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

Optimal. Leaf size=291 \[ -\frac {\log \left (a^2+d^{2/3} \left (x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4\right )^{2/3}+\sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4} \left (\sqrt [3]{d} x-a \sqrt [3]{d}\right )-2 a x+x^2\right )}{2 d^{2/3}}+\frac {\log \left (\sqrt [3]{d} \sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}+a-x\right )}{d^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}}{\sqrt [3]{d} \sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}-2 a+2 x}\right )}{d^{2/3}} \]

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Rubi [F]  time = 8.96, 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 b^2+(4 a-b) b x-3 a x^2+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a^2+\left (2 a+b^2 d\right ) x-(1+2 b d) x^2+d x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(3*(3*a - 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^2 - 2*a*(1 + (b^2*d)/(2*a))*x^3 + (1 + 2*b*d)*x^6 - d*x^9)), x], x, x^(1/3)])/(-((a - x)*(b - x)^2*x)
)^(1/3) + (3*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^2 + 2*a*(1 + (b^2*d)/(2*a))*x^3 - (1 + 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^2 + 2*a*(1 + (b^2*d)/(2*a))*x^3 - (1 + 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 {-a b^2+(4 a-b) b x-3 a x^2+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a^2+\left (2 a+b^2 d\right ) x-(1+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 {-a b^2+(4 a-b) b x-3 a x^2+x^3}{\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3} \left (-a^2+\left (2 a+b^2 d\right ) x-(1+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 (a b+(-3 a+b) x+x^2\right )}{\sqrt [3]{x} \sqrt [3]{-a+x} \left (-a^2+\left (2 a+b^2 d\right ) x-(1+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 (a b+(-3 a+b) x^3+x^6\right )}{\sqrt [3]{-a+x^3} \left (-a^2+\left (2 a+b^2 d\right ) x^3-(1+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 {(3 a-b) x^4 \sqrt [3]{-b+x^3}}{\sqrt [3]{-a+x^3} \left (a^2-2 a \left (1+\frac {b^2 d}{2 a}\right ) x^3+(1+2 b d) x^6-d x^9\right )}+\frac {a b x \sqrt [3]{-b+x^3}}{\sqrt [3]{-a+x^3} \left (-a^2+2 a \left (1+\frac {b^2 d}{2 a}\right ) x^3-(1+2 b d) x^6+d x^9\right )}+\frac {x^7 \sqrt [3]{-b+x^3}}{\sqrt [3]{-a+x^3} \left (-a^2+2 a \left (1+\frac {b^2 d}{2 a}\right ) x^3-(1+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^2+2 a \left (1+\frac {b^2 d}{2 a}\right ) x^3-(1+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 (3 a-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^2-2 a \left (1+\frac {b^2 d}{2 a}\right ) x^3+(1+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 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^2+2 a \left (1+\frac {b^2 d}{2 a}\right ) x^3-(1+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 = 3.63, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-a b^2+(4 a-b) b x-3 a x^2+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a^2+\left (2 a+b^2 d\right ) x-(1+2 b d) x^2+d x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.69, size = 291, 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 a+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 (a-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 (a^2-2 a x+x^2+\left (-a \sqrt [3]{d}+\sqrt [3]{d} x\right ) \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[(-(a*b^2) + (4*a - b)*b*x - 3*a*x^2 + x^3)/((x*(-a + x)*(-b + x)^2)^(1/3)*(-a^2 + (2*a + b^
2*d)*x - (1 + 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*a + 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[a - 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[a^2 - 2*a*x + x^2 + (-(a*d^(1/3)) + 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((-a*b^2+(4*a-b)*b*x-3*a*x^2+x^3)/(x*(-a+x)*(-b+x)^2)^(1/3)/(-a^2+(b^2*d+2*a)*x-(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 {a b^{2} - {\left (4 \, a - b\right )} b x + 3 \, a x^{2} - x^{3}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {1}{3}} {\left (d x^{3} - {\left (2 \, b d + 1\right )} x^{2} - a^{2} + {\left (b^{2} d + 2 \, a\right )} x\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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