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

Optimal. Leaf size=181 \[ \frac {\log \left (\sqrt [3]{d} x \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}+\left (x^3 (-a-b)+a b x^2+x^4\right )^{2/3}+d^{2/3} x^2\right )}{2 \sqrt [3]{d}}-\frac {\log \left (\sqrt [3]{x^3 (-a-b)+a b x^2+x^4}-\sqrt [3]{d} x\right )}{\sqrt [3]{d}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} x}{2 \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}+\sqrt [3]{d} x}\right )}{\sqrt [3]{d}} \]

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Rubi [F]  time = 6.33, 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-x^2}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (a b-(a+b+d) x+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 13.53, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a b-x^2}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (a b-(a+b+d) x+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.40, size = 181, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} x}{\sqrt [3]{d} x+2 \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}\right )}{\sqrt [3]{d}}-\frac {\log \left (-\sqrt [3]{d} x+\sqrt [3]{a b x^2+(-a-b) x^3+x^4}\right )}{\sqrt [3]{d}}+\frac {\log \left (d^{2/3} x^2+\sqrt [3]{d} x \sqrt [3]{a b x^2+(-a-b) x^3+x^4}+\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[(a*b - x^2)/((x^2*(-a + x)*(-b + x))^(1/3)*(a*b - (a + b + d)*x + x^2)),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {a b -x^{2}}{\left (x^{2} \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{3}} \left (a b -\left (a +b +d \right ) x +x^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a\,b-x^2}{{\left (x^2\,\left (a-x\right )\,\left (b-x\right )\right )}^{1/3}\,\left (x^2+\left (-a-b-d\right )\,x+a\,b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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