Integrand size = 76, antiderivative size = 269 \[ \int \frac {2 a b^2-b (2 a+b) x+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a b^2 d+b (2 a+b) d x-(1+a d+2 b d) x^2+d x^3\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{2 x+\sqrt [3]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}\right )}{d^{2/3}}+\frac {\log \left (x-\sqrt [3]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}\right )}{d^{2/3}}-\frac {\log \left (x^2+\sqrt [3]{d} x \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}+d^{2/3} \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{2/3}\right )}{2 d^{2/3}} \]
3^(1/2)*arctan(3^(1/2)*d^(1/3)*(-a*b^2*x+(2*a*b+b^2)*x^2+(-a-2*b)*x^3+x^4) ^(1/3)/(2*x+d^(1/3)*(-a*b^2*x+(2*a*b+b^2)*x^2+(-a-2*b)*x^3+x^4)^(1/3)))/d^ (2/3)+ln(x-d^(1/3)*(-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(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) +d^(2/3)*(-a*b^2*x+(2*a*b+b^2)*x^2+(-a-2*b)*x^3+x^4)^(2/3))/d^(2/3)
Time = 13.98 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.61 \[ \int \frac {2 a b^2-b (2 a+b) x+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a b^2 d+b (2 a+b) d x-(1+a d+2 b d) x^2+d x^3\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{x (-a+x) (-b+x)^2}}{2 x+\sqrt [3]{d} \sqrt [3]{x (-a+x) (-b+x)^2}}\right )+2 \log \left (x-\sqrt [3]{d} \sqrt [3]{x (-a+x) (-b+x)^2}\right )-\log \left (x^2+\sqrt [3]{d} x \sqrt [3]{x (-a+x) (-b+x)^2}+d^{2/3} \left (x (-a+x) (-b+x)^2\right )^{2/3}\right )}{2 d^{2/3}} \]
Integrate[(2*a*b^2 - b*(2*a + b)*x + x^3)/((x*(-a + x)*(-b + x)^2)^(1/3)*( -(a*b^2*d) + b*(2*a + b)*d*x - (1 + a*d + 2*b*d)*x^2 + d*x^3)),x]
(2*Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(x*(-a + x)*(-b + x)^2)^(1/3))/(2*x + d ^(1/3)*(x*(-a + x)*(-b + x)^2)^(1/3))] + 2*Log[x - d^(1/3)*(x*(-a + x)*(-b + x)^2)^(1/3)] - Log[x^2 + d^(1/3)*x*(x*(-a + x)*(-b + x)^2)^(1/3) + d^(2 /3)*(x*(-a + x)*(-b + x)^2)^(2/3)])/(2*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.
\(\displaystyle \int \frac {2 a b^2-b x (2 a+b)+x^3}{\sqrt [3]{x (x-a) (x-b)^2} \left (-a b^2 d-x^2 (a d+2 b d+1)+b d x (2 a+b)+d x^3\right )} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int -\frac {x^3-b (2 a+b) x+2 a b^2}{\sqrt [3]{x} \sqrt [3]{x^3-(a+2 b) x^2+b (2 a+b) x-a b^2} \left (-d x^3+(a d+2 b d+1) x^2-b (2 a+b) d x+a b^2 d\right )}dx}{\sqrt [3]{-\left (x (a-x) (b-x)^2\right )}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt [3]{x} \sqrt [3]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {x^3-b (2 a+b) x+2 a b^2}{\sqrt [3]{x} \sqrt [3]{x^3-(a+2 b) x^2+b (2 a+b) x-a b^2} \left (-d x^3+(a d+2 b d+1) x^2-b (2 a+b) d x+a b^2 d\right )}dx}{\sqrt [3]{-\left (x (a-x) (b-x)^2\right )}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {\sqrt [3]{x} \left (x^3-b (2 a+b) x+2 a b^2\right )}{\sqrt [3]{x^3-(a+2 b) x^2+b (2 a+b) x-a b^2} \left (-d x^3+(a d+2 b d+1) x^2-b (2 a+b) d x+a b^2 d\right )}d\sqrt [3]{x}}{\sqrt [3]{-\left (x (a-x) (b-x)^2\right )}}\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {(b-x) \sqrt [3]{x} \left (-x^2-b x+2 a b\right )}{\sqrt [3]{-\left ((a-x) (x-b)^2\right )} \left (-d x^3+(a d+2 b d+1) x^2-b (2 a+b) d x+a b^2 d\right )}d\sqrt [3]{x}}{\sqrt [3]{-\left (x (a-x) (b-x)^2\right )}}\) |
\(\Big \downarrow \) 2058 |
\(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{a-x} (x-b)^{2/3} \sqrt [3]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {(b-x) \sqrt [3]{x} \left (-x^2-b x+2 a b\right )}{\sqrt [3]{a-x} (x-b)^{2/3} \left (-d x^3+(a d+2 b d+1) x^2-b (2 a+b) d x+a b^2 d\right )}d\sqrt [3]{x}}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )} \sqrt [3]{-\left (x (a-x) (b-x)^2\right )}}\) |
\(\Big \downarrow \) 281 |
\(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{a-x} (x-b)^{2/3} \sqrt [3]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{x-b} \left (-x^2-b x+2 a b\right )}{\sqrt [3]{a-x} \left (-d x^3+(a d+2 b d+1) x^2-b (2 a+b) d x+a b^2 d\right )}d\sqrt [3]{x}}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )} \sqrt [3]{-\left (x (a-x) (b-x)^2\right )}}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{a-x} (x-b)^{2/3} \sqrt [3]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \left (\frac {\sqrt [3]{x-b} x^{7/3}}{\sqrt [3]{a-x} \left (d x^3-((a+2 b) d+1) x^2+2 a b \left (\frac {b}{2 a}+1\right ) d x-a b^2 d\right )}+\frac {b \sqrt [3]{x-b} x^{4/3}}{\sqrt [3]{a-x} \left (d x^3-((a+2 b) d+1) x^2+2 a b \left (\frac {b}{2 a}+1\right ) d x-a b^2 d\right )}+\frac {2 a b \sqrt [3]{x-b} \sqrt [3]{x}}{\sqrt [3]{a-x} \left (-d x^3+((a+2 b) d+1) x^2-2 a b \left (\frac {b}{2 a}+1\right ) d x+a b^2 d\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )} \sqrt [3]{-\left (x (a-x) (b-x)^2\right )}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{a-x} (x-b)^{2/3} \sqrt [3]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \left (2 a b \int \frac {\sqrt [3]{x} \sqrt [3]{x-b}}{\sqrt [3]{a-x} \left (-d x^3+((a+2 b) d+1) x^2-2 a b \left (\frac {b}{2 a}+1\right ) d x+a b^2 d\right )}d\sqrt [3]{x}+b \int \frac {x^{4/3} \sqrt [3]{x-b}}{\sqrt [3]{a-x} \left (d x^3-((a+2 b) d+1) x^2+2 a b \left (\frac {b}{2 a}+1\right ) d x-a b^2 d\right )}d\sqrt [3]{x}+\int \frac {x^{7/3} \sqrt [3]{x-b}}{\sqrt [3]{a-x} \left (d x^3-((a+2 b) d+1) x^2+2 a b \left (\frac {b}{2 a}+1\right ) d x-a b^2 d\right )}d\sqrt [3]{x}\right )}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )} \sqrt [3]{-\left (x (a-x) (b-x)^2\right )}}\) |
Int[(2*a*b^2 - b*(2*a + b)*x + x^3)/((x*(-a + x)*(-b + x)^2)^(1/3)*(-(a*b^ 2*d) + b*(2*a + b)*d*x - (1 + a*d + 2*b*d)*x^2 + d*x^3)),x]
3.28.88.3.1 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_)*(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 = 0.83 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.55
method | result | size |
pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {1}{d}\right )^{\frac {1}{3}} x +2 \left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {1}{d}\right )^{\frac {1}{3}} x}\right )+2 \ln \left (\frac {-\left (\frac {1}{d}\right )^{\frac {1}{3}} x +\left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {\left (\frac {1}{d}\right )^{\frac {2}{3}} x^{2}+\left (\frac {1}{d}\right )^{\frac {1}{3}} \left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{3}} x +\left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {2}{3}}}{x^{2}}\right )}{2 \left (\frac {1}{d}\right )^{\frac {1}{3}} d}\) | \(149\) |
int((2*a*b^2-b*(2*a+b)*x+x^3)/(x*(-a+x)*(-b+x)^2)^(1/3)/(-a*b^2*d+b*(2*a+b )*d*x-(a*d+2*b*d+1)*x^2+d*x^3),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)^2)^(1/3 ))/(1/d)^(1/3)/x)+2*ln((-(1/d)^(1/3)*x+(-x*(a-x)*(b-x)^2)^(1/3))/x)-ln(((1 /d)^(2/3)*x^2+(1/d)^(1/3)*(-x*(a-x)*(b-x)^2)^(1/3)*x+(-x*(a-x)*(b-x)^2)^(2 /3))/x^2))/(1/d)^(1/3)/d
Timed out. \[ \int \frac {2 a b^2-b (2 a+b) x+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a b^2 d+b (2 a+b) d x-(1+a d+2 b d) x^2+d x^3\right )} \, dx=\text {Timed out} \]
integrate((2*a*b^2-b*(2*a+b)*x+x^3)/(x*(-a+x)*(-b+x)^2)^(1/3)/(-a*b^2*d+b* (2*a+b)*d*x-(a*d+2*b*d+1)*x^2+d*x^3),x, algorithm="fricas")
Timed out. \[ \int \frac {2 a b^2-b (2 a+b) x+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a b^2 d+b (2 a+b) d x-(1+a d+2 b d) x^2+d x^3\right )} \, dx=\text {Timed out} \]
integrate((2*a*b**2-b*(2*a+b)*x+x**3)/(x*(-a+x)*(-b+x)**2)**(1/3)/(-a*b**2 *d+b*(2*a+b)*d*x-(a*d+2*b*d+1)*x**2+d*x**3),x)
\[ \int \frac {2 a b^2-b (2 a+b) x+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a b^2 d+b (2 a+b) d x-(1+a d+2 b d) x^2+d x^3\right )} \, dx=\int { -\frac {2 \, a b^{2} - {\left (2 \, a + b\right )} b x + x^{3}}{{\left (a b^{2} d - {\left (2 \, a + b\right )} b d x - d x^{3} + {\left (a d + 2 \, b d + 1\right )} x^{2}\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {1}{3}}} \,d x } \]
integrate((2*a*b^2-b*(2*a+b)*x+x^3)/(x*(-a+x)*(-b+x)^2)^(1/3)/(-a*b^2*d+b* (2*a+b)*d*x-(a*d+2*b*d+1)*x^2+d*x^3),x, algorithm="maxima")
-integrate((2*a*b^2 - (2*a + b)*b*x + x^3)/((a*b^2*d - (2*a + b)*b*d*x - d *x^3 + (a*d + 2*b*d + 1)*x^2)*(-(a - x)*(b - x)^2*x)^(1/3)), x)
\[ \int \frac {2 a b^2-b (2 a+b) x+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a b^2 d+b (2 a+b) d x-(1+a d+2 b d) x^2+d x^3\right )} \, dx=\int { -\frac {2 \, a b^{2} - {\left (2 \, a + b\right )} b x + x^{3}}{{\left (a b^{2} d - {\left (2 \, a + b\right )} b d x - d x^{3} + {\left (a d + 2 \, b d + 1\right )} x^{2}\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {1}{3}}} \,d x } \]
integrate((2*a*b^2-b*(2*a+b)*x+x^3)/(x*(-a+x)*(-b+x)^2)^(1/3)/(-a*b^2*d+b* (2*a+b)*d*x-(a*d+2*b*d+1)*x^2+d*x^3),x, algorithm="giac")
integrate(-(2*a*b^2 - (2*a + b)*b*x + x^3)/((a*b^2*d - (2*a + b)*b*d*x - d *x^3 + (a*d + 2*b*d + 1)*x^2)*(-(a - x)*(b - x)^2*x)^(1/3)), x)
Timed out. \[ \int \frac {2 a b^2-b (2 a+b) x+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a b^2 d+b (2 a+b) d x-(1+a d+2 b d) x^2+d x^3\right )} \, dx=\int \frac {2\,a\,b^2+x^3-b\,x\,\left (2\,a+b\right )}{{\left (-x\,\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/3}\,\left (d\,x^3-x^2\,\left (a\,d+2\,b\,d+1\right )-a\,b^2\,d+b\,d\,x\,\left (2\,a+b\right )\right )} \,d x \]
int((2*a*b^2 + x^3 - b*x*(2*a + b))/((-x*(a - x)*(b - x)^2)^(1/3)*(d*x^3 - x^2*(a*d + 2*b*d + 1) - a*b^2*d + b*d*x*(2*a + b))),x)