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

Optimal. Leaf size=290 \[ \frac {\log \left (\sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}+b \sqrt [3]{d}-\sqrt [3]{d} x\right )}{d^{2/3}}-\frac {\log \left (\sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4} \left (\sqrt [3]{d} x-b \sqrt [3]{d}\right )+\left (x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4\right )^{2/3}+b^2 d^{2/3}-2 b d^{2/3} x+d^{2/3} x^2\right )}{2 d^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} b \sqrt [3]{d}-\sqrt {3} \sqrt [3]{d} x}{-2 \sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}+b \sqrt [3]{d}-\sqrt [3]{d} x}\right )}{d^{2/3}} \]

________________________________________________________________________________________

Rubi [F]  time = 6.18, 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 )}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (b d-(a+d) x+x^2\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)^(2/3)*(b*d - (a + d)*x + x^2)),x]

[Out]

(-3*(b - x)*x*(1 - x/a)^(2/3)*(1 - x/b)^(1/3)*AppellF1[1/3, 2/3, 1/3, 4/3, x/a, x/b])/(-((a - x)*(b - x)^2*x))
^(2/3) + ((a - 2*b + d + Sqrt[a^2 + 2*a*d - 4*b*d + d^2])*x^(2/3)*(-a + x)^(2/3)*(-b + x)^(4/3)*Defer[Int][1/(
x^(2/3)*(-a + x)^(2/3)*(-b + x)^(1/3)*(-a - d - Sqrt[a^2 + 2*a*d - 4*b*d + d^2] + 2*x)), x])/(-((a - x)*(b - x
)^2*x))^(2/3) + ((a - 2*b + d - Sqrt[a^2 + 2*a*d - 4*b*d + d^2])*x^(2/3)*(-a + x)^(2/3)*(-b + x)^(4/3)*Defer[I
nt][1/(x^(2/3)*(-a + x)^(2/3)*(-b + x)^(1/3)*(-a - d + Sqrt[a^2 + 2*a*d - 4*b*d + d^2] + 2*x)), x])/(-((a - x)
*(b - x)^2*x))^(2/3)

Rubi steps

\begin {align*} \int \frac {(-b+x) \left (a b-2 b x+x^2\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (b d-(a+d) x+x^2\right )} \, dx &=\frac {\left (x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {a b-2 b x+x^2}{x^{2/3} (-a+x)^{2/3} \sqrt [3]{-b+x} \left (b d-(a+d) x+x^2\right )} \, dx}{\left (x (-a+x) (-b+x)^2\right )^{2/3}}\\ &=\frac {\left (x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \left (\frac {1}{x^{2/3} (-a+x)^{2/3} \sqrt [3]{-b+x}}+\frac {b (a-d)+(a-2 b+d) x}{x^{2/3} (-a+x)^{2/3} \sqrt [3]{-b+x} \left (b d+(-a-d) x+x^2\right )}\right ) \, dx}{\left (x (-a+x) (-b+x)^2\right )^{2/3}}\\ &=\frac {\left (x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {1}{x^{2/3} (-a+x)^{2/3} \sqrt [3]{-b+x}} \, dx}{\left (x (-a+x) (-b+x)^2\right )^{2/3}}+\frac {\left (x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {b (a-d)+(a-2 b+d) x}{x^{2/3} (-a+x)^{2/3} \sqrt [3]{-b+x} \left (b d+(-a-d) x+x^2\right )} \, dx}{\left (x (-a+x) (-b+x)^2\right )^{2/3}}\\ &=\frac {\left (x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \left (\frac {a-2 b+d+\sqrt {a^2+2 a d-4 b d+d^2}}{x^{2/3} (-a+x)^{2/3} \sqrt [3]{-b+x} \left (-a-d-\sqrt {a^2+2 a d-4 b d+d^2}+2 x\right )}+\frac {a-2 b+d-\sqrt {a^2+2 a d-4 b d+d^2}}{x^{2/3} (-a+x)^{2/3} \sqrt [3]{-b+x} \left (-a-d+\sqrt {a^2+2 a d-4 b d+d^2}+2 x\right )}\right ) \, dx}{\left (x (-a+x) (-b+x)^2\right )^{2/3}}+\frac {\left (x^{2/3} (-b+x)^{4/3} \left (1-\frac {x}{a}\right )^{2/3}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-b+x} \left (1-\frac {x}{a}\right )^{2/3}} \, dx}{\left (x (-a+x) (-b+x)^2\right )^{2/3}}\\ &=\frac {\left (\left (a-2 b+d-\sqrt {a^2+2 a d-4 b d+d^2}\right ) x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {1}{x^{2/3} (-a+x)^{2/3} \sqrt [3]{-b+x} \left (-a-d+\sqrt {a^2+2 a d-4 b d+d^2}+2 x\right )} \, dx}{\left (x (-a+x) (-b+x)^2\right )^{2/3}}+\frac {\left (\left (a-2 b+d+\sqrt {a^2+2 a d-4 b d+d^2}\right ) x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {1}{x^{2/3} (-a+x)^{2/3} \sqrt [3]{-b+x} \left (-a-d-\sqrt {a^2+2 a d-4 b d+d^2}+2 x\right )} \, dx}{\left (x (-a+x) (-b+x)^2\right )^{2/3}}+\frac {\left (x^{2/3} (-b+x) \left (1-\frac {x}{a}\right )^{2/3} \sqrt [3]{1-\frac {x}{b}}\right ) \int \frac {1}{x^{2/3} \left (1-\frac {x}{a}\right )^{2/3} \sqrt [3]{1-\frac {x}{b}}} \, dx}{\left (x (-a+x) (-b+x)^2\right )^{2/3}}\\ &=-\frac {3 (b-x) x \left (1-\frac {x}{a}\right )^{2/3} \sqrt [3]{1-\frac {x}{b}} F_1\left (\frac {1}{3};\frac {2}{3},\frac {1}{3};\frac {4}{3};\frac {x}{a},\frac {x}{b}\right )}{\left (-\left ((a-x) (b-x)^2 x\right )\right )^{2/3}}+\frac {\left (\left (a-2 b+d-\sqrt {a^2+2 a d-4 b d+d^2}\right ) x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {1}{x^{2/3} (-a+x)^{2/3} \sqrt [3]{-b+x} \left (-a-d+\sqrt {a^2+2 a d-4 b d+d^2}+2 x\right )} \, dx}{\left (x (-a+x) (-b+x)^2\right )^{2/3}}+\frac {\left (\left (a-2 b+d+\sqrt {a^2+2 a d-4 b d+d^2}\right ) x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {1}{x^{2/3} (-a+x)^{2/3} \sqrt [3]{-b+x} \left (-a-d-\sqrt {a^2+2 a d-4 b d+d^2}+2 x\right )} \, dx}{\left (x (-a+x) (-b+x)^2\right )^{2/3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 7.30, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(-b+x) \left (a b-2 b x+x^2\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (b d-(a+d) x+x^2\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)^(2/3)*(b*d - (a + d)*x + x^2)),x]

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 3.36, size = 290, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} b \sqrt [3]{d}-\sqrt {3} \sqrt [3]{d} x}{b \sqrt [3]{d}-\sqrt [3]{d} x-2 \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 (b \sqrt [3]{d}-\sqrt [3]{d} x+\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 (b^2 d^{2/3}-2 b d^{2/3} x+d^{2/3} x^2+\left (-b \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}+\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[((-b + x)*(a*b - 2*b*x + x^2))/((x*(-a + x)*(-b + x)^2)^(2/3)*(b*d - (a + d)*x + x^2)),x]

[Out]

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

[Out]

Timed out

________________________________________________________________________________________

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 {2}{3}} {\left (b d - {\left (a + d\right )} x + 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)^(2/3)/(b*d-(a+d)*x+x^2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.16, size = 0, normalized size = 0.00 \[\int \frac {\left (-b +x \right ) \left (a b -2 b x +x^{2}\right )}{\left (x \left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {2}{3}} \left (b d -\left (a +d \right ) x +x^{2}\right )}\, dx\]

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

[Out]

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

________________________________________________________________________________________

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 {2}{3}} {\left (b d - {\left (a + d\right )} x + 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)^(2/3)/(b*d-(a+d)*x+x^2),x, algorithm="maxima")

[Out]

-integrate((a*b - 2*b*x + x^2)*(b - x)/((-(a - x)*(b - x)^2*x)^(2/3)*(b*d - (a + d)*x + x^2)), 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\,b\right )}{\left (x^2+\left (-a-d\right )\,x+b\,d\right )\,{\left (-x\,\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{2/3}} \,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))/((b*d + x^2 - x*(a + d))*(-x*(a - x)*(b - x)^2)^(2/3)),x)

[Out]

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

[Out]

Timed out

________________________________________________________________________________________