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

Optimal. Leaf size=158 \[ -2 \sqrt [4]{d} \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3}}{a-x}\right )+2 \sqrt [4]{d} \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3}}{a-x}\right )+\frac {4 \left (-a b^2+2 a b x-a x^2+b^2 x-2 b x^2+x^3\right )^{3/4}}{(b-x)^2} \]

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Rubi [F]  time = 28.42, 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 {(2 a-3 b+x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{(-b+x) \sqrt [4]{(-a+x) (-b+x)^2} \left (-a^3-b^2 d+\left (3 a^2+2 b d\right ) x-(3 a+d) x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(2*(a - x))/(-((a - x)*(b - x)^2))^(1/4) - (2*(2*a - 3*b)*(a - x))/((a - b)*(-((a - x)*(b - x)^2))^(1/4)) - (2
*(a + d)*(a - x))/((a - b)*(-((a - x)*(b - x)^2))^(1/4)) + (2*(2*a - 3*b)*(b - x)*Sqrt[-a + x])/((a - b)^(3/2)
*(-((a - x)*(b - x)^2))^(1/4)*(1 + Sqrt[-a + x]/Sqrt[a - b])) - (6*(b - x)*Sqrt[-a + x])/(Sqrt[a - b]*(-((a -
x)*(b - x)^2))^(1/4)*(1 + Sqrt[-a + x]/Sqrt[a - b])) + (2*(a + d)*(b - x)*Sqrt[-a + x])/((a - b)^(3/2)*(-((a -
x)*(b - x)^2))^(1/4)*(1 + Sqrt[-a + x]/Sqrt[a - b])) + (2*(2*a - 3*b)*(-a + x)^(1/4)*Sqrt[-((b - x)/((a - b)*
(1 + Sqrt[-a + x]/Sqrt[a - b])^2))]*(1 + Sqrt[-a + x]/Sqrt[a - b])*EllipticE[2*ArcTan[(-a + x)^(1/4)/(a - b)^(
1/4)], 1/2])/((a - b)^(1/4)*(-((a - x)*(b - x)^2))^(1/4)) - (6*(a - b)^(3/4)*(-a + x)^(1/4)*Sqrt[-((b - x)/((a
- b)*(1 + Sqrt[-a + x]/Sqrt[a - b])^2))]*(1 + Sqrt[-a + x]/Sqrt[a - b])*EllipticE[2*ArcTan[(-a + x)^(1/4)/(a
- b)^(1/4)], 1/2])/(-((a - x)*(b - x)^2))^(1/4) + (2*(a + d)*(-a + x)^(1/4)*Sqrt[-((b - x)/((a - b)*(1 + Sqrt[
-a + x]/Sqrt[a - b])^2))]*(1 + Sqrt[-a + x]/Sqrt[a - b])*EllipticE[2*ArcTan[(-a + x)^(1/4)/(a - b)^(1/4)], 1/2
])/((a - b)^(1/4)*(-((a - x)*(b - x)^2))^(1/4)) - ((2*a - 3*b)*(-a + x)^(1/4)*Sqrt[-((b - x)/((a - b)*(1 + Sqr
t[-a + x]/Sqrt[a - b])^2))]*(1 + Sqrt[-a + x]/Sqrt[a - b])*EllipticF[2*ArcTan[(-a + x)^(1/4)/(a - b)^(1/4)], 1
/2])/((a - b)^(1/4)*(-((a - x)*(b - x)^2))^(1/4)) + (3*(a - b)^(3/4)*(-a + x)^(1/4)*Sqrt[-((b - x)/((a - b)*(1
+ Sqrt[-a + x]/Sqrt[a - b])^2))]*(1 + Sqrt[-a + x]/Sqrt[a - b])*EllipticF[2*ArcTan[(-a + x)^(1/4)/(a - b)^(1/
4)], 1/2])/(-((a - x)*(b - x)^2))^(1/4) - ((a + d)*(-a + x)^(1/4)*Sqrt[-((b - x)/((a - b)*(1 + Sqrt[-a + x]/Sq
rt[a - b])^2))]*(1 + Sqrt[-a + x]/Sqrt[a - b])*EllipticF[2*ArcTan[(-a + x)^(1/4)/(a - b)^(1/4)], 1/2])/((a - b
)^(1/4)*(-((a - x)*(b - x)^2))^(1/4)) - (4*(a - b)^2*d*(a + d)*(-a + x)^(1/4)*Sqrt[-b + x]*Defer[Subst][Defer[
Int][x^2/((a - b + x^4)^(3/2)*(a^2*(1 + (b*(-2*a + b))/a^2)*d + 2*a*(1 - b/a)*d*x^4 + d*x^8 - x^12)), x], x, (
-a + x)^(1/4)])/(-((a - x)*(b - x)^2))^(1/4) - (4*(a - b)*d*(3*a - b + 2*d)*(-a + x)^(1/4)*Sqrt[-b + x]*Defer[
Subst][Defer[Int][x^6/((a - b + x^4)^(3/2)*(a^2*(1 + (b*(-2*a + b))/a^2)*d + 2*a*(1 - b/a)*d*x^4 + d*x^8 - x^1
2)), x], x, (-a + x)^(1/4)])/(-((a - x)*(b - x)^2))^(1/4) - (4*d*(3*a - 2*b + d)*(-a + x)^(1/4)*Sqrt[-b + x]*D
efer[Subst][Defer[Int][x^10/((a - b + x^4)^(3/2)*(a^2*(1 + (b*(-2*a + b))/a^2)*d + 2*a*(1 - b/a)*d*x^4 + d*x^8
- x^12)), x], x, (-a + x)^(1/4)])/(-((a - x)*(b - x)^2))^(1/4) - (4*(2*a - 3*b)*d*(-a + x)^(1/4)*Sqrt[-b + x]
*Defer[Subst][Defer[Int][(x^2*Sqrt[a - b + x^4])/(a^2*(1 + b^2/a^2)*d - 2*b*d*x^4 + x^8*(d - x^4) + 2*a*d*(-b
+ x^4)), x], x, (-a + x)^(1/4)])/(-((a - x)*(b - x)^2))^(1/4)

Rubi steps

\begin {align*} \int \frac {(2 a-3 b+x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{(-b+x) \sqrt [4]{(-a+x) (-b+x)^2} \left (-a^3-b^2 d+\left (3 a^2+2 b d\right ) x-(3 a+d) x^2+x^3\right )} \, dx &=\frac {\left (\sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \frac {(2 a-3 b+x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{\sqrt [4]{-a+x} (-b+x)^{3/2} \left (-a^3-b^2 d+\left (3 a^2+2 b d\right ) x-(3 a+d) x^2+x^3\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (\sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \frac {(-a+x)^{3/4} (2 a-3 b+x) \left (a^2-2 a x+x^2\right )}{(-b+x)^{3/2} \left (-a^3-b^2 d+\left (3 a^2+2 b d\right ) x-(3 a+d) x^2+x^3\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (\sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \frac {(-a+x)^{11/4} (2 a-3 b+x)}{(-b+x)^{3/2} \left (-a^3-b^2 d+\left (3 a^2+2 b d\right ) x-(3 a+d) x^2+x^3\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (\sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \left (\frac {3 \left (1-\frac {2 a}{3 b}\right ) b (-a+x)^{11/4}}{(-b+x)^{3/2} \left (a^3+b^2 d-\left (3 a^2+2 b d\right ) x+(3 a+d) x^2-x^3\right )}+\frac {x (-a+x)^{11/4}}{(-b+x)^{3/2} \left (-a^3-b^2 d+\left (3 a^2+2 b d\right ) x-(3 a+d) x^2+x^3\right )}\right ) \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (\sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \frac {x (-a+x)^{11/4}}{(-b+x)^{3/2} \left (-a^3-b^2 d+\left (3 a^2+2 b d\right ) x-(3 a+d) x^2+x^3\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left ((-2 a+3 b) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \frac {(-a+x)^{11/4}}{(-b+x)^{3/2} \left (a^3+b^2 d-\left (3 a^2+2 b d\right ) x+(3 a+d) x^2-x^3\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=-\frac {\left (4 \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^{14} \left (a+x^4\right )}{\left (a-b+x^4\right )^{3/2} \left (a^2 d+b^2 d-2 b d x^4+x^8 \left (d-x^4\right )+2 a d \left (-b+x^4\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (4 (-2 a+3 b) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^{14}}{\left (a-b+x^4\right )^{3/2} \left (a^2 d+b^2 d-2 b d x^4+x^8 \left (d-x^4\right )+2 a d \left (-b+x^4\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=-\frac {\left (4 \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^{14} \left (a+x^4\right )}{\left (a-b+x^4\right )^{3/2} \left (a^2 \left (1+\frac {b^2}{a^2}\right ) d-2 b d x^4+x^8 \left (d-x^4\right )+2 a d \left (-b+x^4\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (4 (-2 a+3 b) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^{14}}{\left (a-b+x^4\right )^{3/2} \left (a^2 \left (1+\frac {b^2}{a^2}\right ) d-2 b d x^4+x^8 \left (d-x^4\right )+2 a d \left (-b+x^4\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=-\frac {\left (4 \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \left (-\frac {(a+d) x^2}{\left (a-b+x^4\right )^{3/2}}-\frac {x^6}{\left (a-b+x^4\right )^{3/2}}+\frac {x^2 \left ((a-b)^2 d (a+d)+(a-b) d (3 a-b+2 d) x^4+d (3 a-2 b+d) x^8\right )}{\left (a-b+x^4\right )^{3/2} \left (a^2 \left (1+\frac {b^2}{a^2}\right ) d-2 b d x^4+x^8 \left (d-x^4\right )+2 a d \left (-b+x^4\right )\right )}\right ) \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (4 (-2 a+3 b) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \left (-\frac {x^2}{\left (a-b+x^4\right )^{3/2}}+\frac {x^2 \left ((a-b)^2 d+2 (a-b) d x^4+d x^8\right )}{\left (a-b+x^4\right )^{3/2} \left (a^2 \left (1+\frac {b^2}{a^2}\right ) d-2 b d x^4+x^8 \left (d-x^4\right )+2 a d \left (-b+x^4\right )\right )}\right ) \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (4 \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (a-b+x^4\right )^{3/2}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (4 \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left ((a-b)^2 d (a+d)+(a-b) d (3 a-b+2 d) x^4+d (3 a-2 b+d) x^8\right )}{\left (a-b+x^4\right )^{3/2} \left (a^2 \left (1+\frac {b^2}{a^2}\right ) d-2 b d x^4+x^8 \left (d-x^4\right )+2 a d \left (-b+x^4\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (4 (-2 a+3 b) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (a-b+x^4\right )^{3/2}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (4 (-2 a+3 b) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left ((a-b)^2 d+2 (a-b) d x^4+d x^8\right )}{\left (a-b+x^4\right )^{3/2} \left (a^2 \left (1+\frac {b^2}{a^2}\right ) d-2 b d x^4+x^8 \left (d-x^4\right )+2 a d \left (-b+x^4\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (4 (-a-d) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (a-b+x^4\right )^{3/2}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {2 (a-x)}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}-\frac {2 (2 a-3 b) (a-x)}{(a-b) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}-\frac {2 (a+d) (a-x)}{(a-b) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}-\frac {\left (4 \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \left (\frac {(a-b)^2 d (a+d) x^2}{\left (a-b+x^4\right )^{3/2} \left (a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) d+2 a \left (1-\frac {b}{a}\right ) d x^4+d x^8-x^{12}\right )}+\frac {(a-b) d (3 a-b+2 d) x^6}{\left (a-b+x^4\right )^{3/2} \left (a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) d+2 a \left (1-\frac {b}{a}\right ) d x^4+d x^8-x^{12}\right )}+\frac {d (3 a-2 b+d) x^{10}}{\left (a-b+x^4\right )^{3/2} \left (a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) d+2 a \left (1-\frac {b}{a}\right ) d x^4+d x^8-x^{12}\right )}\right ) \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (6 \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a-b+x^4}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (2 (-2 a+3 b) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a-b+x^4}} \, dx,x,\sqrt [4]{-a+x}\right )}{(a-b) \sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (2 (-a-d) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a-b+x^4}} \, dx,x,\sqrt [4]{-a+x}\right )}{(a-b) \sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (4 (-2 a+3 b) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left ((a-b) d+d x^4\right )^2}{\left (a-b+x^4\right )^{3/2} \left (a^2 \left (1+\frac {b^2}{a^2}\right ) d-2 b d x^4+x^8 \left (d-x^4\right )+2 a d \left (-b+x^4\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{d \sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {2 (a-x)}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}-\frac {2 (2 a-3 b) (a-x)}{(a-b) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}-\frac {2 (a+d) (a-x)}{(a-b) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}+\frac {\left (6 \sqrt {a-b} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-b+x^4}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (6 \sqrt {a-b} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {x^2}{\sqrt {a-b}}}{\sqrt {a-b+x^4}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (2 (-2 a+3 b) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-b+x^4}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt {a-b} \sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (2 (-2 a+3 b) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {x^2}{\sqrt {a-b}}}{\sqrt {a-b+x^4}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt {a-b} \sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (2 (-a-d) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-b+x^4}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt {a-b} \sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (2 (-a-d) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {x^2}{\sqrt {a-b}}}{\sqrt {a-b+x^4}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt {a-b} \sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (4 (-2 a+3 b) d \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {a-b+x^4}}{a^2 \left (1+\frac {b^2}{a^2}\right ) d-2 b d x^4+x^8 \left (d-x^4\right )+2 a d \left (-b+x^4\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (4 (a-b)^2 d (a+d) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (a-b+x^4\right )^{3/2} \left (a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) d+2 a \left (1-\frac {b}{a}\right ) d x^4+d x^8-x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (4 d (3 a-2 b+d) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{\left (a-b+x^4\right )^{3/2} \left (a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) d+2 a \left (1-\frac {b}{a}\right ) d x^4+d x^8-x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (4 (a-b) d (3 a-b+2 d) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (a-b+x^4\right )^{3/2} \left (a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) d+2 a \left (1-\frac {b}{a}\right ) d x^4+d x^8-x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {2 (a-x)}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}-\frac {2 (2 a-3 b) (a-x)}{(a-b) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}-\frac {2 (a+d) (a-x)}{(a-b) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}+\frac {2 (2 a-3 b) (b-x) \sqrt {-a+x}}{(a-b)^{3/2} \sqrt [4]{-\left ((a-x) (b-x)^2\right )} \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right )}-\frac {6 (b-x) \sqrt {-a+x}}{\sqrt {a-b} \sqrt [4]{-\left ((a-x) (b-x)^2\right )} \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right )}+\frac {2 (a+d) (b-x) \sqrt {-a+x}}{(a-b)^{3/2} \sqrt [4]{-\left ((a-x) (b-x)^2\right )} \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right )}+\frac {2 (2 a-3 b) \sqrt [4]{-a+x} \sqrt {-\frac {b-x}{(a-b) \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right )^2}} \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-a+x}}{\sqrt [4]{a-b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{a-b} \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}-\frac {6 (a-b)^{3/4} \sqrt [4]{-a+x} \sqrt {-\frac {b-x}{(a-b) \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right )^2}} \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-a+x}}{\sqrt [4]{a-b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}+\frac {2 (a+d) \sqrt [4]{-a+x} \sqrt {-\frac {b-x}{(a-b) \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right )^2}} \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-a+x}}{\sqrt [4]{a-b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{a-b} \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}-\frac {(2 a-3 b) \sqrt [4]{-a+x} \sqrt {-\frac {b-x}{(a-b) \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right )^2}} \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-a+x}}{\sqrt [4]{a-b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{a-b} \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}+\frac {3 (a-b)^{3/4} \sqrt [4]{-a+x} \sqrt {-\frac {b-x}{(a-b) \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right )^2}} \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-a+x}}{\sqrt [4]{a-b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}-\frac {(a+d) \sqrt [4]{-a+x} \sqrt {-\frac {b-x}{(a-b) \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right )^2}} \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-a+x}}{\sqrt [4]{a-b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{a-b} \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}+\frac {\left (4 (-2 a+3 b) d \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {a-b+x^4}}{a^2 \left (1+\frac {b^2}{a^2}\right ) d-2 b d x^4+x^8 \left (d-x^4\right )+2 a d \left (-b+x^4\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (4 (a-b)^2 d (a+d) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (a-b+x^4\right )^{3/2} \left (a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) d+2 a \left (1-\frac {b}{a}\right ) d x^4+d x^8-x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (4 d (3 a-2 b+d) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{\left (a-b+x^4\right )^{3/2} \left (a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) d+2 a \left (1-\frac {b}{a}\right ) d x^4+d x^8-x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (4 (a-b) d (3 a-b+2 d) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (a-b+x^4\right )^{3/2} \left (a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) d+2 a \left (1-\frac {b}{a}\right ) d x^4+d x^8-x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ \end {align*}

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Mathematica [C]  time = 12.48, size = 2491, normalized size = 15.77 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(4*(-a + x))/((b - x)^2*(-a + x))^(1/4) + ((2*I)*(a - b)*d*(a - x)^(7/4)*((b - x)/(a - x))^(3/2)*((-(EllipticP
i[-(1/(Sqrt[a - b]*Sqrt[Root[1 + d*#1 + (-2*a*d + 2*b*d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 2]])), I*Ar
cSinh[Sqrt[-Sqrt[a - b]]/(a - x)^(1/4)], -1]*Sqrt[Root[1 + d*#1 + (-2*a*d + 2*b*d)*#1^2 + (a^2*d - 2*a*b*d + b
^2*d)*#1^3 & , 1]]*(1 - 4*(a - b)*Root[1 + d*#1 + (-2*a*d + 2*b*d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 2
] + 3*(a - b)^2*Root[1 + d*#1 + (-2*a*d + 2*b*d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 2]^2)*(Root[1 + d*#
1 + (-2*a*d + 2*b*d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 1] - Root[1 + d*#1 + (-2*a*d + 2*b*d)*#1^2 + (a
^2*d - 2*a*b*d + b^2*d)*#1^3 & , 3])) + EllipticPi[1/(Sqrt[a - b]*Sqrt[Root[1 + d*#1 + (-2*a*d + 2*b*d)*#1^2 +
(a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 2]]), I*ArcSinh[Sqrt[-Sqrt[a - b]]/(a - x)^(1/4)], -1]*Sqrt[Root[1 + d*#1
+ (-2*a*d + 2*b*d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 1]]*(1 - 4*(a - b)*Root[1 + d*#1 + (-2*a*d + 2*b*
d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 2] + 3*(a - b)^2*Root[1 + d*#1 + (-2*a*d + 2*b*d)*#1^2 + (a^2*d -
2*a*b*d + b^2*d)*#1^3 & , 2]^2)*(Root[1 + d*#1 + (-2*a*d + 2*b*d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 1
] - Root[1 + d*#1 + (-2*a*d + 2*b*d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 3]) + (EllipticPi[-(1/(Sqrt[a -
b]*Sqrt[Root[1 + d*#1 + (-2*a*d + 2*b*d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 1]])), I*ArcSinh[Sqrt[-Sqr
t[a - b]]/(a - x)^(1/4)], -1] - EllipticPi[1/(Sqrt[a - b]*Sqrt[Root[1 + d*#1 + (-2*a*d + 2*b*d)*#1^2 + (a^2*d
- 2*a*b*d + b^2*d)*#1^3 & , 1]]), I*ArcSinh[Sqrt[-Sqrt[a - b]]/(a - x)^(1/4)], -1])*(1 - 4*(a - b)*Root[1 + d*
#1 + (-2*a*d + 2*b*d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 1] + 3*(a - b)^2*Root[1 + d*#1 + (-2*a*d + 2*b
*d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 1]^2)*Sqrt[Root[1 + d*#1 + (-2*a*d + 2*b*d)*#1^2 + (a^2*d - 2*a*
b*d + b^2*d)*#1^3 & , 2]]*(Root[1 + d*#1 + (-2*a*d + 2*b*d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 2] - Roo
t[1 + d*#1 + (-2*a*d + 2*b*d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 3]))*Sqrt[Root[1 + d*#1 + (-2*a*d + 2*
b*d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 3]] + EllipticPi[-(1/(Sqrt[a - b]*Sqrt[Root[1 + d*#1 + (-2*a*d
+ 2*b*d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 3]])), I*ArcSinh[Sqrt[-Sqrt[a - b]]/(a - x)^(1/4)], -1]*Sqr
t[Root[1 + d*#1 + (-2*a*d + 2*b*d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 1]]*(Root[1 + d*#1 + (-2*a*d + 2*
b*d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 1] - Root[1 + d*#1 + (-2*a*d + 2*b*d)*#1^2 + (a^2*d - 2*a*b*d +
b^2*d)*#1^3 & , 2])*Sqrt[Root[1 + d*#1 + (-2*a*d + 2*b*d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 2]]*(1 -
4*(a - b)*Root[1 + d*#1 + (-2*a*d + 2*b*d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 3] + 3*(a - b)^2*Root[1 +
d*#1 + (-2*a*d + 2*b*d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 3]^2) - EllipticPi[1/(Sqrt[a - b]*Sqrt[Root
[1 + d*#1 + (-2*a*d + 2*b*d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 3]]), I*ArcSinh[Sqrt[-Sqrt[a - b]]/(a -
x)^(1/4)], -1]*Sqrt[Root[1 + d*#1 + (-2*a*d + 2*b*d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 1]]*(Root[1 +
d*#1 + (-2*a*d + 2*b*d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 1] - Root[1 + d*#1 + (-2*a*d + 2*b*d)*#1^2 +
(a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 2])*Sqrt[Root[1 + d*#1 + (-2*a*d + 2*b*d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)
*#1^3 & , 2]]*(1 - 4*(a - b)*Root[1 + d*#1 + (-2*a*d + 2*b*d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 3] + 3
*(a - b)^2*Root[1 + d*#1 + (-2*a*d + 2*b*d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 3]^2)))/(Sqrt[-Sqrt[a -
b]]*((b - x)^2*(-a + x))^(1/4)*(-b + x)*Sqrt[Root[1 + d*#1 + (-2*a*d + 2*b*d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)
*#1^3 & , 1]]*Sqrt[Root[1 + d*#1 + (-2*a*d + 2*b*d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 2]]*Sqrt[Root[1
+ d*#1 + (-2*a*d + 2*b*d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 3]]*(-3*a + 3*b - 2*d + 2*(a - b)^3*d*Root
[1 + d*#1 + (-2*a*d + 2*b*d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 1]*Root[1 + d*#1 + (-2*a*d + 2*b*d)*#1^
2 + (a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 2]^2 + 2*(a - b)^3*d*Root[1 + d*#1 + (-2*a*d + 2*b*d)*#1^2 + (a^2*d - 2
*a*b*d + b^2*d)*#1^3 & , 1]^2*Root[1 + d*#1 + (-2*a*d + 2*b*d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 3] +
2*(a - b)^3*d*Root[1 + d*#1 + (-2*a*d + 2*b*d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 2]*Root[1 + d*#1 + (-
2*a*d + 2*b*d)*#1^2 + (a^2*d - 2*a*b*d + b^2*d)*#1^3 & , 3]^2))

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IntegrateAlgebraic [F]  time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

$Aborted

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (2 a -3 b +x \right ) \left (-a^{3}+3 a^{2} x -3 a \,x^{2}+x^{3}\right )}{\left (-b +x \right ) \left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{4}} \left (-a^{3}-b^{2} d +\left (3 a^{2}+2 b d \right ) x -\left (3 a +d \right ) x^{2}+x^{3}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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