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

Optimal. Leaf size=109 \[ -2 \sqrt [4]{d} \tan ^{-1}\left (\frac {\sqrt [4]{d} \left (x (-a-b)+a b+x^2\right )^{3/4}}{b-x}\right )+2 \sqrt [4]{d} \tanh ^{-1}\left (\frac {\sqrt [4]{x (-a-b)+a b+x^2}}{\sqrt [4]{d} (a-x)}\right )-\frac {4 (b-x)}{\left (a b-a x-b x+x^2\right )^{3/4}} \]

________________________________________________________________________________________

Rubi [F]  time = 10.44, 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 {(-b+x) (a-3 b+2 x)}{(-a+x) ((-a+x) (-b+x))^{3/4} \left (b-a^3 d-\left (1-3 a^2 d\right ) x-3 a d x^2+d x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-8*a*(b - x))/(3*(a - b)*((a - x)*(b - x))^(3/4)) - (4*(a - 3*b)*(b - x))/(3*(a - b)*((a - x)*(b - x))^(3/4))
+ (8*a*(1 - (a - b)/(a - x))^(3/4)*(-a + x)^(3/2)*EllipticF[ArcCot[Sqrt[-a + x]/Sqrt[a - b]]/2, 2])/(3*(a - b
)^(3/2)*((a - x)*(b - x))^(3/4)) + (4*(a - 3*b)*(1 - (a - b)/(a - x))^(3/4)*(-a + x)^(3/2)*EllipticF[ArcCot[Sq
rt[-a + x]/Sqrt[a - b]]/2, 2])/(3*(a - b)^(3/2)*((a - x)*(b - x))^(3/4)) + (8*b*(-a + x)^(3/4)*(-b + x)^(3/4)*
Defer[Subst][Defer[Int][(a - b + x^4)^(1/4)/(a*(1 - b/a) + x^4 - d*x^12), x], x, (-a + x)^(1/4)])/((a - b)*((a
- x)*(b - x))^(3/4)) - (4*(a - 3*b)*d*(-a + x)^(3/4)*(-b + x)^(3/4)*Defer[Subst][Defer[Int][(x^8*(a - b + x^4
)^(1/4))/(a*(1 - b/a) + x^4 - d*x^12), x], x, (-a + x)^(1/4)])/((a - b)*((a - x)*(b - x))^(3/4)) - (4*(a - 3*b
)*(-a + x)^(3/4)*(-b + x)^(3/4)*Defer[Subst][Defer[Int][(a - b + x^4)^(1/4)/(-(a*(1 - b/a)) - x^4 + d*x^12), x
], x, (-a + x)^(1/4)])/((a - b)*((a - x)*(b - x))^(3/4)) + (8*a*d*(-a + x)^(3/4)*(-b + x)^(3/4)*Defer[Subst][D
efer[Int][(x^8*(a - b + x^4)^(1/4))/(-(a*(1 - b/a)) - x^4 + d*x^12), x], x, (-a + x)^(1/4)])/((a - b)*((a - x)
*(b - x))^(3/4))

Rubi steps

\begin {align*} \int \frac {(-b+x) (a-3 b+2 x)}{(-a+x) ((-a+x) (-b+x))^{3/4} \left (b-a^3 d-\left (1-3 a^2 d\right ) x-3 a d x^2+d x^3\right )} \, dx &=\frac {\left ((-a+x)^{3/4} (-b+x)^{3/4}\right ) \int \frac {\sqrt [4]{-b+x} (a-3 b+2 x)}{(-a+x)^{7/4} \left (b-a^3 d-\left (1-3 a^2 d\right ) x-3 a d x^2+d x^3\right )} \, dx}{((-a+x) (-b+x))^{3/4}}\\ &=\frac {\left ((-a+x)^{3/4} (-b+x)^{3/4}\right ) \int \left (\frac {3 \left (1-\frac {a}{3 b}\right ) b \sqrt [4]{-b+x}}{(-a+x)^{7/4} \left (-b+a^3 d+\left (1-3 a^2 d\right ) x+3 a d x^2-d x^3\right )}+\frac {2 x \sqrt [4]{-b+x}}{(-a+x)^{7/4} \left (b-a^3 d-\left (1-3 a^2 d\right ) x-3 a d x^2+d x^3\right )}\right ) \, dx}{((-a+x) (-b+x))^{3/4}}\\ &=\frac {\left (2 (-a+x)^{3/4} (-b+x)^{3/4}\right ) \int \frac {x \sqrt [4]{-b+x}}{(-a+x)^{7/4} \left (b-a^3 d-\left (1-3 a^2 d\right ) x-3 a d x^2+d x^3\right )} \, dx}{((-a+x) (-b+x))^{3/4}}+\frac {\left ((-a+3 b) (-a+x)^{3/4} (-b+x)^{3/4}\right ) \int \frac {\sqrt [4]{-b+x}}{(-a+x)^{7/4} \left (-b+a^3 d+\left (1-3 a^2 d\right ) x+3 a d x^2-d x^3\right )} \, dx}{((-a+x) (-b+x))^{3/4}}\\ &=\frac {\left (8 (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\left (a+x^4\right ) \sqrt [4]{a-b+x^4}}{x^4 \left (-a+b+x^4 \left (-1+d x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{((-a+x) (-b+x))^{3/4}}+\frac {\left (4 (-a+3 b) (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{a-b+x^4}}{x^4 \left (a-b+x^4-d x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{((-a+x) (-b+x))^{3/4}}\\ &=\frac {\left (8 (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\left (-a-x^4\right ) \sqrt [4]{a-b+x^4}}{x^4 \left (a \left (1-\frac {b}{a}\right )-x^4 \left (-1+d x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{((-a+x) (-b+x))^{3/4}}+\frac {\left (4 (-a+3 b) (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{a-b+x^4}}{x^4 \left (a \left (1-\frac {b}{a}\right )+x^4-d x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{((-a+x) (-b+x))^{3/4}}\\ &=\frac {\left (8 (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (-\frac {a \sqrt [4]{a-b+x^4}}{(a-b) x^4}+\frac {\sqrt [4]{a-b+x^4} \left (b-a d x^8\right )}{(a-b) \left (a \left (1-\frac {b}{a}\right )+x^4-d x^{12}\right )}\right ) \, dx,x,\sqrt [4]{-a+x}\right )}{((-a+x) (-b+x))^{3/4}}+\frac {\left (4 (-a+3 b) (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {\sqrt [4]{a-b+x^4}}{(a-b) x^4}+\frac {\sqrt [4]{a-b+x^4} \left (-1+d x^8\right )}{(a-b) \left (a \left (1-\frac {b}{a}\right )+x^4-d x^{12}\right )}\right ) \, dx,x,\sqrt [4]{-a+x}\right )}{((-a+x) (-b+x))^{3/4}}\\ &=\frac {\left (8 (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{a-b+x^4} \left (b-a d x^8\right )}{a \left (1-\frac {b}{a}\right )+x^4-d x^{12}} \, dx,x,\sqrt [4]{-a+x}\right )}{(a-b) ((-a+x) (-b+x))^{3/4}}-\frac {\left (8 a (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{a-b+x^4}}{x^4} \, dx,x,\sqrt [4]{-a+x}\right )}{(a-b) ((-a+x) (-b+x))^{3/4}}+\frac {\left (4 (-a+3 b) (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{a-b+x^4}}{x^4} \, dx,x,\sqrt [4]{-a+x}\right )}{(a-b) ((-a+x) (-b+x))^{3/4}}+\frac {\left (4 (-a+3 b) (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{a-b+x^4} \left (-1+d x^8\right )}{a \left (1-\frac {b}{a}\right )+x^4-d x^{12}} \, dx,x,\sqrt [4]{-a+x}\right )}{(a-b) ((-a+x) (-b+x))^{3/4}}\\ &=-\frac {8 a (b-x)}{3 (a-b) ((a-x) (b-x))^{3/4}}-\frac {4 (a-3 b) (b-x)}{3 (a-b) ((a-x) (b-x))^{3/4}}+\frac {\left (8 (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {b \sqrt [4]{a-b+x^4}}{a \left (1-\frac {b}{a}\right )+x^4-d x^{12}}+\frac {a d x^8 \sqrt [4]{a-b+x^4}}{-a \left (1-\frac {b}{a}\right )-x^4+d x^{12}}\right ) \, dx,x,\sqrt [4]{-a+x}\right )}{(a-b) ((-a+x) (-b+x))^{3/4}}-\frac {\left (8 a (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a-b+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{-a+x}\right )}{3 (a-b) ((-a+x) (-b+x))^{3/4}}+\frac {\left (4 (-a+3 b) (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a-b+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{-a+x}\right )}{3 (a-b) ((-a+x) (-b+x))^{3/4}}+\frac {\left (4 (-a+3 b) (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {d x^8 \sqrt [4]{a-b+x^4}}{a \left (1-\frac {b}{a}\right )+x^4-d x^{12}}+\frac {\sqrt [4]{a-b+x^4}}{-a \left (1-\frac {b}{a}\right )-x^4+d x^{12}}\right ) \, dx,x,\sqrt [4]{-a+x}\right )}{(a-b) ((-a+x) (-b+x))^{3/4}}\\ &=-\frac {8 a (b-x)}{3 (a-b) ((a-x) (b-x))^{3/4}}-\frac {4 (a-3 b) (b-x)}{3 (a-b) ((a-x) (b-x))^{3/4}}+\frac {\left (8 b (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{a-b+x^4}}{a \left (1-\frac {b}{a}\right )+x^4-d x^{12}} \, dx,x,\sqrt [4]{-a+x}\right )}{(a-b) ((-a+x) (-b+x))^{3/4}}+\frac {\left (4 (-a+3 b) (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{a-b+x^4}}{-a \left (1-\frac {b}{a}\right )-x^4+d x^{12}} \, dx,x,\sqrt [4]{-a+x}\right )}{(a-b) ((-a+x) (-b+x))^{3/4}}+\frac {\left (8 a d (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8 \sqrt [4]{a-b+x^4}}{-a \left (1-\frac {b}{a}\right )-x^4+d x^{12}} \, dx,x,\sqrt [4]{-a+x}\right )}{(a-b) ((-a+x) (-b+x))^{3/4}}+\frac {\left (4 (-a+3 b) d (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8 \sqrt [4]{a-b+x^4}}{a \left (1-\frac {b}{a}\right )+x^4-d x^{12}} \, dx,x,\sqrt [4]{-a+x}\right )}{(a-b) ((-a+x) (-b+x))^{3/4}}-\frac {\left (8 a (-a+x)^{3/2} \left (1+\frac {a-b}{-a+x}\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {a-b}{x^4}\right )^{3/4} x^3} \, dx,x,\sqrt [4]{-a+x}\right )}{3 (a-b) ((-a+x) (-b+x))^{3/4}}+\frac {\left (4 (-a+3 b) (-a+x)^{3/2} \left (1+\frac {a-b}{-a+x}\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {a-b}{x^4}\right )^{3/4} x^3} \, dx,x,\sqrt [4]{-a+x}\right )}{3 (a-b) ((-a+x) (-b+x))^{3/4}}\\ &=-\frac {8 a (b-x)}{3 (a-b) ((a-x) (b-x))^{3/4}}-\frac {4 (a-3 b) (b-x)}{3 (a-b) ((a-x) (b-x))^{3/4}}+\frac {\left (8 b (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{a-b+x^4}}{a \left (1-\frac {b}{a}\right )+x^4-d x^{12}} \, dx,x,\sqrt [4]{-a+x}\right )}{(a-b) ((-a+x) (-b+x))^{3/4}}+\frac {\left (4 (-a+3 b) (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{a-b+x^4}}{-a \left (1-\frac {b}{a}\right )-x^4+d x^{12}} \, dx,x,\sqrt [4]{-a+x}\right )}{(a-b) ((-a+x) (-b+x))^{3/4}}+\frac {\left (8 a d (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8 \sqrt [4]{a-b+x^4}}{-a \left (1-\frac {b}{a}\right )-x^4+d x^{12}} \, dx,x,\sqrt [4]{-a+x}\right )}{(a-b) ((-a+x) (-b+x))^{3/4}}+\frac {\left (4 (-a+3 b) d (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8 \sqrt [4]{a-b+x^4}}{a \left (1-\frac {b}{a}\right )+x^4-d x^{12}} \, dx,x,\sqrt [4]{-a+x}\right )}{(a-b) ((-a+x) (-b+x))^{3/4}}+\frac {\left (8 a (-a+x)^{3/2} \left (1+\frac {a-b}{-a+x}\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+(a-b) x^4\right )^{3/4}} \, dx,x,\frac {1}{\sqrt [4]{-a+x}}\right )}{3 (a-b) ((-a+x) (-b+x))^{3/4}}-\frac {\left (4 (-a+3 b) (-a+x)^{3/2} \left (1+\frac {a-b}{-a+x}\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+(a-b) x^4\right )^{3/4}} \, dx,x,\frac {1}{\sqrt [4]{-a+x}}\right )}{3 (a-b) ((-a+x) (-b+x))^{3/4}}\\ &=-\frac {8 a (b-x)}{3 (a-b) ((a-x) (b-x))^{3/4}}-\frac {4 (a-3 b) (b-x)}{3 (a-b) ((a-x) (b-x))^{3/4}}+\frac {\left (8 b (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{a-b+x^4}}{a \left (1-\frac {b}{a}\right )+x^4-d x^{12}} \, dx,x,\sqrt [4]{-a+x}\right )}{(a-b) ((-a+x) (-b+x))^{3/4}}+\frac {\left (4 (-a+3 b) (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{a-b+x^4}}{-a \left (1-\frac {b}{a}\right )-x^4+d x^{12}} \, dx,x,\sqrt [4]{-a+x}\right )}{(a-b) ((-a+x) (-b+x))^{3/4}}+\frac {\left (8 a d (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8 \sqrt [4]{a-b+x^4}}{-a \left (1-\frac {b}{a}\right )-x^4+d x^{12}} \, dx,x,\sqrt [4]{-a+x}\right )}{(a-b) ((-a+x) (-b+x))^{3/4}}+\frac {\left (4 (-a+3 b) d (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8 \sqrt [4]{a-b+x^4}}{a \left (1-\frac {b}{a}\right )+x^4-d x^{12}} \, dx,x,\sqrt [4]{-a+x}\right )}{(a-b) ((-a+x) (-b+x))^{3/4}}+\frac {\left (4 a (-a+x)^{3/2} \left (1+\frac {a-b}{-a+x}\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+(a-b) x^2\right )^{3/4}} \, dx,x,\frac {1}{\sqrt {-a+x}}\right )}{3 (a-b) ((-a+x) (-b+x))^{3/4}}-\frac {\left (2 (-a+3 b) (-a+x)^{3/2} \left (1+\frac {a-b}{-a+x}\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+(a-b) x^2\right )^{3/4}} \, dx,x,\frac {1}{\sqrt {-a+x}}\right )}{3 (a-b) ((-a+x) (-b+x))^{3/4}}\\ &=-\frac {8 a (b-x)}{3 (a-b) ((a-x) (b-x))^{3/4}}-\frac {4 (a-3 b) (b-x)}{3 (a-b) ((a-x) (b-x))^{3/4}}+\frac {8 a \left (1-\frac {a-b}{a-x}\right )^{3/4} (-a+x)^{3/2} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {a-b}}{\sqrt {-a+x}}\right )\right |2\right )}{3 (a-b)^{3/2} ((a-x) (b-x))^{3/4}}+\frac {4 (a-3 b) \left (1-\frac {a-b}{a-x}\right )^{3/4} (-a+x)^{3/2} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {a-b}}{\sqrt {-a+x}}\right )\right |2\right )}{3 (a-b)^{3/2} ((a-x) (b-x))^{3/4}}+\frac {\left (8 b (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{a-b+x^4}}{a \left (1-\frac {b}{a}\right )+x^4-d x^{12}} \, dx,x,\sqrt [4]{-a+x}\right )}{(a-b) ((-a+x) (-b+x))^{3/4}}+\frac {\left (4 (-a+3 b) (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{a-b+x^4}}{-a \left (1-\frac {b}{a}\right )-x^4+d x^{12}} \, dx,x,\sqrt [4]{-a+x}\right )}{(a-b) ((-a+x) (-b+x))^{3/4}}+\frac {\left (8 a d (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8 \sqrt [4]{a-b+x^4}}{-a \left (1-\frac {b}{a}\right )-x^4+d x^{12}} \, dx,x,\sqrt [4]{-a+x}\right )}{(a-b) ((-a+x) (-b+x))^{3/4}}+\frac {\left (4 (-a+3 b) d (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8 \sqrt [4]{a-b+x^4}}{a \left (1-\frac {b}{a}\right )+x^4-d x^{12}} \, dx,x,\sqrt [4]{-a+x}\right )}{(a-b) ((-a+x) (-b+x))^{3/4}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 3.83, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(-b+x) (a-3 b+2 x)}{(-a+x) ((-a+x) (-b+x))^{3/4} \left (b-a^3 d-\left (1-3 a^2 d\right ) x-3 a d x^2+d x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 7.27, size = 109, normalized size = 1.00 \begin {gather*} -\frac {4 (b-x)}{\left (a b-a x-b x+x^2\right )^{3/4}}-2 \sqrt [4]{d} \tan ^{-1}\left (\frac {\sqrt [4]{d} \left (a b+(-a-b) x+x^2\right )^{3/4}}{b-x}\right )+2 \sqrt [4]{d} \tanh ^{-1}\left (\frac {\sqrt [4]{a b+(-a-b) x+x^2}}{\sqrt [4]{d} (a-x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\left (-b +x \right ) \left (a -3 b +2 x \right )}{\left (-a +x \right ) \left (\left (-a +x \right ) \left (-b +x \right )\right )^{\frac {3}{4}} \left (b -a^{3} d -\left (-3 a^{2} d +1\right ) x -3 a d \,x^{2}+d \,x^{3}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________