3.30.84 \(\int \frac {(-a b+(2 a-b) x) (a^2-2 a x+x^2)}{\sqrt [3]{x (-a+x) (-b+x)} (a^4 d-4 a^3 d x+(-b^2+6 a^2 d) x^2+2 (b-2 a d) x^3+(-1+d) x^4)} \, dx\)

Optimal. Leaf size=387 \[ -\frac {\log \left (a^2 \sqrt [3]{d}+\sqrt [3]{x^2 (-a-b)+a b x+x^3} \left (a \sqrt [6]{d}-\sqrt [6]{d} x\right )+\left (x^2 (-a-b)+a b x+x^3\right )^{2/3}-2 a \sqrt [3]{d} x+\sqrt [3]{d} x^2\right )}{4 d^{2/3}}-\frac {\log \left (a^2 \sqrt [3]{d}+\sqrt [3]{x^2 (-a-b)+a b x+x^3} \left (\sqrt [6]{d} x-a \sqrt [6]{d}\right )+\left (x^2 (-a-b)+a b x+x^3\right )^{2/3}-2 a \sqrt [3]{d} x+\sqrt [3]{d} x^2\right )}{4 d^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {a^2}{\sqrt {3}}+\frac {2 \left (x^2 (-a-b)+a b x+x^3\right )^{2/3}}{\sqrt {3} \sqrt [3]{d}}-\frac {2 a x}{\sqrt {3}}+\frac {x^2}{\sqrt {3}}}{(a-x)^2}\right )}{2 d^{2/3}}+\frac {\log \left (\sqrt [3]{x^2 (-a-b)+a b x+x^3}+a \sqrt [6]{d}-\sqrt [6]{d} x\right )}{2 d^{2/3}}+\frac {\log \left (\sqrt [3]{x^2 (-a-b)+a b x+x^3}+a \left (-\sqrt [6]{d}\right )+\sqrt [6]{d} x\right )}{2 d^{2/3}} \]

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Rubi [F]  time = 13.23, 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 a-b) x) \left (a^2-2 a x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^4 d-4 a^3 d x+\left (-b^2+6 a^2 d\right ) x^2+2 (b-2 a d) x^3+(-1+d) x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 6.93, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(-a b+(2 a-b) x) \left (a^2-2 a x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^4 d-4 a^3 d x+\left (-b^2+6 a^2 d\right ) x^2+2 (b-2 a d) x^3+(-1+d) x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((-(a*b) + (2*a - b)*x)*(a^2 - 2*a*x + x^2))/((x*(-a + x)*(-b + x))^(1/3)*(a^4*d - 4*a^3*d*x + (-b^2
+ 6*a^2*d)*x^2 + 2*(b - 2*a*d)*x^3 + (-1 + d)*x^4)),x]

[Out]

Integrate[((-(a*b) + (2*a - b)*x)*(a^2 - 2*a*x + x^2))/((x*(-a + x)*(-b + x))^(1/3)*(a^4*d - 4*a^3*d*x + (-b^2
+ 6*a^2*d)*x^2 + 2*(b - 2*a*d)*x^3 + (-1 + d)*x^4)), x]

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IntegrateAlgebraic [A]  time = 5.85, size = 387, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {a^2}{\sqrt {3}}-\frac {2 a x}{\sqrt {3}}+\frac {x^2}{\sqrt {3}}+\frac {2 \left (a b x+(-a-b) x^2+x^3\right )^{2/3}}{\sqrt {3} \sqrt [3]{d}}}{(a-x)^2}\right )}{2 d^{2/3}}+\frac {\log \left (a \sqrt [6]{d}-\sqrt [6]{d} x+\sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{2 d^{2/3}}+\frac {\log \left (-a \sqrt [6]{d}+\sqrt [6]{d} x+\sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{2 d^{2/3}}-\frac {\log \left (a^2 \sqrt [3]{d}-2 a \sqrt [3]{d} x+\sqrt [3]{d} x^2+\left (a \sqrt [6]{d}-\sqrt [6]{d} x\right ) \sqrt [3]{a b x+(-a-b) x^2+x^3}+\left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{4 d^{2/3}}-\frac {\log \left (a^2 \sqrt [3]{d}-2 a \sqrt [3]{d} x+\sqrt [3]{d} x^2+\left (-a \sqrt [6]{d}+\sqrt [6]{d} x\right ) \sqrt [3]{a b x+(-a-b) x^2+x^3}+\left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{4 d^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-(a*b) + (2*a - b)*x)*(a^2 - 2*a*x + x^2))/((x*(-a + x)*(-b + x))^(1/3)*(a^4*d - 4*a^3*d*
x + (-b^2 + 6*a^2*d)*x^2 + 2*(b - 2*a*d)*x^3 + (-1 + d)*x^4)),x]

[Out]

-1/2*(Sqrt[3]*ArcTan[(a^2/Sqrt[3] - (2*a*x)/Sqrt[3] + x^2/Sqrt[3] + (2*(a*b*x + (-a - b)*x^2 + x^3)^(2/3))/(Sq
rt[3]*d^(1/3)))/(a - x)^2])/d^(2/3) + Log[a*d^(1/6) - d^(1/6)*x + (a*b*x + (-a - b)*x^2 + x^3)^(1/3)]/(2*d^(2/
3)) + Log[-(a*d^(1/6)) + d^(1/6)*x + (a*b*x + (-a - b)*x^2 + x^3)^(1/3)]/(2*d^(2/3)) - Log[a^2*d^(1/3) - 2*a*d
^(1/3)*x + d^(1/3)*x^2 + (a*d^(1/6) - d^(1/6)*x)*(a*b*x + (-a - b)*x^2 + x^3)^(1/3) + (a*b*x + (-a - b)*x^2 +
x^3)^(2/3)]/(4*d^(2/3)) - Log[a^2*d^(1/3) - 2*a*d^(1/3)*x + d^(1/3)*x^2 + (-(a*d^(1/6)) + d^(1/6)*x)*(a*b*x +
(-a - b)*x^2 + x^3)^(1/3) + (a*b*x + (-a - b)*x^2 + x^3)^(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((-a*b+(2*a-b)*x)*(a^2-2*a*x+x^2)/(x*(-a+x)*(-b+x))^(1/3)/(a^4*d-4*a^3*d*x+(6*a^2*d-b^2)*x^2+2*(-2*a*
d+b)*x^3+(-1+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 (a^{2} - 2 \, a x + x^{2}\right )} {\left (a b - {\left (2 \, a - b\right )} x\right )}}{{\left (a^{4} d - 4 \, a^{3} d x + {\left (d - 1\right )} x^{4} - 2 \, {\left (2 \, a d - b\right )} x^{3} + {\left (6 \, a^{2} d - b^{2}\right )} x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*b+(2*a-b)*x)*(a^2-2*a*x+x^2)/(x*(-a+x)*(-b+x))^(1/3)/(a^4*d-4*a^3*d*x+(6*a^2*d-b^2)*x^2+2*(-2*a*
d+b)*x^3+(-1+d)*x^4),x, algorithm="giac")

[Out]

integrate(-(a^2 - 2*a*x + x^2)*(a*b - (2*a - b)*x)/((a^4*d - 4*a^3*d*x + (d - 1)*x^4 - 2*(2*a*d - b)*x^3 + (6*
a^2*d - b^2)*x^2)*((a - x)*(b - x)*x)^(1/3)), x)

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (-a b +\left (2 a -b \right ) x \right ) \left (a^{2}-2 a x +x^{2}\right )}{\left (x \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{3}} \left (a^{4} d -4 a^{3} d x +\left (6 a^{2} d -b^{2}\right ) x^{2}+2 \left (-2 a d +b \right ) x^{3}+\left (-1+d \right ) x^{4}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a*b+(2*a-b)*x)*(a^2-2*a*x+x^2)/(x*(-a+x)*(-b+x))^(1/3)/(a^4*d-4*a^3*d*x+(6*a^2*d-b^2)*x^2+2*(-2*a*d+b)*x
^3+(-1+d)*x^4),x)

[Out]

int((-a*b+(2*a-b)*x)*(a^2-2*a*x+x^2)/(x*(-a+x)*(-b+x))^(1/3)/(a^4*d-4*a^3*d*x+(6*a^2*d-b^2)*x^2+2*(-2*a*d+b)*x
^3+(-1+d)*x^4),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {{\left (a^{2} - 2 \, a x + x^{2}\right )} {\left (a b - {\left (2 \, a - b\right )} x\right )}}{{\left (a^{4} d - 4 \, a^{3} d x + {\left (d - 1\right )} x^{4} - 2 \, {\left (2 \, a d - b\right )} x^{3} + {\left (6 \, a^{2} d - b^{2}\right )} x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*b+(2*a-b)*x)*(a^2-2*a*x+x^2)/(x*(-a+x)*(-b+x))^(1/3)/(a^4*d-4*a^3*d*x+(6*a^2*d-b^2)*x^2+2*(-2*a*
d+b)*x^3+(-1+d)*x^4),x, algorithm="maxima")

[Out]

-integrate((a^2 - 2*a*x + x^2)*(a*b - (2*a - b)*x)/((a^4*d - 4*a^3*d*x + (d - 1)*x^4 - 2*(2*a*d - b)*x^3 + (6*
a^2*d - b^2)*x^2)*((a - x)*(b - x)*x)^(1/3)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {\left (a\,b-x\,\left (2\,a-b\right )\right )\,\left (a^2-2\,a\,x+x^2\right )}{{\left (x\,\left (a-x\right )\,\left (b-x\right )\right )}^{1/3}\,\left (2\,x^3\,\left (b-2\,a\,d\right )+x^2\,\left (6\,a^2\,d-b^2\right )+a^4\,d+x^4\,\left (d-1\right )-4\,a^3\,d\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((a*b - x*(2*a - b))*(a^2 - 2*a*x + x^2))/((x*(a - x)*(b - x))^(1/3)*(2*x^3*(b - 2*a*d) + x^2*(6*a^2*d -
b^2) + a^4*d + x^4*(d - 1) - 4*a^3*d*x)),x)

[Out]

-int(((a*b - x*(2*a - b))*(a^2 - 2*a*x + x^2))/((x*(a - x)*(b - x))^(1/3)*(2*x^3*(b - 2*a*d) + x^2*(6*a^2*d -
b^2) + a^4*d + x^4*(d - 1) - 4*a^3*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((-a*b+(2*a-b)*x)*(a**2-2*a*x+x**2)/(x*(-a+x)*(-b+x))**(1/3)/(a**4*d-4*a**3*d*x+(6*a**2*d-b**2)*x**2+
2*(-2*a*d+b)*x**3+(-1+d)*x**4),x)

[Out]

Timed out

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