Integrand size = 56, antiderivative size = 77 \[ \int \frac {x^2 \left (3 a b-2 (a+b) x+x^2\right )}{(x (-a+x) (-b+x))^{3/4} \left (-a b+(a+b) x-x^2+d x^3\right )} \, dx=\frac {2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{a b x+(-a-b) x^2+x^3}}\right )}{d^{3/4}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{a b x+(-a-b) x^2+x^3}}\right )}{d^{3/4}} \] Output:
2*arctan(d^(1/4)*x/(a*b*x+(-a-b)*x^2+x^3)^(1/4))/d^(3/4)-2*arctanh(d^(1/4) *x/(a*b*x+(-a-b)*x^2+x^3)^(1/4))/d^(3/4)
\[ \int \frac {x^2 \left (3 a b-2 (a+b) x+x^2\right )}{(x (-a+x) (-b+x))^{3/4} \left (-a b+(a+b) x-x^2+d x^3\right )} \, dx=\int \frac {x^2 \left (3 a b-2 (a+b) x+x^2\right )}{(x (-a+x) (-b+x))^{3/4} \left (-a b+(a+b) x-x^2+d x^3\right )} \, dx \] Input:
Integrate[(x^2*(3*a*b - 2*(a + b)*x + x^2))/((x*(-a + x)*(-b + x))^(3/4)*( -(a*b) + (a + b)*x - x^2 + d*x^3)),x]
Output:
Integrate[(x^2*(3*a*b - 2*(a + b)*x + x^2))/((x*(-a + x)*(-b + x))^(3/4)*( -(a*b) + (a + b)*x - x^2 + d*x^3)), 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.
| \(\displaystyle \int \frac {x^2 \left (-2 x (a+b)+3 a b+x^2\right )}{(x (x-a) (x-b))^{3/4} \left (x (a+b)-a b+d x^3-x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{3/4} \left (-x (a+b)+a b+x^2\right )^{3/4} \int -\frac {x^{5/4} \left (x^2-2 (a+b) x+3 a b\right )}{\left (x^2-(a+b) x+a b\right )^{3/4} \left (-d x^3+x^2-(a+b) x+a b\right )}dx}{(x (a-x) (b-x))^{3/4}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^{3/4} \left (-x (a+b)+a b+x^2\right )^{3/4} \int \frac {x^{5/4} \left (x^2-2 (a+b) x+3 a b\right )}{\left (x^2-(a+b) x+a b\right )^{3/4} \left (-d x^3+x^2-(a+b) x+a b\right )}dx}{(x (a-x) (b-x))^{3/4}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {4 x^{3/4} \left (-x (a+b)+a b+x^2\right )^{3/4} \int \frac {x^2 \left (x^2-2 (a+b) x+3 a b\right )}{\left (x^2-(a+b) x+a b\right )^{3/4} \left (-d x^3+x^2-(a+b) x+a b\right )}d\sqrt [4]{x}}{(x (a-x) (b-x))^{3/4}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {4 x^{3/4} \left (-x (a+b)+a b+x^2\right )^{3/4} \int \left (-\frac {-2 a d-2 b d+1}{d^2 \left (x^2-(a+b) x+a b\right )^{3/4}}-\frac {x}{d \left (x^2-(a+b) x+a b\right )^{3/4}}+\frac {(-3 b d-3 a (1-b d) d+1) x^2+\left (2 d a^2-(1-5 b d) a-b (1-2 b d)\right ) x+a b (-2 a d-2 b d+1)}{d^2 \left (x^2-(a+b) x+a b\right )^{3/4} \left (-d x^3+x^2-(a+b) x+a b\right )}\right )d\sqrt [4]{x}}{(x (a-x) (b-x))^{3/4}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 x^{3/4} \left (-x (a+b)+a b+x^2\right )^{3/4} \left (\frac {\left (2 a^2 d-a (1-5 b d)-b (1-2 b d)\right ) \int \frac {x}{\left (x^2-(a+b) x+a b\right )^{3/4} \left (-d x^3+x^2-a \left (\frac {b}{a}+1\right ) x+a b\right )}d\sqrt [4]{x}}{d^2}+\frac {a b (-2 a d-2 b d+1) \int \frac {1}{\left (x^2-(a+b) x+a b\right )^{3/4} \left (-d x^3+x^2-a \left (\frac {b}{a}+1\right ) x+a b\right )}d\sqrt [4]{x}}{d^2}+\frac {(-3 a d (1-b d)-3 b d+1) \int \frac {x^2}{\left (x^2-(a+b) x+a b\right )^{3/4} \left (-d x^3+x^2-a \left (\frac {b}{a}+1\right ) x+a b\right )}d\sqrt [4]{x}}{d^2}-\frac {\sqrt [4]{x} \left (1-\frac {x}{a}\right )^{3/4} \left (1-\frac {x}{b}\right )^{3/4} (-2 a d-2 b d+1) \operatorname {AppellF1}\left (\frac {1}{4},\frac {3}{4},\frac {3}{4},\frac {5}{4},\frac {2 x}{a+b+\sqrt {a^2-2 b a+b^2}},\frac {2 x}{a+b-\sqrt {a^2-2 b a+b^2}}\right )}{d^2 \left (-x (a+b)+a b+x^2\right )^{3/4}}-\frac {x^{5/4} \left (1-\frac {x}{a}\right )^{3/4} \left (1-\frac {x}{b}\right )^{3/4} \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{4},\frac {3}{4},\frac {9}{4},\frac {2 x}{a+b+\sqrt {a^2-2 b a+b^2}},\frac {2 x}{a+b-\sqrt {a^2-2 b a+b^2}}\right )}{5 d \left (-x (a+b)+a b+x^2\right )^{3/4}}\right )}{(x (a-x) (b-x))^{3/4}}\) |
Input:
Int[(x^2*(3*a*b - 2*(a + b)*x + x^2))/((x*(-a + x)*(-b + x))^(3/4)*(-(a*b) + (a + b)*x - x^2 + d*x^3)),x]
Output:
$Aborted
\[\int \frac {x^{2} \left (3 a b -2 \left (a +b \right ) x +x^{2}\right )}{\left (x \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {3}{4}} \left (-a b +\left (a +b \right ) x -x^{2}+d \,x^{3}\right )}d x\]
Input:
int(x^2*(3*a*b-2*(a+b)*x+x^2)/(x*(-a+x)*(-b+x))^(3/4)/(-a*b+(a+b)*x-x^2+d* x^3),x)
Output:
int(x^2*(3*a*b-2*(a+b)*x+x^2)/(x*(-a+x)*(-b+x))^(3/4)/(-a*b+(a+b)*x-x^2+d* x^3),x)
Timed out. \[ \int \frac {x^2 \left (3 a b-2 (a+b) x+x^2\right )}{(x (-a+x) (-b+x))^{3/4} \left (-a b+(a+b) x-x^2+d x^3\right )} \, dx=\text {Timed out} \] Input:
integrate(x^2*(3*a*b-2*(a+b)*x+x^2)/(x*(-a+x)*(-b+x))^(3/4)/(-a*b+(a+b)*x- x^2+d*x^3),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {x^2 \left (3 a b-2 (a+b) x+x^2\right )}{(x (-a+x) (-b+x))^{3/4} \left (-a b+(a+b) x-x^2+d x^3\right )} \, dx=\text {Timed out} \] Input:
integrate(x**2*(3*a*b-2*(a+b)*x+x**2)/(x*(-a+x)*(-b+x))**(3/4)/(-a*b+(a+b) *x-x**2+d*x**3),x)
Output:
Timed out
\[ \int \frac {x^2 \left (3 a b-2 (a+b) x+x^2\right )}{(x (-a+x) (-b+x))^{3/4} \left (-a b+(a+b) x-x^2+d x^3\right )} \, dx=\int { \frac {{\left (3 \, a b - 2 \, {\left (a + b\right )} x + x^{2}\right )} x^{2}}{{\left (d x^{3} - a b + {\left (a + b\right )} x - x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {3}{4}}} \,d x } \] Input:
integrate(x^2*(3*a*b-2*(a+b)*x+x^2)/(x*(-a+x)*(-b+x))^(3/4)/(-a*b+(a+b)*x- x^2+d*x^3),x, algorithm="maxima")
Output:
integrate((3*a*b - 2*(a + b)*x + x^2)*x^2/((d*x^3 - a*b + (a + b)*x - x^2) *((a - x)*(b - x)*x)^(3/4)), x)
\[ \int \frac {x^2 \left (3 a b-2 (a+b) x+x^2\right )}{(x (-a+x) (-b+x))^{3/4} \left (-a b+(a+b) x-x^2+d x^3\right )} \, dx=\int { \frac {{\left (3 \, a b - 2 \, {\left (a + b\right )} x + x^{2}\right )} x^{2}}{{\left (d x^{3} - a b + {\left (a + b\right )} x - x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {3}{4}}} \,d x } \] Input:
integrate(x^2*(3*a*b-2*(a+b)*x+x^2)/(x*(-a+x)*(-b+x))^(3/4)/(-a*b+(a+b)*x- x^2+d*x^3),x, algorithm="giac")
Output:
integrate((3*a*b - 2*(a + b)*x + x^2)*x^2/((d*x^3 - a*b + (a + b)*x - x^2) *((a - x)*(b - x)*x)^(3/4)), x)
Timed out. \[ \int \frac {x^2 \left (3 a b-2 (a+b) x+x^2\right )}{(x (-a+x) (-b+x))^{3/4} \left (-a b+(a+b) x-x^2+d x^3\right )} \, dx=-\int \frac {x^2\,\left (3\,a\,b+x^2-2\,x\,\left (a+b\right )\right )}{{\left (x\,\left (a-x\right )\,\left (b-x\right )\right )}^{3/4}\,\left (-d\,x^3+x^2+\left (-a-b\right )\,x+a\,b\right )} \,d x \] Input:
int(-(x^2*(3*a*b + x^2 - 2*x*(a + b)))/((x*(a - x)*(b - x))^(3/4)*(a*b - d *x^3 + x^2 - x*(a + b))),x)
Output:
-int((x^2*(3*a*b + x^2 - 2*x*(a + b)))/((x*(a - x)*(b - x))^(3/4)*(a*b - d *x^3 + x^2 - x*(a + b))), x)
\[ \int \frac {x^2 \left (3 a b-2 (a+b) x+x^2\right )}{(x (-a+x) (-b+x))^{3/4} \left (-a b+(a+b) x-x^2+d x^3\right )} \, dx=\int \frac {x^{2} \left (3 a b -2 \left (a +b \right ) x +x^{2}\right )}{\left (x \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {3}{4}} \left (-a b +\left (a +b \right ) x -x^{2}+d \,x^{3}\right )}d x \] Input:
int(x^2*(3*a*b-2*(a+b)*x+x^2)/(x*(-a+x)*(-b+x))^(3/4)/(-a*b+(a+b)*x-x^2+d* x^3),x)
Output:
int(x^2*(3*a*b-2*(a+b)*x+x^2)/(x*(-a+x)*(-b+x))^(3/4)/(-a*b+(a+b)*x-x^2+d* x^3),x)