Integrand size = 52, antiderivative size = 294 \[ \int \frac {x^3}{\sqrt [3]{x^2 (-a+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+(-1+d) x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{d} x^2}{\sqrt [6]{d} x^2-2 \left (-a x^2+x^3\right )^{2/3}}\right )}{4 a d^{5/6}}-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{d} x^2}{\sqrt [6]{d} x^2+2 \left (-a x^2+x^3\right )^{2/3}}\right )}{4 a d^{5/6}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} \left (-a x^2+x^3\right )^{2/3}}{\sqrt [3]{d} x^2+(-a+x) \sqrt [3]{-a x^2+x^3}}\right )}{4 a d^{5/6}}-\frac {\log \left (-\sqrt [12]{d} x+\sqrt [3]{-a x^2+x^3}\right )}{4 a d^{5/6}}-\frac {\log \left (\sqrt [12]{d} x+\sqrt [3]{-a x^2+x^3}\right )}{4 a d^{5/6}}+\frac {\log \left (\sqrt [6]{d} x^2+\left (-a x^2+x^3\right )^{2/3}\right )}{4 a d^{5/6}} \]
1/4*3^(1/2)*arctan(3^(1/2)*d^(1/6)*x^2/(d^(1/6)*x^2-2*(-a*x^2+x^3)^(2/3))) /a/d^(5/6)-1/4*3^(1/2)*arctan(3^(1/2)*d^(1/6)*x^2/(d^(1/6)*x^2+2*(-a*x^2+x ^3)^(2/3)))/a/d^(5/6)+1/4*arctanh(d^(1/6)*(-a*x^2+x^3)^(2/3)/(d^(1/3)*x^2+ (-a+x)*(-a*x^2+x^3)^(1/3)))/a/d^(5/6)-1/4*ln(-d^(1/12)*x+(-a*x^2+x^3)^(1/3 ))/a/d^(5/6)-1/4*ln(d^(1/12)*x+(-a*x^2+x^3)^(1/3))/a/d^(5/6)+1/4*ln(d^(1/6 )*x^2+(-a*x^2+x^3)^(2/3))/a/d^(5/6)
Time = 1.19 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.34 \[ \int \frac {x^3}{\sqrt [3]{x^2 (-a+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+(-1+d) x^4\right )} \, dx=\frac {x^{2/3} \sqrt [3]{-a+x} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{d} x^{2/3}}{\sqrt [6]{d} x^{2/3}-2 (-a+x)^{2/3}}\right )-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{d} x^{2/3}}{\sqrt [6]{d} x^{2/3}+2 (-a+x)^{2/3}}\right )-2 \log \left (-\sqrt [12]{d} \sqrt [3]{x}+\sqrt [3]{-a+x}\right )-2 \log \left (\sqrt [12]{d} \sqrt [3]{x}+\sqrt [3]{-a+x}\right )+2 \log \left (\sqrt [6]{d} x^{2/3}+(-a+x)^{2/3}\right )+\log \left (\sqrt [6]{d} x^{2/3}-\sqrt [12]{d} \sqrt [3]{x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )+\log \left (\sqrt [6]{d} x^{2/3}+\sqrt [12]{d} \sqrt [3]{x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )-\log \left (\sqrt [6]{d} x^{2/3}-\sqrt {3} \sqrt [12]{d} \sqrt [3]{x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )-\log \left (\sqrt [6]{d} x^{2/3}+\sqrt {3} \sqrt [12]{d} \sqrt [3]{x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )\right )}{8 a d^{5/6} \sqrt [3]{x^2 (-a+x)}} \]
(x^(2/3)*(-a + x)^(1/3)*(2*Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/6)*x^(2/3))/(d^(1/ 6)*x^(2/3) - 2*(-a + x)^(2/3))] - 2*Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/6)*x^(2/3 ))/(d^(1/6)*x^(2/3) + 2*(-a + x)^(2/3))] - 2*Log[-(d^(1/12)*x^(1/3)) + (-a + x)^(1/3)] - 2*Log[d^(1/12)*x^(1/3) + (-a + x)^(1/3)] + 2*Log[d^(1/6)*x^ (2/3) + (-a + x)^(2/3)] + Log[d^(1/6)*x^(2/3) - d^(1/12)*x^(1/3)*(-a + x)^ (1/3) + (-a + x)^(2/3)] + Log[d^(1/6)*x^(2/3) + d^(1/12)*x^(1/3)*(-a + x)^ (1/3) + (-a + x)^(2/3)] - Log[d^(1/6)*x^(2/3) - Sqrt[3]*d^(1/12)*x^(1/3)*( -a + x)^(1/3) + (-a + x)^(2/3)] - Log[d^(1/6)*x^(2/3) + Sqrt[3]*d^(1/12)*x ^(1/3)*(-a + x)^(1/3) + (-a + x)^(2/3)]))/(8*a*d^(5/6)*(x^2*(-a + x))^(1/3 ))
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.
| \(\displaystyle \int \frac {x^3}{\sqrt [3]{x^2 (x-a)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+(d-1) x^4\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \sqrt [3]{x-a} \int -\frac {x^{7/3}}{\sqrt [3]{x-a} \left (a^4-4 x a^3+6 x^2 a^2-4 x^3 a+(1-d) x^4\right )}dx}{\sqrt [3]{-\left (x^2 (a-x)\right )}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^{2/3} \sqrt [3]{x-a} \int \frac {x^{7/3}}{\sqrt [3]{x-a} \left (a^4-4 x a^3+6 x^2 a^2-4 x^3 a+(1-d) x^4\right )}dx}{\sqrt [3]{-\left (x^2 (a-x)\right )}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x-a} \int \frac {x^3}{\sqrt [3]{x-a} \left (a^4-4 x a^3+6 x^2 a^2-4 x^3 a+(1-d) x^4\right )}d\sqrt [3]{x}}{\sqrt [3]{-\left (x^2 (a-x)\right )}}\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x-a} \int \frac {x^3}{\sqrt [3]{x-a} \left (a^4-4 x a^3+6 x^2 a^2-4 x^3 a+(1-d) x^4\right )}d\sqrt [3]{x}}{\sqrt [3]{-\left (x^2 (a-x)\right )}}\) |
3.29.51.3.1 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]
Time = 0.93 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.82
| method | result | size |
| pseudoelliptic | \(-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (d^{\frac {1}{6}} x^{2}-2 \left (-x^{2} \left (a -x \right )\right )^{\frac {2}{3}}\right )}{3 d^{\frac {1}{6}} x^{2}}\right )-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (d^{\frac {1}{6}} x^{2}+2 \left (-x^{2} \left (a -x \right )\right )^{\frac {2}{3}}\right )}{3 d^{\frac {1}{6}} x^{2}}\right )-\ln \left (\frac {d^{\frac {1}{6}} x^{2}+\left (-x^{2} \left (a -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )+\ln \left (\frac {d^{\frac {1}{6}} x^{2}-\left (-x^{2} \left (a -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )+\frac {\ln \left (\frac {-d^{\frac {1}{6}} \left (-x^{2} \left (a -x \right )\right )^{\frac {2}{3}}+\left (-a +x \right ) \left (-x^{2} \left (a -x \right )\right )^{\frac {1}{3}}+d^{\frac {1}{3}} x^{2}}{x^{2}}\right )}{2}-\frac {\ln \left (\frac {d^{\frac {1}{6}} \left (-x^{2} \left (a -x \right )\right )^{\frac {2}{3}}+\left (-a +x \right ) \left (-x^{2} \left (a -x \right )\right )^{\frac {1}{3}}+d^{\frac {1}{3}} x^{2}}{x^{2}}\right )}{2}}{4 d^{\frac {5}{6}} a}\) | \(240\) |
int(x^3/(x^2*(-a+x))^(1/3)/(-a^4+4*a^3*x-6*a^2*x^2+4*a*x^3+(-1+d)*x^4),x,m ethod=_RETURNVERBOSE)
-1/4/d^(5/6)*(3^(1/2)*arctan(1/3*3^(1/2)*(d^(1/6)*x^2-2*(-x^2*(a-x))^(2/3) )/d^(1/6)/x^2)-3^(1/2)*arctan(1/3*3^(1/2)*(d^(1/6)*x^2+2*(-x^2*(a-x))^(2/3 ))/d^(1/6)/x^2)-ln((d^(1/6)*x^2+(-x^2*(a-x))^(2/3))/x^2)+ln((d^(1/6)*x^2-( -x^2*(a-x))^(2/3))/x^2)+1/2*ln((-d^(1/6)*(-x^2*(a-x))^(2/3)+(-a+x)*(-x^2*( a-x))^(1/3)+d^(1/3)*x^2)/x^2)-1/2*ln((d^(1/6)*(-x^2*(a-x))^(2/3)+(-a+x)*(- x^2*(a-x))^(1/3)+d^(1/3)*x^2)/x^2))/a
Time = 0.28 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.18 \[ \int \frac {x^3}{\sqrt [3]{x^2 (-a+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+(-1+d) x^4\right )} \, dx=\frac {1}{8} \, {\left (\sqrt {-3} + 1\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (\frac {{\left (\sqrt {-3} a d x^{2} + a d x^{2}\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{2 \, x^{2}}\right ) - \frac {1}{8} \, {\left (\sqrt {-3} + 1\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (-\frac {{\left (\sqrt {-3} a d x^{2} + a d x^{2}\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} - 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{2 \, x^{2}}\right ) + \frac {1}{8} \, {\left (\sqrt {-3} - 1\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (\frac {{\left (\sqrt {-3} a d x^{2} - a d x^{2}\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{2 \, x^{2}}\right ) - \frac {1}{8} \, {\left (\sqrt {-3} - 1\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (-\frac {{\left (\sqrt {-3} a d x^{2} - a d x^{2}\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} - 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{2 \, x^{2}}\right ) + \frac {1}{4} \, \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (\frac {a d x^{2} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{2}}\right ) - \frac {1}{4} \, \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (-\frac {a d x^{2} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} - {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]
integrate(x^3/(x^2*(-a+x))^(1/3)/(-a^4+4*a^3*x-6*a^2*x^2+4*a*x^3+(-1+d)*x^ 4),x, algorithm="fricas")
1/8*(sqrt(-3) + 1)*(1/(a^6*d^5))^(1/6)*log(1/2*((sqrt(-3)*a*d*x^2 + a*d*x^ 2)*(1/(a^6*d^5))^(1/6) + 2*(-a*x^2 + x^3)^(2/3))/x^2) - 1/8*(sqrt(-3) + 1) *(1/(a^6*d^5))^(1/6)*log(-1/2*((sqrt(-3)*a*d*x^2 + a*d*x^2)*(1/(a^6*d^5))^ (1/6) - 2*(-a*x^2 + x^3)^(2/3))/x^2) + 1/8*(sqrt(-3) - 1)*(1/(a^6*d^5))^(1 /6)*log(1/2*((sqrt(-3)*a*d*x^2 - a*d*x^2)*(1/(a^6*d^5))^(1/6) + 2*(-a*x^2 + x^3)^(2/3))/x^2) - 1/8*(sqrt(-3) - 1)*(1/(a^6*d^5))^(1/6)*log(-1/2*((sqr t(-3)*a*d*x^2 - a*d*x^2)*(1/(a^6*d^5))^(1/6) - 2*(-a*x^2 + x^3)^(2/3))/x^2 ) + 1/4*(1/(a^6*d^5))^(1/6)*log((a*d*x^2*(1/(a^6*d^5))^(1/6) + (-a*x^2 + x ^3)^(2/3))/x^2) - 1/4*(1/(a^6*d^5))^(1/6)*log(-(a*d*x^2*(1/(a^6*d^5))^(1/6 ) - (-a*x^2 + x^3)^(2/3))/x^2)
Timed out. \[ \int \frac {x^3}{\sqrt [3]{x^2 (-a+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+(-1+d) x^4\right )} \, dx=\text {Timed out} \]
\[ \int \frac {x^3}{\sqrt [3]{x^2 (-a+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+(-1+d) x^4\right )} \, dx=\int { \frac {x^{3}}{{\left ({\left (d - 1\right )} x^{4} - a^{4} + 4 \, a^{3} x - 6 \, a^{2} x^{2} + 4 \, a x^{3}\right )} \left (-{\left (a - x\right )} x^{2}\right )^{\frac {1}{3}}} \,d x } \]
integrate(x^3/(x^2*(-a+x))^(1/3)/(-a^4+4*a^3*x-6*a^2*x^2+4*a*x^3+(-1+d)*x^ 4),x, algorithm="maxima")
Time = 0.35 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.71 \[ \int \frac {x^3}{\sqrt [3]{x^2 (-a+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+(-1+d) x^4\right )} \, dx=-\frac {\sqrt {3} \log \left (\sqrt {3} \left (-d\right )^{\frac {1}{6}} {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + \left (-d\right )^{\frac {1}{3}}\right )}{8 \, a \left (-d\right )^{\frac {5}{6}}} - \frac {\sqrt {3} \left (-d\right )^{\frac {1}{6}} \log \left (-\sqrt {3} \left (-d\right )^{\frac {1}{6}} {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + \left (-d\right )^{\frac {1}{3}}\right )}{8 \, a d} - \frac {\arctan \left (\frac {\sqrt {3} \left (-d\right )^{\frac {1}{6}} + 2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}}}{\left (-d\right )^{\frac {1}{6}}}\right )}{4 \, a \left (-d\right )^{\frac {5}{6}}} - \frac {\arctan \left (-\frac {\sqrt {3} \left (-d\right )^{\frac {1}{6}} - 2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}}}{\left (-d\right )^{\frac {1}{6}}}\right )}{4 \, a \left (-d\right )^{\frac {5}{6}}} - \frac {\arctan \left (\frac {{\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}}}{\left (-d\right )^{\frac {1}{6}}}\right )}{2 \, a \left (-d\right )^{\frac {5}{6}}} \]
integrate(x^3/(x^2*(-a+x))^(1/3)/(-a^4+4*a^3*x-6*a^2*x^2+4*a*x^3+(-1+d)*x^ 4),x, algorithm="giac")
-1/8*sqrt(3)*log(sqrt(3)*(-d)^(1/6)*(-a/x + 1)^(2/3) + (-a/x + 1)^(4/3) + (-d)^(1/3))/(a*(-d)^(5/6)) - 1/8*sqrt(3)*(-d)^(1/6)*log(-sqrt(3)*(-d)^(1/6 )*(-a/x + 1)^(2/3) + (-a/x + 1)^(4/3) + (-d)^(1/3))/(a*d) - 1/4*arctan((sq rt(3)*(-d)^(1/6) + 2*(-a/x + 1)^(2/3))/(-d)^(1/6))/(a*(-d)^(5/6)) - 1/4*ar ctan(-(sqrt(3)*(-d)^(1/6) - 2*(-a/x + 1)^(2/3))/(-d)^(1/6))/(a*(-d)^(5/6)) - 1/2*arctan((-a/x + 1)^(2/3)/(-d)^(1/6))/(a*(-d)^(5/6))
Timed out. \[ \int \frac {x^3}{\sqrt [3]{x^2 (-a+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+(-1+d) x^4\right )} \, dx=\int \frac {x^3}{{\left (-x^2\,\left (a-x\right )\right )}^{1/3}\,\left (-a^4+4\,a^3\,x-6\,a^2\,x^2+4\,a\,x^3+\left (d-1\right )\,x^4\right )} \,d x \]