Integrand size = 93, antiderivative size = 113 \[ \int \frac {\left (a^2-2 a x+x^2\right ) \left (-((2 a-3 b) b)+2 (a-2 b) x+x^2\right )}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (-b^2-a^3 d+\left (2 b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=\frac {2 \arctan \left (\frac {\sqrt [4]{d} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{3/4}}{(b-x)^2}\right )}{d^{3/4}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{3/4}}{(b-x)^2}\right )}{d^{3/4}} \]
2*arctan(d^(1/4)*(-a*b^2+(2*a*b+b^2)*x+(-a-2*b)*x^2+x^3)^(3/4)/(b-x)^2)/d^ (3/4)-2*arctanh(d^(1/4)*(-a*b^2+(2*a*b+b^2)*x+(-a-2*b)*x^2+x^3)^(3/4)/(b-x )^2)/d^(3/4)
\[ \int \frac {\left (a^2-2 a x+x^2\right ) \left (-((2 a-3 b) b)+2 (a-2 b) x+x^2\right )}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (-b^2-a^3 d+\left (2 b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=\int \frac {\left (a^2-2 a x+x^2\right ) \left (-((2 a-3 b) b)+2 (a-2 b) x+x^2\right )}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (-b^2-a^3 d+\left (2 b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx \]
Integrate[((a^2 - 2*a*x + x^2)*(-((2*a - 3*b)*b) + 2*(a - 2*b)*x + x^2))/( ((-a + x)*(-b + x)^2)^(3/4)*(-b^2 - a^3*d + (2*b + 3*a^2*d)*x - (1 + 3*a*d )*x^2 + d*x^3)),x]
Integrate[((a^2 - 2*a*x + x^2)*(-((2*a - 3*b)*b) + 2*(a - 2*b)*x + x^2))/( ((-a + x)*(-b + x)^2)^(3/4)*(-b^2 - a^3*d + (2*b + 3*a^2*d)*x - (1 + 3*a*d )*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.
| \(\displaystyle \int \frac {\left (a^2-2 a x+x^2\right ) \left (2 x (a-2 b)-b (2 a-3 b)+x^2\right )}{\left ((x-a) (x-b)^2\right )^{3/4} \left (a^3 (-d)+x \left (3 a^2 d+2 b\right )-x^2 (3 a d+1)-b^2+d x^3\right )} \, dx\) |
| \(\Big \downarrow \) 7270 |
| \(\displaystyle \frac {(x-a)^{3/4} (x-b)^{3/2} \int \frac {\left (-x^2-2 (a-2 b) x+(2 a-3 b) b\right ) \left (a^2-2 x a+x^2\right )}{(x-a)^{3/4} (x-b)^{3/2} \left (d a^3-d x^3+b^2+(3 a d+1) x^2-\left (3 d a^2+2 b\right ) x\right )}dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 2004 |
| \(\displaystyle \frac {(x-a)^{3/4} (x-b)^{3/2} \int \frac {(x-a)^{5/4} \left (-x^2-2 (a-2 b) x+(2 a-3 b) b\right )}{(x-b)^{3/2} \left (d a^3-d x^3+b^2+(3 a d+1) x^2-\left (3 d a^2+2 b\right ) x\right )}dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 2004 |
| \(\displaystyle \frac {(x-a)^{3/4} (x-b)^{3/2} \int \frac {(-2 a+3 b-x) (x-a)^{5/4}}{\sqrt {x-b} \left (d a^3-d x^3+b^2+(3 a d+1) x^2-\left (3 d a^2+2 b\right ) x\right )}dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {(x-a)^{3/4} (x-b)^{3/2} \int \left (\frac {3 \left (1-\frac {2 a}{3 b}\right ) b (x-a)^{5/4}}{\sqrt {x-b} \left (d a^3-d x^3+b^2+(3 a d+1) x^2-\left (3 d a^2+2 b\right ) x\right )}+\frac {x (x-a)^{5/4}}{\sqrt {x-b} \left (-d a^3+d x^3-b^2-(3 a d+1) x^2+\left (3 d a^2+2 b\right ) x\right )}\right )dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(x-a)^{3/4} (x-b)^{3/2} \left (-\frac {4 (a-b)^2 \text {Subst}\left (\int \frac {1}{\sqrt {x^4+a-b} \left (-d x^{12}+x^8+2 a \left (1-\frac {b}{a}\right ) x^4+a^2 \left (\frac {b (b-2 a)}{a^2}+1\right )\right )}dx,x,\sqrt [4]{x-a}\right )}{d}-\frac {8 (a-b) \text {Subst}\left (\int \frac {x^4}{\sqrt {x^4+a-b} \left (-d x^{12}+x^8+2 a \left (1-\frac {b}{a}\right ) x^4+a^2 \left (\frac {b (b-2 a)}{a^2}+1\right )\right )}dx,x,\sqrt [4]{x-a}\right )}{d}-4 (2 a-3 b) \text {Subst}\left (\int \frac {x^8}{\sqrt {x^4+a-b} \left (-d x^{12}+x^8+(2 a-2 b) x^4+a^2 \left (\frac {b (b-2 a)}{a^2}+1\right )\right )}dx,x,\sqrt [4]{x-a}\right )-\frac {4 (a d+1) \text {Subst}\left (\int \frac {x^8}{\sqrt {x^4+a-b} \left (-d x^{12}+x^8+2 a \left (1-\frac {b}{a}\right ) x^4+a^2 \left (\frac {b (b-2 a)}{a^2}+1\right )\right )}dx,x,\sqrt [4]{x-a}\right )}{d}+\frac {2 \sqrt [4]{a-b} \sqrt {-\frac {b-x}{(a-b) \left (\frac {\sqrt {x-a}}{\sqrt {a-b}}+1\right )^2}} \left (\frac {\sqrt {x-a}}{\sqrt {a-b}}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{x-a}}{\sqrt [4]{a-b}}\right ),\frac {1}{2}\right )}{d \sqrt {x-b}}\right )}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
Int[((a^2 - 2*a*x + x^2)*(-((2*a - 3*b)*b) + 2*(a - 2*b)*x + x^2))/(((-a + x)*(-b + x)^2)^(3/4)*(-b^2 - a^3*d + (2*b + 3*a^2*d)*x - (1 + 3*a*d)*x^2 + d*x^3)),x]
3.17.85.3.1 Defintions of rubi rules used
Int[(u_)*((d_) + (e_.)*(x_))^(q_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.) , x_Symbol] :> Int[u*(d + e*x)^(p + q)*(a/d + (c/e)*x)^p, x] /; FreeQ[{a, b , c, d, e, q}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p ]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p]))) Int[u*v ^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !Free Q[v, x] && !FreeQ[w, x]
\[\int \frac {\left (a^{2}-2 a x +x^{2}\right ) \left (-\left (2 a -3 b \right ) b +2 \left (a -2 b \right ) x +x^{2}\right )}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {3}{4}} \left (-b^{2}-a^{3} d +\left (3 a^{2} d +2 b \right ) x -\left (3 a d +1\right ) x^{2}+d \,x^{3}\right )}d x\]
int((a^2-2*a*x+x^2)*(-(2*a-3*b)*b+2*(a-2*b)*x+x^2)/((-a+x)*(-b+x)^2)^(3/4) /(-b^2-a^3*d+(3*a^2*d+2*b)*x-(3*a*d+1)*x^2+d*x^3),x)
int((a^2-2*a*x+x^2)*(-(2*a-3*b)*b+2*(a-2*b)*x+x^2)/((-a+x)*(-b+x)^2)^(3/4) /(-b^2-a^3*d+(3*a^2*d+2*b)*x-(3*a*d+1)*x^2+d*x^3),x)
Timed out. \[ \int \frac {\left (a^2-2 a x+x^2\right ) \left (-((2 a-3 b) b)+2 (a-2 b) x+x^2\right )}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (-b^2-a^3 d+\left (2 b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=\text {Timed out} \]
integrate((a^2-2*a*x+x^2)*(-(2*a-3*b)*b+2*(a-2*b)*x+x^2)/((-a+x)*(-b+x)^2) ^(3/4)/(-b^2-a^3*d+(3*a^2*d+2*b)*x-(3*a*d+1)*x^2+d*x^3),x, algorithm="fric as")
Timed out. \[ \int \frac {\left (a^2-2 a x+x^2\right ) \left (-((2 a-3 b) b)+2 (a-2 b) x+x^2\right )}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (-b^2-a^3 d+\left (2 b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=\text {Timed out} \]
integrate((a**2-2*a*x+x**2)*(-(2*a-3*b)*b+2*(a-2*b)*x+x**2)/((-a+x)*(-b+x) **2)**(3/4)/(-b**2-a**3*d+(3*a**2*d+2*b)*x-(3*a*d+1)*x**2+d*x**3),x)
\[ \int \frac {\left (a^2-2 a x+x^2\right ) \left (-((2 a-3 b) b)+2 (a-2 b) x+x^2\right )}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (-b^2-a^3 d+\left (2 b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=\int { \frac {{\left (a^{2} - 2 \, a x + x^{2}\right )} {\left ({\left (2 \, a - 3 \, b\right )} b - 2 \, {\left (a - 2 \, b\right )} x - x^{2}\right )}}{{\left (a^{3} d - d x^{3} + {\left (3 \, a d + 1\right )} x^{2} + b^{2} - {\left (3 \, a^{2} d + 2 \, b\right )} x\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {3}{4}}} \,d x } \]
integrate((a^2-2*a*x+x^2)*(-(2*a-3*b)*b+2*(a-2*b)*x+x^2)/((-a+x)*(-b+x)^2) ^(3/4)/(-b^2-a^3*d+(3*a^2*d+2*b)*x-(3*a*d+1)*x^2+d*x^3),x, algorithm="maxi ma")
integrate((a^2 - 2*a*x + x^2)*((2*a - 3*b)*b - 2*(a - 2*b)*x - x^2)/((a^3* d - d*x^3 + (3*a*d + 1)*x^2 + b^2 - (3*a^2*d + 2*b)*x)*(-(a - x)*(b - x)^2 )^(3/4)), x)
\[ \int \frac {\left (a^2-2 a x+x^2\right ) \left (-((2 a-3 b) b)+2 (a-2 b) x+x^2\right )}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (-b^2-a^3 d+\left (2 b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=\int { \frac {{\left (a^{2} - 2 \, a x + x^{2}\right )} {\left ({\left (2 \, a - 3 \, b\right )} b - 2 \, {\left (a - 2 \, b\right )} x - x^{2}\right )}}{{\left (a^{3} d - d x^{3} + {\left (3 \, a d + 1\right )} x^{2} + b^{2} - {\left (3 \, a^{2} d + 2 \, b\right )} x\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {3}{4}}} \,d x } \]
integrate((a^2-2*a*x+x^2)*(-(2*a-3*b)*b+2*(a-2*b)*x+x^2)/((-a+x)*(-b+x)^2) ^(3/4)/(-b^2-a^3*d+(3*a^2*d+2*b)*x-(3*a*d+1)*x^2+d*x^3),x, algorithm="giac ")
integrate((a^2 - 2*a*x + x^2)*((2*a - 3*b)*b - 2*(a - 2*b)*x - x^2)/((a^3* d - d*x^3 + (3*a*d + 1)*x^2 + b^2 - (3*a^2*d + 2*b)*x)*(-(a - x)*(b - x)^2 )^(3/4)), x)
Timed out. \[ \int \frac {\left (a^2-2 a x+x^2\right ) \left (-((2 a-3 b) b)+2 (a-2 b) x+x^2\right )}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (-b^2-a^3 d+\left (2 b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=\int -\frac {\left (a^2-2\,a\,x+x^2\right )\,\left (2\,x\,\left (a-2\,b\right )-b\,\left (2\,a-3\,b\right )+x^2\right )}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{3/4}\,\left (a^3\,d-x\,\left (3\,d\,a^2+2\,b\right )-d\,x^3+x^2\,\left (3\,a\,d+1\right )+b^2\right )} \,d x \]
int(-((a^2 - 2*a*x + x^2)*(2*x*(a - 2*b) - b*(2*a - 3*b) + x^2))/((-(a - x )*(b - x)^2)^(3/4)*(a^3*d - x*(2*b + 3*a^2*d) - d*x^3 + x^2*(3*a*d + 1) + b^2)),x)