3.20.36 \(\int \frac {(a^2-2 a x+x^2) (-a b-a c+3 b c+2 (a-b-c) x+x^2)}{((-a+x) (-b+x) (-c+x))^{3/4} (-b c-a^3 d+(b+c+3 a^2 d) x-(1+3 a d) x^2+d x^3)} \, dx\)

Optimal. Leaf size=135 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \left (x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3\right )^{3/4}}{(b-x) (x-c)}\right )}{d^{3/4}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \left (x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3\right )^{3/4}}{(b-x) (x-c)}\right )}{d^{3/4}} \]

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Rubi [F]  time = 67.84, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (a^2-2 a x+x^2\right ) \left (-a b-a c+3 b c+2 (a-b-c) x+x^2\right )}{((-a+x) (-b+x) (-c+x))^{3/4} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-8*(a - b - c)*(a - x)*(-((b - x)/(a - b)))^(3/4)*(-((c - x)/(a - c)))^(3/4)*AppellF1[1/4, 3/4, 3/4, 5/4, (a
- x)/(a - b), (a - x)/(a - c)])/(d*(-((a - x)*(b - x)*(c - x)))^(3/4)) - (4*(1 + 2*a*d)*(a - x)*(-((b - x)/(a
- b)))^(3/4)*(-((c - x)/(a - c)))^(3/4)*AppellF1[1/4, 3/4, 3/4, 5/4, (a - x)/(a - b), (a - x)/(a - c)])/(d^2*(
-((a - x)*(b - x)*(c - x)))^(3/4)) + (4*(a - x)^2*(-((b - x)/(a - b)))^(3/4)*(-((c - x)/(a - c)))^(3/4)*Appell
F1[5/4, 3/4, 3/4, 9/4, (a - x)/(a - b), (a - x)/(a - c)])/(5*d*(-((a - x)*(b - x)*(c - x)))^(3/4)) - (8*(a - b
)*(a - c)*(a - b - c)*(-a + x)^(3/4)*(-b + x)^(3/4)*(-c + x)^(3/4)*Defer[Subst][Defer[Int][1/((a - b + x^4)^(3
/4)*(a - c + x^4)^(3/4)*(a^2*(1 + (b*c - a*(b + c))/a^2) + 2*a*(1 - (b + c)/(2*a))*x^4 + x^8 - d*x^12)), x], x
, (-a + x)^(1/4)])/(d*(-((a - x)*(b - x)*(c - x)))^(3/4)) - (4*(a - b)*(a - c)*(1 + 2*a*d)*(-a + x)^(3/4)*(-b
+ x)^(3/4)*(-c + x)^(3/4)*Defer[Subst][Defer[Int][1/((a - b + x^4)^(3/4)*(a - c + x^4)^(3/4)*(a^2*(1 + (b*c -
a*(b + c))/a^2) + 2*a*(1 - (b + c)/(2*a))*x^4 + x^8 - d*x^12)), x], x, (-a + x)^(1/4)])/(d^2*(-((a - x)*(b - x
)*(c - x)))^(3/4)) - (8*(a - b - c)*(2*a - b - c)*(-a + x)^(3/4)*(-b + x)^(3/4)*(-c + x)^(3/4)*Defer[Subst][De
fer[Int][x^4/((a - b + x^4)^(3/4)*(a - c + x^4)^(3/4)*(a^2*(1 + (b*c - a*(b + c))/a^2) + 2*a*(1 - (b + c)/(2*a
))*x^4 + x^8 - d*x^12)), x], x, (-a + x)^(1/4)])/(d*(-((a - x)*(b - x)*(c - x)))^(3/4)) + (4*(b + c - 5*a^2*d
- b*c*d - a*(2 - 3*b*d - 3*c*d))*(-a + x)^(3/4)*(-b + x)^(3/4)*(-c + x)^(3/4)*Defer[Subst][Defer[Int][x^4/((a
- b + x^4)^(3/4)*(a - c + x^4)^(3/4)*(a^2*(1 + (b*c - a*(b + c))/a^2) + 2*a*(1 - (b + c)/(2*a))*x^4 + x^8 - d*
x^12)), x], x, (-a + x)^(1/4)])/(d^2*(-((a - x)*(b - x)*(c - x)))^(3/4)) - (8*(a - b - c)*(1 + a*d)*(-a + x)^(
3/4)*(-b + x)^(3/4)*(-c + x)^(3/4)*Defer[Subst][Defer[Int][x^8/((a - b + x^4)^(3/4)*(a - c + x^4)^(3/4)*(a^2*(
1 + (b*c - a*(b + c))/a^2) + 2*a*(1 - (b + c)/(2*a))*x^4 + x^8 - d*x^12)), x], x, (-a + x)^(1/4)])/(d*(-((a -
x)*(b - x)*(c - x)))^(3/4)) - (4*(1 + 4*a*d - b*d - c*d + a^2*d^2)*(-a + x)^(3/4)*(-b + x)^(3/4)*(-c + x)^(3/4
)*Defer[Subst][Defer[Int][x^8/((a - b + x^4)^(3/4)*(a - c + x^4)^(3/4)*(a^2*(1 + (b*c - a*(b + c))/a^2) + 2*a*
(1 - (b + c)/(2*a))*x^4 + x^8 - d*x^12)), x], x, (-a + x)^(1/4)])/(d^2*(-((a - x)*(b - x)*(c - x)))^(3/4)) + (
4*(3*b*c - a*(b + c))*(-a + x)^(3/4)*(-b + x)^(3/4)*(-c + x)^(3/4)*Defer[Subst][Defer[Int][x^8/((a - b + x^4)^
(3/4)*(a - c + x^4)^(3/4)*(-a^2 + a*(b + c - 2*x^4) + b*(-c + x^4) + x^4*(c - x^4 + d*x^8))), x], x, (-a + x)^
(1/4)])/(-((a - x)*(b - x)*(c - x)))^(3/4)

Rubi steps

\begin {align*} \int \frac {\left (a^2-2 a x+x^2\right ) \left (-a b-a c+3 b c+2 (a-b-c) x+x^2\right )}{((-a+x) (-b+x) (-c+x))^{3/4} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx &=\int \frac {(-a+x)^2 \left (-a b-a c+3 b c+2 (a-b-c) x+x^2\right )}{((-a+x) (-b+x) (-c+x))^{3/4} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx\\ &=\frac {\left ((-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \int \frac {(-a+x)^{5/4} \left (-a b-a c+3 b c+2 (a-b-c) x+x^2\right )}{(-b+x)^{3/4} (-c+x)^{3/4} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx}{((-a+x) (-b+x) (-c+x))^{3/4}}\\ &=\frac {\left ((-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \int \frac {(-a+x)^{5/4} \left (-3 b c+a (b+c)-2 (a-b-c) x-x^2\right )}{(-b+x)^{3/4} (-c+x)^{3/4} \left (b c+a^3 d-\left (b+c+3 a^2 d\right ) x+(1+3 a d) x^2-d x^3\right )} \, dx}{((-a+x) (-b+x) (-c+x))^{3/4}}\\ &=\frac {\left ((-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \int \left (\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right ) (-a+x)^{5/4}}{(-b+x)^{3/4} (-c+x)^{3/4} \left (b c+a^3 d-\left (b+c+3 a^2 d\right ) x+(1+3 a d) x^2-d x^3\right )}+\frac {2 (-a+b+c) x (-a+x)^{5/4}}{(-b+x)^{3/4} (-c+x)^{3/4} \left (b c+a^3 d-\left (b+c+3 a^2 d\right ) x+(1+3 a d) x^2-d x^3\right )}+\frac {x^2 (-a+x)^{5/4}}{(-b+x)^{3/4} (-c+x)^{3/4} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )}\right ) \, dx}{((-a+x) (-b+x) (-c+x))^{3/4}}\\ &=\frac {\left ((-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \int \frac {x^2 (-a+x)^{5/4}}{(-b+x)^{3/4} (-c+x)^{3/4} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx}{((-a+x) (-b+x) (-c+x))^{3/4}}-\frac {\left (2 (a-b-c) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \int \frac {x (-a+x)^{5/4}}{(-b+x)^{3/4} (-c+x)^{3/4} \left (b c+a^3 d-\left (b+c+3 a^2 d\right ) x+(1+3 a d) x^2-d x^3\right )} \, dx}{((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left ((-3 b c+a (b+c)) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \int \frac {(-a+x)^{5/4}}{(-b+x)^{3/4} (-c+x)^{3/4} \left (b c+a^3 d-\left (b+c+3 a^2 d\right ) x+(1+3 a d) x^2-d x^3\right )} \, dx}{((-a+x) (-b+x) (-c+x))^{3/4}}\\ &=\frac {\left (4 (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8 \left (a+x^4\right )^2}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (-a^2+a \left (b+c-2 x^4\right )+b \left (-c+x^4\right )+x^4 \left (c-x^4+d x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (8 (a-b-c) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8 \left (a+x^4\right )}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (-a^2+a \left (b+c-2 x^4\right )+b \left (-c+x^4\right )+x^4 \left (c-x^4+d x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{((-a+x) (-b+x) (-c+x))^{3/4}}-\frac {\left (4 (-3 b c+a (b+c)) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (-a^2+a \left (b+c-2 x^4\right )+b \left (-c+x^4\right )+x^4 \left (c-x^4+d x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{((-a+x) (-b+x) (-c+x))^{3/4}}\\ &=\frac {\left (4 (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {1+2 a d}{d^2 \left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4}}+\frac {x^4}{d \left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4}}+\frac {(a-b) (a-c) (1+2 a d)-\left (b+c-5 a^2 d-b c d-a (2-3 b d-3 c d)\right ) x^4+\left (1+4 a d-b d-c d+a^2 d^2\right ) x^8}{d^2 \left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (-a^2+a \left (b+c-2 x^4\right )+b \left (-c+x^4\right )+x^4 \left (c-x^4+d x^8\right )\right )}\right ) \, dx,x,\sqrt [4]{-a+x}\right )}{((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (8 (a-b-c) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{d \left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4}}+\frac {(a-b) (a-c)+(2 a-b-c) x^4+(1+a d) x^8}{d \left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (-a^2+a \left (b+c-2 x^4\right )+b \left (-c+x^4\right )+x^4 \left (c-x^4+d x^8\right )\right )}\right ) \, dx,x,\sqrt [4]{-a+x}\right )}{((-a+x) (-b+x) (-c+x))^{3/4}}-\frac {\left (4 (-3 b c+a (b+c)) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (-a^2+a \left (b+c-2 x^4\right )+b \left (-c+x^4\right )+x^4 \left (c-x^4+d x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{((-a+x) (-b+x) (-c+x))^{3/4}}\\ &=-\frac {\left (4 (-3 b c+a (b+c)) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (-a^2+a \left (b+c-2 x^4\right )+b \left (-c+x^4\right )+x^4 \left (c-x^4+d x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (4 (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {(a-b) (a-c) (1+2 a d)-\left (b+c-5 a^2 d-b c d-a (2-3 b d-3 c d)\right ) x^4+\left (1+4 a d-b d-c d+a^2 d^2\right ) x^8}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (-a^2+a \left (b+c-2 x^4\right )+b \left (-c+x^4\right )+x^4 \left (c-x^4+d x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{d^2 ((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (4 (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{-a+x}\right )}{d ((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (8 (a-b-c) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{-a+x}\right )}{d ((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (8 (a-b-c) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {(a-b) (a-c)+(2 a-b-c) x^4+(1+a d) x^8}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (-a^2+a \left (b+c-2 x^4\right )+b \left (-c+x^4\right )+x^4 \left (c-x^4+d x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{d ((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (4 (1+2 a d) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{-a+x}\right )}{d^2 ((-a+x) (-b+x) (-c+x))^{3/4}}\\ &=-\frac {\left (4 (-3 b c+a (b+c)) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (-a^2+a \left (b+c-2 x^4\right )+b \left (-c+x^4\right )+x^4 \left (c-x^4+d x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (4 (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {(a-b) (a-c) (-1-2 a d)}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )+2 a \left (1-\frac {b+c}{2 a}\right ) x^4+x^8-d x^{12}\right )}+\frac {\left (c-5 a^2 d-a (2-3 b d-3 c d)+b (1-c d)\right ) x^4}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )+2 a \left (1-\frac {b+c}{2 a}\right ) x^4+x^8-d x^{12}\right )}+\frac {\left (-1-4 a d+b d+c d-a^2 d^2\right ) x^8}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )+2 a \left (1-\frac {b+c}{2 a}\right ) x^4+x^8-d x^{12}\right )}\right ) \, dx,x,\sqrt [4]{-a+x}\right )}{d^2 ((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (8 (a-b-c) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {(a-b) (-a+c)}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )+2 a \left (1-\frac {b+c}{2 a}\right ) x^4+x^8-d x^{12}\right )}+\frac {(-2 a+b+c) x^4}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )+2 a \left (1-\frac {b+c}{2 a}\right ) x^4+x^8-d x^{12}\right )}+\frac {(-1-a d) x^8}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )+2 a \left (1-\frac {b+c}{2 a}\right ) x^4+x^8-d x^{12}\right )}\right ) \, dx,x,\sqrt [4]{-a+x}\right )}{d ((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (4 (-a+x)^{3/4} \left (\frac {-b+x}{a-b}\right )^{3/4} (-c+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (a-c+x^4\right )^{3/4} \left (1+\frac {x^4}{a-b}\right )^{3/4}} \, dx,x,\sqrt [4]{-a+x}\right )}{d ((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (8 (a-b-c) (-a+x)^{3/4} \left (\frac {-b+x}{a-b}\right )^{3/4} (-c+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a-c+x^4\right )^{3/4} \left (1+\frac {x^4}{a-b}\right )^{3/4}} \, dx,x,\sqrt [4]{-a+x}\right )}{d ((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (4 (1+2 a d) (-a+x)^{3/4} \left (\frac {-b+x}{a-b}\right )^{3/4} (-c+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a-c+x^4\right )^{3/4} \left (1+\frac {x^4}{a-b}\right )^{3/4}} \, dx,x,\sqrt [4]{-a+x}\right )}{d^2 ((-a+x) (-b+x) (-c+x))^{3/4}}\\ &=-\frac {\left (4 (-3 b c+a (b+c)) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (-a^2+a \left (b+c-2 x^4\right )+b \left (-c+x^4\right )+x^4 \left (c-x^4+d x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{((-a+x) (-b+x) (-c+x))^{3/4}}-\frac {\left (8 (a-b) (a-c) (a-b-c) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )+2 a \left (1-\frac {b+c}{2 a}\right ) x^4+x^8-d x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{d ((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (8 (a-b-c) (-2 a+b+c) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )+2 a \left (1-\frac {b+c}{2 a}\right ) x^4+x^8-d x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{d ((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (8 (a-b-c) (-1-a d) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )+2 a \left (1-\frac {b+c}{2 a}\right ) x^4+x^8-d x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{d ((-a+x) (-b+x) (-c+x))^{3/4}}-\frac {\left (4 (a-b) (a-c) (1+2 a d) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )+2 a \left (1-\frac {b+c}{2 a}\right ) x^4+x^8-d x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{d^2 ((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (4 \left (-1-4 a d+b d+c d-a^2 d^2\right ) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )+2 a \left (1-\frac {b+c}{2 a}\right ) x^4+x^8-d x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{d^2 ((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (4 \left (b+c-5 a^2 d-b c d-a (2-3 b d-3 c d)\right ) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )+2 a \left (1-\frac {b+c}{2 a}\right ) x^4+x^8-d x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{d^2 ((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (4 (-a+x)^{3/4} \left (\frac {-b+x}{a-b}\right )^{3/4} \left (\frac {-c+x}{a-c}\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (1+\frac {x^4}{a-b}\right )^{3/4} \left (1+\frac {x^4}{a-c}\right )^{3/4}} \, dx,x,\sqrt [4]{-a+x}\right )}{d ((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (8 (a-b-c) (-a+x)^{3/4} \left (\frac {-b+x}{a-b}\right )^{3/4} \left (\frac {-c+x}{a-c}\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {x^4}{a-b}\right )^{3/4} \left (1+\frac {x^4}{a-c}\right )^{3/4}} \, dx,x,\sqrt [4]{-a+x}\right )}{d ((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (4 (1+2 a d) (-a+x)^{3/4} \left (\frac {-b+x}{a-b}\right )^{3/4} \left (\frac {-c+x}{a-c}\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {x^4}{a-b}\right )^{3/4} \left (1+\frac {x^4}{a-c}\right )^{3/4}} \, dx,x,\sqrt [4]{-a+x}\right )}{d^2 ((-a+x) (-b+x) (-c+x))^{3/4}}\\ &=-\frac {8 (a-b-c) (a-x) \left (-\frac {b-x}{a-b}\right )^{3/4} \left (-\frac {c-x}{a-c}\right )^{3/4} F_1\left (\frac {1}{4};\frac {3}{4},\frac {3}{4};\frac {5}{4};\frac {a-x}{a-b},\frac {a-x}{a-c}\right )}{d (-((a-x) (b-x) (c-x)))^{3/4}}-\frac {4 (1+2 a d) (a-x) \left (-\frac {b-x}{a-b}\right )^{3/4} \left (-\frac {c-x}{a-c}\right )^{3/4} F_1\left (\frac {1}{4};\frac {3}{4},\frac {3}{4};\frac {5}{4};\frac {a-x}{a-b},\frac {a-x}{a-c}\right )}{d^2 (-((a-x) (b-x) (c-x)))^{3/4}}+\frac {4 (a-x)^2 \left (-\frac {b-x}{a-b}\right )^{3/4} \left (-\frac {c-x}{a-c}\right )^{3/4} F_1\left (\frac {5}{4};\frac {3}{4},\frac {3}{4};\frac {9}{4};\frac {a-x}{a-b},\frac {a-x}{a-c}\right )}{5 d (-((a-x) (b-x) (c-x)))^{3/4}}-\frac {\left (4 (-3 b c+a (b+c)) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (-a^2+a \left (b+c-2 x^4\right )+b \left (-c+x^4\right )+x^4 \left (c-x^4+d x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{((-a+x) (-b+x) (-c+x))^{3/4}}-\frac {\left (8 (a-b) (a-c) (a-b-c) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )+2 a \left (1-\frac {b+c}{2 a}\right ) x^4+x^8-d x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{d ((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (8 (a-b-c) (-2 a+b+c) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )+2 a \left (1-\frac {b+c}{2 a}\right ) x^4+x^8-d x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{d ((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (8 (a-b-c) (-1-a d) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )+2 a \left (1-\frac {b+c}{2 a}\right ) x^4+x^8-d x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{d ((-a+x) (-b+x) (-c+x))^{3/4}}-\frac {\left (4 (a-b) (a-c) (1+2 a d) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )+2 a \left (1-\frac {b+c}{2 a}\right ) x^4+x^8-d x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{d^2 ((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (4 \left (-1-4 a d+b d+c d-a^2 d^2\right ) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )+2 a \left (1-\frac {b+c}{2 a}\right ) x^4+x^8-d x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{d^2 ((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (4 \left (b+c-5 a^2 d-b c d-a (2-3 b d-3 c d)\right ) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )+2 a \left (1-\frac {b+c}{2 a}\right ) x^4+x^8-d x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{d^2 ((-a+x) (-b+x) (-c+x))^{3/4}}\\ \end {align*}

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Mathematica [F]  time = 1.39, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a^2-2 a x+x^2\right ) \left (-a b-a c+3 b c+2 (a-b-c) x+x^2\right )}{((-a+x) (-b+x) (-c+x))^{3/4} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 4.05, size = 135, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \left (-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3\right )^{3/4}}{(b-x) (-c+x)}\right )}{d^{3/4}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \left (-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3\right )^{3/4}}{(b-x) (-c+x)}\right )}{d^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*ArcTan[(d^(1/4)*(-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3)^(3/4))/((b - x)*(-c + x))])/d^(3
/4) + (2*ArcTanh[(d^(1/4)*(-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3)^(3/4))/((b - x)*(-c + x))]
)/d^(3/4)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a^{2} - 2 \, a x + x^{2}\right )} {\left (a b + a c - 3 \, b c - 2 \, {\left (a - b - c\right )} x - x^{2}\right )}}{{\left (a^{3} d - d x^{3} + {\left (3 \, a d + 1\right )} x^{2} + b c - {\left (3 \, a^{2} d + b + c\right )} x\right )} \left (-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )}\right )^{\frac {3}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a^{2}-2 a x +x^{2}\right ) \left (-a b -a c +3 b c +2 \left (a -b -c \right ) x +x^{2}\right )}{\left (\left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )\right )^{\frac {3}{4}} \left (-b c -a^{3} d +\left (3 a^{2} d +b +c \right ) x -\left (3 a d +1\right ) x^{2}+d \,x^{3}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a^{2} - 2 \, a x + x^{2}\right )} {\left (a b + a c - 3 \, b c - 2 \, {\left (a - b - c\right )} x - x^{2}\right )}}{{\left (a^{3} d - d x^{3} + {\left (3 \, a d + 1\right )} x^{2} + b c - {\left (3 \, a^{2} d + b + c\right )} x\right )} \left (-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )}\right )^{\frac {3}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (a^2-2\,a\,x+x^2\right )\,\left (a\,b+a\,c-3\,b\,c+2\,x\,\left (b-a+c\right )-x^2\right )}{{\left (-\left (a-x\right )\,\left (b-x\right )\,\left (c-x\right )\right )}^{3/4}\,\left (b\,c-x\,\left (3\,d\,a^2+b+c\right )+a^3\,d-d\,x^3+x^2\,\left (3\,a\,d+1\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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