3.28.23 \(\int \frac {(-b+x) (-a (a-2 b)-2 b x+x^2)}{((-a+x) (-b+x))^{2/3} (a^4-b^2 d-2 (2 a^3-b d) x+(6 a^2-d) x^2-4 a x^3+x^4)} \, dx\)

Optimal. Leaf size=250 \[ \frac {\log \left (a^2-\sqrt [3]{d} \left (x (-a-b)+a b+x^2\right )^{2/3}-2 a x+x^2\right )}{2 d^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \left (x (-a-b)+a b+x^2\right )^{2/3}}{2 a^2+\sqrt [3]{d} \left (x (-a-b)+a b+x^2\right )^{2/3}-4 a x+2 x^2}\right )}{2 d^{2/3}}-\frac {\log \left (a^4-4 a^3 x+\left (x (-a-b)+a b+x^2\right )^{2/3} \left (a^2 \sqrt [3]{d}-2 a \sqrt [3]{d} x+\sqrt [3]{d} x^2\right )+6 a^2 x^2+d^{2/3} \left (x (-a-b)+a b+x^2\right )^{4/3}-4 a x^3+x^4\right )}{4 d^{2/3}} \]

________________________________________________________________________________________

Rubi [F]  time = 7.40, 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 (a-2 b)-2 b x+x^2\right )}{((-a+x) (-b+x))^{2/3} \left (a^4-b^2 d-2 \left (2 a^3-b d\right ) x+\left (6 a^2-d\right ) x^2-4 a x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [F]  time = 1.91, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(-b+x) \left (-a (a-2 b)-2 b x+x^2\right )}{((-a+x) (-b+x))^{2/3} \left (a^4-b^2 d-2 \left (2 a^3-b d\right ) x+\left (6 a^2-d\right ) x^2-4 a x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 1.05, size = 250, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \left (a b+(-a-b) x+x^2\right )^{2/3}}{2 a^2-4 a x+2 x^2+\sqrt [3]{d} \left (a b+(-a-b) x+x^2\right )^{2/3}}\right )}{2 d^{2/3}}+\frac {\log \left (a^2-2 a x+x^2-\sqrt [3]{d} \left (a b+(-a-b) x+x^2\right )^{2/3}\right )}{2 d^{2/3}}-\frac {\log \left (a^4-4 a^3 x+6 a^2 x^2-4 a x^3+x^4+d^{2/3} \left (a b+(-a-b) x+x^2\right )^{4/3}+\left (a b+(-a-b) x+x^2\right )^{2/3} \left (a^2 \sqrt [3]{d}-2 a \sqrt [3]{d} x+\sqrt [3]{d} x^2\right )\right )}{4 d^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(a*b + (-a - b)*x + x^2)^(2/3))/(2*a^2 - 4*a*x + 2*x^2 + d^(1/3)*(a*b + (-a -
b)*x + x^2)^(2/3))])/(2*d^(2/3)) + Log[a^2 - 2*a*x + x^2 - d^(1/3)*(a*b + (-a - b)*x + x^2)^(2/3)]/(2*d^(2/3)
) - Log[a^4 - 4*a^3*x + 6*a^2*x^2 - 4*a*x^3 + x^4 + d^(2/3)*(a*b + (-a - b)*x + x^2)^(4/3) + (a*b + (-a - b)*x
+ x^2)^(2/3)*(a^2*d^(1/3) - 2*a*d^(1/3)*x + d^(1/3)*x^2)]/(4*d^(2/3))

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {\left (-b +x \right ) \left (-a \left (a -2 b \right )-2 b x +x^{2}\right )}{\left (\left (-a +x \right ) \left (-b +x \right )\right )^{\frac {2}{3}} \left (a^{4}-b^{2} d -2 \left (2 a^{3}-b d \right ) x +\left (6 a^{2}-d \right ) x^{2}-4 a \,x^{3}+x^{4}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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\,\left (a-2\,b\right )\right )}{{\left (\left (a-x\right )\,\left (b-x\right )\right )}^{2/3}\,\left (x^2\,\left (d-6\,a^2\right )-2\,x\,\left (b\,d-2\,a^3\right )+b^2\,d+4\,a\,x^3-a^4-x^4\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________