Optimal. Leaf size=226 \[ -\frac {\log \left (a^2+d^{2/3} \left (x^3 (-a-b)+a b x^2+x^4\right )^{2/3}+\sqrt [3]{x^3 (-a-b)+a b x^2+x^4} \left (\sqrt [3]{d} x-a \sqrt [3]{d}\right )-2 a x+x^2\right )}{2 d^{2/3}}+\frac {\log \left (\sqrt [3]{d} \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}+a-x\right )}{d^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}}{\sqrt [3]{d} \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}-2 a+2 x}\right )}{d^{2/3}} \]
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Rubi [F] time = 13.26, 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+2 a x-(1+b d) x^2+d x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\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+2 a x-(1+b d) x^2+d 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+2 a x-(1+b d) x^2+d 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+2 a x-(1+b d) x^2+d 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 ) \operatorname {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+2 a x^3-(1+b d) x^6+d 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 ) \operatorname {Subst}\left (\int \left (\frac {1}{d \sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3}}+\frac {a^2-2 a (1-b d) x^3+(1-3 a d+b d) x^6}{d \sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (-a^2+2 a x^3-(1+b d) x^6+d 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 ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3}} \, dx,x,\sqrt [3]{x}\right )}{d \sqrt [3]{x^2 (-a+x) (-b+x)}}+\frac {\left (3 x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {a^2-2 a (1-b d) x^3+(1-3 a d+b d) x^6}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (-a^2+2 a x^3-(1+b d) x^6+d x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{d \sqrt [3]{x^2 (-a+x) (-b+x)}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \left (\frac {2 a (1-b d) x^3}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a^2-2 a x^3+(1+b d) x^6-d x^9\right )}+\frac {(-1+3 a d-b d) x^6}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a^2-2 a x^3+(1+b d) x^6-d x^9\right )}+\frac {a^2}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (-a^2+2 a x^3-(1+b d) x^6+d x^9\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{d \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 ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-b+x^3} \sqrt [3]{1-\frac {x^3}{a}}} \, dx,x,\sqrt [3]{x}\right )}{d \sqrt [3]{x^2 (-a+x) (-b+x)}}\\ &=\frac {\left (3 a^2 x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (-a^2+2 a x^3-(1+b d) x^6+d x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{d \sqrt [3]{x^2 (-a+x) (-b+x)}}+\frac {\left (6 a (1-b d) x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a^2-2 a x^3+(1+b d) x^6-d x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{d \sqrt [3]{x^2 (-a+x) (-b+x)}}+\frac {\left (3 (-1+3 a d-b d) x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a^2-2 a x^3+(1+b d) x^6-d x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{d \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 ) \operatorname {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 )}{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 (3 a^2 x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (-a^2+2 a x^3-(1+b d) x^6+d x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{d \sqrt [3]{x^2 (-a+x) (-b+x)}}+\frac {\left (6 a (1-b d) x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a^2-2 a x^3+(1+b d) x^6-d x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{d \sqrt [3]{x^2 (-a+x) (-b+x)}}+\frac {\left (3 (-1+3 a d-b d) x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a^2-2 a x^3+(1+b d) x^6-d x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{d \sqrt [3]{x^2 (-a+x) (-b+x)}}\\ \end {align*}
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Mathematica [F] time = 4.24, 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+2 a x-(1+b d) x^2+d x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.37, size = 226, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}{-2 a+2 x+\sqrt [3]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}\right )}{d^{2/3}}+\frac {\log \left (a-x+\sqrt [3]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}\right )}{d^{2/3}}-\frac {\log \left (a^2-2 a x+x^2+\left (-a \sqrt [3]{d}+\sqrt [3]{d} x\right ) \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.
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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.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, a b x - 3 \, a x^{2} + x^{3}}{\left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {1}{3}} {\left (d x^{3} - {\left (b d + 1\right )} x^{2} - a^{2} + 2 \, a x\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {2 a b x -3 a \,x^{2}+x^{3}}{\left (x^{2} \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{3}} \left (-a^{2}+2 a x -\left (b d +1\right ) x^{2}+d \,x^{3}\right )}\, dx\]
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*} \int \frac {2 \, a b x - 3 \, a x^{2} + x^{3}}{\left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {1}{3}} {\left (d x^{3} - {\left (b d + 1\right )} x^{2} - a^{2} + 2 \, a x\right )}}\,{d x} \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 (-a^2+2\,a\,x+d\,x^3+\left (-b\,d-1\right )\,x^2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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