Optimal. Leaf size=249 \[ -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} x}{\sqrt [6]{d} x-2 \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}}\right )}{2 d^{5/6}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} x}{2 \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}+\sqrt [6]{d} x}\right )}{2 d^{5/6}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{d} x}{\sqrt [3]{x^3 (-a-b)+a b x^2+x^4}}\right )}{d^{5/6}}-\frac {\tanh ^{-1}\left (\frac {\frac {\left (x^3 (-a-b)+a b x^2+x^4\right )^{2/3}}{\sqrt [6]{d}}+\sqrt [6]{d} x^2}{x \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}}\right )}{2 d^{5/6}} \]
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Rubi [F] time = 19.26, 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 {-a b x^2+x^4}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {-a b x^2+x^4}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx &=\int \frac {x^2 \left (-a b+x^2\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx\\ &=\frac {\left (x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {x^{2/3} \left (-a b+x^2\right )}{(-a+x)^{2/3} (-b+x)^{2/3} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+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^4 \left (-a b+x^6\right )}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (a^2 b^2-2 a b (a+b) x^3+\left (a^2+4 a b+b^2-d\right ) x^6-2 (a+b) x^9+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 {a b x^4}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (-a^2 b^2+2 a^2 b \left (1+\frac {b}{a}\right ) x^3-a^2 \left (1+\frac {4 a b+b^2-d}{a^2}\right ) x^6+2 a \left (1+\frac {b}{a}\right ) x^9-x^{12}\right )}+\frac {x^{10}}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (a^2 b^2-2 a^2 b \left (1+\frac {b}{a}\right ) x^3+a^2 \left (1+\frac {4 a b+b^2-d}{a^2}\right ) x^6-2 a \left (1+\frac {b}{a}\right ) x^9+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^{10}}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \left (a^2 b^2-2 a^2 b \left (1+\frac {b}{a}\right ) x^3+a^2 \left (1+\frac {4 a b+b^2-d}{a^2}\right ) x^6-2 a \left (1+\frac {b}{a}\right ) x^9+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3}}+\frac {\left (3 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^2 b^2+2 a^2 b \left (1+\frac {b}{a}\right ) x^3-a^2 \left (1+\frac {4 a b+b^2-d}{a^2}\right ) x^6+2 a \left (1+\frac {b}{a}\right ) x^9-x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3}}\\ \end {align*}
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Mathematica [F] time = 3.25, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-a b x^2+x^4}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.42, size = 249, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} x}{\sqrt [6]{d} x-2 \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} x}{\sqrt [6]{d} x+2 \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} x}{\sqrt [3]{a b x^2+(-a-b) x^3+x^4}}\right )}{d^{5/6}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{d} x^2+\frac {\left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}}{\sqrt [6]{d}}}{x \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}\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] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {a b x^{2} - x^{4}}{{\left (a^{2} b^{2} - 2 \, {\left (a + b\right )} a b x - 2 \, {\left (a + b\right )} x^{3} + x^{4} + {\left (a^{2} + 4 \, a b + b^{2} - d\right )} 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 \[\int \frac {-a b \,x^{2}+x^{4}}{\left (x^{2} \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {2}{3}} \left (a^{2} b^{2}-2 a b \left (a +b \right ) x +\left (a^{2}+4 a b +b^{2}-d \right ) x^{2}-2 \left (a +b \right ) x^{3}+x^{4}\right )}\, dx\]
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 {a b x^{2} - x^{4}}{{\left (a^{2} b^{2} - 2 \, {\left (a + b\right )} a b x - 2 \, {\left (a + b\right )} x^{3} + x^{4} + {\left (a^{2} + 4 \, a b + b^{2} - d\right )} 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^4-a\,b\,x^2}{{\left (x^2\,\left (a-x\right )\,\left (b-x\right )\right )}^{2/3}\,\left (x^4-2\,x^3\,\left (a+b\right )+a^2\,b^2+x^2\,\left (a^2+4\,a\,b+b^2-d\right )-2\,a\,b\,x\,\left (a+b\right )\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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