\(\int \frac {x^7 (-4 a+3 x)}{(x^2 (-a+x))^{2/3} (-a^2+2 a x-x^2+d x^8)} \, dx\) [2497]

Optimal result

Integrand size = 46, antiderivative size = 207 \[ \int \frac {x^7 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x-x^2+d x^8\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{d} x^2}{\sqrt [6]{d} x^2-2 \sqrt [3]{-a x^2+x^3}}\right )}{2 d^{5/6}}+\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{d} x^2}{\sqrt [6]{d} x^2+2 \sqrt [3]{-a x^2+x^3}}\right )}{2 d^{5/6}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} \left (-a x^2+x^3\right )^{2/3}}{a-x}\right )}{d^{5/6}}-\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} x^4+\frac {\left (-a x^2+x^3\right )^{2/3}}{\sqrt [6]{d}}}{x^2 \sqrt [3]{-a x^2+x^3}}\right )}{2 d^{5/6}} \]

[Out]

-1/2*3^(1/2)*arctan(3^(1/2)*d^(1/6)*x^2/(d^(1/6)*x^2-2*(-a*x^2+x^3)^(1/3)))/d^(5/6)+1/2*3^(1/2)*arctan(3^(1/2)
*d^(1/6)*x^2/(d^(1/6)*x^2+2*(-a*x^2+x^3)^(1/3)))/d^(5/6)+arctanh(d^(1/6)*(-a*x^2+x^3)^(2/3)/(a-x))/d^(5/6)-1/2
*arctanh((d^(1/6)*x^4+(-a*x^2+x^3)^(2/3)/d^(1/6))/x^2/(-a*x^2+x^3)^(1/3))/d^(5/6)

Rubi [F]

\[ \int \frac {x^7 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x-x^2+d x^8\right )} \, dx=\int \frac {x^7 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x-x^2+d x^8\right )} \, dx \]

[In]

Int[(x^7*(-4*a + 3*x))/((x^2*(-a + x))^(2/3)*(-a^2 + 2*a*x - x^2 + d*x^8)),x]

[Out]

(12*a*x^(4/3)*(-a + x)^(2/3)*Defer[Subst][Defer[Int][x^19/((-a + x^3)^(2/3)*(a^2 - 2*a*x^3 + x^6 - d*x^24)), x
], x, x^(1/3)])/(-((a - x)*x^2))^(2/3) + (9*x^(4/3)*(-a + x)^(2/3)*Defer[Subst][Defer[Int][x^22/((-a + x^3)^(2
/3)*(-a^2 + 2*a*x^3 - x^6 + d*x^24)), x], x, x^(1/3)])/(-((a - x)*x^2))^(2/3)

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{4/3} (-a+x)^{2/3}\right ) \int \frac {x^{17/3} (-4 a+3 x)}{(-a+x)^{2/3} \left (-a^2+2 a x-x^2+d x^8\right )} \, dx}{\left (x^2 (-a+x)\right )^{2/3}} \\ & = \frac {\left (3 x^{4/3} (-a+x)^{2/3}\right ) \text {Subst}\left (\int \frac {x^{19} \left (-4 a+3 x^3\right )}{\left (-a+x^3\right )^{2/3} \left (-a^2+2 a x^3-x^6+d x^{24}\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (x^2 (-a+x)\right )^{2/3}} \\ & = \frac {\left (3 x^{4/3} (-a+x)^{2/3}\right ) \text {Subst}\left (\int \left (\frac {4 a x^{19}}{\left (-a+x^3\right )^{2/3} \left (a^2-2 a x^3+x^6-d x^{24}\right )}+\frac {3 x^{22}}{\left (-a+x^3\right )^{2/3} \left (-a^2+2 a x^3-x^6+d x^{24}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (x^2 (-a+x)\right )^{2/3}} \\ & = \frac {\left (9 x^{4/3} (-a+x)^{2/3}\right ) \text {Subst}\left (\int \frac {x^{22}}{\left (-a+x^3\right )^{2/3} \left (-a^2+2 a x^3-x^6+d x^{24}\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (x^2 (-a+x)\right )^{2/3}}+\frac {\left (12 a x^{4/3} (-a+x)^{2/3}\right ) \text {Subst}\left (\int \frac {x^{19}}{\left (-a+x^3\right )^{2/3} \left (a^2-2 a x^3+x^6-d x^{24}\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (x^2 (-a+x)\right )^{2/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.85 \[ \int \frac {x^7 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x-x^2+d x^8\right )} \, dx=\frac {x^{4/3} (-a+x)^{2/3} \left (\sqrt {3} \left (\arctan \left (\frac {1-\frac {2 \sqrt [3]{-a+x}}{\sqrt [6]{d} x^{4/3}}}{\sqrt {3}}\right )-\arctan \left (\frac {1+\frac {2 \sqrt [3]{-a+x}}{\sqrt [6]{d} x^{4/3}}}{\sqrt {3}}\right )\right )-2 \text {arctanh}\left (\frac {\sqrt [3]{-a+x}}{\sqrt [6]{d} x^{4/3}}\right )-\text {arctanh}\left (\frac {\sqrt [6]{d} x^{4/3} \sqrt [3]{-a+x}}{\sqrt [3]{d} x^{8/3}+(-a+x)^{2/3}}\right )\right )}{2 d^{5/6} \left (x^2 (-a+x)\right )^{2/3}} \]

[In]

Integrate[(x^7*(-4*a + 3*x))/((x^2*(-a + x))^(2/3)*(-a^2 + 2*a*x - x^2 + d*x^8)),x]

[Out]

(x^(4/3)*(-a + x)^(2/3)*(Sqrt[3]*(ArcTan[(1 - (2*(-a + x)^(1/3))/(d^(1/6)*x^(4/3)))/Sqrt[3]] - ArcTan[(1 + (2*
(-a + x)^(1/3))/(d^(1/6)*x^(4/3)))/Sqrt[3]]) - 2*ArcTanh[(-a + x)^(1/3)/(d^(1/6)*x^(4/3))] - ArcTanh[(d^(1/6)*
x^(4/3)*(-a + x)^(1/3))/(d^(1/3)*x^(8/3) + (-a + x)^(2/3))]))/(2*d^(5/6)*(x^2*(-a + x))^(2/3))

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.25 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.42

method	result	size
pseudoelliptic	\(-\frac {a^{5} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{24}-6 \textit {\_Z}^{21}+15 \textit {\_Z}^{18}-20 \textit {\_Z}^{15}+15 \textit {\_Z}^{12}-6 \textit {\_Z}^{9}-a^{6} d +\textit {\_Z}^{6}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (-x^{2} \left (a -x \right )\right )^{\frac {1}{3}}}{x}\right )}{\left (\textit {\_R} -1\right )^{5} \left (\textit {\_R}^{2}+\textit {\_R} +1\right )^{5} \textit {\_R}^{5}}\right )}{2}\)	\(87\)

[In]

int(x^7*(-4*a+3*x)/(x^2*(-a+x))^(2/3)/(d*x^8-a^2+2*a*x-x^2),x,method=_RETURNVERBOSE)

[Out]

-1/2*a^5*sum(ln((-_R*x+(-x^2*(a-x))^(1/3))/x)/(_R-1)^5/(_R^2+_R+1)^5/_R^5,_R=RootOf(_Z^24-6*_Z^21+15*_Z^18-20*
_Z^15+15*_Z^12-6*_Z^9-a^6*d+_Z^6))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.39 \[ \int \frac {x^7 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x-x^2+d x^8\right )} \, dx=-\frac {1}{4} \, {\left (\sqrt {-3} + 1\right )} \frac {1}{d^{5}}^{\frac {1}{6}} \log \left (-\frac {{\left (\sqrt {-3} d x^{2} + d x^{2}\right )} \frac {1}{d^{5}}^{\frac {1}{6}} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{2 \, x^{2}}\right ) + \frac {1}{4} \, {\left (\sqrt {-3} + 1\right )} \frac {1}{d^{5}}^{\frac {1}{6}} \log \left (\frac {{\left (\sqrt {-3} d x^{2} + d x^{2}\right )} \frac {1}{d^{5}}^{\frac {1}{6}} - 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{2 \, x^{2}}\right ) - \frac {1}{4} \, {\left (\sqrt {-3} - 1\right )} \frac {1}{d^{5}}^{\frac {1}{6}} \log \left (-\frac {{\left (\sqrt {-3} d x^{2} - d x^{2}\right )} \frac {1}{d^{5}}^{\frac {1}{6}} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{2 \, x^{2}}\right ) + \frac {1}{4} \, {\left (\sqrt {-3} - 1\right )} \frac {1}{d^{5}}^{\frac {1}{6}} \log \left (\frac {{\left (\sqrt {-3} d x^{2} - d x^{2}\right )} \frac {1}{d^{5}}^{\frac {1}{6}} - 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{2 \, x^{2}}\right ) - \frac {1}{2} \, \frac {1}{d^{5}}^{\frac {1}{6}} \log \left (-\frac {d \frac {1}{d^{5}}^{\frac {1}{6}} x^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x^{2}}\right ) + \frac {1}{2} \, \frac {1}{d^{5}}^{\frac {1}{6}} \log \left (\frac {d \frac {1}{d^{5}}^{\frac {1}{6}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x^{2}}\right ) \]

[In]

integrate(x^7*(-4*a+3*x)/(x^2*(-a+x))^(2/3)/(d*x^8-a^2+2*a*x-x^2),x, algorithm="fricas")

[Out]

-1/4*(sqrt(-3) + 1)*(d^(-5))^(1/6)*log(-1/2*((sqrt(-3)*d*x^2 + d*x^2)*(d^(-5))^(1/6) + 2*(-a*x^2 + x^3)^(1/3))
/x^2) + 1/4*(sqrt(-3) + 1)*(d^(-5))^(1/6)*log(1/2*((sqrt(-3)*d*x^2 + d*x^2)*(d^(-5))^(1/6) - 2*(-a*x^2 + x^3)^
(1/3))/x^2) - 1/4*(sqrt(-3) - 1)*(d^(-5))^(1/6)*log(-1/2*((sqrt(-3)*d*x^2 - d*x^2)*(d^(-5))^(1/6) + 2*(-a*x^2
+ x^3)^(1/3))/x^2) + 1/4*(sqrt(-3) - 1)*(d^(-5))^(1/6)*log(1/2*((sqrt(-3)*d*x^2 - d*x^2)*(d^(-5))^(1/6) - 2*(-
a*x^2 + x^3)^(1/3))/x^2) - 1/2*(d^(-5))^(1/6)*log(-(d*(d^(-5))^(1/6)*x^2 + (-a*x^2 + x^3)^(1/3))/x^2) + 1/2*(d
^(-5))^(1/6)*log((d*(d^(-5))^(1/6)*x^2 - (-a*x^2 + x^3)^(1/3))/x^2)

Sympy [F]

\[ \int \frac {x^7 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x-x^2+d x^8\right )} \, dx=\int \frac {x^{7} \left (- 4 a + 3 x\right )}{\left (x^{2} \left (- a + x\right )\right )^{\frac {2}{3}} \left (- a^{2} + 2 a x + d x^{8} - x^{2}\right )}\, dx \]

[In]

integrate(x**7*(-4*a+3*x)/(x**2*(-a+x))**(2/3)/(d*x**8-a**2+2*a*x-x**2),x)

[Out]

Integral(x**7*(-4*a + 3*x)/((x**2*(-a + x))**(2/3)*(-a**2 + 2*a*x + d*x**8 - x**2)), x)

Maxima [F]

\[ \int \frac {x^7 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x-x^2+d x^8\right )} \, dx=\int { -\frac {{\left (4 \, a - 3 \, x\right )} x^{7}}{{\left (d x^{8} - a^{2} + 2 \, a x - x^{2}\right )} \left (-{\left (a - x\right )} x^{2}\right )^{\frac {2}{3}}} \,d x } \]

[In]

integrate(x^7*(-4*a+3*x)/(x^2*(-a+x))^(2/3)/(d*x^8-a^2+2*a*x-x^2),x, algorithm="maxima")

[Out]

-integrate((4*a - 3*x)*x^7/((d*x^8 - a^2 + 2*a*x - x^2)*(-(a - x)*x^2)^(2/3)), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 399 vs. \(2 (161) = 322\).

Time = 0.50 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.93 \[ \int \frac {x^7 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x-x^2+d x^8\right )} \, dx=\frac {1}{2} \, \sqrt {3} \frac {1}{d^{5}}^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3} {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} - \sqrt {3} a d^{\frac {1}{6}} - \sqrt {3} {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}{{\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + a d^{\frac {1}{6}} - {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}\right ) + \frac {1}{2} \, \sqrt {3} \frac {1}{d^{5}}^{\frac {1}{6}} \arctan \left (-\frac {\sqrt {3} {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + \sqrt {3} a d^{\frac {1}{6}} - \sqrt {3} {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}{{\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} - a d^{\frac {1}{6}} - {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}\right ) + \frac {1}{4} \, \frac {1}{d^{5}}^{\frac {1}{6}} \log \left ({\left (\sqrt {3} {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + \sqrt {3} a d^{\frac {1}{6}} - \sqrt {3} {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}\right )}^{2} + {\left ({\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} - a d^{\frac {1}{6}} - {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}\right )}^{2}\right ) - \frac {1}{4} \, \frac {1}{d^{5}}^{\frac {1}{6}} \log \left ({\left (\sqrt {3} {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} - \sqrt {3} a d^{\frac {1}{6}} - \sqrt {3} {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}\right )}^{2} + {\left ({\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + a d^{\frac {1}{6}} - {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}\right )}^{2}\right ) + \frac {1}{2} \, \frac {1}{d^{5}}^{\frac {1}{6}} \log \left ({\left | {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + a d^{\frac {1}{6}} - {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} \right |}\right ) - \frac {1}{2} \, \frac {1}{d^{5}}^{\frac {1}{6}} \log \left ({\left | {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} - a d^{\frac {1}{6}} - {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} \right |}\right ) \]

[In]

integrate(x^7*(-4*a+3*x)/(x^2*(-a+x))^(2/3)/(d*x^8-a^2+2*a*x-x^2),x, algorithm="giac")

[Out]

1/2*sqrt(3)*(d^(-5))^(1/6)*arctan((sqrt(3)*(-a/x + 1)^(4/3) - sqrt(3)*a*d^(1/6) - sqrt(3)*(-a/x + 1)^(1/3))/((
-a/x + 1)^(4/3) + a*d^(1/6) - (-a/x + 1)^(1/3))) + 1/2*sqrt(3)*(d^(-5))^(1/6)*arctan(-(sqrt(3)*(-a/x + 1)^(4/3
) + sqrt(3)*a*d^(1/6) - sqrt(3)*(-a/x + 1)^(1/3))/((-a/x + 1)^(4/3) - a*d^(1/6) - (-a/x + 1)^(1/3))) + 1/4*(d^
(-5))^(1/6)*log((sqrt(3)*(-a/x + 1)^(4/3) + sqrt(3)*a*d^(1/6) - sqrt(3)*(-a/x + 1)^(1/3))^2 + ((-a/x + 1)^(4/3
) - a*d^(1/6) - (-a/x + 1)^(1/3))^2) - 1/4*(d^(-5))^(1/6)*log((sqrt(3)*(-a/x + 1)^(4/3) - sqrt(3)*a*d^(1/6) -
sqrt(3)*(-a/x + 1)^(1/3))^2 + ((-a/x + 1)^(4/3) + a*d^(1/6) - (-a/x + 1)^(1/3))^2) + 1/2*(d^(-5))^(1/6)*log(ab
s((-a/x + 1)^(4/3) + a*d^(1/6) - (-a/x + 1)^(1/3))) - 1/2*(d^(-5))^(1/6)*log(abs((-a/x + 1)^(4/3) - a*d^(1/6)
- (-a/x + 1)^(1/3)))

Mupad [F(-1)]

Timed out. \[ \int \frac {x^7 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x-x^2+d x^8\right )} \, dx=\int -\frac {x^7\,\left (4\,a-3\,x\right )}{{\left (-x^2\,\left (a-x\right )\right )}^{2/3}\,\left (-a^2+2\,a\,x+d\,x^8-x^2\right )} \,d x \]

[In]

int(-(x^7*(4*a - 3*x))/((-x^2*(a - x))^(2/3)*(2*a*x + d*x^8 - a^2 - x^2)),x)

[Out]

int(-(x^7*(4*a - 3*x))/((-x^2*(a - x))^(2/3)*(2*a*x + d*x^8 - a^2 - x^2)), x)