Optimal. Leaf size=227 \[ \frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x-2 \sqrt [6]{d} \sqrt [3]{a b x+(-a-b) x^2+x^3}}\right )}{2 d^{5/6}}-\frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [6]{d} \sqrt [3]{a b x+(-a-b) x^2+x^3}}\right )}{2 d^{5/6}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [6]{d} \sqrt [3]{a b x+(-a-b) x^2+x^3}}\right )}{d^{5/6}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt [6]{d}}+\sqrt [6]{d} \left (a b x+(-a-b) x^2+x^3\right )^{2/3}}{x \sqrt [3]{a b x+(-a-b) x^2+x^3}}\right )}{2 d^{5/6}} \]
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Rubi [F]
time = 26.74, 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+x) (-b+x) (-2 a b+(a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^2 b^2 d-2 a b (a+b) d x+\left (a^2+4 a b+b^2\right ) d x^2-2 (a+b) d x^3+(-1+d) x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(-a+x) (-b+x) (-2 a b+(a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^2 b^2 d-2 a b (a+b) d x+\left (a^2+4 a b+b^2\right ) d x^2-2 (a+b) d x^3+(-1+d) x^4\right )} \, dx &=\frac {\left (\sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {(-a+x)^{2/3} (-b+x)^{2/3} (-2 a b+(a+b) x)}{\sqrt [3]{x} \left (a^2 b^2 d-2 a b (a+b) d x+\left (a^2+4 a b+b^2\right ) d x^2-2 (a+b) d x^3+(-1+d) x^4\right )} \, dx}{\sqrt [3]{x (-a+x) (-b+x)}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \text {Subst}\left (\int \frac {x \left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (-2 a b+(a+b) x^3\right )}{a^2 b^2 d-2 a b (a+b) d x^3+\left (a^2+4 a b+b^2\right ) d x^6-2 (a+b) d x^9+(-1+d) x^{12}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x (-a+x) (-b+x)}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \text {Subst}\left (\int \left (\frac {(a+b) x^4 \left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3}}{a^2 b^2 d-2 a^2 b \left (1+\frac {b}{a}\right ) d x^3+a^2 \left (1+\frac {b (4 a+b)}{a^2}\right ) d x^6-2 a \left (1+\frac {b}{a}\right ) d x^9-(1-d) x^{12}}+\frac {2 a b x \left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3}}{-a^2 b^2 d+2 a^2 b \left (1+\frac {b}{a}\right ) d x^3-a^2 \left (1+\frac {b (4 a+b)}{a^2}\right ) d x^6+2 a \left (1+\frac {b}{a}\right ) d x^9+(1-d) x^{12}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x (-a+x) (-b+x)}}\\ &=\frac {\left (6 a b \sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \text {Subst}\left (\int \frac {x \left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3}}{-a^2 b^2 d+2 a^2 b \left (1+\frac {b}{a}\right ) d x^3-a^2 \left (1+\frac {b (4 a+b)}{a^2}\right ) d x^6+2 a \left (1+\frac {b}{a}\right ) d x^9+(1-d) x^{12}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x (-a+x) (-b+x)}}+\frac {\left (3 (a+b) \sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \text {Subst}\left (\int \frac {x^4 \left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3}}{a^2 b^2 d-2 a^2 b \left (1+\frac {b}{a}\right ) d x^3+a^2 \left (1+\frac {b (4 a+b)}{a^2}\right ) d x^6-2 a \left (1+\frac {b}{a}\right ) d x^9-(1-d) x^{12}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x (-a+x) (-b+x)}}\\ \end {align*}
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Mathematica [F]
time = 20.19, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(-a+x) (-b+x) (-2 a b+(a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^2 b^2 d-2 a b (a+b) d x+\left (a^2+4 a b+b^2\right ) d x^2-2 (a+b) d x^3+(-1+d) x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (-a +x \right ) \left (-b +x \right ) \left (-2 a b +\left (a +b \right ) x \right )}{\left (x \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{3}} \left (a^{2} b^{2} d -2 a b \left (a +b \right ) d x +\left (a^{2}+4 a b +b^{2}\right ) d \,x^{2}-2 \left (a +b \right ) d \,x^{3}+\left (-1+d \right ) x^{4}\right )}\, dx\]
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 [A]
time = 0.58, size = 315, normalized size = 1.39 \begin {gather*} -\left (-\frac {1}{d}\right )^{\frac {5}{6}} \arctan \left (\frac {{\left (\frac {a b}{x^{2}} - \frac {a}{x} - \frac {b}{x} + 1\right )}^{\frac {1}{3}}}{\left (-\frac {1}{d}\right )^{\frac {1}{6}}}\right ) + \frac {\sqrt {3} \left (-d^{5}\right )^{\frac {5}{6}} \log \left (\sqrt {3} {\left (\frac {a b}{x^{2}} - \frac {a}{x} - \frac {b}{x} + 1\right )}^{\frac {1}{3}} \left (-\frac {1}{d}\right )^{\frac {1}{6}} + {\left (\frac {a b}{x^{2}} - \frac {a}{x} - \frac {b}{x} + 1\right )}^{\frac {2}{3}} + \left (-\frac {1}{d}\right )^{\frac {1}{3}}\right )}{4 \, d^{5}} - \frac {\sqrt {3} \left (-d^{5}\right )^{\frac {5}{6}} \log \left (-\sqrt {3} {\left (\frac {a b}{x^{2}} - \frac {a}{x} - \frac {b}{x} + 1\right )}^{\frac {1}{3}} \left (-\frac {1}{d}\right )^{\frac {1}{6}} + {\left (\frac {a b}{x^{2}} - \frac {a}{x} - \frac {b}{x} + 1\right )}^{\frac {2}{3}} + \left (-\frac {1}{d}\right )^{\frac {1}{3}}\right )}{4 \, d^{5}} - \frac {\left (-d^{5}\right )^{\frac {5}{6}} \arctan \left (\frac {\sqrt {3} \left (-\frac {1}{d}\right )^{\frac {1}{6}} + 2 \, {\left (\frac {a b}{x^{2}} - \frac {a}{x} - \frac {b}{x} + 1\right )}^{\frac {1}{3}}}{\left (-\frac {1}{d}\right )^{\frac {1}{6}}}\right )}{2 \, d^{5}} - \frac {\left (-d^{5}\right )^{\frac {5}{6}} \arctan \left (-\frac {\sqrt {3} \left (-\frac {1}{d}\right )^{\frac {1}{6}} - 2 \, {\left (\frac {a b}{x^{2}} - \frac {a}{x} - \frac {b}{x} + 1\right )}^{\frac {1}{3}}}{\left (-\frac {1}{d}\right )^{\frac {1}{6}}}\right )}{2 \, d^{5}} \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 {\left (2\,a\,b-x\,\left (a+b\right )\right )\,\left (a-x\right )\,\left (b-x\right )}{{\left (x\,\left (a-x\right )\,\left (b-x\right )\right )}^{1/3}\,\left (x^4\,\left (d-1\right )+a^2\,b^2\,d+d\,x^2\,\left (a^2+4\,a\,b+b^2\right )-2\,d\,x^3\,\left (a+b\right )-2\,a\,b\,d\,x\,\left (a+b\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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