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

Optimal. Leaf size=111 \[ -2 \sqrt [4]{d} \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{x^3 (-a-b)+a b x^2+x^4}}\right )+2 \sqrt [4]{d} \tanh ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{x^3 (-a-b)+a b x^2+x^4}}\right )-\frac {4 \sqrt [4]{x^3 (-a-b)+a b x^2+x^4}}{x} \]

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Rubi [F]  time = 6.94, 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+x) (-b+x) (-2 a b+(a+b) x)}{\left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 24.68, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(-a+x) (-b+x) (-2 a b+(a+b) x)}{\left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.84, size = 111, normalized size = 1.00 \begin {gather*} -\frac {4 \sqrt [4]{a b x^2+(-a-b) x^3+x^4}}{x}-2 \sqrt [4]{d} \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{a b x^2+(-a-b) x^3+x^4}}\right )+2 \sqrt [4]{d} \tanh ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{a b x^2+(-a-b) x^3+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (2 \, a b - {\left (a + b\right )} x\right )} {\left (a - x\right )} {\left (b - x\right )}}{\left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {3}{4}} {\left ({\left (d - 1\right )} x^{2} - a b + {\left (a + b\right )} x\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (-a +x \right ) \left (-b +x \right ) \left (-2 a b +\left (a +b \right ) x \right )}{\left (x^{2} \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {3}{4}} \left (-a b +\left (a +b \right ) x +\left (-1+d \right ) x^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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