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

Optimal. Leaf size=111 \[ -2 \sqrt [4]{d} \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{x (-a-b)+a b+x^2}}{a-x}\right )+2 \sqrt [4]{d} \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{x (-a-b)+a b+x^2}}{a-x}\right )-\frac {4 \left (a b-a x-b x+x^2\right )^{3/4}}{b-x} \]

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Rubi [F]  time = 18.10, 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 {(a-3 b+2 x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{(-b+x) \sqrt [4]{(-a+x) (-b+x)} \left (-a^3+b d-\left (-3 a^2+d\right ) x-3 a x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-4*(a - x))/((a - x)*(b - x))^(1/4) - (8*a*(1 - (a - b)/(a - x))^(1/4)*Sqrt[-a + x]*EllipticE[ArcCot[Sqrt[-a
+ x]/Sqrt[a - b]]/2, 2])/(Sqrt[a - b]*((a - x)*(b - x))^(1/4)) - (4*(a - 3*b)*(1 - (a - b)/(a - x))^(1/4)*Sqrt
[-a + x]*EllipticE[ArcCot[Sqrt[-a + x]/Sqrt[a - b]]/2, 2])/(Sqrt[a - b]*((a - x)*(b - x))^(1/4)) + (12*Sqrt[a
- b]*(1 - (a - b)/(a - x))^(1/4)*Sqrt[-a + x]*EllipticE[ArcCot[Sqrt[-a + x]/Sqrt[a - b]]/2, 2])/((a - x)*(b -
x))^(1/4) + (8*a*d*(-a + x)^(1/4)*(-b + x)^(1/4)*Defer[Subst][Defer[Int][x^2/((a - b + x^4)^(1/4)*(-(a*(1 - b/
a)*d) - d*x^4 + x^12)), x], x, (-a + x)^(1/4)])/((a - x)*(b - x))^(1/4) + (8*d*(-a + x)^(1/4)*(-b + x)^(1/4)*D
efer[Subst][Defer[Int][x^6/((a - b + x^4)^(1/4)*(-(a*(1 - b/a)*d) - d*x^4 + x^12)), x], x, (-a + x)^(1/4)])/((
a - x)*(b - x))^(1/4) - (4*(a - 3*b)*d*(-a + x)^(1/4)*(-b + x)^(1/4)*Defer[Subst][Defer[Int][x^2/((a - b + x^4
)^(1/4)*(a*(1 - b/a)*d + x^4*(d - x^8))), x], x, (-a + x)^(1/4)])/((a - x)*(b - x))^(1/4)

Rubi steps

\begin {align*} \int \frac {(a-3 b+2 x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{(-b+x) \sqrt [4]{(-a+x) (-b+x)} \left (-a^3+b d-\left (-3 a^2+d\right ) x-3 a x^2+x^3\right )} \, dx &=\frac {\left (\sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \int \frac {(a-3 b+2 x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{\sqrt [4]{-a+x} (-b+x)^{5/4} \left (-a^3+b d-\left (-3 a^2+d\right ) x-3 a x^2+x^3\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)}}\\ &=\frac {\left (\sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \int \frac {(-a+x)^{3/4} (a-3 b+2 x) \left (a^2-2 a x+x^2\right )}{(-b+x)^{5/4} \left (-a^3+b d-\left (-3 a^2+d\right ) x-3 a x^2+x^3\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)}}\\ &=\frac {\left (\sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \int \frac {(-a+x)^{11/4} (a-3 b+2 x)}{(-b+x)^{5/4} \left (-a^3+b d-\left (-3 a^2+d\right ) x-3 a x^2+x^3\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)}}\\ &=\frac {\left (\sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \int \frac {(-a+3 b-2 x) (-a+x)^{11/4}}{(-b+x)^{5/4} \left (a^3-b d-\left (3 a^2-d\right ) x+3 a x^2-x^3\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)}}\\ &=\frac {\left (\sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \int \left (\frac {3 \left (1-\frac {a}{3 b}\right ) b (-a+x)^{11/4}}{(-b+x)^{5/4} \left (a^3-b d-\left (3 a^2-d\right ) x+3 a x^2-x^3\right )}+\frac {2 x (-a+x)^{11/4}}{(-b+x)^{5/4} \left (-a^3+b d+\left (3 a^2-d\right ) x-3 a x^2+x^3\right )}\right ) \, dx}{\sqrt [4]{(-a+x) (-b+x)}}\\ &=\frac {\left (2 \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \int \frac {x (-a+x)^{11/4}}{(-b+x)^{5/4} \left (-a^3+b d+\left (3 a^2-d\right ) x-3 a x^2+x^3\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)}}+\frac {\left ((-a+3 b) \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \int \frac {(-a+x)^{11/4}}{(-b+x)^{5/4} \left (a^3-b d-\left (3 a^2-d\right ) x+3 a x^2-x^3\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)}}\\ &=\frac {\left (8 \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^{14} \left (a+x^4\right )}{\left (a-b+x^4\right )^{5/4} \left (-a d+b d-d x^4+x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}+\frac {\left (4 (-a+3 b) \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^{14}}{\left (a-b+x^4\right )^{5/4} \left (a d-b d+x^4 \left (d-x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}\\ &=\frac {\left (8 \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^{14} \left (-a-x^4\right )}{\left (a-b+x^4\right )^{5/4} \left (a \left (1-\frac {b}{a}\right ) d+d x^4-x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}+\frac {\left (4 (-a+3 b) \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^{14}}{\left (a-b+x^4\right )^{5/4} \left (a \left (1-\frac {b}{a}\right ) d+x^4 \left (d-x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}\\ &=\frac {\left (8 \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \left (\frac {a x^2}{\left (a-b+x^4\right )^{5/4}}+\frac {x^6}{\left (a-b+x^4\right )^{5/4}}+\frac {x^2 \left (-a (a-b) d-(2 a-b) d x^4-d x^8\right )}{\left (a-b+x^4\right )^{5/4} \left (a \left (1-\frac {b}{a}\right ) d+d x^4-x^{12}\right )}\right ) \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}+\frac {\left (4 (-a+3 b) \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \left (-\frac {x^2}{\left (a-b+x^4\right )^{5/4}}+\frac {x^2 \left ((a-b) d+d x^4\right )}{\left (a-b+x^4\right )^{5/4} \left (a \left (1-\frac {b}{a}\right ) d+x^4 \left (d-x^8\right )\right )}\right ) \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}\\ &=\frac {\left (8 \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (a-b+x^4\right )^{5/4}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}+\frac {\left (8 \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-a (a-b) d-(2 a-b) d x^4-d x^8\right )}{\left (a-b+x^4\right )^{5/4} \left (a \left (1-\frac {b}{a}\right ) d+d x^4-x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}+\frac {\left (8 a \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (a-b+x^4\right )^{5/4}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}-\frac {\left (4 (-a+3 b) \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (a-b+x^4\right )^{5/4}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}+\frac {\left (4 (-a+3 b) \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left ((a-b) d+d x^4\right )}{\left (a-b+x^4\right )^{5/4} \left (a \left (1-\frac {b}{a}\right ) d+x^4 \left (d-x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}\\ &=-\frac {4 (a-x)}{\sqrt [4]{(a-x) (b-x)}}+\frac {\left (8 \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-a d-d x^4\right )}{\sqrt [4]{a-b+x^4} \left (a \left (1-\frac {b}{a}\right ) d+d x^4-x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}-\frac {\left (12 (a-b) \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (a-b+x^4\right )^{5/4}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}+\frac {\left (4 (-a+3 b) d \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{a-b+x^4} \left (a \left (1-\frac {b}{a}\right ) d+x^4 \left (d-x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}+\frac {\left (8 a \sqrt {-a+x} \sqrt [4]{1+\frac {a-b}{-a+x}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {a-b}{x^4}\right )^{5/4} x^3} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}-\frac {\left (4 (-a+3 b) \sqrt {-a+x} \sqrt [4]{1+\frac {a-b}{-a+x}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {a-b}{x^4}\right )^{5/4} x^3} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}\\ &=-\frac {4 (a-x)}{\sqrt [4]{(a-x) (b-x)}}+\frac {\left (8 \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \left (\frac {a d x^2}{\sqrt [4]{a-b+x^4} \left (-a \left (1-\frac {b}{a}\right ) d-d x^4+x^{12}\right )}+\frac {d x^6}{\sqrt [4]{a-b+x^4} \left (-a \left (1-\frac {b}{a}\right ) d-d x^4+x^{12}\right )}\right ) \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}+\frac {\left (4 (-a+3 b) d \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{a-b+x^4} \left (a \left (1-\frac {b}{a}\right ) d+x^4 \left (d-x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}-\frac {\left (8 a \sqrt {-a+x} \sqrt [4]{1+\frac {a-b}{-a+x}}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+(a-b) x^4\right )^{5/4}} \, dx,x,\frac {1}{\sqrt [4]{-a+x}}\right )}{\sqrt [4]{(-a+x) (-b+x)}}-\frac {\left (12 (a-b) \sqrt {-a+x} \sqrt [4]{1+\frac {a-b}{-a+x}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {a-b}{x^4}\right )^{5/4} x^3} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}+\frac {\left (4 (-a+3 b) \sqrt {-a+x} \sqrt [4]{1+\frac {a-b}{-a+x}}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+(a-b) x^4\right )^{5/4}} \, dx,x,\frac {1}{\sqrt [4]{-a+x}}\right )}{\sqrt [4]{(-a+x) (-b+x)}}\\ &=-\frac {4 (a-x)}{\sqrt [4]{(a-x) (b-x)}}+\frac {\left (8 d \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\sqrt [4]{a-b+x^4} \left (-a \left (1-\frac {b}{a}\right ) d-d x^4+x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}+\frac {\left (8 a d \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{a-b+x^4} \left (-a \left (1-\frac {b}{a}\right ) d-d x^4+x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}+\frac {\left (4 (-a+3 b) d \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{a-b+x^4} \left (a \left (1-\frac {b}{a}\right ) d+x^4 \left (d-x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}-\frac {\left (4 a \sqrt {-a+x} \sqrt [4]{1+\frac {a-b}{-a+x}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+(a-b) x^2\right )^{5/4}} \, dx,x,\frac {1}{\sqrt {-a+x}}\right )}{\sqrt [4]{(-a+x) (-b+x)}}+\frac {\left (12 (a-b) \sqrt {-a+x} \sqrt [4]{1+\frac {a-b}{-a+x}}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+(a-b) x^4\right )^{5/4}} \, dx,x,\frac {1}{\sqrt [4]{-a+x}}\right )}{\sqrt [4]{(-a+x) (-b+x)}}+\frac {\left (2 (-a+3 b) \sqrt {-a+x} \sqrt [4]{1+\frac {a-b}{-a+x}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+(a-b) x^2\right )^{5/4}} \, dx,x,\frac {1}{\sqrt {-a+x}}\right )}{\sqrt [4]{(-a+x) (-b+x)}}\\ &=-\frac {4 (a-x)}{\sqrt [4]{(a-x) (b-x)}}-\frac {8 a \sqrt [4]{1-\frac {a-b}{a-x}} \sqrt {-a+x} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {a-b}}{\sqrt {-a+x}}\right )\right |2\right )}{\sqrt {a-b} \sqrt [4]{(a-x) (b-x)}}-\frac {4 (a-3 b) \sqrt [4]{1-\frac {a-b}{a-x}} \sqrt {-a+x} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {a-b}}{\sqrt {-a+x}}\right )\right |2\right )}{\sqrt {a-b} \sqrt [4]{(a-x) (b-x)}}+\frac {\left (8 d \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\sqrt [4]{a-b+x^4} \left (-a \left (1-\frac {b}{a}\right ) d-d x^4+x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}+\frac {\left (8 a d \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{a-b+x^4} \left (-a \left (1-\frac {b}{a}\right ) d-d x^4+x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}+\frac {\left (4 (-a+3 b) d \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{a-b+x^4} \left (a \left (1-\frac {b}{a}\right ) d+x^4 \left (d-x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}+\frac {\left (6 (a-b) \sqrt {-a+x} \sqrt [4]{1+\frac {a-b}{-a+x}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+(a-b) x^2\right )^{5/4}} \, dx,x,\frac {1}{\sqrt {-a+x}}\right )}{\sqrt [4]{(-a+x) (-b+x)}}\\ &=-\frac {4 (a-x)}{\sqrt [4]{(a-x) (b-x)}}-\frac {8 a \sqrt [4]{1-\frac {a-b}{a-x}} \sqrt {-a+x} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {a-b}}{\sqrt {-a+x}}\right )\right |2\right )}{\sqrt {a-b} \sqrt [4]{(a-x) (b-x)}}-\frac {4 (a-3 b) \sqrt [4]{1-\frac {a-b}{a-x}} \sqrt {-a+x} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {a-b}}{\sqrt {-a+x}}\right )\right |2\right )}{\sqrt {a-b} \sqrt [4]{(a-x) (b-x)}}+\frac {12 \sqrt {a-b} \sqrt [4]{1-\frac {a-b}{a-x}} \sqrt {-a+x} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {a-b}}{\sqrt {-a+x}}\right )\right |2\right )}{\sqrt [4]{(a-x) (b-x)}}+\frac {\left (8 d \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\sqrt [4]{a-b+x^4} \left (-a \left (1-\frac {b}{a}\right ) d-d x^4+x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}+\frac {\left (8 a d \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{a-b+x^4} \left (-a \left (1-\frac {b}{a}\right ) d-d x^4+x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}+\frac {\left (4 (-a+3 b) d \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{a-b+x^4} \left (a \left (1-\frac {b}{a}\right ) d+x^4 \left (d-x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}\\ \end {align*}

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Mathematica [F]  time = 4.76, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a-3 b+2 x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{(-b+x) \sqrt [4]{(-a+x) (-b+x)} \left (-a^3+b d-\left (-3 a^2+d\right ) x-3 a x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.07, size = 111, normalized size = 1.00 \begin {gather*} -\frac {4 \left (a b-a x-b x+x^2\right )^{3/4}}{b-x}-2 \sqrt [4]{d} \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a b+(-a-b) x+x^2}}{a-x}\right )+2 \sqrt [4]{d} \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a b+(-a-b) x+x^2}}{a-x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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-3*b+2*x)*(-a^3+3*a^2*x-3*a*x^2+x^3)/(-b+x)/((-a+x)*(-b+x))^(1/4)/(-a^3+b*d-(-3*a^2+d)*x-3*a*x^2+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^{3} - 3 \, a^{2} x + 3 \, a x^{2} - x^{3}\right )} {\left (a - 3 \, b + 2 \, x\right )}}{{\left (a^{3} + 3 \, a x^{2} - x^{3} - b d - {\left (3 \, a^{2} - d\right )} x\right )} \left ({\left (a - x\right )} {\left (b - x\right )}\right )^{\frac {1}{4}} {\left (b - x\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {\left (a -3 b +2 x \right ) \left (-a^{3}+3 a^{2} x -3 a \,x^{2}+x^{3}\right )}{\left (-b +x \right ) \left (\left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{4}} \left (-a^{3}+b d -\left (-3 a^{2}+d \right ) x -3 a \,x^{2}+x^{3}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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