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

Optimal. Leaf size=269 \[ \frac {\log \left (x-\sqrt [3]{d} \sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}\right )}{d^{2/3}}-\frac {\log \left (d^{2/3} \left (x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4\right )^{2/3}+\sqrt [3]{d} x \sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}+x^2\right )}{2 d^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}}{\sqrt [3]{d} \sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}+2 x}\right )}{d^{2/3}} \]

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Rubi [F]  time = 15.31, 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 d+b (2 a+b) d x-(1+a d+2 b d) x^2+d 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*d) + b*(2*a + b)*d*x - (1 + a*d +
2*b*d)*x^2 + d*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*d - 2*a*b*(1 + b/(2*a))*d*x^3 + (1 + (a + 2*b)*d)*x^6 - d*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*d) + 2*a*b*(1 + b/(2*a))*d*x^3 - (1 + (a + 2*b)*d)*x^6 + d*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*d) + 2*a*b*(1 + b/(2*a))*d*x^3 - (1 + (a + 2*b)*d)*x^6 + d*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 d+b (2 a+b) d x-(1+a d+2 b d) x^2+d 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 d+b (2 a+b) d x-(1+a d+2 b d) x^2+d 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 d+b (2 a+b) d x-(1+a d+2 b d) x^2+d 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 d+b (2 a+b) d x^3-(1+a d+2 b d) x^6+d 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 d-2 a b \left (1+\frac {b}{2 a}\right ) d x^3+(1+(a+2 b) d) x^6-d x^9\right )}+\frac {b x^4 \sqrt [3]{-b+x^3}}{\sqrt [3]{-a+x^3} \left (-a b^2 d+2 a b \left (1+\frac {b}{2 a}\right ) d x^3-(1+(a+2 b) d) x^6+d x^9\right )}+\frac {x^7 \sqrt [3]{-b+x^3}}{\sqrt [3]{-a+x^3} \left (-a b^2 d+2 a b \left (1+\frac {b}{2 a}\right ) d x^3-(1+(a+2 b) d) x^6+d 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 d+2 a b \left (1+\frac {b}{2 a}\right ) d x^3-(1+(a+2 b) d) x^6+d 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 d+2 a b \left (1+\frac {b}{2 a}\right ) d x^3-(1+(a+2 b) d) x^6+d 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 d-2 a b \left (1+\frac {b}{2 a}\right ) d x^3+(1+(a+2 b) d) x^6-d 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 = 3.39, 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 d+b (2 a+b) d x-(1+a d+2 b d) x^2+d 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*d) + b*(2*a + b)*d*x - (1 +
a*d + 2*b*d)*x^2 + d*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*d) + b*(2*a + b)*d*x - (1 +
a*d + 2*b*d)*x^2 + d*x^3)), x]

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IntegrateAlgebraic [A]  time = 0.71, size = 269, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{2 x+\sqrt [3]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}\right )}{d^{2/3}}+\frac {\log \left (x-\sqrt [3]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}\right )}{d^{2/3}}-\frac {\log \left (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}+d^{2/3} \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 d^{2/3}} \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*d) + b*(2*a + b)*d*
x - (1 + a*d + 2*b*d)*x^2 + d*x^3)),x]

[Out]

(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3))/(2*x + d^(1/3)
*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3))])/d^(2/3) + Log[x - d^(1/3)*(-(a*b^2*x) + (2*a
*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3)]/d^(2/3) - Log[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) + d^(2/3)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(2/3)]/(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((2*a*b^2-b*(2*a+b)*x+x^3)/(x*(-a+x)*(-b+x)^2)^(1/3)/(-a*b^2*d+b*(2*a+b)*d*x-(a*d+2*b*d+1)*x^2+d*x^3)
,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^{2} - {\left (2 \, a + b\right )} b x + x^{3}}{{\left (a b^{2} d - {\left (2 \, a + b\right )} b d x - d x^{3} + {\left (a d + 2 \, b d + 1\right )} x^{2}\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {1}{3}}}\,{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*d+b*(2*a+b)*d*x-(a*d+2*b*d+1)*x^2+d*x^3)
,x, algorithm="giac")

[Out]

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

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

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*d+b*(2*a+b)*d*x-(a*d+2*b*d+1)*x^2+d*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*d+b*(2*a+b)*d*x-(a*d+2*b*d+1)*x^2+d*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 (a b^{2} d - {\left (2 \, a + b\right )} b d x - d x^{3} + {\left (a d + 2 \, b d + 1\right )} x^{2}\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {1}{3}}}\,{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*d+b*(2*a+b)*d*x-(a*d+2*b*d+1)*x^2+d*x^3)
,x, algorithm="maxima")

[Out]

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

[Out]

Timed out

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