/3}}+\frac {\left ((a+b) \left (b k^2+b+2\right )-(a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right ) x\right ) \sqrt [3]{x}}{(b k-1)^2 (1-x)^{4/3} (1-k x)^{4/3} \left ((b k-1) x^2-b (k+1) x+b\right )}-\frac {\left (b \left (k^2+1\right )+a \left ((1-2 b) k^2+4 k+1\right )+2\right ) \sqrt [3]{x}}{(1-b k)^2 (1-x)^{4/3} (1-k x)^{4/3}}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \frac {\sqrt [3]{x} (2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x)^{4/3} (1-k x)^{4/3} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \left (\frac {(k+1) (a k+1) x^{4/3}}{(1-b k) (1-x)^{4/3} (1-k x)^{4/3}}+\frac {\left ((a+b) \left (b k^2+b+2\right )-(a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right ) x\right ) \sqrt [3]{x}}{(b k-1)^2 (1-x)^{4/3} (1-k x)^{4/3} \left ((b k-1) x^2-b (k+1) x+b\right )}-\frac {\left (b \left (k^2+1\right )+a \left ((1-2 b) k^2+4 k+1\right )+2\right ) \sqrt [3]{x}}{(1-b k)^2 (1-x)^{4/3} (1-k x)^{4/3}}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \frac {\sqrt [3]{x} (2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x)^{4/3} (1-k x)^{4/3} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \left (\frac {(k+1) (a k+1) x^{4/3}}{(1-b k) (1-x)^{4/3} (1-k x)^{4/3}}+\frac {\left ((a+b) \left (b k^2+b+2\right )-(a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right ) x\right ) \sqrt [3]{x}}{(b k-1)^2 (1-x)^{4/3} (1-k x)^{4/3} \left ((b k-1) x^2-b (k+1) x+b\right )}-\frac {\left (b \left (k^2+1\right )+a \left ((1-2 b) k^2+4 k+1\right )+2\right ) \sqrt [3]{x}}{(1-b k)^2 (1-x)^{4/3} (1-k x)^{4/3}}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \frac {\sqrt [3]{x} (2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x)^{4/3} (1-k x)^{4/3} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \left (\frac {(k+1) (a k+1) x^{4/3}}{(1-b k) (1-x)^{4/3} (1-k x)^{4/3}}+\frac {\left ((a+b) \left (b k^2+b+2\right )-(a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right ) x\right ) \sqrt [3]{x}}{(b k-1)^2 (1-x)^{4/3} (1-k x)^{4/3} \left ((b k-1) x^2-b (k+1) x+b\right )}-\frac {\left (b \left (k^2+1\right )+a \left ((1-2 b) k^2+4 k+1\right )+2\right ) \sqrt [3]{x}}{(1-b k)^2 (1-x)^{4/3} (1-k x)^{4/3}}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \frac {\sqrt [3]{x} (2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x)^{4/3} (1-k x)^{4/3} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \left (\frac {(k+1) (a k+1) x^{4/3}}{(1-b k) (1-x)^{4/3} (1-k x)^{4/3}}+\frac {\left ((a+b) \left (b k^2+b+2\right )-(a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right ) x\right ) \sqrt [3]{x}}{(b k-1)^2 (1-x)^{4/3} (1-k x)^{4/3} \left ((b k-1) x^2-b (k+1) x+b\right )}-\frac {\left (b \left (k^2+1\right )+a \left ((1-2 b) k^2+4 k+1\right )+2\right ) \sqrt [3]{x}}{(1-b k)^2 (1-x)^{4/3} (1-k x)^{4/3}}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \frac {\sqrt [3]{x} (2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x)^{4/3} (1-k x)^{4/3} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \left (\frac {(k+1) (a k+1) x^{4/3}}{(1-b k) (1-x)^{4/3} (1-k x)^{4/3}}+\frac {\left ((a+b) \left (b k^2+b+2\right )-(a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right ) x\right ) \sqrt [3]{x}}{(b k-1)^2 (1-x)^{4/3} (1-k x)^{4/3} \left ((b k-1) x^2-b (k+1) x+b\right )}-\frac {\left (b \left (k^2+1\right )+a \left ((1-2 b) k^2+4 k+1\right )+2\right ) \sqrt [3]{x}}{(1-b k)^2 (1-x)^{4/3} (1-k x)^{4/3}}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \frac {\sqrt [3]{x} (2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x)^{4/3} (1-k x)^{4/3} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \left (\frac {(k+1) (a k+1) x^{4/3}}{(1-b k) (1-x)^{4/3} (1-k x)^{4/3}}+\frac {\left ((a+b) \left (b k^2+b+2\right )-(a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right ) x\right ) \sqrt [3]{x}}{(b k-1)^2 (1-x)^{4/3} (1-k x)^{4/3} \left ((b k-1) x^2-b (k+1) x+b\right )}-\frac {\left (b \left (k^2+1\right )+a \left ((1-2 b) k^2+4 k+1\right )+2\right ) \sqrt [3]{x}}{(1-b k)^2 (1-x)^{4/3} (1-k x)^{4/3}}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \frac {\sqrt [3]{x} (2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x)^{4/3} (1-k x)^{4/3} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \left (\frac {(k+1) (a k+1) x^{4/3}}{(1-b k) (1-x)^{4/3} (1-k x)^{4/3}}+\frac {\left ((a+b) \left (b k^2+b+2\right )-(a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right ) x\right ) \sqrt [3]{x}}{(b k-1)^2 (1-x)^{4/3} (1-k x)^{4/3} \left ((b k-1) x^2-b (k+1) x+b\right )}-\frac {\left (b \left (k^2+1\right )+a \left ((1-2 b) k^2+4 k+1\right )+2\right ) \sqrt [3]{x}}{(1-b k)^2 (1-x)^{4/3} (1-k x)^{4/3}}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \frac {\sqrt [3]{x} (2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x)^{4/3} (1-k x)^{4/3} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \left (\frac {(k+1) (a k+1) x^{4/3}}{(1-b k) (1-x)^{4/3} (1-k x)^{4/3}}+\frac {\left ((a+b) \left (b k^2+b+2\right )-(a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right ) x\right ) \sqrt [3]{x}}{(b k-1)^2 (1-x)^{4/3} (1-k x)^{4/3} \left ((b k-1) x^2-b (k+1) x+b\right )}-\frac {\left (b \left (k^2+1\right )+a \left ((1-2 b) k^2+4 k+1\right )+2\right ) \sqrt [3]{x}}{(1-b k)^2 (1-x)^{4/3} (1-k x)^{4/3}}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \frac {\sqrt [3]{x} (2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x)^{4/3} (1-k x)^{4/3} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \left (\frac {(k+1) (a k+1) x^{4/3}}{(1-b k) (1-x)^{4/3} (1-k x)^{4/3}}+\frac {\left ((a+b) \left (b k^2+b+2\right )-(a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right ) x\right ) \sqrt [3]{x}}{(b k-1)^2 (1-x)^{4/3} (1-k x)^{4/3} \left ((b k-1) x^2-b (k+1) x+b\right )}-\frac {\left (b \left (k^2+1\right )+a \left ((1-2 b) k^2+4 k+1\right )+2\right ) \sqrt [3]{x}}{(1-b k)^2 (1-x)^{4/3} (1-k x)^{4/3}}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \frac {\sqrt [3]{x} (2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x)^{4/3} (1-k x)^{4/3} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \left (\frac {(k+1) (a k+1) x^{4/3}}{(1-b k) (1-x)^{4/3} (1-k x)^{4/3}}+\frac {\left ((a+b) \left (b k^2+b+2\right )-(a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right ) x\right ) \sqrt [3]{x}}{(b k-1)^2 (1-x)^{4/3} (1-k x)^{4/3} \left ((b k-1) x^2-b (k+1) x+b\right )}-\frac {\left (b \left (k^2+1\right )+a \left ((1-2 b) k^2+4 k+1\right )+2\right ) \sqrt [3]{x}}{(1-b k)^2 (1-x)^{4/3} (1-k x)^{4/3}}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \frac {\sqrt [3]{x} (2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x)^{4/3} (1-k x)^{4/3} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \left (\frac {(k+1) (a k+1) x^{4/3}}{(1-b k) (1-x)^{4/3} (1-k x)^{4/3}}+\frac {\left ((a+b) \left (b k^2+b+2\right )-(a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right ) x\right ) \sqrt [3]{x}}{(b k-1)^2 (1-x)^{4/3} (1-k x)^{4/3} \left ((b k-1) x^2-b (k+1) x+b\right )}-\frac {\left (b \left (k^2+1\right )+a \left ((1-2 b) k^2+4 k+1\right )+2\right ) \sqrt [3]{x}}{(1-b k)^2 (1-x)^{4/3} (1-k x)^{4/3}}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \frac {\sqrt [3]{x} (2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x)^{4/3} (1-k x)^{4/3} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \left (\frac {(k+1) (a k+1) x^{4/3}}{(1-b k) (1-x)^{4/3} (1-k x)^{4/3}}+\frac {\left ((a+b) \left (b k^2+b+2\right )-(a+b) (k+1) \left (b \left (k^2-k+1\right )+3\right ) x\right ) \sqrt [3]{x}}{(b k-1)^2 (1-x)^{4/3} (1-k x)^{4/3} \left ((b k-1) x^2-b (k+1) x+b\right )}-\frac {\left (b \left (k^2+1\right )+a \left ((1-2 b) k^2+4 k+1\right )+2\right ) \sqrt [3]{x}}{(1-b k)^2 (1-x)^{4/3} (1-k x)^{4/3}}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \frac {\sqrt [3]{x} (2-(k+1) x) \left ((a k+1) x^2-a (k+1) x+a\right )}{(1-x)^{4/3} (1-k x)^{4/3} \left (-\left ((1-b k) x^2\right )-b (k+1) x+b\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
Int[((-2 + (1 + k)*x)*(a - a*(1 + k)*x + (1 + a*k)*x^2))/((-1 + x)*((1 - x )*x*(1 - k*x))^(1/3)*(-1 + k*x)*(b - b*(1 + k)*x + (-1 + b*k)*x^2)),x]
3.26.28.3.1 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_)*(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 = 1.47 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.78
| method | result | size |
| pseudoelliptic | \(-\frac {\left (a +b \right ) \left (-2 \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {1}{b}\right )^{\frac {1}{3}} x +2 \left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {1}{3}}\right )}{3 \left (\frac {1}{b}\right )^{\frac {1}{3}} x}\right ) \sqrt {3}+\ln \left (\frac {\left (\frac {1}{b}\right )^{\frac {2}{3}} x^{2}+\left (\frac {1}{b}\right )^{\frac {1}{3}} \left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {1}{3}} x +\left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-\left (\frac {1}{b}\right )^{\frac {1}{3}} x +\left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {1}{3}}}{x}\right )\right ) \left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {1}{3}}-6 x b \left (\frac {1}{b}\right )^{\frac {1}{3}}}{2 \left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {1}{3}} \left (\frac {1}{b}\right )^{\frac {1}{3}} b}\) | \(165\) |
int((-2+(1+k)*x)*(a-a*(1+k)*x+(a*k+1)*x^2)/(-1+x)/((1-x)*x*(-k*x+1))^(1/3) /(k*x-1)/(b-b*(1+k)*x+(b*k-1)*x^2),x,method=_RETURNVERBOSE)
-1/2*((a+b)*(-2*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)^(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)-2*ln((-(1/b)^(1/3)*x+((-1+x)*x* (k*x-1))^(1/3))/x))*((-1+x)*x*(k*x-1))^(1/3)-6*x*b*(1/b)^(1/3))/((-1+x)*x* (k*x-1))^(1/3)/(1/b)^(1/3)/b
Timed out. \[ \int \frac {(-2+(1+k) x) \left (a-a (1+k) x+(1+a k) x^2\right )}{(-1+x) \sqrt [3]{(1-x) x (1-k x)} (-1+k x) \left (b-b (1+k) x+(-1+b k) x^2\right )} \, dx=\text {Timed out} \]
integrate((-2+(1+k)*x)*(a-a*(1+k)*x+(a*k+1)*x^2)/(-1+x)/((1-x)*x*(-k*x+1)) ^(1/3)/(k*x-1)/(b-b*(1+k)*x+(b*k-1)*x^2),x, algorithm="fricas")
Timed out. \[ \int \frac {(-2+(1+k) x) \left (a-a (1+k) x+(1+a k) x^2\right )}{(-1+x) \sqrt [3]{(1-x) x (1-k x)} (-1+k x) \left (b-b (1+k) x+(-1+b k) x^2\right )} \, dx=\text {Timed out} \]
integrate((-2+(1+k)*x)*(a-a*(1+k)*x+(a*k+1)*x**2)/(-1+x)/((1-x)*x*(-k*x+1) )**(1/3)/(k*x-1)/(b-b*(1+k)*x+(b*k-1)*x**2),x)
\[ \int \frac {(-2+(1+k) x) \left (a-a (1+k) x+(1+a k) x^2\right )}{(-1+x) \sqrt [3]{(1-x) x (1-k x)} (-1+k x) \left (b-b (1+k) x+(-1+b k) x^2\right )} \, dx=\int { \frac {{\left (a {\left (k + 1\right )} x - {\left (a k + 1\right )} x^{2} - a\right )} {\left ({\left (k + 1\right )} x - 2\right )}}{{\left (b {\left (k + 1\right )} x - {\left (b k - 1\right )} x^{2} - b\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}} {\left (k x - 1\right )} {\left (x - 1\right )}} \,d x } \]
integrate((-2+(1+k)*x)*(a-a*(1+k)*x+(a*k+1)*x^2)/(-1+x)/((1-x)*x*(-k*x+1)) ^(1/3)/(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)/((b*(k + 1)*x - (b*k - 1)*x^2 - b)*((k*x - 1)*(x - 1)*x)^(1/3)*(k*x - 1)*(x - 1)), x)
\[ \int \frac {(-2+(1+k) x) \left (a-a (1+k) x+(1+a k) x^2\right )}{(-1+x) \sqrt [3]{(1-x) x (1-k x)} (-1+k x) \left (b-b (1+k) x+(-1+b k) x^2\right )} \, dx=\int { \frac {{\left (a {\left (k + 1\right )} x - {\left (a k + 1\right )} x^{2} - a\right )} {\left ({\left (k + 1\right )} x - 2\right )}}{{\left (b {\left (k + 1\right )} x - {\left (b k - 1\right )} x^{2} - b\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}} {\left (k x - 1\right )} {\left (x - 1\right )}} \,d x } \]
integrate((-2+(1+k)*x)*(a-a*(1+k)*x+(a*k+1)*x^2)/(-1+x)/((1-x)*x*(-k*x+1)) ^(1/3)/(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)/((b*(k + 1)*x - (b*k - 1)*x^2 - b)*((k*x - 1)*(x - 1)*x)^(1/3)*(k*x - 1)*(x - 1)), x)
Timed out. \[ \int \frac {(-2+(1+k) x) \left (a-a (1+k) x+(1+a k) x^2\right )}{(-1+x) \sqrt [3]{(1-x) x (1-k x)} (-1+k x) \left (b-b (1+k) x+(-1+b k) x^2\right )} \, dx=\int \frac {\left (x\,\left (k+1\right )-2\right )\,\left (\left (a\,k+1\right )\,x^2-a\,\left (k+1\right )\,x+a\right )}{\left (k\,x-1\right )\,\left (x-1\right )\,{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{1/3}\,\left (\left (b\,k-1\right )\,x^2-b\,\left (k+1\right )\,x+b\right )} \,d x \]
int(((x*(k + 1) - 2)*(a + x^2*(a*k + 1) - a*x*(k + 1)))/((k*x - 1)*(x - 1) *(x*(k*x - 1)*(x - 1))^(1/3)*(b + x^2*(b*k - 1) - b*x*(k + 1))),x)