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

Optimal. Leaf size=83 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \left (x (-a-b)+a b+x^2\right )^{3/4}}{b-x}\right )}{d^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \left (x (-a-b)+a b+x^2\right )^{3/4}}{b-x}\right )}{d^{3/4}} \]

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Rubi [F]  time = 7.56, 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^2-2 a x+x^2\right )}{((-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[((a - 3*b + 2*x)*(a^2 - 2*a*x + x^2))/(((-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*(1 - (a - b)/(a - x))^(3/4)*(-a + x)^(3/2)*EllipticF[ArcCot[Sqrt[-a + x]/Sqrt[a - b]]/2, 2])/(Sqrt[a - b]*d
*((a - x)*(b - x))^(3/4)) + (8*(a - b)*(-a + x)^(3/4)*(-b + x)^(3/4)*Defer[Subst][Defer[Int][1/((a - b + x^4)^
(3/4)*(a*(1 - b/a) + x^4 - d*x^12)), x], x, (-a + x)^(1/4)])/(d*((a - x)*(b - x))^(3/4)) + (8*(-a + x)^(3/4)*(
-b + x)^(3/4)*Defer[Subst][Defer[Int][x^4/((a - b + x^4)^(3/4)*(a*(1 - b/a) + x^4 - d*x^12)), x], x, (-a + x)^
(1/4)])/(d*((a - x)*(b - x))^(3/4)) + (8*a*(-a + x)^(3/4)*(-b + x)^(3/4)*Defer[Subst][Defer[Int][x^8/((a - b +
x^4)^(3/4)*(a*(1 - b/a) + x^4 - d*x^12)), x], x, (-a + x)^(1/4)])/((a - x)*(b - x))^(3/4) + (4*(a - 3*b)*(-a
+ x)^(3/4)*(-b + x)^(3/4)*Defer[Subst][Defer[Int][x^8/((a - b + x^4)^(3/4)*(a*(1 - b/a) + x^4 - d*x^12)), x],
x, (-a + x)^(1/4)])/((a - x)*(b - x))^(3/4)

Rubi steps

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

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Mathematica [F]  time = 0.77, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a-3 b+2 x) \left (a^2-2 a x+x^2\right )}{((-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[((a - 3*b + 2*x)*(a^2 - 2*a*x + x^2))/(((-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[((a - 3*b + 2*x)*(a^2 - 2*a*x + x^2))/(((-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]

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IntegrateAlgebraic [A]  time = 0.60, size = 83, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \left (a b+(-a-b) x+x^2\right )^{3/4}}{b-x}\right )}{d^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \left (a b+(-a-b) x+x^2\right )^{3/4}}{b-x}\right )}{d^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((a - 3*b + 2*x)*(a^2 - 2*a*x + x^2))/(((-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]

(2*ArcTan[(d^(1/4)*(a*b + (-a - b)*x + x^2)^(3/4))/(b - x)])/d^(3/4) - (2*ArcTanh[(d^(1/4)*(a*b + (-a - b)*x +
x^2)^(3/4))/(b - 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-3*b+2*x)*(a^2-2*a*x+x^2)/((-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, algor
ithm="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 - 3 \, b + 2 \, 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}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-3*b+2*x)*(a^2-2*a*x+x^2)/((-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, algor
ithm="giac")

[Out]

integrate((a^2 - 2*a*x + x^2)*(a - 3*b + 2*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)), x)

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maple [F]  time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (a -3 b +2 x \right ) \left (a^{2}-2 a x +x^{2}\right )}{\left (\left (-a +x \right ) \left (-b +x \right )\right )^{\frac {3}{4}} \left (-b +a^{3} d +\left (-3 a^{2} d +1\right ) x +3 a d \,x^{2}-d \,x^{3}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a-3*b+2*x)*(a^2-2*a*x+x^2)/((-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((a-3*b+2*x)*(a^2-2*a*x+x^2)/((-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)

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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 - 3 \, b + 2 \, 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}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-3*b+2*x)*(a^2-2*a*x+x^2)/((-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, algor
ithm="maxima")

[Out]

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

[Out]

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

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