Integrand size = 72, antiderivative size = 232 \[ \int \frac {2 a b^2 x-b (2 a+b) x^2+x^4}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} x}{\sqrt [3]{d} x+2 \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}\right )}{d^{2/3}}+\frac {\log \left (-\sqrt [3]{d} x+\sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}\right )}{d^{2/3}}-\frac {\log \left (d^{2/3} x^2+\sqrt [3]{d} x \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}+\left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{2/3}\right )}{2 d^{2/3}} \]
3^(1/2)*arctan(3^(1/2)*d^(1/3)*x/(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(-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(d^(2/3)*x^2+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)
\[ \int \frac {2 a b^2 x-b (2 a+b) x^2+x^4}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3\right )} \, dx=\int \frac {2 a b^2 x-b (2 a+b) x^2+x^4}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3\right )} \, dx \]
Integrate[(2*a*b^2*x - b*(2*a + b)*x^2 + x^4)/((x*(-a + x)*(-b + x)^2)^(2/ 3)*(-(a*b^2) + b*(2*a + b)*x - (a + 2*b + d)*x^2 + x^3)),x]
Integrate[(2*a*b^2*x - b*(2*a + b)*x^2 + x^4)/((x*(-a + x)*(-b + x)^2)^(2/ 3)*(-(a*b^2) + b*(2*a + b)*x - (a + 2*b + d)*x^2 + 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.
| \(\displaystyle \int \frac {2 a b^2 x-b x^2 (2 a+b)+x^4}{\left (x (x-a) (x-b)^2\right )^{2/3} \left (-a b^2-x^2 (a+2 b+d)+b x (2 a+b)+x^3\right )} \, dx\) |
| \(\Big \downarrow \) 2028 |
| \(\displaystyle \int \frac {x \left (2 a b^2-b x (2 a+b)+x^3\right )}{\left (x (x-a) (x-b)^2\right )^{2/3} \left (-a b^2-x^2 (a+2 b+d)+b x (2 a+b)+x^3\right )}dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{2/3} \int -\frac {\sqrt [3]{x} \left (x^3-b (2 a+b) x+2 a b^2\right )}{\left (-x^3+(a+2 b+d) x^2-b (2 a+b) x+a b^2\right ) \left (x^3-(a+2 b) x^2+b (2 a+b) x-a b^2\right )^{2/3}}dx}{\left (-\left (x (a-x) (b-x)^2\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^{2/3} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{2/3} \int \frac {\sqrt [3]{x} \left (x^3-b (2 a+b) x+2 a b^2\right )}{\left (-x^3+(a+2 b+d) x^2-b (2 a+b) x+a b^2\right ) \left (x^3-(a+2 b) x^2+b (2 a+b) x-a b^2\right )^{2/3}}dx}{\left (-\left (x (a-x) (b-x)^2\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 x^{2/3} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{2/3} \int \frac {x \left (x^3-b (2 a+b) x+2 a b^2\right )}{\left (-x^3+(a+2 b+d) x^2-b (2 a+b) x+a b^2\right ) \left (x^3-(a+2 b) x^2+b (2 a+b) x-a b^2\right )^{2/3}}d\sqrt [3]{x}}{\left (-\left (x (a-x) (b-x)^2\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -\frac {3 x^{2/3} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{2/3} \int \frac {(b-x) x \left (-x^2-b x+2 a b\right )}{\left (-\left ((a-x) (x-b)^2\right )\right )^{2/3} \left (-x^3+(a+2 b+d) x^2-b (2 a+b) x+a b^2\right )}d\sqrt [3]{x}}{\left (-\left (x (a-x) (b-x)^2\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle -\frac {3 x^{2/3} (a-x)^{2/3} (x-b)^{4/3} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{2/3} \int \frac {(b-x) x \left (-x^2-b x+2 a b\right )}{(a-x)^{2/3} (x-b)^{4/3} \left (-x^3+(a+2 b+d) x^2-b (2 a+b) x+a b^2\right )}d\sqrt [3]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3} \left (-\left (x (a-x) (b-x)^2\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 281 |
| \(\displaystyle \frac {3 x^{2/3} (a-x)^{2/3} (x-b)^{4/3} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{2/3} \int \frac {x \left (-x^2-b x+2 a b\right )}{(a-x)^{2/3} \sqrt [3]{x-b} \left (-x^3+(a+2 b+d) x^2-b (2 a+b) x+a b^2\right )}d\sqrt [3]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3} \left (-\left (x (a-x) (b-x)^2\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3 x^{2/3} (a-x)^{2/3} (x-b)^{4/3} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{2/3} \int \left (\frac {1}{(a-x)^{2/3} \sqrt [3]{x-b}}-\frac {a b^2-(4 a+b) x b+(a+3 b+d) x^2}{(a-x)^{2/3} \sqrt [3]{x-b} \left (-x^3+(a+2 b+d) x^2-b (2 a+b) x+a b^2\right )}\right )d\sqrt [3]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3} \left (-\left (x (a-x) (b-x)^2\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 x^{2/3} (a-x)^{2/3} (x-b)^{4/3} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{2/3} \left (-a b^2 \int \frac {1}{(a-x)^{2/3} \sqrt [3]{x-b} \left (-x^3+a \left (\frac {2 b+d}{a}+1\right ) x^2-2 a b \left (\frac {b}{2 a}+1\right ) x+a b^2\right )}d\sqrt [3]{x}+b (4 a+b) \int \frac {x}{(a-x)^{2/3} \sqrt [3]{x-b} \left (-x^3+a \left (\frac {2 b+d}{a}+1\right ) x^2-2 a b \left (\frac {b}{2 a}+1\right ) x+a b^2\right )}d\sqrt [3]{x}-(a+3 b+d) \int \frac {x^2}{(a-x)^{2/3} \sqrt [3]{x-b} \left (-x^3+a \left (\frac {2 b+d}{a}+1\right ) x^2-2 a b \left (\frac {b}{2 a}+1\right ) x+a b^2\right )}d\sqrt [3]{x}+\frac {\sqrt [3]{x} \left (1-\frac {x}{a}\right )^{2/3} \sqrt [3]{1-\frac {x}{b}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},\frac {1}{3},\frac {4}{3},\frac {x}{a},\frac {x}{b}\right )}{(a-x)^{2/3} \sqrt [3]{x-b}}\right )}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3} \left (-\left (x (a-x) (b-x)^2\right )\right )^{2/3}}\) |
Int[(2*a*b^2*x - b*(2*a + b)*x^2 + x^4)/((x*(-a + x)*(-b + x)^2)^(2/3)*(-( a*b^2) + b*(2*a + b)*x - (a + 2*b + d)*x^2 + x^3)),x]
3.27.31.3.1 Defintions of rubi rules used
Int[(u_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_ Symbol] :> Simp[(b/d)^p Int[u*(c + d*x^n)^(p + q), x], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && EqQ[b*c - a*d, 0] && IntegerQ[p] && !(IntegerQ[q] & & SimplerQ[a + b*x^n, c + d*x^n])
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.) + (c_.)*(x_)^(t_.))^(p_.), x_Symbol] :> Int[x^(p*r)*(a + b*x^(s - r) + c*x^(t - r))^p*Fx, x] /; FreeQ[ {a, b, c, r, s, t}, x] && IntegerQ[p] && PosQ[s - r] && PosQ[t - r] && !(E qQ[p, 1] && EqQ[u, 1])
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]
Time = 0.76 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.58
| method | result | size |
| pseudoelliptic | \(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (d^{\frac {1}{3}} x +2 \left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{3}}\right )}{3 d^{\frac {1}{3}} x}\right )-\ln \left (\frac {d^{\frac {2}{3}} x^{2}+d^{\frac {1}{3}} \left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{3}} x +\left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {2}{3}}}{x^{2}}\right )+2 \ln \left (\frac {-d^{\frac {1}{3}} x +\left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{3}}}{x}\right )}{2 d^{\frac {2}{3}}}\) | \(134\) |
int((2*a*b^2*x-b*(2*a+b)*x^2+x^4)/(x*(-a+x)*(-b+x)^2)^(2/3)/(-a*b^2+b*(2*a +b)*x-(a+2*b+d)*x^2+x^3),x,method=_RETURNVERBOSE)
1/2*(-2*3^(1/2)*arctan(1/3*3^(1/2)*(d^(1/3)*x+2*(-x*(a-x)*(b-x)^2)^(1/3))/ d^(1/3)/x)-ln((d^(2/3)*x^2+d^(1/3)*(-x*(a-x)*(b-x)^2)^(1/3)*x+(-x*(a-x)*(b -x)^2)^(2/3))/x^2)+2*ln((-d^(1/3)*x+(-x*(a-x)*(b-x)^2)^(1/3))/x))/d^(2/3)
Timed out. \[ \int \frac {2 a b^2 x-b (2 a+b) x^2+x^4}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3\right )} \, dx=\text {Timed out} \]
integrate((2*a*b^2*x-b*(2*a+b)*x^2+x^4)/(x*(-a+x)*(-b+x)^2)^(2/3)/(-a*b^2+ b*(2*a+b)*x-(a+2*b+d)*x^2+x^3),x, algorithm="fricas")
Timed out. \[ \int \frac {2 a b^2 x-b (2 a+b) x^2+x^4}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3\right )} \, dx=\text {Timed out} \]
integrate((2*a*b**2*x-b*(2*a+b)*x**2+x**4)/(x*(-a+x)*(-b+x)**2)**(2/3)/(-a *b**2+b*(2*a+b)*x-(a+2*b+d)*x**2+x**3),x)
\[ \int \frac {2 a b^2 x-b (2 a+b) x^2+x^4}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3\right )} \, dx=\int { -\frac {2 \, a b^{2} x - {\left (2 \, a + b\right )} b x^{2} + x^{4}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {2}{3}} {\left (a b^{2} - {\left (2 \, a + b\right )} b x + {\left (a + 2 \, b + d\right )} x^{2} - x^{3}\right )}} \,d x } \]
integrate((2*a*b^2*x-b*(2*a+b)*x^2+x^4)/(x*(-a+x)*(-b+x)^2)^(2/3)/(-a*b^2+ b*(2*a+b)*x-(a+2*b+d)*x^2+x^3),x, algorithm="maxima")
-integrate((2*a*b^2*x - (2*a + b)*b*x^2 + x^4)/((-(a - x)*(b - x)^2*x)^(2/ 3)*(a*b^2 - (2*a + b)*b*x + (a + 2*b + d)*x^2 - x^3)), x)
\[ \int \frac {2 a b^2 x-b (2 a+b) x^2+x^4}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3\right )} \, dx=\int { -\frac {2 \, a b^{2} x - {\left (2 \, a + b\right )} b x^{2} + x^{4}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {2}{3}} {\left (a b^{2} - {\left (2 \, a + b\right )} b x + {\left (a + 2 \, b + d\right )} x^{2} - x^{3}\right )}} \,d x } \]
integrate((2*a*b^2*x-b*(2*a+b)*x^2+x^4)/(x*(-a+x)*(-b+x)^2)^(2/3)/(-a*b^2+ b*(2*a+b)*x-(a+2*b+d)*x^2+x^3),x, algorithm="giac")
integrate(-(2*a*b^2*x - (2*a + b)*b*x^2 + x^4)/((-(a - x)*(b - x)^2*x)^(2/ 3)*(a*b^2 - (2*a + b)*b*x + (a + 2*b + d)*x^2 - x^3)), x)
Timed out. \[ \int \frac {2 a b^2 x-b (2 a+b) x^2+x^4}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3\right )} \, dx=\int -\frac {x^4-b\,x^2\,\left (2\,a+b\right )+2\,a\,b^2\,x}{{\left (-x\,\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{2/3}\,\left (x^2\,\left (a+2\,b+d\right )+a\,b^2-x^3-b\,x\,\left (2\,a+b\right )\right )} \,d x \]
int(-(x^4 - b*x^2*(2*a + b) + 2*a*b^2*x)/((-x*(a - x)*(b - x)^2)^(2/3)*(x^ 2*(a + 2*b + d) + a*b^2 - x^3 - b*x*(2*a + b))),x)