Integrand size = 50, antiderivative size = 290 \[ \int \frac {(-b+x) \left (a b-2 b x+x^2\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (b d-(a+d) x+x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} b \sqrt [3]{d}-\sqrt {3} \sqrt [3]{d} x}{b \sqrt [3]{d}-\sqrt [3]{d} x-2 \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}\right )}{d^{2/3}}+\frac {\log \left (b \sqrt [3]{d}-\sqrt [3]{d} x+\sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}\right )}{d^{2/3}}-\frac {\log \left (b^2 d^{2/3}-2 b d^{2/3} x+d^{2/3} x^2+\left (-b \sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}+\left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{2/3}\right )}{2 d^{2/3}} \]
3^(1/2)*arctan((3^(1/2)*b*d^(1/3)-3^(1/2)*d^(1/3)*x)/(b*d^(1/3)-d^(1/3)*x- 2*(-a*b^2*x+(2*a*b+b^2)*x^2+(-a-2*b)*x^3+x^4)^(1/3)))/d^(2/3)+ln(b*d^(1/3) -d^(1/3)*x+(-a*b^2*x+(2*a*b+b^2)*x^2+(-a-2*b)*x^3+x^4)^(1/3))/d^(2/3)-1/2* ln(b^2*d^(2/3)-2*b*d^(2/3)*x+d^(2/3)*x^2+(-b*d^(1/3)+d^(1/3)*x)*(-a*b^2*x+ (2*a*b+b^2)*x^2+(-a-2*b)*x^3+x^4)^(1/3)+(-a*b^2*x+(2*a*b+b^2)*x^2+(-a-2*b) *x^3+x^4)^(2/3))/d^(2/3)
\[ \int \frac {(-b+x) \left (a b-2 b x+x^2\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (b d-(a+d) x+x^2\right )} \, dx=\int \frac {(-b+x) \left (a b-2 b x+x^2\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (b d-(a+d) x+x^2\right )} \, dx \]
Integrate[((-b + x)*(a*b - 2*b*x + x^2))/((x*(-a + x)*(-b + x)^2)^(2/3)*(b *d - (a + d)*x + x^2)),x]
Integrate[((-b + x)*(a*b - 2*b*x + x^2))/((x*(-a + x)*(-b + x)^2)^(2/3)*(b *d - (a + d)*x + 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.
| \(\displaystyle \int \frac {(x-b) \left (a b-2 b x+x^2\right )}{\left (x (x-a) (x-b)^2\right )^{2/3} \left (-x (a+d)+b d+x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{2/3} \int -\frac {(b-x) \left (x^2-2 b x+a b\right )}{x^{2/3} \left (x^2-(a+d) x+b d\right ) \left (x^3-(a+2 b) x^2+b (2 a+b) x-a b^2\right )^{2/3}}dx}{\left (-\left (x (a-x) (b-x)^2\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^{2/3} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{2/3} \int \frac {(b-x) \left (x^2-2 b x+a b\right )}{x^{2/3} \left (x^2-(a+d) x+b d\right ) \left (x^3-(a+2 b) x^2+b (2 a+b) x-a b^2\right )^{2/3}}dx}{\left (-\left (x (a-x) (b-x)^2\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 x^{2/3} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{2/3} \int \frac {(b-x) \left (x^2-2 b x+a b\right )}{\left (x^2-(a+d) x+b d\right ) \left (x^3-(a+2 b) x^2+b (2 a+b) x-a b^2\right )^{2/3}}d\sqrt [3]{x}}{\left (-\left (x (a-x) (b-x)^2\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 x^{2/3} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{2/3} \int \left (-\frac {a}{\left (x^3-(a+2 b) x^2+b (2 a+b) x-a b^2\right )^{2/3}}+\frac {3 b}{\left (x^3-(a+2 b) x^2+b (2 a+b) x-a b^2\right )^{2/3}}-\frac {d}{\left (x^3-(a+2 b) x^2+b (2 a+b) x-a b^2\right )^{2/3}}-\frac {x}{\left (x^3-(a+2 b) x^2+b (2 a+b) x-a b^2\right )^{2/3}}+\frac {-b ((3 b-d) d-a (b+d))-\left (a^2-2 b a+2 d a+2 b^2+d^2-4 b d\right ) x}{\left (x^2+(-a-d) x+b d\right ) \left (x^3-(a+2 b) x^2+b (2 a+b) x-a b^2\right )^{2/3}}\right )d\sqrt [3]{x}}{\left (-\left (x (a-x) (b-x)^2\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle -\frac {3 x^{2/3} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{2/3} \int \left (\frac {3 b-a}{\left (x^3-(a+2 b) x^2+b (2 a+b) x-a b^2\right )^{2/3}}-\frac {d}{\left (x^3-(a+2 b) x^2+b (2 a+b) x-a b^2\right )^{2/3}}-\frac {x}{\left (x^3-(a+2 b) x^2+b (2 a+b) x-a b^2\right )^{2/3}}+\frac {-b ((3 b-d) d-a (b+d))-\left (a^2-2 b a+2 d a+2 b^2+d^2-4 b d\right ) x}{\left (x^2+(-a-d) x+b d\right ) \left (x^3-(a+2 b) x^2+b (2 a+b) x-a b^2\right )^{2/3}}\right )d\sqrt [3]{x}}{\left (-\left (x (a-x) (b-x)^2\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle -\frac {3 x^{2/3} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{2/3} \int \left (\frac {-a+3 b-d}{\left (x^3-(a+2 b) x^2+b (2 a+b) x-a b^2\right )^{2/3}}-\frac {x}{\left (x^3-(a+2 b) x^2+b (2 a+b) x-a b^2\right )^{2/3}}+\frac {-b ((3 b-d) d-a (b+d))-\left (a^2-2 b a+2 d a+2 b^2+d^2-4 b d\right ) x}{\left (x^2+(-a-d) x+b d\right ) \left (x^3-(a+2 b) x^2+b (2 a+b) x-a b^2\right )^{2/3}}\right )d\sqrt [3]{x}}{\left (-\left (x (a-x) (b-x)^2\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3 x^{2/3} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{2/3} \int \frac {(b-x) \left (x^2-2 b x+a b\right )}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3} \left (x^2-(a+d) x+b d\right )}d\sqrt [3]{x}}{\left (-\left (x (a-x) (b-x)^2\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle -\frac {3 x^{2/3} (a-x)^{2/3} (b-x)^{4/3} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{2/3} \int \frac {x^2-2 b x+a b}{(a-x)^{2/3} \sqrt [3]{b-x} \left (x^2-(a+d) x+b d\right )}d\sqrt [3]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3} \left (-\left (x (a-x) (b-x)^2\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 x^{2/3} (a-x)^{2/3} (b-x)^{4/3} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{2/3} \int \left (\frac {b (a-d)+(a-2 b+d) x}{(a-x)^{2/3} \sqrt [3]{b-x} \left (x^2+(-a-d) x+b d\right )}+\frac {1}{(a-x)^{2/3} \sqrt [3]{b-x}}\right )d\sqrt [3]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3} \left (-\left (x (a-x) (b-x)^2\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 (a-x)^{2/3} (b-x)^{4/3} x^{2/3} \left (x^3-(a+2 b) x^2+b (2 a+b) x-a b^2\right )^{2/3} \left (\frac {\sqrt [3]{x} \left (1-\frac {x}{a}\right )^{2/3} \sqrt [3]{1-\frac {x}{b}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},\frac {1}{3},\frac {4}{3},\frac {x}{a},\frac {x}{b}\right )}{(a-x)^{2/3} \sqrt [3]{b-x}}-\frac {\left (a-2 b+d-\sqrt {a^2+2 d a+d^2-4 b d}\right ) \int \frac {1}{\left (\sqrt [3]{a+d-\sqrt {a^2+2 d a+d^2-4 b d}}+\sqrt [3]{-2} \sqrt [3]{x}\right ) (a-x)^{2/3} \sqrt [3]{b-x}}d\sqrt [3]{x}}{3 \left (a+d-\sqrt {a^2+2 d a+d^2-4 b d}\right )^{2/3}}-\frac {\left (a-2 b+d+\sqrt {a^2+2 d a+d^2-4 b d}\right ) \int \frac {1}{\left (\sqrt [3]{a+d+\sqrt {a^2+2 d a+d^2-4 b d}}+\sqrt [3]{-2} \sqrt [3]{x}\right ) (a-x)^{2/3} \sqrt [3]{b-x}}d\sqrt [3]{x}}{3 \left (a+d+\sqrt {a^2+2 d a+d^2-4 b d}\right )^{2/3}}-\frac {\left (a-2 b+d-\sqrt {a^2+2 d a+d^2-4 b d}\right ) \int \frac {1}{\left (\sqrt [3]{a+d-\sqrt {a^2+2 d a+d^2-4 b d}}-\sqrt [3]{2} \sqrt [3]{x}\right ) (a-x)^{2/3} \sqrt [3]{b-x}}d\sqrt [3]{x}}{3 \left (a+d-\sqrt {a^2+2 d a+d^2-4 b d}\right )^{2/3}}-\frac {\left (a-2 b+d+\sqrt {a^2+2 d a+d^2-4 b d}\right ) \int \frac {1}{\left (\sqrt [3]{a+d+\sqrt {a^2+2 d a+d^2-4 b d}}-\sqrt [3]{2} \sqrt [3]{x}\right ) (a-x)^{2/3} \sqrt [3]{b-x}}d\sqrt [3]{x}}{3 \left (a+d+\sqrt {a^2+2 d a+d^2-4 b d}\right )^{2/3}}-\frac {\left (a-2 b+d-\sqrt {a^2+2 d a+d^2-4 b d}\right ) \int \frac {1}{\left (\sqrt [3]{a+d-\sqrt {a^2+2 d a+d^2-4 b d}}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{x}\right ) (a-x)^{2/3} \sqrt [3]{b-x}}d\sqrt [3]{x}}{3 \left (a+d-\sqrt {a^2+2 d a+d^2-4 b d}\right )^{2/3}}-\frac {\left (a-2 b+d+\sqrt {a^2+2 d a+d^2-4 b d}\right ) \int \frac {1}{\left (\sqrt [3]{a+d+\sqrt {a^2+2 d a+d^2-4 b d}}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{x}\right ) (a-x)^{2/3} \sqrt [3]{b-x}}d\sqrt [3]{x}}{3 \left (a+d+\sqrt {a^2+2 d a+d^2-4 b d}\right )^{2/3}}\right )}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3} \left (-\left ((a-x) (b-x)^2 x\right )\right )^{2/3}}\) |
Int[((-b + x)*(a*b - 2*b*x + x^2))/((x*(-a + x)*(-b + x)^2)^(2/3)*(b*d - ( a + d)*x + x^2)),x]
3.29.40.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
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]
\[\int \frac {\left (-b +x \right ) \left (a b -2 b x +x^{2}\right )}{\left (x \left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {2}{3}} \left (b d -\left (a +d \right ) x +x^{2}\right )}d x\]
Timed out. \[ \int \frac {(-b+x) \left (a b-2 b x+x^2\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (b d-(a+d) x+x^2\right )} \, dx=\text {Timed out} \]
integrate((-b+x)*(a*b-2*b*x+x^2)/(x*(-a+x)*(-b+x)^2)^(2/3)/(b*d-(a+d)*x+x^ 2),x, algorithm="fricas")
Timed out. \[ \int \frac {(-b+x) \left (a b-2 b x+x^2\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (b d-(a+d) x+x^2\right )} \, dx=\text {Timed out} \]
\[ \int \frac {(-b+x) \left (a b-2 b x+x^2\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (b d-(a+d) x+x^2\right )} \, dx=\int { -\frac {{\left (a b - 2 \, b x + x^{2}\right )} {\left (b - x\right )}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {2}{3}} {\left (b d - {\left (a + d\right )} x + x^{2}\right )}} \,d x } \]
integrate((-b+x)*(a*b-2*b*x+x^2)/(x*(-a+x)*(-b+x)^2)^(2/3)/(b*d-(a+d)*x+x^ 2),x, algorithm="maxima")
-integrate((a*b - 2*b*x + x^2)*(b - x)/((-(a - x)*(b - x)^2*x)^(2/3)*(b*d - (a + d)*x + x^2)), x)
\[ \int \frac {(-b+x) \left (a b-2 b x+x^2\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (b d-(a+d) x+x^2\right )} \, dx=\int { -\frac {{\left (a b - 2 \, b x + x^{2}\right )} {\left (b - x\right )}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {2}{3}} {\left (b d - {\left (a + d\right )} x + x^{2}\right )}} \,d x } \]
integrate((-b+x)*(a*b-2*b*x+x^2)/(x*(-a+x)*(-b+x)^2)^(2/3)/(b*d-(a+d)*x+x^ 2),x, algorithm="giac")
integrate(-(a*b - 2*b*x + x^2)*(b - x)/((-(a - x)*(b - x)^2*x)^(2/3)*(b*d - (a + d)*x + x^2)), x)
Timed out. \[ \int \frac {(-b+x) \left (a b-2 b x+x^2\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (b d-(a+d) x+x^2\right )} \, dx=\int -\frac {\left (b-x\right )\,\left (x^2-2\,b\,x+a\,b\right )}{\left (x^2+\left (-a-d\right )\,x+b\,d\right )\,{\left (-x\,\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{2/3}} \,d x \]