Optimal. Leaf size=241 \[ \frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}{-2 a \sqrt [3]{d}+2 \sqrt [3]{d} x+\sqrt [3]{a b x^2+(-a-b) x^3+x^4}}\right )}{\sqrt [3]{d}}+\frac {\log \left (a \sqrt [3]{d}-\sqrt [3]{d} x+\sqrt [3]{a b x^2+(-a-b) x^3+x^4}\right )}{\sqrt [3]{d}}-\frac {\log \left (a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left (-a \sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{a b x^2+(-a-b) x^3+x^4}+\left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}\right )}{2 \sqrt [3]{d}} \]
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Rubi [F]
time = 10.97, 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 b x-3 a x^2+x^3}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a^2 d+2 a d x-(b+d) x^2+x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {2 a b x-3 a x^2+x^3}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a^2 d+2 a d x-(b+d) x^2+x^3\right )} \, dx &=\int \frac {x \left (2 a b-3 a x+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a^2 d+2 a d x-(b+d) x^2+x^3\right )} \, dx\\ &=\frac {\left (x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {\sqrt [3]{x} \left (2 a b-3 a x+x^2\right )}{\sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (-a^2 d+2 a d x-(b+d) x^2+x^3\right )} \, dx}{\sqrt [3]{x^2 (-a+x) (-b+x)}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \text {Subst}\left (\int \frac {x^3 \left (2 a b-3 a x^3+x^6\right )}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (-a^2 d+2 a d x^3-(b+d) x^6+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \text {Subst}\left (\int \left (\frac {1}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3}}+\frac {a^2 d+2 a (b-d) x^3-(3 a-b-d) x^6}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (-a^2 d+2 a d x^3-(b+d) x^6+x^9\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)}}+\frac {\left (3 x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \text {Subst}\left (\int \frac {a^2 d+2 a (b-d) x^3-(3 a-b-d) x^6}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (-a^2 d+2 a d x^3-(b+d) x^6+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \text {Subst}\left (\int \left (\frac {2 a (-b+d) x^3}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a^2 d-2 a d x^3+b \left (1+\frac {d}{b}\right ) x^6-x^9\right )}+\frac {(3 a-b-d) x^6}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a^2 d-2 a d x^3+b \left (1+\frac {d}{b}\right ) x^6-x^9\right )}+\frac {a^2 d}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (-a^2 d+2 a d x^3-b \left (1+\frac {d}{b}\right ) x^6+x^9\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)}}+\frac {\left (3 x^{2/3} \sqrt [3]{-b+x} \sqrt [3]{1-\frac {x}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-b+x^3} \sqrt [3]{1-\frac {x^3}{a}}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)}}\\ &=\frac {\left (3 (3 a-b-d) x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \text {Subst}\left (\int \frac {x^6}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a^2 d-2 a d x^3+b \left (1+\frac {d}{b}\right ) x^6-x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)}}-\frac {\left (6 a (b-d) x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a^2 d-2 a d x^3+b \left (1+\frac {d}{b}\right ) x^6-x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)}}+\frac {\left (3 a^2 d x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (-a^2 d+2 a d x^3-b \left (1+\frac {d}{b}\right ) x^6+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)}}+\frac {\left (3 x^{2/3} \sqrt [3]{1-\frac {x}{a}} \sqrt [3]{1-\frac {x}{b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-\frac {x^3}{a}} \sqrt [3]{1-\frac {x^3}{b}}} \, dx,x,\sqrt [3]{x}\right )}{\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 (3 (3 a-b-d) x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \text {Subst}\left (\int \frac {x^6}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a^2 d-2 a d x^3+b \left (1+\frac {d}{b}\right ) x^6-x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)}}-\frac {\left (6 a (b-d) x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a^2 d-2 a d x^3+b \left (1+\frac {d}{b}\right ) x^6-x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)}}+\frac {\left (3 a^2 d x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (-a^2 d+2 a d x^3-b \left (1+\frac {d}{b}\right ) x^6+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)}}\\ \end {align*}
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Mathematica [F]
time = 33.09, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 a b x-3 a x^2+x^3}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a^2 d+2 a d x-(b+d) x^2+x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {2 a b x -3 a \,x^{2}+x^{3}}{\left (x^{2} \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{3}} \left (-a^{2} d +2 a d x -\left (b +d \right ) x^{2}+x^{3}\right )}\, dx\]
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 = -0.00 \begin {gather*} \int -\frac {x^3-3\,a\,x^2+2\,a\,b\,x}{{\left (x^2\,\left (a-x\right )\,\left (b-x\right )\right )}^{1/3}\,\left (d\,a^2-2\,d\,a\,x-x^3+\left (b+d\right )\,x^2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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