Integrand size = 90, antiderivative size = 324 \[ \int \frac {x^3 (-b+x) \left (2 a b-3 a x+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+\left (-1+b^2 d\right ) x^4-2 b d x^5+d x^6\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}{2 a-2 x+\sqrt [6]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}\right )}{2 d^{5/6}}+\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}{-2 a+2 x+\sqrt [6]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}\right )}{2 d^{5/6}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}{a-x}\right )}{d^{5/6}}+\frac {\text {arctanh}\left (\frac {\left (a \sqrt [6]{d}-\sqrt [6]{d} x\right ) \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}{a^2-2 a x+x^2+\sqrt [3]{d} \left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}}\right )}{2 d^{5/6}} \]
-1/2*3^(1/2)*arctan(3^(1/2)*d^(1/6)*(a*b*x^2+(-a-b)*x^3+x^4)^(1/3)/(2*a-2* x+d^(1/6)*(a*b*x^2+(-a-b)*x^3+x^4)^(1/3)))/d^(5/6)+1/2*3^(1/2)*arctan(3^(1 /2)*d^(1/6)*(a*b*x^2+(-a-b)*x^3+x^4)^(1/3)/(-2*a+2*x+d^(1/6)*(a*b*x^2+(-a- b)*x^3+x^4)^(1/3)))/d^(5/6)+arctanh(d^(1/6)*(a*b*x^2+(-a-b)*x^3+x^4)^(1/3) /(a-x))/d^(5/6)+1/2*arctanh((a*d^(1/6)-d^(1/6)*x)*(a*b*x^2+(-a-b)*x^3+x^4) ^(1/3)/(a^2-2*a*x+x^2+d^(1/3)*(a*b*x^2+(-a-b)*x^3+x^4)^(2/3)))/d^(5/6)
\[ \int \frac {x^3 (-b+x) \left (2 a b-3 a x+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+\left (-1+b^2 d\right ) x^4-2 b d x^5+d x^6\right )} \, dx=\int \frac {x^3 (-b+x) \left (2 a b-3 a x+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+\left (-1+b^2 d\right ) x^4-2 b d x^5+d x^6\right )} \, dx \]
Integrate[(x^3*(-b + x)*(2*a*b - 3*a*x + x^2))/((x^2*(-a + x)*(-b + x))^(1 /3)*(-a^4 + 4*a^3*x - 6*a^2*x^2 + 4*a*x^3 + (-1 + b^2*d)*x^4 - 2*b*d*x^5 + d*x^6)),x]
Integrate[(x^3*(-b + x)*(2*a*b - 3*a*x + x^2))/((x^2*(-a + x)*(-b + x))^(1 /3)*(-a^4 + 4*a^3*x - 6*a^2*x^2 + 4*a*x^3 + (-1 + b^2*d)*x^4 - 2*b*d*x^5 + d*x^6)), x]
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.
| \(\displaystyle \int \frac {x^3 (x-b) \left (2 a b-3 a x+x^2\right )}{\sqrt [3]{x^2 (x-a) (x-b)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+x^4 \left (b^2 d-1\right )-2 b d x^5+d x^6\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \sqrt [3]{-x (a+b)+a b+x^2} \int \frac {(b-x) x^{7/3} \left (x^2-3 a x+2 a b\right )}{\sqrt [3]{x^2-(a+b) x+a b} \left (-d x^6+2 b d x^5+\left (1-b^2 d\right ) x^4-4 a x^3+6 a^2 x^2-4 a^3 x+a^4\right )}dx}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{-x (a+b)+a b+x^2} \int \frac {(b-x) x^3 \left (x^2-3 a x+2 a b\right )}{\sqrt [3]{x^2-(a+b) x+a b} \left (-d x^6+2 b d x^5+\left (1-b^2 d\right ) x^4-4 a x^3+6 a^2 x^2-4 a^3 x+a^4\right )}d\sqrt [3]{x}}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 1395 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a-x} \sqrt [3]{b-x} \int \frac {(b-x)^{2/3} x^3 \left (x^2-3 a x+2 a b\right )}{\sqrt [3]{a-x} \left (-d x^6+2 b d x^5+\left (1-b^2 d\right ) x^4-4 a x^3+6 a^2 x^2-4 a^3 x+a^4\right )}d\sqrt [3]{x}}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a-x} \sqrt [3]{b-x} \int \left (\frac {(b-x)^{2/3} x^5}{\sqrt [3]{a-x} \left (-d x^6+2 b d x^5+\left (1-b^2 d\right ) x^4-4 a x^3+6 a^2 x^2-4 a^3 x+a^4\right )}+\frac {3 a (b-x)^{2/3} x^4}{\sqrt [3]{a-x} \left (d x^6-2 b d x^5-\left (1-b^2 d\right ) x^4+4 a x^3-6 a^2 x^2+4 a^3 x-a^4\right )}+\frac {2 a b (b-x)^{2/3} x^3}{\sqrt [3]{a-x} \left (-d x^6+2 b d x^5+\left (1-b^2 d\right ) x^4-4 a x^3+6 a^2 x^2-4 a^3 x+a^4\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a-x} \sqrt [3]{b-x} \int \frac {(b-x)^{2/3} x^3 \left (x^2-3 a x+2 a b\right )}{\sqrt [3]{a-x} \left (a^4-4 x a^3+6 x^2 a^2-4 x^3 a-x^4 \left (d b^2-2 d x b+d x^2-1\right )\right )}d\sqrt [3]{x}}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a-x} \sqrt [3]{b-x} \int \left (\frac {(b-x)^{2/3} x^5}{\sqrt [3]{a-x} \left (-d x^6+2 b d x^5+\left (1-b^2 d\right ) x^4-4 a x^3+6 a^2 x^2-4 a^3 x+a^4\right )}+\frac {3 a (b-x)^{2/3} x^4}{\sqrt [3]{a-x} \left (d x^6-2 b d x^5-\left (1-b^2 d\right ) x^4+4 a x^3-6 a^2 x^2+4 a^3 x-a^4\right )}+\frac {2 a b (b-x)^{2/3} x^3}{\sqrt [3]{a-x} \left (-d x^6+2 b d x^5+\left (1-b^2 d\right ) x^4-4 a x^3+6 a^2 x^2-4 a^3 x+a^4\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a-x} \sqrt [3]{b-x} \int \frac {(b-x)^{2/3} x^3 \left (x^2-3 a x+2 a b\right )}{\sqrt [3]{a-x} \left (a^4-4 x a^3+6 x^2 a^2-4 x^3 a-x^4 \left (d b^2-2 d x b+d x^2-1\right )\right )}d\sqrt [3]{x}}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a-x} \sqrt [3]{b-x} \int \left (\frac {(b-x)^{2/3} x^5}{\sqrt [3]{a-x} \left (-d x^6+2 b d x^5+\left (1-b^2 d\right ) x^4-4 a x^3+6 a^2 x^2-4 a^3 x+a^4\right )}+\frac {3 a (b-x)^{2/3} x^4}{\sqrt [3]{a-x} \left (d x^6-2 b d x^5-\left (1-b^2 d\right ) x^4+4 a x^3-6 a^2 x^2+4 a^3 x-a^4\right )}+\frac {2 a b (b-x)^{2/3} x^3}{\sqrt [3]{a-x} \left (-d x^6+2 b d x^5+\left (1-b^2 d\right ) x^4-4 a x^3+6 a^2 x^2-4 a^3 x+a^4\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a-x} \sqrt [3]{b-x} \int \frac {(b-x)^{2/3} x^3 \left (x^2-3 a x+2 a b\right )}{\sqrt [3]{a-x} \left (a^4-4 x a^3+6 x^2 a^2-4 x^3 a-x^4 \left (d b^2-2 d x b+d x^2-1\right )\right )}d\sqrt [3]{x}}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a-x} \sqrt [3]{b-x} \int \left (\frac {(b-x)^{2/3} x^5}{\sqrt [3]{a-x} \left (-d x^6+2 b d x^5+\left (1-b^2 d\right ) x^4-4 a x^3+6 a^2 x^2-4 a^3 x+a^4\right )}+\frac {3 a (b-x)^{2/3} x^4}{\sqrt [3]{a-x} \left (d x^6-2 b d x^5-\left (1-b^2 d\right ) x^4+4 a x^3-6 a^2 x^2+4 a^3 x-a^4\right )}+\frac {2 a b (b-x)^{2/3} x^3}{\sqrt [3]{a-x} \left (-d x^6+2 b d x^5+\left (1-b^2 d\right ) x^4-4 a x^3+6 a^2 x^2-4 a^3 x+a^4\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a-x} \sqrt [3]{b-x} \int \frac {(b-x)^{2/3} x^3 \left (x^2-3 a x+2 a b\right )}{\sqrt [3]{a-x} \left (a^4-4 x a^3+6 x^2 a^2-4 x^3 a-x^4 \left (d b^2-2 d x b+d x^2-1\right )\right )}d\sqrt [3]{x}}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a-x} \sqrt [3]{b-x} \int \left (\frac {(b-x)^{2/3} x^5}{\sqrt [3]{a-x} \left (-d x^6+2 b d x^5+\left (1-b^2 d\right ) x^4-4 a x^3+6 a^2 x^2-4 a^3 x+a^4\right )}+\frac {3 a (b-x)^{2/3} x^4}{\sqrt [3]{a-x} \left (d x^6-2 b d x^5-\left (1-b^2 d\right ) x^4+4 a x^3-6 a^2 x^2+4 a^3 x-a^4\right )}+\frac {2 a b (b-x)^{2/3} x^3}{\sqrt [3]{a-x} \left (-d x^6+2 b d x^5+\left (1-b^2 d\right ) x^4-4 a x^3+6 a^2 x^2-4 a^3 x+a^4\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a-x} \sqrt [3]{b-x} \int \frac {(b-x)^{2/3} x^3 \left (x^2-3 a x+2 a b\right )}{\sqrt [3]{a-x} \left (a^4-4 x a^3+6 x^2 a^2-4 x^3 a-x^4 \left (d b^2-2 d x b+d x^2-1\right )\right )}d\sqrt [3]{x}}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a-x} \sqrt [3]{b-x} \int \left (\frac {(b-x)^{2/3} x^5}{\sqrt [3]{a-x} \left (-d x^6+2 b d x^5+\left (1-b^2 d\right ) x^4-4 a x^3+6 a^2 x^2-4 a^3 x+a^4\right )}+\frac {3 a (b-x)^{2/3} x^4}{\sqrt [3]{a-x} \left (d x^6-2 b d x^5-\left (1-b^2 d\right ) x^4+4 a x^3-6 a^2 x^2+4 a^3 x-a^4\right )}+\frac {2 a b (b-x)^{2/3} x^3}{\sqrt [3]{a-x} \left (-d x^6+2 b d x^5+\left (1-b^2 d\right ) x^4-4 a x^3+6 a^2 x^2-4 a^3 x+a^4\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a-x} \sqrt [3]{b-x} \int \frac {(b-x)^{2/3} x^3 \left (x^2-3 a x+2 a b\right )}{\sqrt [3]{a-x} \left (a^4-4 x a^3+6 x^2 a^2-4 x^3 a-x^4 \left (d b^2-2 d x b+d x^2-1\right )\right )}d\sqrt [3]{x}}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a-x} \sqrt [3]{b-x} \int \left (\frac {(b-x)^{2/3} x^5}{\sqrt [3]{a-x} \left (-d x^6+2 b d x^5+\left (1-b^2 d\right ) x^4-4 a x^3+6 a^2 x^2-4 a^3 x+a^4\right )}+\frac {3 a (b-x)^{2/3} x^4}{\sqrt [3]{a-x} \left (d x^6-2 b d x^5-\left (1-b^2 d\right ) x^4+4 a x^3-6 a^2 x^2+4 a^3 x-a^4\right )}+\frac {2 a b (b-x)^{2/3} x^3}{\sqrt [3]{a-x} \left (-d x^6+2 b d x^5+\left (1-b^2 d\right ) x^4-4 a x^3+6 a^2 x^2-4 a^3 x+a^4\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a-x} \sqrt [3]{b-x} \int \frac {(b-x)^{2/3} x^3 \left (x^2-3 a x+2 a b\right )}{\sqrt [3]{a-x} \left (a^4-4 x a^3+6 x^2 a^2-4 x^3 a-x^4 \left (d b^2-2 d x b+d x^2-1\right )\right )}d\sqrt [3]{x}}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a-x} \sqrt [3]{b-x} \int \left (\frac {(b-x)^{2/3} x^5}{\sqrt [3]{a-x} \left (-d x^6+2 b d x^5+\left (1-b^2 d\right ) x^4-4 a x^3+6 a^2 x^2-4 a^3 x+a^4\right )}+\frac {3 a (b-x)^{2/3} x^4}{\sqrt [3]{a-x} \left (d x^6-2 b d x^5-\left (1-b^2 d\right ) x^4+4 a x^3-6 a^2 x^2+4 a^3 x-a^4\right )}+\frac {2 a b (b-x)^{2/3} x^3}{\sqrt [3]{a-x} \left (-d x^6+2 b d x^5+\left (1-b^2 d\right ) x^4-4 a x^3+6 a^2 x^2-4 a^3 x+a^4\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a-x} \sqrt [3]{b-x} \int \frac {(b-x)^{2/3} x^3 \left (x^2-3 a x+2 a b\right )}{\sqrt [3]{a-x} \left (a^4-4 x a^3+6 x^2 a^2-4 x^3 a-x^4 \left (d b^2-2 d x b+d x^2-1\right )\right )}d\sqrt [3]{x}}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a-x} \sqrt [3]{b-x} \int \left (\frac {(b-x)^{2/3} x^5}{\sqrt [3]{a-x} \left (-d x^6+2 b d x^5+\left (1-b^2 d\right ) x^4-4 a x^3+6 a^2 x^2-4 a^3 x+a^4\right )}+\frac {3 a (b-x)^{2/3} x^4}{\sqrt [3]{a-x} \left (d x^6-2 b d x^5-\left (1-b^2 d\right ) x^4+4 a x^3-6 a^2 x^2+4 a^3 x-a^4\right )}+\frac {2 a b (b-x)^{2/3} x^3}{\sqrt [3]{a-x} \left (-d x^6+2 b d x^5+\left (1-b^2 d\right ) x^4-4 a x^3+6 a^2 x^2-4 a^3 x+a^4\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a-x} \sqrt [3]{b-x} \int \frac {(b-x)^{2/3} x^3 \left (x^2-3 a x+2 a b\right )}{\sqrt [3]{a-x} \left (a^4-4 x a^3+6 x^2 a^2-4 x^3 a-x^4 \left (d b^2-2 d x b+d x^2-1\right )\right )}d\sqrt [3]{x}}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a-x} \sqrt [3]{b-x} \int \left (\frac {(b-x)^{2/3} x^5}{\sqrt [3]{a-x} \left (-d x^6+2 b d x^5+\left (1-b^2 d\right ) x^4-4 a x^3+6 a^2 x^2-4 a^3 x+a^4\right )}+\frac {3 a (b-x)^{2/3} x^4}{\sqrt [3]{a-x} \left (d x^6-2 b d x^5-\left (1-b^2 d\right ) x^4+4 a x^3-6 a^2 x^2+4 a^3 x-a^4\right )}+\frac {2 a b (b-x)^{2/3} x^3}{\sqrt [3]{a-x} \left (-d x^6+2 b d x^5+\left (1-b^2 d\right ) x^4-4 a x^3+6 a^2 x^2-4 a^3 x+a^4\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a-x} \sqrt [3]{b-x} \int \frac {(b-x)^{2/3} x^3 \left (x^2-3 a x+2 a b\right )}{\sqrt [3]{a-x} \left (a^4-4 x a^3+6 x^2 a^2-4 x^3 a-x^4 \left (d b^2-2 d x b+d x^2-1\right )\right )}d\sqrt [3]{x}}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a-x} \sqrt [3]{b-x} \int \left (\frac {(b-x)^{2/3} x^5}{\sqrt [3]{a-x} \left (-d x^6+2 b d x^5+\left (1-b^2 d\right ) x^4-4 a x^3+6 a^2 x^2-4 a^3 x+a^4\right )}+\frac {3 a (b-x)^{2/3} x^4}{\sqrt [3]{a-x} \left (d x^6-2 b d x^5-\left (1-b^2 d\right ) x^4+4 a x^3-6 a^2 x^2+4 a^3 x-a^4\right )}+\frac {2 a b (b-x)^{2/3} x^3}{\sqrt [3]{a-x} \left (-d x^6+2 b d x^5+\left (1-b^2 d\right ) x^4-4 a x^3+6 a^2 x^2-4 a^3 x+a^4\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a-x} \sqrt [3]{b-x} \int \frac {(b-x)^{2/3} x^3 \left (x^2-3 a x+2 a b\right )}{\sqrt [3]{a-x} \left (a^4-4 x a^3+6 x^2 a^2-4 x^3 a-x^4 \left (d b^2-2 d x b+d x^2-1\right )\right )}d\sqrt [3]{x}}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a-x} \sqrt [3]{b-x} \int \left (\frac {(b-x)^{2/3} x^5}{\sqrt [3]{a-x} \left (-d x^6+2 b d x^5+\left (1-b^2 d\right ) x^4-4 a x^3+6 a^2 x^2-4 a^3 x+a^4\right )}+\frac {3 a (b-x)^{2/3} x^4}{\sqrt [3]{a-x} \left (d x^6-2 b d x^5-\left (1-b^2 d\right ) x^4+4 a x^3-6 a^2 x^2+4 a^3 x-a^4\right )}+\frac {2 a b (b-x)^{2/3} x^3}{\sqrt [3]{a-x} \left (-d x^6+2 b d x^5+\left (1-b^2 d\right ) x^4-4 a x^3+6 a^2 x^2-4 a^3 x+a^4\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a-x} \sqrt [3]{b-x} \int \frac {(b-x)^{2/3} x^3 \left (x^2-3 a x+2 a b\right )}{\sqrt [3]{a-x} \left (a^4-4 x a^3+6 x^2 a^2-4 x^3 a-x^4 \left (d b^2-2 d x b+d x^2-1\right )\right )}d\sqrt [3]{x}}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a-x} \sqrt [3]{b-x} \int \left (\frac {(b-x)^{2/3} x^5}{\sqrt [3]{a-x} \left (-d x^6+2 b d x^5+\left (1-b^2 d\right ) x^4-4 a x^3+6 a^2 x^2-4 a^3 x+a^4\right )}+\frac {3 a (b-x)^{2/3} x^4}{\sqrt [3]{a-x} \left (d x^6-2 b d x^5-\left (1-b^2 d\right ) x^4+4 a x^3-6 a^2 x^2+4 a^3 x-a^4\right )}+\frac {2 a b (b-x)^{2/3} x^3}{\sqrt [3]{a-x} \left (-d x^6+2 b d x^5+\left (1-b^2 d\right ) x^4-4 a x^3+6 a^2 x^2-4 a^3 x+a^4\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a-x} \sqrt [3]{b-x} \int \frac {(b-x)^{2/3} x^3 \left (x^2-3 a x+2 a b\right )}{\sqrt [3]{a-x} \left (a^4-4 x a^3+6 x^2 a^2-4 x^3 a-x^4 \left (d b^2-2 d x b+d x^2-1\right )\right )}d\sqrt [3]{x}}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a-x} \sqrt [3]{b-x} \int \left (\frac {(b-x)^{2/3} x^5}{\sqrt [3]{a-x} \left (-d x^6+2 b d x^5+\left (1-b^2 d\right ) x^4-4 a x^3+6 a^2 x^2-4 a^3 x+a^4\right )}+\frac {3 a (b-x)^{2/3} x^4}{\sqrt [3]{a-x} \left (d x^6-2 b d x^5-\left (1-b^2 d\right ) x^4+4 a x^3-6 a^2 x^2+4 a^3 x-a^4\right )}+\frac {2 a b (b-x)^{2/3} x^3}{\sqrt [3]{a-x} \left (-d x^6+2 b d x^5+\left (1-b^2 d\right ) x^4-4 a x^3+6 a^2 x^2-4 a^3 x+a^4\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
Int[(x^3*(-b + x)*(2*a*b - 3*a*x + x^2))/((x^2*(-a + x)*(-b + x))^(1/3)*(- a^4 + 4*a^3*x - 6*a^2*x^2 + 4*a*x^3 + (-1 + b^2*d)*x^4 - 2*b*d*x^5 + d*x^6 )),x]
3.30.7.3.1 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]]
\[\int \frac {x^{3} \left (-b +x \right ) \left (2 a b -3 a x +x^{2}\right )}{\left (x^{2} \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{3}} \left (-a^{4}+4 a^{3} x -6 a^{2} x^{2}+4 a \,x^{3}+\left (b^{2} d -1\right ) x^{4}-2 b d \,x^{5}+d \,x^{6}\right )}d x\]
int(x^3*(-b+x)*(2*a*b-3*a*x+x^2)/(x^2*(-a+x)*(-b+x))^(1/3)/(-a^4+4*a^3*x-6 *a^2*x^2+4*a*x^3+(b^2*d-1)*x^4-2*b*d*x^5+d*x^6),x)
int(x^3*(-b+x)*(2*a*b-3*a*x+x^2)/(x^2*(-a+x)*(-b+x))^(1/3)/(-a^4+4*a^3*x-6 *a^2*x^2+4*a*x^3+(b^2*d-1)*x^4-2*b*d*x^5+d*x^6),x)
Timed out. \[ \int \frac {x^3 (-b+x) \left (2 a b-3 a x+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+\left (-1+b^2 d\right ) x^4-2 b d x^5+d x^6\right )} \, dx=\text {Timed out} \]
integrate(x^3*(-b+x)*(2*a*b-3*a*x+x^2)/(x^2*(-a+x)*(-b+x))^(1/3)/(-a^4+4*a ^3*x-6*a^2*x^2+4*a*x^3+(b^2*d-1)*x^4-2*b*d*x^5+d*x^6),x, algorithm="fricas ")
\[ \int \frac {x^3 (-b+x) \left (2 a b-3 a x+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+\left (-1+b^2 d\right ) x^4-2 b d x^5+d x^6\right )} \, dx=\int \frac {x^{3} \left (- b + x\right ) \left (2 a b - 3 a x + x^{2}\right )}{\sqrt [3]{x^{2} \left (- a + x\right ) \left (- b + x\right )} \left (- a^{4} + 4 a^{3} x - 6 a^{2} x^{2} + 4 a x^{3} + b^{2} d x^{4} - 2 b d x^{5} + d x^{6} - x^{4}\right )}\, dx \]
integrate(x**3*(-b+x)*(2*a*b-3*a*x+x**2)/(x**2*(-a+x)*(-b+x))**(1/3)/(-a** 4+4*a**3*x-6*a**2*x**2+4*a*x**3+(b**2*d-1)*x**4-2*b*d*x**5+d*x**6),x)
Integral(x**3*(-b + x)*(2*a*b - 3*a*x + x**2)/((x**2*(-a + x)*(-b + x))**( 1/3)*(-a**4 + 4*a**3*x - 6*a**2*x**2 + 4*a*x**3 + b**2*d*x**4 - 2*b*d*x**5 + d*x**6 - x**4)), x)
\[ \int \frac {x^3 (-b+x) \left (2 a b-3 a x+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+\left (-1+b^2 d\right ) x^4-2 b d x^5+d x^6\right )} \, dx=\int { \frac {{\left (2 \, a b - 3 \, a x + x^{2}\right )} {\left (b - x\right )} x^{3}}{{\left (2 \, b d x^{5} - d x^{6} - {\left (b^{2} d - 1\right )} x^{4} + a^{4} - 4 \, a^{3} x + 6 \, a^{2} x^{2} - 4 \, a x^{3}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {1}{3}}} \,d x } \]
integrate(x^3*(-b+x)*(2*a*b-3*a*x+x^2)/(x^2*(-a+x)*(-b+x))^(1/3)/(-a^4+4*a ^3*x-6*a^2*x^2+4*a*x^3+(b^2*d-1)*x^4-2*b*d*x^5+d*x^6),x, algorithm="maxima ")
integrate((2*a*b - 3*a*x + x^2)*(b - x)*x^3/((2*b*d*x^5 - d*x^6 - (b^2*d - 1)*x^4 + a^4 - 4*a^3*x + 6*a^2*x^2 - 4*a*x^3)*((a - x)*(b - x)*x^2)^(1/3) ), x)
\[ \int \frac {x^3 (-b+x) \left (2 a b-3 a x+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+\left (-1+b^2 d\right ) x^4-2 b d x^5+d x^6\right )} \, dx=\int { \frac {{\left (2 \, a b - 3 \, a x + x^{2}\right )} {\left (b - x\right )} x^{3}}{{\left (2 \, b d x^{5} - d x^{6} - {\left (b^{2} d - 1\right )} x^{4} + a^{4} - 4 \, a^{3} x + 6 \, a^{2} x^{2} - 4 \, a x^{3}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {1}{3}}} \,d x } \]
integrate(x^3*(-b+x)*(2*a*b-3*a*x+x^2)/(x^2*(-a+x)*(-b+x))^(1/3)/(-a^4+4*a ^3*x-6*a^2*x^2+4*a*x^3+(b^2*d-1)*x^4-2*b*d*x^5+d*x^6),x, algorithm="giac")
integrate((2*a*b - 3*a*x + x^2)*(b - x)*x^3/((2*b*d*x^5 - d*x^6 - (b^2*d - 1)*x^4 + a^4 - 4*a^3*x + 6*a^2*x^2 - 4*a*x^3)*((a - x)*(b - x)*x^2)^(1/3) ), x)
Timed out. \[ \int \frac {x^3 (-b+x) \left (2 a b-3 a x+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+\left (-1+b^2 d\right ) x^4-2 b d x^5+d x^6\right )} \, dx=\int -\frac {x^3\,\left (b-x\right )\,\left (x^2-3\,a\,x+2\,a\,b\right )}{{\left (x^2\,\left (a-x\right )\,\left (b-x\right )\right )}^{1/3}\,\left (-a^4+4\,a^3\,x-6\,a^2\,x^2+4\,a\,x^3+d\,x^6-2\,b\,d\,x^5+\left (b^2\,d-1\right )\,x^4\right )} \,d x \]
int(-(x^3*(b - x)*(2*a*b - 3*a*x + x^2))/((x^2*(a - x)*(b - x))^(1/3)*(x^4 *(b^2*d - 1) + 4*a*x^3 + 4*a^3*x + d*x^6 - a^4 - 6*a^2*x^2 - 2*b*d*x^5)),x )