3.11.71 \(\int \frac {-2 a b+(a+b) x}{\sqrt [4]{x^2 (-a+x) (-b+x)} (a b d-(a+b) d x+(-1+d) x^2)} \, dx\)

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

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Rubi [F]  time = 5.91, 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 b+(a+b) x}{\sqrt [4]{x^2 (-a+x) (-b+x)} \left (a b d-(a+b) d x+(-1+d) x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

((a + b + Sqrt[2*a*b*(2 - d) + a^2*d + b^2*d]/Sqrt[d])*Sqrt[x]*(-a + x)^(1/4)*(-b + x)^(1/4)*Defer[Int][1/(Sqr
t[x]*(-a + x)^(1/4)*(-b + x)^(1/4)*(-((a + b)*d) - Sqrt[d]*Sqrt[4*a*b + a^2*d - 2*a*b*d + b^2*d] + 2*(-1 + d)*
x)), x])/((a - x)*(b - x)*x^2)^(1/4) + ((a + b - Sqrt[2*a*b*(2 - d) + a^2*d + b^2*d]/Sqrt[d])*Sqrt[x]*(-a + x)
^(1/4)*(-b + x)^(1/4)*Defer[Int][1/(Sqrt[x]*(-a + x)^(1/4)*(-b + x)^(1/4)*(-((a + b)*d) + Sqrt[d]*Sqrt[4*a*b +
a^2*d - 2*a*b*d + b^2*d] + 2*(-1 + d)*x)), x])/((a - x)*(b - x)*x^2)^(1/4)

Rubi steps

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

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Mathematica [F]  time = 8.40, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-2 a b+(a+b) x}{\sqrt [4]{x^2 (-a+x) (-b+x)} \left (a b d-(a+b) d x+(-1+d) x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.37, size = 81, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{d} \sqrt [4]{a b x^2+(-a-b) x^3+x^4}}\right )}{d^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{d} \sqrt [4]{a b x^2+(-a-b) x^3+x^4}}\right )}{d^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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