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

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

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Rubi [F]  time = 6.76, 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}{\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 + x)*(-b + x))^(1/4)*(-a^3 + b*d - (-3*a^2 + d)*x - 3*a*x^2 + x^3)),x]

[Out]

(8*a*(-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*(-a + x)^(1/4)*(-b + x)^(1/4)*Defer[Subst][Defe
r[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)*(-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}{\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}{\sqrt [4]{-a+x} \sqrt [4]{-b+x} \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}{\sqrt [4]{-a+x} \sqrt [4]{-b+x} \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}{\sqrt [4]{-a+x} \sqrt [4]{-b+x} \left (a^3-b d-\left (3 a^2-d\right ) x+3 a x^2-x^3\right )}+\frac {2 x}{\sqrt [4]{-a+x} \sqrt [4]{-b+x} \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}{\sqrt [4]{-a+x} \sqrt [4]{-b+x} \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 {1}{\sqrt [4]{-a+x} \sqrt [4]{-b+x} \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^2 \left (a+x^4\right )}{\sqrt [4]{a-b+x^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^2}{\sqrt [4]{a-b+x^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^2 \left (-a-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 (4 (-a+3 b) \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 \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \left (\frac {a x^2}{\sqrt [4]{a-b+x^4} \left (-a \left (1-\frac {b}{a}\right ) d-d x^4+x^{12}\right )}+\frac {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) \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 \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 \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) \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 = 1.84, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a-3 b+2 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 + 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 + 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 = 0.36, size = 83, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a b+(-a-b) x+x^2}}{a-x}\right )}{d^{3/4}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a b+(-a-b) x+x^2}}{a-x}\right )}{d^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a-3*b+2*x)/((-a+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+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 {a - 3 \, b + 2 \, x}{{\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}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

[Out]

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