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

Optimal. Leaf size=263 \[ -\frac {\log \left (\sqrt [3]{d} x \sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}+\left (x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4\right )^{2/3}+d^{2/3} x^2\right )}{2 \sqrt [3]{d}}+\frac {\log \left (\sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}-\sqrt [3]{d} x\right )}{\sqrt [3]{d}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}}{\sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}+2 \sqrt [3]{d} x}\right )}{\sqrt [3]{d}} \]

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Rubi [F]  time = 11.20, 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^2-b (2 a+b) x+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 3.60, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 a b^2-b (2 a+b) x+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.68, size = 263, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{2 \sqrt [3]{d} x+\sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}\right )}{\sqrt [3]{d}}+\frac {\log \left (-\sqrt [3]{d} x+\sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}\right )}{\sqrt [3]{d}}-\frac {\log \left (d^{2/3} x^2+\sqrt [3]{d} x \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}+\left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{2/3}\right )}{2 \sqrt [3]{d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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((2*a*b^2-b*(2*a+b)*x+x^3)/(x*(-a+x)*(-b+x)^2)^(1/3)/(-a*b^2+b*(2*a+b)*x-(a+2*b+d)*x^2+x^3),x, algori
thm="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^{2} - {\left (2 \, a + b\right )} b x + x^{3}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {1}{3}} {\left (a b^{2} - {\left (2 \, a + b\right )} b x + {\left (a + 2 \, b + d\right )} x^{2} - x^{3}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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