Optimal. Leaf size=324 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt [3]{x^3 (-a-b)+a b x^2+x^4} \left (a \sqrt [6]{d}-\sqrt [6]{d} x\right )}{a^2+\sqrt [3]{d} \left (x^3 (-a-b)+a b x^2+x^4\right )^{2/3}-2 a x+x^2}\right )}{2 d^{5/6}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}}{\sqrt [6]{d} \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}+2 a-2 x}\right )}{2 d^{5/6}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}}{\sqrt [6]{d} \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}-2 a+2 x}\right )}{2 d^{5/6}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}}{a-x}\right )}{d^{5/6}} \]
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Rubi [F] time = 52.34, 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 {x^3 (-b+x) \left (2 a b-3 a x+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+\left (-1+b^2 d\right ) x^4-2 b d x^5+d x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {x^3 (-b+x) \left (2 a b-3 a x+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+\left (-1+b^2 d\right ) x^4-2 b d x^5+d x^6\right )} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {x^{7/3} (-b+x)^{2/3} \left (2 a b-3 a x+x^2\right )}{\sqrt [3]{-a+x} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+\left (-1+b^2 d\right ) x^4-2 b d x^5+d x^6\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^9 \left (-b+x^3\right )^{2/3} \left (2 a b-3 a x^3+x^6\right )}{\sqrt [3]{-a+x^3} \left (-a^4+4 a^3 x^3-6 a^2 x^6+4 a x^9+\left (-1+b^2 d\right ) x^{12}-2 b d x^{15}+d x^{18}\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 {3 a x^{12} \left (-b+x^3\right )^{2/3}}{\sqrt [3]{-a+x^3} \left (a^4-4 a^3 x^3+6 a^2 x^6-4 a x^9+\left (1-b^2 d\right ) x^{12}+2 b d x^{15}-d x^{18}\right )}+\frac {2 a b x^9 \left (-b+x^3\right )^{2/3}}{\sqrt [3]{-a+x^3} \left (-a^4+4 a^3 x^3-6 a^2 x^6+4 a x^9-\left (1-b^2 d\right ) x^{12}-2 b d x^{15}+d x^{18}\right )}+\frac {x^{15} \left (-b+x^3\right )^{2/3}}{\sqrt [3]{-a+x^3} \left (-a^4+4 a^3 x^3-6 a^2 x^6+4 a x^9-\left (1-b^2 d\right ) x^{12}-2 b d x^{15}+d x^{18}\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 {x^{15} \left (-b+x^3\right )^{2/3}}{\sqrt [3]{-a+x^3} \left (-a^4+4 a^3 x^3-6 a^2 x^6+4 a x^9-\left (1-b^2 d\right ) x^{12}-2 b d x^{15}+d x^{18}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)}}+\frac {\left (9 a x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^{12} \left (-b+x^3\right )^{2/3}}{\sqrt [3]{-a+x^3} \left (a^4-4 a^3 x^3+6 a^2 x^6-4 a x^9+\left (1-b^2 d\right ) x^{12}+2 b d x^{15}-d x^{18}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)}}+\frac {\left (6 a b x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^9 \left (-b+x^3\right )^{2/3}}{\sqrt [3]{-a+x^3} \left (-a^4+4 a^3 x^3-6 a^2 x^6+4 a x^9-\left (1-b^2 d\right ) x^{12}-2 b d x^{15}+d x^{18}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)}}\\ \end {align*}
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Mathematica [F] time = 3.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3 (-b+x) \left (2 a b-3 a x+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+\left (-1+b^2 d\right ) x^4-2 b d x^5+d x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 4.35, size = 324, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}{2 a-2 x+\sqrt [6]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}\right )}{2 d^{5/6}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}{-2 a+2 x+\sqrt [6]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}\right )}{2 d^{5/6}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}{a-x}\right )}{d^{5/6}}+\frac {\tanh ^{-1}\left (\frac {\left (a \sqrt [6]{d}-\sqrt [6]{d} x\right ) \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}{a^2-2 a x+x^2+\sqrt [3]{d} \left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}}\right )}{2 d^{5/6}} \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(-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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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{3} \left (-b +x \right ) \left (2 a b -3 a x +x^{2}\right )}{\left (x^{2} \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{3}} \left (-a^{4}+4 a^{3} x -6 a^{2} x^{2}+4 a \,x^{3}+\left (b^{2} d -1\right ) x^{4}-2 b d \,x^{5}+d \,x^{6}\right )}\, dx\]
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 {{\left (2 \, a b - 3 \, a x + x^{2}\right )} {\left (b - x\right )} x^{3}}{{\left (2 \, b d x^{5} - d x^{6} - {\left (b^{2} d - 1\right )} x^{4} + a^{4} - 4 \, a^{3} x + 6 \, a^{2} x^{2} - 4 \, a x^{3}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {1}{3}}}\,{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\,\left (b-x\right )\,\left (x^2-3\,a\,x+2\,a\,b\right )}{{\left (x^2\,\left (a-x\right )\,\left (b-x\right )\right )}^{1/3}\,\left (-a^4+4\,a^3\,x-6\,a^2\,x^2+4\,a\,x^3+d\,x^6-2\,b\,d\,x^5+\left (b^2\,d-1\right )\,x^4\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \left (- b + x\right ) \left (2 a b - 3 a x + x^{2}\right )}{\sqrt [3]{x^{2} \left (- a + x\right ) \left (- b + x\right )} \left (- a^{4} + 4 a^{3} x - 6 a^{2} x^{2} + 4 a x^{3} + b^{2} d x^{4} - 2 b d x^{5} + d x^{6} - x^{4}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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