Integrand size = 48, antiderivative size = 167 \[ \int \frac {-2 a b+(a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (a b d-(a+b) d x+(-1+d) x^2\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{d} \sqrt [3]{a b x+(-a-b) x^2+x^3}}\right )}{d^{2/3}}+\frac {\log \left (x-\sqrt [3]{d} \sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{d^{2/3}}-\frac {\log \left (x^2+\sqrt [3]{d} x \sqrt [3]{a b x+(-a-b) x^2+x^3}+d^{2/3} \left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{2 d^{2/3}} \]
-3^(1/2)*arctan(3^(1/2)*x/(x+2*d^(1/3)*(a*b*x+(-a-b)*x^2+x^3)^(1/3)))/d^(2 /3)+ln(x-d^(1/3)*(a*b*x+(-a-b)*x^2+x^3)^(1/3))/d^(2/3)-1/2*ln(x^2+d^(1/3)* x*(a*b*x+(-a-b)*x^2+x^3)^(1/3)+d^(2/3)*(a*b*x+(-a-b)*x^2+x^3)^(2/3))/d^(2/ 3)
Time = 15.36 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.78 \[ \int \frac {-2 a b+(a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (a b d-(a+b) d x+(-1+d) x^2\right )} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{d} \sqrt [3]{x (-a+x) (-b+x)}}\right )-2 \log \left (x-\sqrt [3]{d} \sqrt [3]{x (-a+x) (-b+x)}\right )+\log \left (x^2+\sqrt [3]{d} x \sqrt [3]{x (-a+x) (-b+x)}+d^{2/3} (x (-a+x) (-b+x))^{2/3}\right )}{2 d^{2/3}} \]
Integrate[(-2*a*b + (a + b)*x)/((x*(-a + x)*(-b + x))^(1/3)*(a*b*d - (a + b)*d*x + (-1 + d)*x^2)),x]
-1/2*(2*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*d^(1/3)*(x*(-a + x)*(-b + x))^(1 /3))] - 2*Log[x - d^(1/3)*(x*(-a + x)*(-b + x))^(1/3)] + Log[x^2 + d^(1/3) *x*(x*(-a + x)*(-b + x))^(1/3) + d^(2/3)*(x*(-a + x)*(-b + x))^(2/3)])/d^( 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.
| \(\displaystyle \int \frac {x (a+b)-2 a b}{\sqrt [3]{x (x-a) (x-b)} \left (-d x (a+b)+a b d+(d-1) x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{-x (a+b)+a b+x^2} \int -\frac {2 a b-(a+b) x}{\sqrt [3]{x} \sqrt [3]{x^2-(a+b) x+a b} \left (-\left ((1-d) x^2\right )-(a+b) d x+a b d\right )}dx}{\sqrt [3]{x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{x} \sqrt [3]{-x (a+b)+a b+x^2} \int \frac {2 a b-(a+b) x}{\sqrt [3]{x} \sqrt [3]{x^2-(a+b) x+a b} \left (-\left ((1-d) x^2\right )-(a+b) d x+a b d\right )}dx}{\sqrt [3]{x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{-x (a+b)+a b+x^2} \int \frac {\sqrt [3]{x} (2 a b-(a+b) x)}{\sqrt [3]{x^2-(a+b) x+a b} \left (-\left ((1-d) x^2\right )-(a+b) d x+a b d\right )}d\sqrt [3]{x}}{\sqrt [3]{x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{-x (a+b)+a b+x^2} \int \left (\frac {(-a-b) x^{4/3}}{\sqrt [3]{x^2-(a+b) x+a b} \left (-\left ((1-d) x^2\right )-(a+b) d x+a b d\right )}+\frac {2 a b \sqrt [3]{x}}{\sqrt [3]{x^2-(a+b) x+a b} \left (-\left ((1-d) x^2\right )-(a+b) d x+a b d\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{-x (a+b)+a b+x^2} \left (-\frac {4 a b (1-d) \int \frac {\sqrt [3]{x}}{\left (-a d-b d-\sqrt {d a^2+4 b a-2 b d a+b^2 d} \sqrt {d}-2 (1-d) x\right ) \sqrt [3]{x^2+(-a-b) x+a b}}d\sqrt [3]{x}}{\sqrt {d} \sqrt {a^2 d+2 a b (2-d)+b^2 d}}-\frac {(a+b) \left (-\sqrt {a^2 d+2 a b (2-d)+b^2 d}+a \sqrt {d}+b \sqrt {d}\right ) \int \frac {\sqrt [3]{x}}{\left (a d+b d-\sqrt {d a^2+4 b a-2 b d a+b^2 d} \sqrt {d}+2 (1-d) x\right ) \sqrt [3]{x^2+(-a-b) x+a b}}d\sqrt [3]{x}}{\sqrt {a^2 d+2 a b (2-d)+b^2 d}}-\frac {4 a b (1-d) \int \frac {\sqrt [3]{x}}{\left (a d+b d-\sqrt {d a^2+4 b a-2 b d a+b^2 d} \sqrt {d}+2 (1-d) x\right ) \sqrt [3]{x^2+(-a-b) x+a b}}d\sqrt [3]{x}}{\sqrt {d} \sqrt {a^2 d+2 a b (2-d)+b^2 d}}+\frac {(a+b) \left (\sqrt {a^2 d+2 a b (2-d)+b^2 d}+a \sqrt {d}+b \sqrt {d}\right ) \int \frac {\sqrt [3]{x}}{\left (a d+b d+\sqrt {d a^2+4 b a-2 b d a+b^2 d} \sqrt {d}+2 (1-d) x\right ) \sqrt [3]{x^2+(-a-b) x+a b}}d\sqrt [3]{x}}{\sqrt {a^2 d+2 a b (2-d)+b^2 d}}\right )}{\sqrt [3]{x (a-x) (b-x)}}\) |
Int[(-2*a*b + (a + b)*x)/((x*(-a + x)*(-b + x))^(1/3)*(a*b*d - (a + b)*d*x + (-1 + d)*x^2)),x]
3.23.34.3.1 Defintions of rubi rules used
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[(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_)/((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]
Time = 0.56 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.82
| method | result | size |
| pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {1}{d}\right )^{\frac {1}{3}} x +2 \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}\right )}{3 \left (\frac {1}{d}\right )^{\frac {1}{3}} x}\right )+2 \ln \left (\frac {-\left (\frac {1}{d}\right )^{\frac {1}{3}} x +\left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {\left (\frac {1}{d}\right )^{\frac {2}{3}} x^{2}+\left (\frac {1}{d}\right )^{\frac {1}{3}} \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}} x +\left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{2 \left (\frac {1}{d}\right )^{\frac {1}{3}} d}\) | \(137\) |
int((-2*a*b+(a+b)*x)/(x*(-a+x)*(-b+x))^(1/3)/(a*b*d-(a+b)*d*x+(-1+d)*x^2), x,method=_RETURNVERBOSE)
1/2*(2*3^(1/2)*arctan(1/3*3^(1/2)*((1/d)^(1/3)*x+2*(x*(a-x)*(b-x))^(1/3))/ (1/d)^(1/3)/x)+2*ln((-(1/d)^(1/3)*x+(x*(a-x)*(b-x))^(1/3))/x)-ln(((1/d)^(2 /3)*x^2+(1/d)^(1/3)*(x*(a-x)*(b-x))^(1/3)*x+(x*(a-x)*(b-x))^(2/3))/x^2))/( 1/d)^(1/3)/d
Timed out. \[ \int \frac {-2 a b+(a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (a b d-(a+b) d x+(-1+d) x^2\right )} \, dx=\text {Timed out} \]
integrate((-2*a*b+(a+b)*x)/(x*(-a+x)*(-b+x))^(1/3)/(a*b*d-(a+b)*d*x+(-1+d) *x^2),x, algorithm="fricas")
Timed out. \[ \int \frac {-2 a b+(a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (a b d-(a+b) d x+(-1+d) x^2\right )} \, dx=\text {Timed out} \]
\[ \int \frac {-2 a b+(a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (a b d-(a+b) d x+(-1+d) x^2\right )} \, dx=\int { -\frac {2 \, a b - {\left (a + b\right )} x}{{\left (a b d - {\left (a + b\right )} d x + {\left (d - 1\right )} x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {1}{3}}} \,d x } \]
integrate((-2*a*b+(a+b)*x)/(x*(-a+x)*(-b+x))^(1/3)/(a*b*d-(a+b)*d*x+(-1+d) *x^2),x, algorithm="maxima")
-integrate((2*a*b - (a + b)*x)/((a*b*d - (a + b)*d*x + (d - 1)*x^2)*((a - x)*(b - x)*x)^(1/3)), x)
\[ \int \frac {-2 a b+(a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (a b d-(a+b) d x+(-1+d) x^2\right )} \, dx=\int { -\frac {2 \, a b - {\left (a + b\right )} x}{{\left (a b d - {\left (a + b\right )} d x + {\left (d - 1\right )} x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {1}{3}}} \,d x } \]
integrate((-2*a*b+(a+b)*x)/(x*(-a+x)*(-b+x))^(1/3)/(a*b*d-(a+b)*d*x+(-1+d) *x^2),x, algorithm="giac")
integrate(-(2*a*b - (a + b)*x)/((a*b*d - (a + b)*d*x + (d - 1)*x^2)*((a - x)*(b - x)*x)^(1/3)), x)
Timed out. \[ \int \frac {-2 a b+(a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (a b d-(a+b) d x+(-1+d) x^2\right )} \, dx=\int -\frac {2\,a\,b-x\,\left (a+b\right )}{{\left (x\,\left (a-x\right )\,\left (b-x\right )\right )}^{1/3}\,\left (\left (d-1\right )\,x^2-d\,\left (a+b\right )\,x+a\,b\,d\right )} \,d x \]