Optimal. Leaf size=175 \[ \frac {\log \left (x-\sqrt [3]{d} \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}\right )}{d^{2/3}}-\frac {\log \left (d^{2/3} \left (x^3 (-a-b)+a b x^2+x^4\right )^{2/3}+\sqrt [3]{d} x \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}+x^2\right )}{2 d^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{d} \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}+x}\right )}{d^{2/3}} \]
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Rubi [F] time = 9.15, 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}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (a b d-(1+a d+b d) x+d x^2\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}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (a b d-(1+a d+b d) x+d x^2\right )} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {-a b+x^2}{x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (a b d-(1+a d+b d) x+d x^2\right )} \, dx}{\sqrt [3]{x^2 (-a+x) (-b+x)}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \left (\frac {1}{d x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}}-\frac {2 a b d-(1+a d+b d) x}{d x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (a b d+(-1-a d-b d) x+d x^2\right )}\right ) \, dx}{\sqrt [3]{x^2 (-a+x) (-b+x)}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}} \, dx}{d \sqrt [3]{x^2 (-a+x) (-b+x)}}-\frac {\left (x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {2 a b d-(1+a d+b d) x}{x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (a b d+(-1-a d-b d) x+d x^2\right )} \, dx}{d \sqrt [3]{x^2 (-a+x) (-b+x)}}\\ &=-\frac {\left (x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \left (\frac {-1-a d-b d-\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}}{x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (-1-a d-b d-\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}+2 d x\right )}+\frac {-1-a d-b d+\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}}{x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (-1-a d-b d+\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}+2 d x\right )}\right ) \, dx}{d \sqrt [3]{x^2 (-a+x) (-b+x)}}+\frac {\left (x^{2/3} \sqrt [3]{-b+x} \sqrt [3]{1-\frac {x}{a}}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-b+x} \sqrt [3]{1-\frac {x}{a}}} \, dx}{d \sqrt [3]{x^2 (-a+x) (-b+x)}}\\ &=-\frac {\left (\left (-1-a d-b d-\sqrt {a^2 d^2+2 a d (1-b d)+(1+b d)^2}\right ) x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (-1-a d-b d-\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt [3]{x^2 (-a+x) (-b+x)}}-\frac {\left (\left (-1-a d-b d+\sqrt {a^2 d^2+2 a d (1-b d)+(1+b d)^2}\right ) x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (-1-a d-b d+\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt [3]{x^2 (-a+x) (-b+x)}}+\frac {\left (x^{2/3} \sqrt [3]{1-\frac {x}{a}} \sqrt [3]{1-\frac {x}{b}}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{1-\frac {x}{a}} \sqrt [3]{1-\frac {x}{b}}} \, dx}{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 (\left (-1-a d-b d-\sqrt {a^2 d^2+2 a d (1-b d)+(1+b d)^2}\right ) x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (-1-a d-b d-\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt [3]{x^2 (-a+x) (-b+x)}}-\frac {\left (\left (-1-a d-b d+\sqrt {a^2 d^2+2 a d (1-b d)+(1+b d)^2}\right ) x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (-1-a d-b d+\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt [3]{x^2 (-a+x) (-b+x)}}\\ \end {align*}
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Mathematica [F] time = 14.41, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-a b+x^2}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (a b d-(1+a d+b d) x+d x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.42, size = 175, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}\right )}{d^{2/3}}+\frac {\log \left (x-\sqrt [3]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}\right )}{d^{2/3}}-\frac {\log \left (x^2+\sqrt [3]{d} x \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 {a b - x^{2}}{\left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {1}{3}} {\left (a b d + d x^{2} - {\left (a d + b d + 1\right )} x\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {-a b +x^{2}}{\left (x^{2} \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{3}} \left (a b d -\left (a d +b d +1\right ) x +d \,x^{2}\right )}\, dx\]
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}}{\left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {1}{3}} {\left (a b d + d x^{2} - {\left (a d + b d + 1\right )} 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 = -0.01 \begin {gather*} \int -\frac {a\,b-x^2}{\left (d\,x^2+\left (-a\,d-b\,d-1\right )\,x+a\,b\,d\right )\,{\left (x^2\,\left (a-x\right )\,\left (b-x\right )\right )}^{1/3}} \,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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