\(\int \frac {-((2 a-b) b^2)+(4 a-b) b x-(2 a+b) x^2+x^3}{(-a+x) \sqrt [4]{(-a+x) (-b+x)^2} (b^2+a d-(2 b+d) x+x^2)} \, dx\) [2217]

Optimal result

Integrand size = 81, antiderivative size = 165 \[ \int \frac {-\left ((2 a-b) b^2\right )+(4 a-b) b x-(2 a+b) x^2+x^3}{(-a+x) \sqrt [4]{(-a+x) (-b+x)^2} \left (b^2+a d-(2 b+d) x+x^2\right )} \, dx=-\frac {4 \left (-a b^2+2 a b x+b^2 x-a x^2-2 b x^2+x^3\right )^{3/4}}{(b-x) (-a+x)}-2 \sqrt [4]{d} \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}{b-x}\right )+2 \sqrt [4]{d} \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}{b-x}\right ) \]

[Out]

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

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 2.03 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.97, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {6820, 6851, 6860, 142, 141} \[ \int \frac {-\left ((2 a-b) b^2\right )+(4 a-b) b x-(2 a+b) x^2+x^3}{(-a+x) \sqrt [4]{(-a+x) (-b+x)^2} \left (b^2+a d-(2 b+d) x+x^2\right )} \, dx=\frac {4 (a-b) (b-x) \left (\sqrt {-4 a+4 b+d}+\sqrt {d}\right ) \operatorname {AppellF1}\left (-\frac {1}{4},-\frac {3}{2},1,\frac {3}{4},\frac {a-x}{a-b},\frac {2 (a-x)}{2 a-2 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right )}{\sqrt {d} \left (-\sqrt {d} \sqrt {-4 a+4 b+d}+2 a-2 b-d\right ) \sqrt {-\frac {b-x}{a-b}} \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}+\frac {4 (a-b) (b-x) \left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \operatorname {AppellF1}\left (-\frac {1}{4},-\frac {3}{2},1,\frac {3}{4},\frac {a-x}{a-b},\frac {2 (a-x)}{2 a-2 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right )}{\left (\sqrt {d} \sqrt {-4 a+4 b+d}+2 a-2 b-d\right ) \sqrt {-\frac {b-x}{a-b}} \sqrt [4]{-\left ((a-x) (b-x)^2\right )}} \]

[In]

Int[(-((2*a - b)*b^2) + (4*a - b)*b*x - (2*a + b)*x^2 + x^3)/((-a + x)*((-a + x)*(-b + x)^2)^(1/4)*(b^2 + a*d
- (2*b + d)*x + x^2)),x]

[Out]

(4*(a - b)*(Sqrt[d] + Sqrt[-4*a + 4*b + d])*(b - x)*AppellF1[-1/4, -3/2, 1, 3/4, (a - x)/(a - b), (2*(a - x))/
(2*a - 2*b - d - Sqrt[d]*Sqrt[-4*a + 4*b + d])])/(Sqrt[d]*(2*a - 2*b - d - Sqrt[d]*Sqrt[-4*a + 4*b + d])*Sqrt[
-((b - x)/(a - b))]*(-((a - x)*(b - x)^2))^(1/4)) + (4*(a - b)*(1 - Sqrt[-4*a + 4*b + d]/Sqrt[d])*(b - x)*Appe
llF1[-1/4, -3/2, 1, 3/4, (a - x)/(a - b), (2*(a - x))/(2*a - 2*b - d + Sqrt[d]*Sqrt[-4*a + 4*b + d])])/((2*a -
2*b - d + Sqrt[d]*Sqrt[-4*a + 4*b + d])*Sqrt[-((b - x)/(a - b))]*(-((a - x)*(b - x)^2))^(1/4))

Rule 141

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[(b*e - a*f
)^p*((a + b*x)^(m + 1)/(b^(p + 1)*(m + 1)*(b/(b*c - a*d))^n))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/(
b*c - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] &&  !IntegerQ[m] &&  !Int
egerQ[n] && IntegerQ[p] && GtQ[b/(b*c - a*d), 0] &&  !(GtQ[d/(d*a - c*b), 0] && SimplerQ[c + d*x, a + b*x])

Rule 142

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(c + d*x)^
FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d)))^FracPart[n]), Int[(a + b*x)^m*(b*(c/(b*c -
a*d)) + b*d*(x/(b*c - a*d)))^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] &&  !IntegerQ[m] &&
!IntegerQ[n] && IntegerQ[p] &&  !GtQ[b/(b*c - a*d), 0] &&  !SimplerQ[c + d*x, a + b*x]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6851

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a*v^m*w^n)^FracPart[p]/(v^(m*Fr
acPart[p])*w^(n*FracPart[p]))), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(b-x)^4 (-2 a+b+x)}{\left (-\left ((a-x) (b-x)^2\right )\right )^{5/4} \left (b^2+a d-(2 b+d) x+x^2\right )} \, dx \\ & = -\frac {\left (\sqrt [4]{a-x} \sqrt {b-x}\right ) \int \frac {(b-x)^{3/2} (-2 a+b+x)}{(a-x)^{5/4} \left (b^2+a d-(2 b+d) x+x^2\right )} \, dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}} \\ & = -\frac {\left (\sqrt [4]{a-x} \sqrt {b-x}\right ) \int \left (\frac {\left (1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) (b-x)^{3/2}}{(a-x)^{5/4} \left (-2 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}+2 x\right )}+\frac {\left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) (b-x)^{3/2}}{(a-x)^{5/4} \left (-2 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}+2 x\right )}\right ) \, dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}} \\ & = -\frac {\left (\left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \sqrt [4]{a-x} \sqrt {b-x}\right ) \int \frac {(b-x)^{3/2}}{(a-x)^{5/4} \left (-2 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}+2 x\right )} \, dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}-\frac {\left (\left (1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \sqrt [4]{a-x} \sqrt {b-x}\right ) \int \frac {(b-x)^{3/2}}{(a-x)^{5/4} \left (-2 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}+2 x\right )} \, dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}} \\ & = \frac {\left ((a-b) \left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \sqrt [4]{a-x} (b-x)\right ) \int \frac {\left (-\frac {b}{a-b}+\frac {x}{a-b}\right )^{3/2}}{(a-x)^{5/4} \left (-2 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}+2 x\right )} \, dx}{\sqrt {-\frac {b-x}{a-b}} \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}+\frac {\left ((a-b) \left (1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \sqrt [4]{a-x} (b-x)\right ) \int \frac {\left (-\frac {b}{a-b}+\frac {x}{a-b}\right )^{3/2}}{(a-x)^{5/4} \left (-2 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}+2 x\right )} \, dx}{\sqrt {-\frac {b-x}{a-b}} \sqrt [4]{-\left ((a-x) (b-x)^2\right )}} \\ & = \frac {4 (a-b) \left (\sqrt {d}+\sqrt {-4 a+4 b+d}\right ) (b-x) \operatorname {AppellF1}\left (-\frac {1}{4},-\frac {3}{2},1,\frac {3}{4},\frac {a-x}{a-b},\frac {2 (a-x)}{2 a-2 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right )}{\sqrt {d} \left (2 a-2 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt {-\frac {b-x}{a-b}} \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}+\frac {4 (a-b) \left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) (b-x) \operatorname {AppellF1}\left (-\frac {1}{4},-\frac {3}{2},1,\frac {3}{4},\frac {a-x}{a-b},\frac {2 (a-x)}{2 a-2 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right )}{\left (2 a-2 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt {-\frac {b-x}{a-b}} \sqrt [4]{-\left ((a-x) (b-x)^2\right )}} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.29 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.13 \[ \int \frac {-\left ((2 a-b) b^2\right )+(4 a-b) b x-(2 a+b) x^2+x^3}{(-a+x) \sqrt [4]{(-a+x) (-b+x)^2} \left (b^2+a d-(2 b+d) x+x^2\right )} \, dx=-\frac {4 b-4 x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{a-x} \sqrt {b-x} \arctan \left (\frac {-b+\sqrt {d} \sqrt {a-x}+x}{\sqrt {2} \sqrt [4]{d} \sqrt [4]{a-x} \sqrt {b-x}}\right )-\sqrt {2} \sqrt [4]{d} \sqrt [4]{a-x} \sqrt {b-x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt [4]{a-x} \sqrt {b-x}}{b+\sqrt {d} \sqrt {a-x}-x}\right )}{\sqrt [4]{(b-x)^2 (-a+x)}} \]

[In]

Integrate[(-((2*a - b)*b^2) + (4*a - b)*b*x - (2*a + b)*x^2 + x^3)/((-a + x)*((-a + x)*(-b + x)^2)^(1/4)*(b^2
+ a*d - (2*b + d)*x + x^2)),x]

[Out]

-((4*b - 4*x + Sqrt[2]*d^(1/4)*(a - x)^(1/4)*Sqrt[b - x]*ArcTan[(-b + Sqrt[d]*Sqrt[a - x] + x)/(Sqrt[2]*d^(1/4
)*(a - x)^(1/4)*Sqrt[b - x])] - Sqrt[2]*d^(1/4)*(a - x)^(1/4)*Sqrt[b - x]*ArcTanh[(Sqrt[2]*d^(1/4)*(a - x)^(1/
4)*Sqrt[b - x])/(b + Sqrt[d]*Sqrt[a - x] - x)])/((b - x)^2*(-a + x))^(1/4))

Maple [F]

\[\int \frac {-\left (2 a -b \right ) b^{2}+\left (4 a -b \right ) b x -\left (2 a +b \right ) x^{2}+x^{3}}{\left (-a +x \right ) \left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{4}} \left (b^{2}+a d -\left (2 b +d \right ) x +x^{2}\right )}d x\]

[In]

int((-(2*a-b)*b^2+(4*a-b)*b*x-(2*a+b)*x^2+x^3)/(-a+x)/((-a+x)*(-b+x)^2)^(1/4)/(b^2+a*d-(2*b+d)*x+x^2),x)

[Out]

int((-(2*a-b)*b^2+(4*a-b)*b*x-(2*a+b)*x^2+x^3)/(-a+x)/((-a+x)*(-b+x)^2)^(1/4)/(b^2+a*d-(2*b+d)*x+x^2),x)

Fricas [F(-1)]

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

[In]

integrate((-(2*a-b)*b^2+(4*a-b)*b*x-(2*a+b)*x^2+x^3)/(-a+x)/((-a+x)*(-b+x)^2)^(1/4)/(b^2+a*d-(2*b+d)*x+x^2),x,
algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

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

[In]

integrate((-(2*a-b)*b**2+(4*a-b)*b*x-(2*a+b)*x**2+x**3)/(-a+x)/((-a+x)*(-b+x)**2)**(1/4)/(b**2+a*d-(2*b+d)*x+x
**2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {-\left ((2 a-b) b^2\right )+(4 a-b) b x-(2 a+b) x^2+x^3}{(-a+x) \sqrt [4]{(-a+x) (-b+x)^2} \left (b^2+a d-(2 b+d) x+x^2\right )} \, dx=\int { \frac {{\left (2 \, a - b\right )} b^{2} - {\left (4 \, a - b\right )} b x + {\left (2 \, a + b\right )} x^{2} - x^{3}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{4}} {\left (b^{2} + a d - {\left (2 \, b + d\right )} x + x^{2}\right )} {\left (a - x\right )}} \,d x } \]

[In]

integrate((-(2*a-b)*b^2+(4*a-b)*b*x-(2*a+b)*x^2+x^3)/(-a+x)/((-a+x)*(-b+x)^2)^(1/4)/(b^2+a*d-(2*b+d)*x+x^2),x,
algorithm="maxima")

[Out]

integrate(((2*a - b)*b^2 - (4*a - b)*b*x + (2*a + b)*x^2 - x^3)/((-(a - x)*(b - x)^2)^(1/4)*(b^2 + a*d - (2*b
+ d)*x + x^2)*(a - x)), x)

Giac [F]

\[ \int \frac {-\left ((2 a-b) b^2\right )+(4 a-b) b x-(2 a+b) x^2+x^3}{(-a+x) \sqrt [4]{(-a+x) (-b+x)^2} \left (b^2+a d-(2 b+d) x+x^2\right )} \, dx=\int { \frac {{\left (2 \, a - b\right )} b^{2} - {\left (4 \, a - b\right )} b x + {\left (2 \, a + b\right )} x^{2} - x^{3}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{4}} {\left (b^{2} + a d - {\left (2 \, b + d\right )} x + x^{2}\right )} {\left (a - x\right )}} \,d x } \]

[In]

integrate((-(2*a-b)*b^2+(4*a-b)*b*x-(2*a+b)*x^2+x^3)/(-a+x)/((-a+x)*(-b+x)^2)^(1/4)/(b^2+a*d-(2*b+d)*x+x^2),x,
algorithm="giac")

[Out]

integrate(((2*a - b)*b^2 - (4*a - b)*b*x + (2*a + b)*x^2 - x^3)/((-(a - x)*(b - x)^2)^(1/4)*(b^2 + a*d - (2*b
+ d)*x + x^2)*(a - x)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {-\left ((2 a-b) b^2\right )+(4 a-b) b x-(2 a+b) x^2+x^3}{(-a+x) \sqrt [4]{(-a+x) (-b+x)^2} \left (b^2+a d-(2 b+d) x+x^2\right )} \, dx=-\int -\frac {b^2\,\left (2\,a-b\right )+x^2\,\left (2\,a+b\right )-x^3-b\,x\,\left (4\,a-b\right )}{\left (a-x\right )\,{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/4}\,\left (a\,d-x\,\left (2\,b+d\right )+b^2+x^2\right )} \,d x \]

[In]

int((b^2*(2*a - b) + x^2*(2*a + b) - x^3 - b*x*(4*a - b))/((a - x)*(-(a - x)*(b - x)^2)^(1/4)*(a*d - x*(2*b +
d) + b^2 + x^2)),x)

[Out]

-int(-(b^2*(2*a - b) + x^2*(2*a + b) - x^3 - b*x*(4*a - b))/((a - x)*(-(a - x)*(b - x)^2)^(1/4)*(a*d - x*(2*b
+ d) + b^2 + x^2)), x)