Integrand size = 80, antiderivative size = 291 \[ \int \frac {-a b^2+(4 a-b) b x-3 a x^2+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a^2+\left (2 a+b^2 d\right ) x-(1+2 b d) x^2+d x^3\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{-2 a+2 x+\sqrt [3]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}\right )}{d^{2/3}}+\frac {\log \left (a-x+\sqrt [3]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}\right )}{d^{2/3}}-\frac {\log \left (a^2-2 a x+x^2+\left (-a \sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}+d^{2/3} \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{2/3}\right )}{2 d^{2/3}} \]
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*a+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(a-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(a^2-2*a*x+x^2+(-a*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)*(-a*b^2*x+(2*a*b+b^2)*x^2+(-a-2*b)* x^3+x^4)^(2/3))/d^(2/3)
Time = 10.97 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.69 \[ \int \frac {-a b^2+(4 a-b) b x-3 a x^2+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a^2+\left (2 a+b^2 d\right ) x-(1+2 b d) x^2+d x^3\right )} \, dx=-\frac {-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{(b-x)^2 x (-a+x)}}{-2 a+2 x+\sqrt [3]{d} \sqrt [3]{x (-a+x) (-b+x)^2}}\right )-2 \log \left (-a+x-\sqrt [3]{d} \sqrt [3]{x (-a+x) (-b+x)^2}\right )+\log \left (a^2-2 a x+x^2-a \sqrt [3]{d} \sqrt [3]{x (-a+x) (-b+x)^2}+\sqrt [3]{d} x \sqrt [3]{x (-a+x) (-b+x)^2}+d^{2/3} \left (x (-a+x) (-b+x)^2\right )^{2/3}\right )}{2 d^{2/3}} \]
Integrate[(-(a*b^2) + (4*a - b)*b*x - 3*a*x^2 + x^3)/((x*(-a + x)*(-b + x) ^2)^(1/3)*(-a^2 + (2*a + b^2*d)*x - (1 + 2*b*d)*x^2 + d*x^3)),x]
-1/2*(-2*Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*((b - x)^2*x*(-a + x))^(1/3))/(-2 *a + 2*x + d^(1/3)*(x*(-a + x)*(-b + x)^2)^(1/3))] - 2*Log[-a + x - d^(1/3 )*(x*(-a + x)*(-b + x)^2)^(1/3)] + Log[a^2 - 2*a*x + x^2 - a*d^(1/3)*(x*(- a + x)*(-b + x)^2)^(1/3) + d^(1/3)*x*(x*(-a + x)*(-b + x)^2)^(1/3) + d^(2/ 3)*(x*(-a + x)*(-b + x)^2)^(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.
\(\displaystyle \int \frac {-a b^2+b x (4 a-b)-3 a x^2+x^3}{\sqrt [3]{x (x-a) (x-b)^2} \left (-a^2+x \left (2 a+b^2 d\right )-x^2 (2 b d+1)+d x^3\right )} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {-x^3+3 a x^2-(4 a-b) b x+a b^2}{\sqrt [3]{x} \sqrt [3]{x^3-(a+2 b) x^2+b (2 a+b) x-a b^2} \left (-d x^3+(2 b d+1) x^2-\left (d b^2+2 a\right ) x+a^2\right )}dx}{\sqrt [3]{-\left (x (a-x) (b-x)^2\right )}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {\sqrt [3]{x} \left (-x^3+3 a x^2-(4 a-b) b x+a b^2\right )}{\sqrt [3]{x^3-(a+2 b) x^2+b (2 a+b) x-a b^2} \left (-d x^3+(2 b d+1) x^2-\left (d b^2+2 a\right ) x+a^2\right )}d\sqrt [3]{x}}{\sqrt [3]{-\left (x (a-x) (b-x)^2\right )}}\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {(b-x) \sqrt [3]{x} \left (x^2-(3 a-b) x+a b\right )}{\sqrt [3]{-\left ((a-x) (x-b)^2\right )} \left (-d x^3+(2 b d+1) x^2-\left (d b^2+2 a\right ) x+a^2\right )}d\sqrt [3]{x}}{\sqrt [3]{-\left (x (a-x) (b-x)^2\right )}}\) |
\(\Big \downarrow \) 2058 |
\(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{a-x} (x-b)^{2/3} \sqrt [3]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {(b-x) \sqrt [3]{x} \left (x^2-(3 a-b) x+a b\right )}{\sqrt [3]{a-x} (x-b)^{2/3} \left (-d x^3+(2 b d+1) x^2-\left (d b^2+2 a\right ) x+a^2\right )}d\sqrt [3]{x}}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )} \sqrt [3]{-\left (x (a-x) (b-x)^2\right )}}\) |
\(\Big \downarrow \) 281 |
\(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{a-x} (x-b)^{2/3} \sqrt [3]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{x-b} \left (x^2-(3 a-b) x+a b\right )}{\sqrt [3]{a-x} \left (-d x^3+(2 b d+1) x^2-\left (d b^2+2 a\right ) x+a^2\right )}d\sqrt [3]{x}}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )} \sqrt [3]{-\left (x (a-x) (b-x)^2\right )}}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{a-x} (x-b)^{2/3} \sqrt [3]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \left (\frac {\sqrt [3]{x-b} x^{7/3}}{\sqrt [3]{a-x} \left (-d x^3+(2 b d+1) x^2-2 a \left (\frac {d b^2}{2 a}+1\right ) x+a^2\right )}+\frac {(b-3 a) \sqrt [3]{x-b} x^{4/3}}{\sqrt [3]{a-x} \left (-d x^3+(2 b d+1) x^2-2 a \left (\frac {d b^2}{2 a}+1\right ) x+a^2\right )}+\frac {a b \sqrt [3]{x-b} \sqrt [3]{x}}{\sqrt [3]{a-x} \left (-d x^3+(2 b d+1) x^2-2 a \left (\frac {d b^2}{2 a}+1\right ) x+a^2\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )} \sqrt [3]{-\left (x (a-x) (b-x)^2\right )}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{a-x} (x-b)^{2/3} \sqrt [3]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \left (a b \int \frac {\sqrt [3]{x} \sqrt [3]{x-b}}{\sqrt [3]{a-x} \left (-d x^3+(2 b d+1) x^2-2 a \left (\frac {d b^2}{2 a}+1\right ) x+a^2\right )}d\sqrt [3]{x}-(3 a-b) \int \frac {x^{4/3} \sqrt [3]{x-b}}{\sqrt [3]{a-x} \left (-d x^3+(2 b d+1) x^2-2 a \left (\frac {d b^2}{2 a}+1\right ) x+a^2\right )}d\sqrt [3]{x}+\int \frac {x^{7/3} \sqrt [3]{x-b}}{\sqrt [3]{a-x} \left (-d x^3+(2 b d+1) x^2-2 a \left (\frac {d b^2}{2 a}+1\right ) x+a^2\right )}d\sqrt [3]{x}\right )}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )} \sqrt [3]{-\left (x (a-x) (b-x)^2\right )}}\) |
Int[(-(a*b^2) + (4*a - b)*b*x - 3*a*x^2 + x^3)/((x*(-a + x)*(-b + x)^2)^(1 /3)*(-a^2 + (2*a + b^2*d)*x - (1 + 2*b*d)*x^2 + d*x^3)),x]
3.29.43.3.1 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]
\[\int \frac {-a \,b^{2}+\left (4 a -b \right ) b x -3 a \,x^{2}+x^{3}}{\left (x \left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{3}} \left (-a^{2}+\left (b^{2} d +2 a \right ) x -\left (2 b d +1\right ) x^{2}+d \,x^{3}\right )}d x\]
int((-a*b^2+(4*a-b)*b*x-3*a*x^2+x^3)/(x*(-a+x)*(-b+x)^2)^(1/3)/(-a^2+(b^2* d+2*a)*x-(2*b*d+1)*x^2+d*x^3),x)
int((-a*b^2+(4*a-b)*b*x-3*a*x^2+x^3)/(x*(-a+x)*(-b+x)^2)^(1/3)/(-a^2+(b^2* d+2*a)*x-(2*b*d+1)*x^2+d*x^3),x)
Timed out. \[ \int \frac {-a b^2+(4 a-b) b x-3 a x^2+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a^2+\left (2 a+b^2 d\right ) x-(1+2 b d) x^2+d x^3\right )} \, dx=\text {Timed out} \]
integrate((-a*b^2+(4*a-b)*b*x-3*a*x^2+x^3)/(x*(-a+x)*(-b+x)^2)^(1/3)/(-a^2 +(b^2*d+2*a)*x-(2*b*d+1)*x^2+d*x^3),x, algorithm="fricas")
Timed out. \[ \int \frac {-a b^2+(4 a-b) b x-3 a x^2+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a^2+\left (2 a+b^2 d\right ) x-(1+2 b d) x^2+d x^3\right )} \, dx=\text {Timed out} \]
integrate((-a*b**2+(4*a-b)*b*x-3*a*x**2+x**3)/(x*(-a+x)*(-b+x)**2)**(1/3)/ (-a**2+(b**2*d+2*a)*x-(2*b*d+1)*x**2+d*x**3),x)
\[ \int \frac {-a b^2+(4 a-b) b x-3 a x^2+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a^2+\left (2 a+b^2 d\right ) x-(1+2 b d) x^2+d x^3\right )} \, dx=\int { -\frac {a b^{2} - {\left (4 \, a - b\right )} b x + 3 \, a x^{2} - x^{3}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {1}{3}} {\left (d x^{3} - {\left (2 \, b d + 1\right )} x^{2} - a^{2} + {\left (b^{2} d + 2 \, a\right )} x\right )}} \,d x } \]
integrate((-a*b^2+(4*a-b)*b*x-3*a*x^2+x^3)/(x*(-a+x)*(-b+x)^2)^(1/3)/(-a^2 +(b^2*d+2*a)*x-(2*b*d+1)*x^2+d*x^3),x, algorithm="maxima")
-integrate((a*b^2 - (4*a - b)*b*x + 3*a*x^2 - x^3)/((-(a - x)*(b - x)^2*x) ^(1/3)*(d*x^3 - (2*b*d + 1)*x^2 - a^2 + (b^2*d + 2*a)*x)), x)
\[ \int \frac {-a b^2+(4 a-b) b x-3 a x^2+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a^2+\left (2 a+b^2 d\right ) x-(1+2 b d) x^2+d x^3\right )} \, dx=\int { -\frac {a b^{2} - {\left (4 \, a - b\right )} b x + 3 \, a x^{2} - x^{3}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {1}{3}} {\left (d x^{3} - {\left (2 \, b d + 1\right )} x^{2} - a^{2} + {\left (b^{2} d + 2 \, a\right )} x\right )}} \,d x } \]
integrate((-a*b^2+(4*a-b)*b*x-3*a*x^2+x^3)/(x*(-a+x)*(-b+x)^2)^(1/3)/(-a^2 +(b^2*d+2*a)*x-(2*b*d+1)*x^2+d*x^3),x, algorithm="giac")
integrate(-(a*b^2 - (4*a - b)*b*x + 3*a*x^2 - x^3)/((-(a - x)*(b - x)^2*x) ^(1/3)*(d*x^3 - (2*b*d + 1)*x^2 - a^2 + (b^2*d + 2*a)*x)), x)
Timed out. \[ \int \frac {-a b^2+(4 a-b) b x-3 a x^2+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a^2+\left (2 a+b^2 d\right ) x-(1+2 b d) x^2+d x^3\right )} \, dx=\int -\frac {a\,b^2+3\,a\,x^2-x^3-b\,x\,\left (4\,a-b\right )}{{\left (-x\,\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/3}\,\left (x\,\left (d\,b^2+2\,a\right )+d\,x^3-x^2\,\left (2\,b\,d+1\right )-a^2\right )} \,d x \]
int(-(a*b^2 + 3*a*x^2 - x^3 - b*x*(4*a - b))/((-x*(a - x)*(b - x)^2)^(1/3) *(x*(2*a + b^2*d) + d*x^3 - x^2*(2*b*d + 1) - a^2)),x)