Optimal. Leaf size=181 \[ \frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{d} x}{\sqrt [3]{d} x+2 \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}\right )}{\sqrt [3]{d}}-\frac {\log \left (-\sqrt [3]{d} x+\sqrt [3]{a b x^2+(-a-b) x^3+x^4}\right )}{\sqrt [3]{d}}+\frac {\log \left (d^{2/3} x^2+\sqrt [3]{d} x \sqrt [3]{a b x^2+(-a-b) x^3+x^4}+\left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}\right )}{2 \sqrt [3]{d}} \]
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Rubi [F]
time = 4.51, 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.
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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 = 32.49, 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.
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Maple [F]
time = 0.10, 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.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
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.
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Sympy [F(-1)] Timed out
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.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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