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

Optimal. Leaf size=81 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{x^2 (-a-b)+a b x+x^3}}{x}\right )}{d^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{x^2 (-a-b)+a b x+x^3}}{x}\right )}{d^{3/4}} \]

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Rubi [F]  time = 16.81, 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 {3 a b-2 (a+b) x+x^2}{\sqrt [4]{x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(8*(a + b)*x^(1/4)*(-a + x)^(1/4)*(-b + x)^(1/4)*Defer[Subst][Defer[Int][x^6/((-a + x^4)^(1/4)*(-b + x^4)^(1/4
)*(a*b*d - a*(1 + b/a)*d*x^4 + d*x^8 - x^12)), x], x, x^(1/4)])/((a - x)*(b - x)*x)^(1/4) + (12*a*b*x^(1/4)*(-
a + x)^(1/4)*(-b + x)^(1/4)*Defer[Subst][Defer[Int][x^2/((-a + x^4)^(1/4)*(-b + x^4)^(1/4)*(-(a*b*d) + a*(1 +
b/a)*d*x^4 - d*x^8 + x^12)), x], x, x^(1/4)])/((a - x)*(b - x)*x)^(1/4) + (4*x^(1/4)*(-a + x)^(1/4)*(-b + x)^(
1/4)*Defer[Subst][Defer[Int][x^10/((-a + x^4)^(1/4)*(-b + x^4)^(1/4)*(-(a*b*d) + a*(1 + b/a)*d*x^4 - d*x^8 + x
^12)), x], x, x^(1/4)])/((a - x)*(b - x)*x)^(1/4)

Rubi steps

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

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Mathematica [F]  time = 2.93, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 a b-2 (a+b) x+x^2}{\sqrt [4]{x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.28, size = 81, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a b x+(-a-b) x^2+x^3}}{x}\right )}{d^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a b x+(-a-b) x^2+x^3}}{x}\right )}{d^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {3 a b -2 \left (a +b \right ) x +x^{2}}{\left (x \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{4}} \left (-a b d +\left (a +b \right ) d x -d \,x^{2}+x^{3}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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