3.19.35 \(\int \frac {-2 a b x^2+(a+b) x^3}{(-a+x) (-b+x) \sqrt [4]{x^2 (-a+x) (-b+x)} (-a b d+(a+b) d x+(1-d) x^2)} \, dx\)

Optimal. Leaf size=125 \[ 2 \sqrt [4]{d} \tan ^{-1}\left (\frac {x}{\sqrt [4]{d} \sqrt [4]{x^3 (-a-b)+a b x^2+x^4}}\right )+2 \sqrt [4]{d} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{d} \sqrt [4]{x^3 (-a-b)+a b x^2+x^4}}\right )-\frac {4 \left (x^3 (-a-b)+a b x^2+x^4\right )^{3/4}}{x (x-a) (x-b)} \]

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Rubi [F]  time = 19.79, 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 x^2+(a+b) x^3}{(-a+x) (-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*x^2 + (a + b)*x^3)/((-a + x)*(-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][x^(3/2
)/((-a + x)^(5/4)*(-b + x)^(5/4)*((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][x^(3/2)/((-a + x)^(5/4)*(-b + x)^(5/4)*((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 x^2+(a+b) x^3}{(-a+x) (-b+x) \sqrt [4]{x^2 (-a+x) (-b+x)} \left (-a b d+(a+b) d x+(1-d) x^2\right )} \, dx &=\int \frac {x^2 (-2 a b+(a+b) x)}{(-a+x) (-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 {x^{3/2} (-2 a b+(a+b) x)}{(-a+x)^{5/4} (-b+x)^{5/4} \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 {\left (a+b-\frac {\sqrt {4 a b+a^2 d-2 a b d+b^2 d}}{\sqrt {d}}\right ) x^{3/2}}{(-a+x)^{5/4} (-b+x)^{5/4} \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 {\left (a+b+\frac {\sqrt {4 a b+a^2 d-2 a b d+b^2 d}}{\sqrt {d}}\right ) x^{3/2}}{(-a+x)^{5/4} (-b+x)^{5/4} \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 {x^{3/2}}{(-a+x)^{5/4} (-b+x)^{5/4} \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 {x^{3/2}}{(-a+x)^{5/4} (-b+x)^{5/4} \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 = 15.03, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-2 a b x^2+(a+b) x^3}{(-a+x) (-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*x^2 + (a + b)*x^3)/((-a + x)*(-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*x^2 + (a + b)*x^3)/((-a + x)*(-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 = 0.72, size = 125, normalized size = 1.00 \begin {gather*} -\frac {4 \left (a b x^2+(-a-b) x^3+x^4\right )^{3/4}}{x (-a+x) (-b+x)}+2 \sqrt [4]{d} \tan ^{-1}\left (\frac {x}{\sqrt [4]{d} \sqrt [4]{a b x^2+(-a-b) x^3+x^4}}\right )+2 \sqrt [4]{d} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{d} \sqrt [4]{a b x^2+(-a-b) x^3+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.16, size = 0, normalized size = 0.00 \[\int \frac {-2 a b \,x^{2}+\left (a +b \right ) x^{3}}{\left (-a +x \right ) \left (-b +x \right ) \left (x^{2} \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{4}} \left (-a b d +\left (a +b \right ) d x +\left (1-d \right ) x^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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