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

Optimal. Leaf size=315 \[ -\frac {\log \left (a^2 b^2-2 a^2 b x+a^2 x^2-2 a b^2 x+d^{2/3} \left (x^3 (-a-b)+a b x^2+x^4\right )^{4/3}+\left (x^3 (-a-b)+a b x^2+x^4\right )^{2/3} \left (a b \sqrt [3]{d}-a \sqrt [3]{d} x-b \sqrt [3]{d} x+\sqrt [3]{d} x^2\right )+4 a b x^2-2 a x^3+b^2 x^2-2 b x^3+x^4\right )}{2 d^{2/3}}+\frac {\log \left (-\sqrt [3]{d} \left (x^3 (-a-b)+a b x^2+x^4\right )^{2/3}+a b-a x-b x+x^2\right )}{d^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \left (x^3 (-a-b)+a b x^2+x^4\right )^{2/3}}{\sqrt [3]{d} \left (x^3 (-a-b)+a b x^2+x^4\right )^{2/3}+2 a b-2 a x-2 b x+2 x^2}\right )}{d^{2/3}} \]

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Rubi [F]  time = 19.10, 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 {x^3 \left (4 a b-3 (a+b) x+2 x^2\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \left (-a b+(a+b) x-x^2+d x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(3*x^2*(1 - x/a)^(2/3)*(1 - x/b)^(2/3)*AppellF1[2/3, 2/3, 2/3, 5/3, x/a, x/b])/(d*((a - x)*(b - x)*x^2)^(2/3))
+ (6*(a + b)*x^(4/3)*(-a + x)^(2/3)*(-b + x)^(2/3)*Defer[Subst][Defer[Int][x^4/((-a + x^3)^(2/3)*(-b + x^3)^(
2/3)*(a*b - a*(1 + b/a)*x^3 + x^6 - d*x^12)), x], x, x^(1/3)])/(d*((a - x)*(b - x)*x^2)^(2/3)) - (6*(1 + 2*a*b
*d)*x^(4/3)*(-a + x)^(2/3)*(-b + x)^(2/3)*Defer[Subst][Defer[Int][x^7/((-a + x^3)^(2/3)*(-b + x^3)^(2/3)*(a*b
- a*(1 + b/a)*x^3 + x^6 - d*x^12)), x], x, x^(1/3)])/(d*((a - x)*(b - x)*x^2)^(2/3)) + (9*(a + b)*x^(4/3)*(-a
+ x)^(2/3)*(-b + x)^(2/3)*Defer[Subst][Defer[Int][x^10/((-a + x^3)^(2/3)*(-b + x^3)^(2/3)*(a*b - a*(1 + b/a)*x
^3 + x^6 - d*x^12)), x], x, x^(1/3)])/((a - x)*(b - x)*x^2)^(2/3) + (6*a*b*x^(4/3)*(-a + x)^(2/3)*(-b + x)^(2/
3)*Defer[Subst][Defer[Int][x/((-a + x^3)^(2/3)*(-b + x^3)^(2/3)*(-(a*b) + a*(1 + b/a)*x^3 - x^6 + d*x^12)), x]
, x, x^(1/3)])/(d*((a - x)*(b - x)*x^2)^(2/3))

Rubi steps

\begin {align*} \int \frac {x^3 \left (4 a b-3 (a+b) x+2 x^2\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \left (-a b+(a+b) x-x^2+d x^4\right )} \, dx &=\frac {\left (x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {x^{5/3} \left (4 a b-3 (a+b) x+2 x^2\right )}{(-a+x)^{2/3} (-b+x)^{2/3} \left (-a b+(a+b) x-x^2+d x^4\right )} \, dx}{\left (x^2 (-a+x) (-b+x)\right )^{2/3}}\\ &=\frac {\left (3 x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^7 \left (4 a b-3 (a+b) x^3+2 x^6\right )}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (-a b+(a+b) x^3-x^6+d x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3}}\\ &=\frac {\left (3 x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \left (\frac {2 x}{d \left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3}}+\frac {x \left (2 a b-2 (a+b) x^3+2 (1+2 a b d) x^6-3 (a+b) d x^9\right )}{d \left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (-a b+(a+b) x^3-x^6+d x^{12}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3}}\\ &=\frac {\left (3 x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x \left (2 a b-2 (a+b) x^3+2 (1+2 a b d) x^6-3 (a+b) d x^9\right )}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (-a b+(a+b) x^3-x^6+d x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{d \left (x^2 (-a+x) (-b+x)\right )^{2/3}}+\frac {\left (6 x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{d \left (x^2 (-a+x) (-b+x)\right )^{2/3}}\\ &=\frac {\left (3 x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \left (\frac {2 (a+b) x^4}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (a b-a \left (1+\frac {b}{a}\right ) x^3+x^6-d x^{12}\right )}+\frac {2 (-1-2 a b d) x^7}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (a b-a \left (1+\frac {b}{a}\right ) x^3+x^6-d x^{12}\right )}+\frac {3 (a+b) d x^{10}}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (a b-a \left (1+\frac {b}{a}\right ) x^3+x^6-d x^{12}\right )}+\frac {2 a b x}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (-a b+a \left (1+\frac {b}{a}\right ) x^3-x^6+d x^{12}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{d \left (x^2 (-a+x) (-b+x)\right )^{2/3}}+\frac {\left (6 x^{4/3} (-b+x)^{2/3} \left (1-\frac {x}{a}\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (-b+x^3\right )^{2/3} \left (1-\frac {x^3}{a}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{d \left (x^2 (-a+x) (-b+x)\right )^{2/3}}\\ &=\frac {\left (9 (a+b) x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (a b-a \left (1+\frac {b}{a}\right ) x^3+x^6-d x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3}}+\frac {\left (6 a b x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (-a b+a \left (1+\frac {b}{a}\right ) x^3-x^6+d x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{d \left (x^2 (-a+x) (-b+x)\right )^{2/3}}+\frac {\left (6 (a+b) x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (a b-a \left (1+\frac {b}{a}\right ) x^3+x^6-d x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{d \left (x^2 (-a+x) (-b+x)\right )^{2/3}}-\frac {\left (6 (1+2 a b d) x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^7}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (a b-a \left (1+\frac {b}{a}\right ) x^3+x^6-d x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{d \left (x^2 (-a+x) (-b+x)\right )^{2/3}}+\frac {\left (6 x^{4/3} \left (1-\frac {x}{a}\right )^{2/3} \left (1-\frac {x}{b}\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1-\frac {x^3}{a}\right )^{2/3} \left (1-\frac {x^3}{b}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{d \left (x^2 (-a+x) (-b+x)\right )^{2/3}}\\ &=\frac {3 x^2 \left (1-\frac {x}{a}\right )^{2/3} \left (1-\frac {x}{b}\right )^{2/3} F_1\left (\frac {2}{3};\frac {2}{3},\frac {2}{3};\frac {5}{3};\frac {x}{a},\frac {x}{b}\right )}{d \left ((a-x) (b-x) x^2\right )^{2/3}}+\frac {\left (9 (a+b) x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (a b-a \left (1+\frac {b}{a}\right ) x^3+x^6-d x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3}}+\frac {\left (6 a b x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (-a b+a \left (1+\frac {b}{a}\right ) x^3-x^6+d x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{d \left (x^2 (-a+x) (-b+x)\right )^{2/3}}+\frac {\left (6 (a+b) x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (a b-a \left (1+\frac {b}{a}\right ) x^3+x^6-d x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{d \left (x^2 (-a+x) (-b+x)\right )^{2/3}}-\frac {\left (6 (1+2 a b d) x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^7}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (a b-a \left (1+\frac {b}{a}\right ) x^3+x^6-d x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{d \left (x^2 (-a+x) (-b+x)\right )^{2/3}}\\ \end {align*}

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Mathematica [F]  time = 3.38, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3 \left (4 a b-3 (a+b) x+2 x^2\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \left (-a b+(a+b) x-x^2+d x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.89, size = 315, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}}{2 a b-2 a x-2 b x+2 x^2+\sqrt [3]{d} \left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}}\right )}{d^{2/3}}+\frac {\log \left (a b-a x-b x+x^2-\sqrt [3]{d} \left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}\right )}{d^{2/3}}-\frac {\log \left (a^2 b^2-2 a^2 b x-2 a b^2 x+a^2 x^2+4 a b x^2+b^2 x^2-2 a x^3-2 b x^3+x^4+\left (a b \sqrt [3]{d}-a \sqrt [3]{d} x-b \sqrt [3]{d} x+\sqrt [3]{d} x^2\right ) \left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}+d^{2/3} \left (a b x^2+(-a-b) x^3+x^4\right )^{4/3}\right )}{2 d^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

[Out]

Timed out

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