Optimal. Leaf size=315 \[ -\frac {\log \left (a^2 b^2-2 a^2 b x+a^2 x^2-2 a b^2 x+d^{2/3} \left (x^3 (-a-b)+a b x^2+x^4\right )^{4/3}+\left (x^3 (-a-b)+a b x^2+x^4\right )^{2/3} \left (a b \sqrt [3]{d}-a \sqrt [3]{d} x-b \sqrt [3]{d} x+\sqrt [3]{d} x^2\right )+4 a b x^2-2 a x^3+b^2 x^2-2 b x^3+x^4\right )}{2 d^{2/3}}+\frac {\log \left (-\sqrt [3]{d} \left (x^3 (-a-b)+a b x^2+x^4\right )^{2/3}+a b-a x-b x+x^2\right )}{d^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \left (x^3 (-a-b)+a b x^2+x^4\right )^{2/3}}{\sqrt [3]{d} \left (x^3 (-a-b)+a b x^2+x^4\right )^{2/3}+2 a b-2 a x-2 b x+2 x^2}\right )}{d^{2/3}} \]
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Rubi [F] time = 19.10, 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 \left (4 a b-3 (a+b) x+2 x^2\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \left (-a b+(a+b) x-x^2+d x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {x^3 \left (4 a b-3 (a+b) x+2 x^2\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \left (-a b+(a+b) x-x^2+d x^4\right )} \, dx &=\frac {\left (x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {x^{5/3} \left (4 a b-3 (a+b) x+2 x^2\right )}{(-a+x)^{2/3} (-b+x)^{2/3} \left (-a b+(a+b) x-x^2+d x^4\right )} \, dx}{\left (x^2 (-a+x) (-b+x)\right )^{2/3}}\\ &=\frac {\left (3 x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^7 \left (4 a b-3 (a+b) x^3+2 x^6\right )}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (-a b+(a+b) x^3-x^6+d x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3}}\\ &=\frac {\left (3 x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \left (\frac {2 x}{d \left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3}}+\frac {x \left (2 a b-2 (a+b) x^3+2 (1+2 a b d) x^6-3 (a+b) d x^9\right )}{d \left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (-a b+(a+b) x^3-x^6+d x^{12}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3}}\\ &=\frac {\left (3 x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x \left (2 a b-2 (a+b) x^3+2 (1+2 a b d) x^6-3 (a+b) d x^9\right )}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (-a b+(a+b) x^3-x^6+d x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{d \left (x^2 (-a+x) (-b+x)\right )^{2/3}}+\frac {\left (6 x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{d \left (x^2 (-a+x) (-b+x)\right )^{2/3}}\\ &=\frac {\left (3 x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \left (\frac {2 (a+b) x^4}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (a b-a \left (1+\frac {b}{a}\right ) x^3+x^6-d x^{12}\right )}+\frac {2 (-1-2 a b d) x^7}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (a b-a \left (1+\frac {b}{a}\right ) x^3+x^6-d x^{12}\right )}+\frac {3 (a+b) d x^{10}}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (a b-a \left (1+\frac {b}{a}\right ) x^3+x^6-d x^{12}\right )}+\frac {2 a b x}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (-a b+a \left (1+\frac {b}{a}\right ) x^3-x^6+d x^{12}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{d \left (x^2 (-a+x) (-b+x)\right )^{2/3}}+\frac {\left (6 x^{4/3} (-b+x)^{2/3} \left (1-\frac {x}{a}\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (-b+x^3\right )^{2/3} \left (1-\frac {x^3}{a}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{d \left (x^2 (-a+x) (-b+x)\right )^{2/3}}\\ &=\frac {\left (9 (a+b) x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (a b-a \left (1+\frac {b}{a}\right ) x^3+x^6-d x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3}}+\frac {\left (6 a b x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (-a b+a \left (1+\frac {b}{a}\right ) x^3-x^6+d x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{d \left (x^2 (-a+x) (-b+x)\right )^{2/3}}+\frac {\left (6 (a+b) x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (a b-a \left (1+\frac {b}{a}\right ) x^3+x^6-d x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{d \left (x^2 (-a+x) (-b+x)\right )^{2/3}}-\frac {\left (6 (1+2 a b d) x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^7}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (a b-a \left (1+\frac {b}{a}\right ) x^3+x^6-d x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{d \left (x^2 (-a+x) (-b+x)\right )^{2/3}}+\frac {\left (6 x^{4/3} \left (1-\frac {x}{a}\right )^{2/3} \left (1-\frac {x}{b}\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1-\frac {x^3}{a}\right )^{2/3} \left (1-\frac {x^3}{b}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{d \left (x^2 (-a+x) (-b+x)\right )^{2/3}}\\ &=\frac {3 x^2 \left (1-\frac {x}{a}\right )^{2/3} \left (1-\frac {x}{b}\right )^{2/3} F_1\left (\frac {2}{3};\frac {2}{3},\frac {2}{3};\frac {5}{3};\frac {x}{a},\frac {x}{b}\right )}{d \left ((a-x) (b-x) x^2\right )^{2/3}}+\frac {\left (9 (a+b) x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (a b-a \left (1+\frac {b}{a}\right ) x^3+x^6-d x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3}}+\frac {\left (6 a b x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (-a b+a \left (1+\frac {b}{a}\right ) x^3-x^6+d x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{d \left (x^2 (-a+x) (-b+x)\right )^{2/3}}+\frac {\left (6 (a+b) x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (a b-a \left (1+\frac {b}{a}\right ) x^3+x^6-d x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{d \left (x^2 (-a+x) (-b+x)\right )^{2/3}}-\frac {\left (6 (1+2 a b d) x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^7}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (a b-a \left (1+\frac {b}{a}\right ) x^3+x^6-d x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{d \left (x^2 (-a+x) (-b+x)\right )^{2/3}}\\ \end {align*}
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Mathematica [F] time = 3.38, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3 \left (4 a b-3 (a+b) x+2 x^2\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \left (-a b+(a+b) x-x^2+d x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.89, size = 315, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}}{2 a b-2 a x-2 b x+2 x^2+\sqrt [3]{d} \left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}}\right )}{d^{2/3}}+\frac {\log \left (a b-a x-b x+x^2-\sqrt [3]{d} \left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}\right )}{d^{2/3}}-\frac {\log \left (a^2 b^2-2 a^2 b x-2 a b^2 x+a^2 x^2+4 a b x^2+b^2 x^2-2 a x^3-2 b x^3+x^4+\left (a b \sqrt [3]{d}-a \sqrt [3]{d} x-b \sqrt [3]{d} x+\sqrt [3]{d} x^2\right ) \left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}+d^{2/3} \left (a b x^2+(-a-b) x^3+x^4\right )^{4/3}\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 {{\left (4 \, a b - 3 \, {\left (a + b\right )} x + 2 \, x^{2}\right )} x^{3}}{{\left (d x^{4} - a b + {\left (a + b\right )} x - x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {2}{3}}}\,{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 \[\int \frac {x^{3} \left (4 a b -3 \left (a +b \right ) x +2 x^{2}\right )}{\left (x^{2} \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {2}{3}} \left (-a b +\left (a +b \right ) x -x^{2}+d \,x^{4}\right )}\, dx\]
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*} \int \frac {{\left (4 \, a b - 3 \, {\left (a + b\right )} x + 2 \, x^{2}\right )} x^{3}}{{\left (d x^{4} - a b + {\left (a + b\right )} x - x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {2}{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 (4\,a\,b+2\,x^2-3\,x\,\left (a+b\right )\right )}{{\left (x^2\,\left (a-x\right )\,\left (b-x\right )\right )}^{2/3}\,\left (-d\,x^4+x^2+\left (-a-b\right )\,x+a\,b\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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