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

Optimal. Leaf size=313 \[ -\frac {\log \left (a^2 x^2+d^{2/3} \left (x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4\right )^{2/3}+\sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4} \left (\sqrt [3]{d} x^2-a \sqrt [3]{d} x\right )-2 a x^3+x^4\right )}{4 d^{2/3}}+\frac {\log \left (\sqrt [3]{d} \sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}+a x-x^2\right )}{2 d^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}}{\sqrt [3]{d} \sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}-2 a x+2 x^2}\right )}{2 d^{2/3}} \]

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Rubi [F]  time = 13.53, 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 {(-b+x) \left (a b-2 b x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-b^2 d+2 b d x-\left (-a^2+d\right ) x^2-2 a x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(6*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)*(b
^2*d - 2*b*d*x^3 - a^2*(1 - d/a^2)*x^6 + 2*a*x^9 - x^12)), 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)*(-(
b^2*d) + 2*b*d*x^3 + a^2*(1 - d/a^2)*x^6 - 2*a*x^9 + x^12)), 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)*(-(
b^2*d) + 2*b*d*x^3 + a^2*(1 - d/a^2)*x^6 - 2*a*x^9 + x^12)), x], x, x^(1/3)])/(-((a - x)*(b - x)^2*x))^(1/3)

Rubi steps

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

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Mathematica [F]  time = 3.57, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(-b+x) \left (a b-2 b x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-b^2 d+2 b d x-\left (-a^2+d\right ) x^2-2 a x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.69, size = 313, 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 x+2 x^2+\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 )}{2 d^{2/3}}+\frac {\log \left (a x-x^2+\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 )}{2 d^{2/3}}-\frac {\log \left (a^2 x^2-2 a x^3+x^4+\left (-a \sqrt [3]{d} x+\sqrt [3]{d} x^2\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 )}{4 d^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a b - 2 \, b x + x^{2}\right )} {\left (b - x\right )}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {1}{3}} {\left (2 \, a x^{3} - x^{4} + b^{2} d - 2 \, b d x - {\left (a^{2} - d\right )} x^{2}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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