Integrand size = 37, antiderivative size = 151 \[ \int \frac {x^3 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a d-d x+x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{d} \sqrt [3]{-a x^2+x^3}}\right )}{\sqrt [3]{d}}+\frac {\log \left (a x^2-a \sqrt [3]{d} \sqrt [3]{-a x^2+x^3}\right )}{\sqrt [3]{d}}-\frac {\log \left (a^2 x^4+a^2 \sqrt [3]{d} x^2 \sqrt [3]{-a x^2+x^3}+a^2 d^{2/3} \left (-a x^2+x^3\right )^{2/3}\right )}{2 \sqrt [3]{d}} \]
3^(1/2)*arctan(3^(1/2)*x^2/(x^2+2*d^(1/3)*(-a*x^2+x^3)^(1/3)))/d^(1/3)+ln( a*x^2-a*d^(1/3)*(-a*x^2+x^3)^(1/3))/d^(1/3)-1/2*ln(a^2*x^4+a^2*d^(1/3)*x^2 *(-a*x^2+x^3)^(1/3)+a^2*d^(2/3)*(-a*x^2+x^3)^(2/3))/d^(1/3)
Time = 0.62 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.03 \[ \int \frac {x^3 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a d-d x+x^4\right )} \, dx=\frac {x^{4/3} (-a+x)^{2/3} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{4/3}}{x^{4/3}+2 \sqrt [3]{d} \sqrt [3]{-a+x}}\right )+2 \log \left (a \left (x^{4/3}-\sqrt [3]{d} \sqrt [3]{-a+x}\right )\right )-\log \left (a^2 \left (x^{8/3}+\sqrt [3]{d} x^{4/3} \sqrt [3]{-a+x}+d^{2/3} (-a+x)^{2/3}\right )\right )\right )}{2 \sqrt [3]{d} \left (x^2 (-a+x)\right )^{2/3}} \]
(x^(4/3)*(-a + x)^(2/3)*(2*Sqrt[3]*ArcTan[(Sqrt[3]*x^(4/3))/(x^(4/3) + 2*d ^(1/3)*(-a + x)^(1/3))] + 2*Log[a*(x^(4/3) - d^(1/3)*(-a + x)^(1/3))] - Lo g[a^2*(x^(8/3) + d^(1/3)*x^(4/3)*(-a + x)^(1/3) + d^(2/3)*(-a + x)^(2/3))] ))/(2*d^(1/3)*(x^2*(-a + x))^(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.
| \(\displaystyle \int \frac {x^3 (3 x-4 a)}{\left (x^2 (x-a)\right )^{2/3} \left (a d-d x+x^4\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{4/3} (x-a)^{2/3} \int -\frac {(4 a-3 x) x^{5/3}}{(x-a)^{2/3} \left (x^4-d x+a d\right )}dx}{\left (-\left (x^2 (a-x)\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^{4/3} (x-a)^{2/3} \int \frac {(4 a-3 x) x^{5/3}}{(x-a)^{2/3} \left (x^4-d x+a d\right )}dx}{\left (-\left (x^2 (a-x)\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 x^{4/3} (x-a)^{2/3} \int \frac {(4 a-3 x) x^{7/3}}{(x-a)^{2/3} \left (x^4-d x+a d\right )}d\sqrt [3]{x}}{\left (-\left (x^2 (a-x)\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3 x^{4/3} (x-a)^{2/3} \int \left (\frac {4 a x^{7/3}}{(x-a)^{2/3} \left (x^4-d x+a d\right )}-\frac {3 x^{10/3}}{(x-a)^{2/3} \left (x^4-d x+a d\right )}\right )d\sqrt [3]{x}}{\left (-\left (x^2 (a-x)\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 x^{4/3} (x-a)^{2/3} \left (4 a \int \frac {x^{7/3}}{(x-a)^{2/3} \left (x^4-d x+a d\right )}d\sqrt [3]{x}-3 \int \frac {x^{10/3}}{(x-a)^{2/3} \left (x^4-d x+a d\right )}d\sqrt [3]{x}\right )}{\left (-\left (x^2 (a-x)\right )\right )^{2/3}}\) |
3.21.87.3.1 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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.30 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.50
| method | result | size |
| pseudoelliptic | \(\frac {a^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{12}-3 d \,\textit {\_Z}^{9}+3 d \,\textit {\_Z}^{6}-d \,\textit {\_Z}^{3}+a^{3}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (-\left (a -x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}^{8}-2 \textit {\_R}^{5}+\textit {\_R}^{2}}\right )}{d}\) | \(76\) |
a^2*sum(ln((-_R*x+(-(a-x)*x^2)^(1/3))/x)/(_R^8-2*_R^5+_R^2),_R=RootOf(_Z^1 2*d-3*_Z^9*d+3*_Z^6*d-_Z^3*d+a^3))/d
Time = 0.25 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.24 \[ \int \frac {x^3 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a d-d x+x^4\right )} \, dx=\left [\frac {\sqrt {3} d \sqrt {-\frac {1}{d^{\frac {2}{3}}}} \log \left (-\frac {x^{4} - 3 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{\frac {1}{3}} x^{2} - 2 \, a d + 2 \, d x + \sqrt {3} {\left (d^{\frac {1}{3}} x^{4} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{\frac {2}{3}} x^{2} - 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d\right )} \sqrt {-\frac {1}{d^{\frac {2}{3}}}}}{x^{4} + a d - d x}\right ) + 2 \, d^{\frac {2}{3}} \log \left (\frac {d^{\frac {2}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d}{x^{2}}\right ) - d^{\frac {2}{3}} \log \left (\frac {d^{\frac {1}{3}} x^{4} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{\frac {2}{3}} x^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d}{x^{4}}\right )}{2 \, d}, -\frac {2 \, \sqrt {3} d^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (d^{\frac {1}{3}} x^{2} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{\frac {2}{3}}\right )}}{3 \, d^{\frac {1}{3}} x^{2}}\right ) - 2 \, d^{\frac {2}{3}} \log \left (\frac {d^{\frac {2}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d}{x^{2}}\right ) + d^{\frac {2}{3}} \log \left (\frac {d^{\frac {1}{3}} x^{4} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{\frac {2}{3}} x^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d}{x^{4}}\right )}{2 \, d}\right ] \]
[1/2*(sqrt(3)*d*sqrt(-1/d^(2/3))*log(-(x^4 - 3*(-a*x^2 + x^3)^(1/3)*d^(1/3 )*x^2 - 2*a*d + 2*d*x + sqrt(3)*(d^(1/3)*x^4 + (-a*x^2 + x^3)^(1/3)*d^(2/3 )*x^2 - 2*(-a*x^2 + x^3)^(2/3)*d)*sqrt(-1/d^(2/3)))/(x^4 + a*d - d*x)) + 2 *d^(2/3)*log((d^(2/3)*x^2 - (-a*x^2 + x^3)^(1/3)*d)/x^2) - d^(2/3)*log((d^ (1/3)*x^4 + (-a*x^2 + x^3)^(1/3)*d^(2/3)*x^2 + (-a*x^2 + x^3)^(2/3)*d)/x^4 ))/d, -1/2*(2*sqrt(3)*d^(2/3)*arctan(1/3*sqrt(3)*(d^(1/3)*x^2 + 2*(-a*x^2 + x^3)^(1/3)*d^(2/3))/(d^(1/3)*x^2)) - 2*d^(2/3)*log((d^(2/3)*x^2 - (-a*x^ 2 + x^3)^(1/3)*d)/x^2) + d^(2/3)*log((d^(1/3)*x^4 + (-a*x^2 + x^3)^(1/3)*d ^(2/3)*x^2 + (-a*x^2 + x^3)^(2/3)*d)/x^4))/d]
Timed out. \[ \int \frac {x^3 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a d-d x+x^4\right )} \, dx=\text {Timed out} \]
\[ \int \frac {x^3 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a d-d x+x^4\right )} \, dx=\int { -\frac {{\left (4 \, a - 3 \, x\right )} x^{3}}{{\left (x^{4} + a d - d x\right )} \left (-{\left (a - x\right )} x^{2}\right )^{\frac {2}{3}}} \,d x } \]
Time = 0.33 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.56 \[ \int \frac {x^3 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a d-d x+x^4\right )} \, dx=-\sqrt {3} \left (-\frac {1}{d}\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} d {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + \sqrt {3} a {\left | d \right |}^{\frac {2}{3}} - \sqrt {3} d {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}{d {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + a {\left | d \right |}^{\frac {2}{3}} - d {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}\right ) - \frac {1}{2} \, \left (-\frac {1}{d}\right )^{\frac {1}{3}} \log \left (\frac {3}{4} \, a^{2} d^{\frac {4}{3}} + \frac {1}{4} \, {\left (2 \, d {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} - a d^{\frac {2}{3}} - 2 \, d {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}\right )}^{2}\right ) + \frac {1}{2} \, \left (-\frac {1}{d}\right )^{\frac {1}{3}} \log \left ({\left (\sqrt {3} d {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + \sqrt {3} a {\left | d \right |}^{\frac {2}{3}} - \sqrt {3} d {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}\right )}^{2} + {\left (d {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + a {\left | d \right |}^{\frac {2}{3}} - d {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}\right )}^{2}\right ) \]
-sqrt(3)*(-1/d)^(1/3)*arctan(-(sqrt(3)*d*(-a/x + 1)^(4/3) + sqrt(3)*a*abs( d)^(2/3) - sqrt(3)*d*(-a/x + 1)^(1/3))/(d*(-a/x + 1)^(4/3) + a*abs(d)^(2/3 ) - d*(-a/x + 1)^(1/3))) - 1/2*(-1/d)^(1/3)*log(3/4*a^2*d^(4/3) + 1/4*(2*d *(-a/x + 1)^(4/3) - a*d^(2/3) - 2*d*(-a/x + 1)^(1/3))^2) + 1/2*(-1/d)^(1/3 )*log((sqrt(3)*d*(-a/x + 1)^(4/3) + sqrt(3)*a*abs(d)^(2/3) - sqrt(3)*d*(-a /x + 1)^(1/3))^2 + (d*(-a/x + 1)^(4/3) + a*abs(d)^(2/3) - d*(-a/x + 1)^(1/ 3))^2)
Timed out. \[ \int \frac {x^3 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a d-d x+x^4\right )} \, dx=\int -\frac {x^3\,\left (4\,a-3\,x\right )}{{\left (-x^2\,\left (a-x\right )\right )}^{2/3}\,\left (x^4-d\,x+a\,d\right )} \,d x \]