3.25.90 \(\int \frac {-2 a^2 b x+a (3 a+2 b) x^2-4 a x^3+x^4}{(x^2 (-a+x) (-b+x))^{2/3} (-a^2+2 a x-(1+b d) x^2+d x^3)} \, dx\)

Optimal. Leaf size=206 \[ -\frac {\log \left (a^2+d^{2/3} \left (x^3 (-a-b)+a b x^2+x^4\right )^{2/3}+\sqrt [3]{x^3 (-a-b)+a b x^2+x^4} \left (\sqrt [3]{d} x-a \sqrt [3]{d}\right )-2 a x+x^2\right )}{2 \sqrt [3]{d}}+\frac {\log \left (\sqrt [3]{d} \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}+a-x\right )}{\sqrt [3]{d}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} a-\sqrt {3} x}{-2 \sqrt [3]{d} \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}+a-x}\right )}{\sqrt [3]{d}} \]

________________________________________________________________________________________

Rubi [F]  time = 11.50, 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^2 b x+a (3 a+2 b) x^2-4 a x^3+x^4}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \left (-a^2+2 a x-(1+b d) x^2+d x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(9*a*x^(4/3)*(-a + x)^(2/3)*(-b + x)^(2/3)*Defer[Subst][Defer[Int][(x^4*(-a + x^3)^(1/3))/((-b + x^3)^(2/3)*(a
^2 - 2*a*x^3 + (1 + b*d)*x^6 - d*x^9)), x], x, x^(1/3)])/((a - x)*(b - x)*x^2)^(2/3) + (6*a*b*x^(4/3)*(-a + x)
^(2/3)*(-b + x)^(2/3)*Defer[Subst][Defer[Int][(x*(-a + x^3)^(1/3))/((-b + x^3)^(2/3)*(-a^2 + 2*a*x^3 - (1 + b*
d)*x^6 + d*x^9)), x], x, x^(1/3)])/((a - x)*(b - x)*x^2)^(2/3) + (3*x^(4/3)*(-a + x)^(2/3)*(-b + x)^(2/3)*Defe
r[Subst][Defer[Int][(x^7*(-a + x^3)^(1/3))/((-b + x^3)^(2/3)*(-a^2 + 2*a*x^3 - (1 + b*d)*x^6 + d*x^9)), x], x,
x^(1/3)])/((a - x)*(b - x)*x^2)^(2/3)

Rubi steps

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

________________________________________________________________________________________

Mathematica [F]  time = 4.49, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-2 a^2 b x+a (3 a+2 b) x^2-4 a x^3+x^4}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \left (-a^2+2 a x-(1+b d) x^2+d x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 4.09, size = 206, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} a-\sqrt {3} x}{a-x-2 \sqrt [3]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}\right )}{\sqrt [3]{d}}+\frac {\log \left (a-x+\sqrt [3]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}\right )}{\sqrt [3]{d}}-\frac {\log \left (a^2-2 a x+x^2+\left (-a \sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{a b x^2+(-a-b) x^3+x^4}+d^{2/3} \left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}\right )}{2 \sqrt [3]{d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

________________________________________________________________________________________