Integrand size = 88, antiderivative size = 276 \[ \int \frac {(-4 a+b+3 x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{2/3}}{-2 a+2 x+\sqrt [3]{d} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{2/3}}\right )}{d^{2/3}}+\frac {\log \left (a-x+\sqrt [3]{d} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{2/3}\right )}{d^{2/3}}-\frac {\log \left (a^2-2 a x+x^2+\left (-a \sqrt [3]{d}+\sqrt [3]{d} x\right ) \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{2/3}+d^{2/3} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{4/3}\right )}{2 d^{2/3}} \]
3^(1/2)*arctan(3^(1/2)*d^(1/3)*(-a*b^2+(2*a*b+b^2)*x+(-a-2*b)*x^2+x^3)^(2/ 3)/(-2*a+2*x+d^(1/3)*(-a*b^2+(2*a*b+b^2)*x+(-a-2*b)*x^2+x^3)^(2/3)))/d^(2/ 3)+ln(a-x+d^(1/3)*(-a*b^2+(2*a*b+b^2)*x+(-a-2*b)*x^2+x^3)^(2/3))/d^(2/3)-1 /2*ln(a^2-2*a*x+x^2+(-a*d^(1/3)+d^(1/3)*x)*(-a*b^2+(2*a*b+b^2)*x+(-a-2*b)* x^2+x^3)^(2/3)+d^(2/3)*(-a*b^2+(2*a*b+b^2)*x+(-a-2*b)*x^2+x^3)^(4/3))/d^(2 /3)
Time = 0.64 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.00 \[ \int \frac {(-4 a+b+3 x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )} \, dx=-\frac {(a-x)^{2/3} (b-x)^{4/3} \left (2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a-x}}{\sqrt [3]{d} (b-x)^{4/3}}}{\sqrt {3}}\right )-2 \log \left (a \sqrt [3]{d}-b \sqrt [3]{d}+\frac {(a-x)^{4/3}}{(b-x)^{4/3}}-\frac {\sqrt [3]{a-x}}{\sqrt [3]{b-x}}\right )+\log \left (\frac {(a-b)^2 \left ((a-x)^{2/3}+b^2 d^{2/3} (b-x)^{2/3}+\sqrt [3]{d} \sqrt [3]{a-x} \sqrt [3]{b-x} x+d^{2/3} (b-x)^{2/3} x^2-b \sqrt [3]{d} \sqrt [3]{b-x} \left (\sqrt [3]{a-x}+2 \sqrt [3]{d} \sqrt [3]{b-x} x\right )\right )}{(b-x)^{8/3}}\right )\right )}{2 d^{2/3} \left ((b-x)^2 (-a+x)\right )^{2/3}} \]
Integrate[((-4*a + b + 3*x)*(-b^3 + 3*b^2*x - 3*b*x^2 + x^3))/(((-a + x)*( -b + x)^2)^(2/3)*(a + b^4*d - (1 + 4*b^3*d)*x + 6*b^2*d*x^2 - 4*b*d*x^3 + d*x^4)),x]
-1/2*((a - x)^(2/3)*(b - x)^(4/3)*(2*Sqrt[3]*ArcTan[(1 - (2*(a - x)^(1/3)) /(d^(1/3)*(b - x)^(4/3)))/Sqrt[3]] - 2*Log[a*d^(1/3) - b*d^(1/3) + (a - x) ^(4/3)/(b - x)^(4/3) - (a - x)^(1/3)/(b - x)^(1/3)] + Log[((a - b)^2*((a - x)^(2/3) + b^2*d^(2/3)*(b - x)^(2/3) + d^(1/3)*(a - x)^(1/3)*(b - x)^(1/3 )*x + d^(2/3)*(b - x)^(2/3)*x^2 - b*d^(1/3)*(b - x)^(1/3)*((a - x)^(1/3) + 2*d^(1/3)*(b - x)^(1/3)*x)))/(b - x)^(8/3)]))/(d^(2/3)*((b - x)^2*(-a + x ))^(2/3))
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.
| \(\displaystyle \int \frac {\left (-b^3+3 b^2 x-3 b x^2+x^3\right ) (-4 a+b+3 x)}{\left ((x-a) (x-b)^2\right )^{2/3} \left (a+b^4 d-x \left (4 b^3 d+1\right )+6 b^2 d x^2-4 b d x^3+d x^4\right )} \, dx\) |
| \(\Big \downarrow \) 2006 |
| \(\displaystyle \int \frac {(x-b)^3 (-4 a+b+3 x)}{\left ((x-a) (x-b)^2\right )^{2/3} \left (a+b^4 d-x \left (4 b^3 d+1\right )+6 b^2 d x^2-4 b d x^3+d x^4\right )}dx\) |
| \(\Big \downarrow \) 7270 |
| \(\displaystyle \frac {(x-a)^{2/3} (x-b)^{4/3} \int -\frac {(4 a-b-3 x) (x-b)^{5/3}}{(x-a)^{2/3} \left (d b^4+6 d x^2 b^2-4 d x^3 b+d x^4+a-\left (4 d b^3+1\right ) x\right )}dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {(x-a)^{2/3} (x-b)^{4/3} \int \frac {(4 a-b-3 x) (x-b)^{5/3}}{(x-a)^{2/3} \left (d b^4+6 d x^2 b^2-4 d x^3 b+d x^4+a-\left (4 d b^3+1\right ) x\right )}dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {(x-a)^{2/3} (x-b)^{4/3} \int \left (\frac {\left (1-\frac {4 a}{b}\right ) b (x-b)^{5/3}}{(x-a)^{2/3} \left (-d b^4-6 d x^2 b^2+4 d x^3 b-d x^4-a+\left (4 d b^3+1\right ) x\right )}+\frac {3 x (x-b)^{5/3}}{(x-a)^{2/3} \left (-d b^4-6 d x^2 b^2+4 d x^3 b-d x^4-a+\left (4 d b^3+1\right ) x\right )}\right )dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {(x-a)^{2/3} (x-b)^{4/3} \left (-9 a \text {Subst}\left (\int \frac {\left (x^3+a-b\right )^{5/3}}{d x^{12}+4 a \left (1-\frac {b}{a}\right ) d x^9+6 a^2 \left (\frac {b (b-2 a)}{a^2}+1\right ) d x^6-\left (1-4 (a-b)^3 d\right ) x^3+a^4 \left (\frac {b \left (-4 a^3+6 b a^2-4 b^2 a+b^3\right )}{a^4}+1\right ) d}dx,x,\sqrt [3]{x-a}\right )-9 \text {Subst}\left (\int \frac {x^3 \left (x^3+a-b\right )^{5/3}}{d x^{12}+4 a \left (1-\frac {b}{a}\right ) d x^9+6 a^2 \left (\frac {b (b-2 a)}{a^2}+1\right ) d x^6-\left (1-4 (a-b)^3 d\right ) x^3+a^4 \left (\frac {b \left (-4 a^3+6 b a^2-4 b^2 a+b^3\right )}{a^4}+1\right ) d}dx,x,\sqrt [3]{x-a}\right )+3 (4 a-b) \text {Subst}\left (\int \frac {\left (x^3+a-b\right )^{5/3}}{d \left (x^3+a\right )^4-4 b d \left (x^3+a\right )^3+6 b^2 d \left (x^3+a\right )^2-\left (4 d b^3+1\right ) \left (x^3+a\right )+a \left (\frac {d b^4}{a}+1\right )}dx,x,\sqrt [3]{x-a}\right )\right )}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}\) |
Int[((-4*a + b + 3*x)*(-b^3 + 3*b^2*x - 3*b*x^2 + x^3))/(((-a + x)*(-b + x )^2)^(2/3)*(a + b^4*d - (1 + 4*b^3*d)*x + 6*b^2*d*x^2 - 4*b*d*x^3 + d*x^4) ),x]
3.29.12.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^Expon[Px , x], x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; PolyQ[Px, x] && GtQ[Expon[P x, x], 1] && NeQ[Coeff[Px, x, 0], 0] && !MatchQ[Px, (a_.)*(v_)^Expon[Px, x ] /; FreeQ[a, x] && LinearQ[v, x]]
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]
\[\int \frac {\left (-4 a +b +3 x \right ) \left (-b^{3}+3 b^{2} x -3 b \,x^{2}+x^{3}\right )}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {2}{3}} \left (a +b^{4} d -\left (4 b^{3} d +1\right ) x +6 b^{2} d \,x^{2}-4 b d \,x^{3}+d \,x^{4}\right )}d x\]
int((-4*a+b+3*x)*(-b^3+3*b^2*x-3*b*x^2+x^3)/((-a+x)*(-b+x)^2)^(2/3)/(a+b^4 *d-(4*b^3*d+1)*x+6*b^2*d*x^2-4*b*d*x^3+d*x^4),x)
int((-4*a+b+3*x)*(-b^3+3*b^2*x-3*b*x^2+x^3)/((-a+x)*(-b+x)^2)^(2/3)/(a+b^4 *d-(4*b^3*d+1)*x+6*b^2*d*x^2-4*b*d*x^3+d*x^4),x)
Time = 0.43 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.27 \[ \int \frac {(-4 a+b+3 x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )} \, dx=-\frac {2 \, \sqrt {3} {\left (d^{2}\right )}^{\frac {1}{6}} d \arctan \left (\frac {\sqrt {3} {\left (d^{2}\right )}^{\frac {1}{6}} {\left ({\left (b^{2} d - 2 \, b d x + d x^{2}\right )} {\left (d^{2}\right )}^{\frac {1}{3}} + 2 \, {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (d^{2}\right )}^{\frac {2}{3}}\right )}}{3 \, {\left (b^{2} d^{2} - 2 \, b d^{2} x + d^{2} x^{2}\right )}}\right ) + {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (\frac {{\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (b^{2} - 2 \, b x + x^{2}\right )} {\left (d^{2}\right )}^{\frac {2}{3}} + {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}} d + {\left (b^{4} d - 4 \, b^{3} d x + 6 \, b^{2} d x^{2} - 4 \, b d x^{3} + d x^{4}\right )} {\left (d^{2}\right )}^{\frac {1}{3}}}{b^{4} - 4 \, b^{3} x + 6 \, b^{2} x^{2} - 4 \, b x^{3} + x^{4}}\right ) - 2 \, {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (-\frac {{\left (b^{2} - 2 \, b x + x^{2}\right )} {\left (d^{2}\right )}^{\frac {2}{3}} - {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} d}{b^{2} - 2 \, b x + x^{2}}\right )}{2 \, d^{2}} \]
integrate((-4*a+b+3*x)*(-b^3+3*b^2*x-3*b*x^2+x^3)/((-a+x)*(-b+x)^2)^(2/3)/ (a+b^4*d-(4*b^3*d+1)*x+6*b^2*d*x^2-4*b*d*x^3+d*x^4),x, algorithm="fricas")
-1/2*(2*sqrt(3)*(d^2)^(1/6)*d*arctan(1/3*sqrt(3)*(d^2)^(1/6)*((b^2*d - 2*b *d*x + d*x^2)*(d^2)^(1/3) + 2*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2 )*x)^(1/3)*(d^2)^(2/3))/(b^2*d^2 - 2*b*d^2*x + d^2*x^2)) + (d^2)^(2/3)*log (((-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*(b^2 - 2*b*x + x^ 2)*(d^2)^(2/3) + (-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(2/3)*d + (b^4*d - 4*b^3*d*x + 6*b^2*d*x^2 - 4*b*d*x^3 + d*x^4)*(d^2)^(1/3))/(b^4 - 4*b^3*x + 6*b^2*x^2 - 4*b*x^3 + x^4)) - 2*(d^2)^(2/3)*log(-((b^2 - 2*b*x + x^2)*(d^2)^(2/3) - (-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/ 3)*d)/(b^2 - 2*b*x + x^2)))/d^2
Timed out. \[ \int \frac {(-4 a+b+3 x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )} \, dx=\text {Timed out} \]
integrate((-4*a+b+3*x)*(-b**3+3*b**2*x-3*b*x**2+x**3)/((-a+x)*(-b+x)**2)** (2/3)/(a+b**4*d-(4*b**3*d+1)*x+6*b**2*d*x**2-4*b*d*x**3+d*x**4),x)
\[ \int \frac {(-4 a+b+3 x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )} \, dx=\int { \frac {{\left (b^{3} - 3 \, b^{2} x + 3 \, b x^{2} - x^{3}\right )} {\left (4 \, a - b - 3 \, x\right )}}{{\left (b^{4} d + 6 \, b^{2} d x^{2} - 4 \, b d x^{3} + d x^{4} - {\left (4 \, b^{3} d + 1\right )} x + a\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {2}{3}}} \,d x } \]
integrate((-4*a+b+3*x)*(-b^3+3*b^2*x-3*b*x^2+x^3)/((-a+x)*(-b+x)^2)^(2/3)/ (a+b^4*d-(4*b^3*d+1)*x+6*b^2*d*x^2-4*b*d*x^3+d*x^4),x, algorithm="maxima")
integrate((b^3 - 3*b^2*x + 3*b*x^2 - x^3)*(4*a - b - 3*x)/((b^4*d + 6*b^2* d*x^2 - 4*b*d*x^3 + d*x^4 - (4*b^3*d + 1)*x + a)*(-(a - x)*(b - x)^2)^(2/3 )), x)
\[ \int \frac {(-4 a+b+3 x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )} \, dx=\int { \frac {{\left (b^{3} - 3 \, b^{2} x + 3 \, b x^{2} - x^{3}\right )} {\left (4 \, a - b - 3 \, x\right )}}{{\left (b^{4} d + 6 \, b^{2} d x^{2} - 4 \, b d x^{3} + d x^{4} - {\left (4 \, b^{3} d + 1\right )} x + a\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {2}{3}}} \,d x } \]
integrate((-4*a+b+3*x)*(-b^3+3*b^2*x-3*b*x^2+x^3)/((-a+x)*(-b+x)^2)^(2/3)/ (a+b^4*d-(4*b^3*d+1)*x+6*b^2*d*x^2-4*b*d*x^3+d*x^4),x, algorithm="giac")
integrate((b^3 - 3*b^2*x + 3*b*x^2 - x^3)*(4*a - b - 3*x)/((b^4*d + 6*b^2* d*x^2 - 4*b*d*x^3 + d*x^4 - (4*b^3*d + 1)*x + a)*(-(a - x)*(b - x)^2)^(2/3 )), x)
Timed out. \[ \int \frac {(-4 a+b+3 x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )} \, dx=\int -\frac {\left (b-4\,a+3\,x\right )\,\left (b^3-3\,b^2\,x+3\,b\,x^2-x^3\right )}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{2/3}\,\left (a+b^4\,d+d\,x^4-x\,\left (4\,d\,b^3+1\right )+6\,b^2\,d\,x^2-4\,b\,d\,x^3\right )} \,d x \]
int(-((b - 4*a + 3*x)*(3*b*x^2 - 3*b^2*x + b^3 - x^3))/((-(a - x)*(b - x)^ 2)^(2/3)*(a + b^4*d + d*x^4 - x*(4*b^3*d + 1) + 6*b^2*d*x^2 - 4*b*d*x^3)), x)