Integrand size = 59, antiderivative size = 186 \[ \int \frac {x (-a+x) (a b+(a-2 b) x)}{\left (x (-a+x) (-b+x)^2\right )^{3/4} \left (b^2 d+(a-2 b d) x+(-1+d) x^2\right )} \, dx=\frac {4 \sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{b-x}+2 \sqrt [4]{d} \arctan \left (\frac {\sqrt [4]{d} \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{3/4}}{(b-x) x (-a+x)}\right )-2 \sqrt [4]{d} \text {arctanh}\left (\frac {\sqrt [4]{d} \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{3/4}}{(b-x) x (-a+x)}\right ) \]
4*(-a*b^2*x+(2*a*b+b^2)*x^2+(-a-2*b)*x^3+x^4)^(1/4)/(b-x)+2*d^(1/4)*arctan (d^(1/4)*(-a*b^2*x+(2*a*b+b^2)*x^2+(-a-2*b)*x^3+x^4)^(3/4)/(b-x)/x/(-a+x)) -2*d^(1/4)*arctanh(d^(1/4)*(-a*b^2*x+(2*a*b+b^2)*x^2+(-a-2*b)*x^3+x^4)^(3/ 4)/(b-x)/x/(-a+x))
\[ \int \frac {x (-a+x) (a b+(a-2 b) x)}{\left (x (-a+x) (-b+x)^2\right )^{3/4} \left (b^2 d+(a-2 b d) x+(-1+d) x^2\right )} \, dx=\int \frac {x (-a+x) (a b+(a-2 b) x)}{\left (x (-a+x) (-b+x)^2\right )^{3/4} \left (b^2 d+(a-2 b d) x+(-1+d) x^2\right )} \, dx \]
Integrate[(x*(-a + x)*(a*b + (a - 2*b)*x))/((x*(-a + x)*(-b + x)^2)^(3/4)* (b^2*d + (a - 2*b*d)*x + (-1 + d)*x^2)),x]
Integrate[(x*(-a + x)*(a*b + (a - 2*b)*x))/((x*(-a + x)*(-b + x)^2)^(3/4)* (b^2*d + (a - 2*b*d)*x + (-1 + d)*x^2)), 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.
| \(\displaystyle \int \frac {x (x-a) (x (a-2 b)+a b)}{\left (x (x-a) (x-b)^2\right )^{3/4} \left (x (a-2 b d)+b^2 d+(d-1) x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{3/4} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int -\frac {(a-x) \sqrt [4]{x} (a b+(a-2 b) x)}{\left (d b^2-(1-d) x^2+(a-2 b d) x\right ) \left (x^3-(a+2 b) x^2+b (2 a+b) x-a b^2\right )^{3/4}}dx}{\left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^{3/4} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \frac {(a-x) \sqrt [4]{x} (a b+(a-2 b) x)}{\left (d b^2-(1-d) x^2+(a-2 b d) x\right ) \left (x^3-(a+2 b) x^2+b (2 a+b) x-a b^2\right )^{3/4}}dx}{\left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {4 x^{3/4} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \frac {(a-x) x (a b+(a-2 b) x)}{\left (d b^2-(1-d) x^2+(a-2 b d) x\right ) \left (x^3-(a+2 b) x^2+b (2 a+b) x-a b^2\right )^{3/4}}d\sqrt [4]{x}}{\left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {4 x^{3/4} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \left (\frac {d a^2+(b-5 b d) a+4 b^2 d}{(1-d)^2 \left (x^3-(a+2 b) x^2+b (2 a+b) x-a b^2\right )^{3/4}}+\frac {(a-2 b) x}{(1-d) \left (x^3-(a+2 b) x^2+b (2 a+b) x-a b^2\right )^{3/4}}-\frac {d \left (d a^2+(b-5 b d) a+4 b^2 d\right ) b^2+(a-2 b) d \left (a^2-(3 d b+b) a+b^2 (3 d+1)\right ) x}{(d-1)^2 \left (d b^2+(d-1) x^2+(a-2 b d) x\right ) \left (x^3-(a+2 b) x^2+b (2 a+b) x-a b^2\right )^{3/4}}\right )d\sqrt [4]{x}}{\left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 x^{3/4} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \frac {(a-x) x (a b+(a-2 b) x)}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (d b^2-(1-d) x^2+(a-2 b d) x\right )}d\sqrt [4]{x}}{\left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/4} (b-x)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \frac {\sqrt [4]{a-x} x (a b+(a-2 b) x)}{(b-x)^{3/2} \left (d b^2-(1-d) x^2+(a-2 b d) x\right )}d\sqrt [4]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/4} (b-x)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \left (-\frac {\sqrt [4]{a-x} (a-2 b)}{(1-d) (b-x)^{3/2}}-\frac {\sqrt [4]{a-x} \left ((a-2 b) d b^2+\left (a^2-(3 d b+b) a+4 b^2 d\right ) x\right )}{(d-1) (b-x)^{3/2} \left (d b^2+(d-1) x^2+(a-2 b d) x\right )}\right )d\sqrt [4]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/4} (b-x)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \frac {\sqrt [4]{a-x} x (a b+(a-2 b) x)}{(b-x)^{3/2} \left (d b^2-(1-d) x^2+(a-2 b d) x\right )}d\sqrt [4]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/4} (b-x)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \left (-\frac {\sqrt [4]{a-x} (a-2 b)}{(1-d) (b-x)^{3/2}}-\frac {\sqrt [4]{a-x} \left ((a-2 b) d b^2+\left (a^2-(3 d b+b) a+4 b^2 d\right ) x\right )}{(d-1) (b-x)^{3/2} \left (d b^2+(d-1) x^2+(a-2 b d) x\right )}\right )d\sqrt [4]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/4} (b-x)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \frac {\sqrt [4]{a-x} x (a b+(a-2 b) x)}{(b-x)^{3/2} \left (d b^2-(1-d) x^2+(a-2 b d) x\right )}d\sqrt [4]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/4} (b-x)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \left (-\frac {\sqrt [4]{a-x} (a-2 b)}{(1-d) (b-x)^{3/2}}-\frac {\sqrt [4]{a-x} \left ((a-2 b) d b^2+\left (a^2-(3 d b+b) a+4 b^2 d\right ) x\right )}{(d-1) (b-x)^{3/2} \left (d b^2+(d-1) x^2+(a-2 b d) x\right )}\right )d\sqrt [4]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/4} (b-x)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \frac {\sqrt [4]{a-x} x (a b+(a-2 b) x)}{(b-x)^{3/2} \left (d b^2-(1-d) x^2+(a-2 b d) x\right )}d\sqrt [4]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/4} (b-x)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \left (-\frac {\sqrt [4]{a-x} (a-2 b)}{(1-d) (b-x)^{3/2}}-\frac {\sqrt [4]{a-x} \left ((a-2 b) d b^2+\left (a^2-(3 d b+b) a+4 b^2 d\right ) x\right )}{(d-1) (b-x)^{3/2} \left (d b^2+(d-1) x^2+(a-2 b d) x\right )}\right )d\sqrt [4]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/4} (b-x)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \frac {\sqrt [4]{a-x} x (a b+(a-2 b) x)}{(b-x)^{3/2} \left (d b^2-(1-d) x^2+(a-2 b d) x\right )}d\sqrt [4]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/4} (b-x)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \left (-\frac {\sqrt [4]{a-x} (a-2 b)}{(1-d) (b-x)^{3/2}}-\frac {\sqrt [4]{a-x} \left ((a-2 b) d b^2+\left (a^2-(3 d b+b) a+4 b^2 d\right ) x\right )}{(d-1) (b-x)^{3/2} \left (d b^2+(d-1) x^2+(a-2 b d) x\right )}\right )d\sqrt [4]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/4} (b-x)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \frac {\sqrt [4]{a-x} x (a b+(a-2 b) x)}{(b-x)^{3/2} \left (d b^2-(1-d) x^2+(a-2 b d) x\right )}d\sqrt [4]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/4} (b-x)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \left (-\frac {\sqrt [4]{a-x} (a-2 b)}{(1-d) (b-x)^{3/2}}-\frac {\sqrt [4]{a-x} \left ((a-2 b) d b^2+\left (a^2-(3 d b+b) a+4 b^2 d\right ) x\right )}{(d-1) (b-x)^{3/2} \left (d b^2+(d-1) x^2+(a-2 b d) x\right )}\right )d\sqrt [4]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/4} (b-x)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \frac {\sqrt [4]{a-x} x (a b+(a-2 b) x)}{(b-x)^{3/2} \left (d b^2-(1-d) x^2+(a-2 b d) x\right )}d\sqrt [4]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/4} (b-x)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \left (-\frac {\sqrt [4]{a-x} (a-2 b)}{(1-d) (b-x)^{3/2}}-\frac {\sqrt [4]{a-x} \left ((a-2 b) d b^2+\left (a^2-(3 d b+b) a+4 b^2 d\right ) x\right )}{(d-1) (b-x)^{3/2} \left (d b^2+(d-1) x^2+(a-2 b d) x\right )}\right )d\sqrt [4]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/4} (b-x)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \frac {\sqrt [4]{a-x} x (a b+(a-2 b) x)}{(b-x)^{3/2} \left (d b^2-(1-d) x^2+(a-2 b d) x\right )}d\sqrt [4]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/4} (b-x)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \left (-\frac {\sqrt [4]{a-x} (a-2 b)}{(1-d) (b-x)^{3/2}}-\frac {\sqrt [4]{a-x} \left ((a-2 b) d b^2+\left (a^2-(3 d b+b) a+4 b^2 d\right ) x\right )}{(d-1) (b-x)^{3/2} \left (d b^2+(d-1) x^2+(a-2 b d) x\right )}\right )d\sqrt [4]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/4} (b-x)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \frac {\sqrt [4]{a-x} x (a b+(a-2 b) x)}{(b-x)^{3/2} \left (d b^2-(1-d) x^2+(a-2 b d) x\right )}d\sqrt [4]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/4} (b-x)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \left (-\frac {\sqrt [4]{a-x} (a-2 b)}{(1-d) (b-x)^{3/2}}-\frac {\sqrt [4]{a-x} \left ((a-2 b) d b^2+\left (a^2-(3 d b+b) a+4 b^2 d\right ) x\right )}{(d-1) (b-x)^{3/2} \left (d b^2+(d-1) x^2+(a-2 b d) x\right )}\right )d\sqrt [4]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/4} (b-x)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \frac {\sqrt [4]{a-x} x (a b+(a-2 b) x)}{(b-x)^{3/2} \left (d b^2-(1-d) x^2+(a-2 b d) x\right )}d\sqrt [4]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/4} (b-x)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \left (-\frac {\sqrt [4]{a-x} (a-2 b)}{(1-d) (b-x)^{3/2}}-\frac {\sqrt [4]{a-x} \left ((a-2 b) d b^2+\left (a^2-(3 d b+b) a+4 b^2 d\right ) x\right )}{(d-1) (b-x)^{3/2} \left (d b^2+(d-1) x^2+(a-2 b d) x\right )}\right )d\sqrt [4]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/4} (b-x)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \frac {\sqrt [4]{a-x} x (a b+(a-2 b) x)}{(b-x)^{3/2} \left (d b^2-(1-d) x^2+(a-2 b d) x\right )}d\sqrt [4]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/4} (b-x)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \left (-\frac {\sqrt [4]{a-x} (a-2 b)}{(1-d) (b-x)^{3/2}}-\frac {\sqrt [4]{a-x} \left ((a-2 b) d b^2+\left (a^2-(3 d b+b) a+4 b^2 d\right ) x\right )}{(d-1) (b-x)^{3/2} \left (d b^2+(d-1) x^2+(a-2 b d) x\right )}\right )d\sqrt [4]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/4} (b-x)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \frac {\sqrt [4]{a-x} x (a b+(a-2 b) x)}{(b-x)^{3/2} \left (d b^2-(1-d) x^2+(a-2 b d) x\right )}d\sqrt [4]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/4} (b-x)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \left (-\frac {\sqrt [4]{a-x} (a-2 b)}{(1-d) (b-x)^{3/2}}-\frac {\sqrt [4]{a-x} \left ((a-2 b) d b^2+\left (a^2-(3 d b+b) a+4 b^2 d\right ) x\right )}{(d-1) (b-x)^{3/2} \left (d b^2+(d-1) x^2+(a-2 b d) x\right )}\right )d\sqrt [4]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/4} (b-x)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \frac {\sqrt [4]{a-x} x (a b+(a-2 b) x)}{(b-x)^{3/2} \left (d b^2-(1-d) x^2+(a-2 b d) x\right )}d\sqrt [4]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
Int[(x*(-a + x)*(a*b + (a - 2*b)*x))/((x*(-a + x)*(-b + x)^2)^(3/4)*(b^2*d + (a - 2*b*d)*x + (-1 + d)*x^2)),x]
3.24.47.3.1 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[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ (r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + b*x^n)^(p*q)*(c + d*x^n)^(p*r))] Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, 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]
\[\int \frac {x \left (-a +x \right ) \left (a b +\left (a -2 b \right ) x \right )}{\left (x \left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {3}{4}} \left (b^{2} d +\left (-2 b d +a \right ) x +\left (-1+d \right ) x^{2}\right )}d x\]
Timed out. \[ \int \frac {x (-a+x) (a b+(a-2 b) x)}{\left (x (-a+x) (-b+x)^2\right )^{3/4} \left (b^2 d+(a-2 b d) x+(-1+d) x^2\right )} \, dx=\text {Timed out} \]
integrate(x*(-a+x)*(a*b+(a-2*b)*x)/(x*(-a+x)*(-b+x)^2)^(3/4)/(b^2*d+(-2*b* d+a)*x+(-1+d)*x^2),x, algorithm="fricas")
Timed out. \[ \int \frac {x (-a+x) (a b+(a-2 b) x)}{\left (x (-a+x) (-b+x)^2\right )^{3/4} \left (b^2 d+(a-2 b d) x+(-1+d) x^2\right )} \, dx=\text {Timed out} \]
integrate(x*(-a+x)*(a*b+(a-2*b)*x)/(x*(-a+x)*(-b+x)**2)**(3/4)/(b**2*d+(-2 *b*d+a)*x+(-1+d)*x**2),x)
\[ \int \frac {x (-a+x) (a b+(a-2 b) x)}{\left (x (-a+x) (-b+x)^2\right )^{3/4} \left (b^2 d+(a-2 b d) x+(-1+d) x^2\right )} \, dx=\int { -\frac {{\left (a b + {\left (a - 2 \, b\right )} x\right )} {\left (a - x\right )} x}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {3}{4}} {\left (b^{2} d + {\left (d - 1\right )} x^{2} - {\left (2 \, b d - a\right )} x\right )}} \,d x } \]
integrate(x*(-a+x)*(a*b+(a-2*b)*x)/(x*(-a+x)*(-b+x)^2)^(3/4)/(b^2*d+(-2*b* d+a)*x+(-1+d)*x^2),x, algorithm="maxima")
-integrate((a*b + (a - 2*b)*x)*(a - x)*x/((-(a - x)*(b - x)^2*x)^(3/4)*(b^ 2*d + (d - 1)*x^2 - (2*b*d - a)*x)), x)
\[ \int \frac {x (-a+x) (a b+(a-2 b) x)}{\left (x (-a+x) (-b+x)^2\right )^{3/4} \left (b^2 d+(a-2 b d) x+(-1+d) x^2\right )} \, dx=\int { -\frac {{\left (a b + {\left (a - 2 \, b\right )} x\right )} {\left (a - x\right )} x}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {3}{4}} {\left (b^{2} d + {\left (d - 1\right )} x^{2} - {\left (2 \, b d - a\right )} x\right )}} \,d x } \]
integrate(x*(-a+x)*(a*b+(a-2*b)*x)/(x*(-a+x)*(-b+x)^2)^(3/4)/(b^2*d+(-2*b* d+a)*x+(-1+d)*x^2),x, algorithm="giac")
integrate(-(a*b + (a - 2*b)*x)*(a - x)*x/((-(a - x)*(b - x)^2*x)^(3/4)*(b^ 2*d + (d - 1)*x^2 - (2*b*d - a)*x)), x)
Timed out. \[ \int \frac {x (-a+x) (a b+(a-2 b) x)}{\left (x (-a+x) (-b+x)^2\right )^{3/4} \left (b^2 d+(a-2 b d) x+(-1+d) x^2\right )} \, dx=\int -\frac {x\,\left (a-x\right )\,\left (a\,b+x\,\left (a-2\,b\right )\right )}{{\left (-x\,\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{3/4}\,\left (b^2\,d+x\,\left (a-2\,b\,d\right )+x^2\,\left (d-1\right )\right )} \,d x \]
int(-(x*(a - x)*(a*b + x*(a - 2*b)))/((-x*(a - x)*(b - x)^2)^(3/4)*(b^2*d + x*(a - 2*b*d) + x^2*(d - 1))),x)