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

Optimal. Leaf size=175 \[ \frac {\log \left (x-\sqrt [3]{d} \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}\right )}{d^{2/3}}-\frac {\log \left (d^{2/3} \left (x^3 (-a-b)+a b x^2+x^4\right )^{2/3}+\sqrt [3]{d} x \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}+x^2\right )}{2 d^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{d} \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}+x}\right )}{d^{2/3}} \]

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Rubi [F]  time = 9.15, 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 d-(1+a d+b d) x+d 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*d - (1 + a*d + b*d)*x + d*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])/(d*((a - x)*(b - x)*x^2)^(1/3)) +
((1 + a*d + b*d + Sqrt[a^2*d^2 + 2*a*d*(1 - b*d) + (1 + 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)*(-1 - a*d - b*d - Sqrt[1 + 2*a*d + 2*b*d + a^2*d^2 - 2*a*b*d^2 +
b^2*d^2] + 2*d*x)), x])/(d*((a - x)*(b - x)*x^2)^(1/3)) + ((1 + a*d + b*d - Sqrt[a^2*d^2 + 2*a*d*(1 - b*d) +
(1 + 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)*(-1 -
a*d - b*d + Sqrt[1 + 2*a*d + 2*b*d + a^2*d^2 - 2*a*b*d^2 + b^2*d^2] + 2*d*x)), x])/(d*((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 d-(1+a d+b d) x+d 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 d-(1+a d+b d) x+d 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}{d x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}}-\frac {2 a b d-(1+a d+b d) x}{d x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (a b d+(-1-a d-b d) x+d 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}{d \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 d-(1+a d+b d) x}{x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (a b d+(-1-a d-b d) x+d x^2\right )} \, dx}{d \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-a d-b d-\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}}{x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (-1-a d-b d-\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}+2 d x\right )}+\frac {-1-a d-b d+\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}}{x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (-1-a d-b d+\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}+2 d x\right )}\right ) \, dx}{d \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}{d \sqrt [3]{x^2 (-a+x) (-b+x)}}\\ &=-\frac {\left (\left (-1-a d-b d-\sqrt {a^2 d^2+2 a d (1-b d)+(1+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 (-1-a d-b d-\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt [3]{x^2 (-a+x) (-b+x)}}-\frac {\left (\left (-1-a d-b d+\sqrt {a^2 d^2+2 a d (1-b d)+(1+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 (-1-a d-b d+\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}+2 d x\right )} \, dx}{d \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}{d \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 )}{d \sqrt [3]{(a-x) (b-x) x^2}}-\frac {\left (\left (-1-a d-b d-\sqrt {a^2 d^2+2 a d (1-b d)+(1+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 (-1-a d-b d-\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt [3]{x^2 (-a+x) (-b+x)}}-\frac {\left (\left (-1-a d-b d+\sqrt {a^2 d^2+2 a d (1-b d)+(1+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 (-1-a d-b d+\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt [3]{x^2 (-a+x) (-b+x)}}\\ \end {align*}

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Mathematica [F]  time = 14.41, 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 d-(1+a d+b d) x+d 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*d - (1 + a*d + b*d)*x + d*x^2)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.42, size = 175, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}\right )}{d^{2/3}}+\frac {\log \left (x-\sqrt [3]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}\right )}{d^{2/3}}-\frac {\log \left (x^2+\sqrt [3]{d} x \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 d^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*d^(1/3)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))])/d^(2/3)) + Log[x - d^(1/3
)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3)]/d^(2/3) - Log[x^2 + 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^(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+x^2)/(x^2*(-a+x)*(-b+x))^(1/3)/(a*b*d-(a*d+b*d+1)*x+d*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 d + d x^{2} - {\left (a d + b d + 1\right )} x\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*d-(a*d+b*d+1)*x+d*x^2),x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {-a b +x^{2}}{\left (x^{2} \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{3}} \left (a b d -\left (a d +b d +1\right ) x +d \,x^{2}\right )}\, dx\]

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

[Out]

int((-a*b+x^2)/(x^2*(-a+x)*(-b+x))^(1/3)/(a*b*d-(a*d+b*d+1)*x+d*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 d + d x^{2} - {\left (a d + b d + 1\right )} x\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*d-(a*d+b*d+1)*x+d*x^2),x, algorithm="maxima")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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