Optimal. Leaf size=289 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt [3]{x (-a-b)+a b+x^2} \left (a \sqrt [6]{d}-\sqrt [6]{d} x\right )}{a^2+\sqrt [3]{d} \left (x (-a-b)+a b+x^2\right )^{2/3}-2 a x+x^2}\right )}{2 d^{5/6}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{x (-a-b)+a b+x^2}}{\sqrt [6]{d} \sqrt [3]{x (-a-b)+a b+x^2}+2 a-2 x}\right )}{2 d^{5/6}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{x (-a-b)+a b+x^2}}{\sqrt [6]{d} \sqrt [3]{x (-a-b)+a b+x^2}-2 a+2 x}\right )}{2 d^{5/6}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [3]{x (-a-b)+a b+x^2}}{a-x}\right )}{d^{5/6}} \]
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Rubi [F] time = 6.62, 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-2 b+x) (-b+x)}{\sqrt [3]{(-a+x) (-b+x)} \left (a^4-b^2 d-2 \left (2 a^3-b d\right ) x+\left (6 a^2-d\right ) x^2-4 a 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-2 b+x) (-b+x)}{\sqrt [3]{(-a+x) (-b+x)} \left (a^4-b^2 d-2 \left (2 a^3-b d\right ) x+\left (6 a^2-d\right ) x^2-4 a x^3+x^4\right )} \, dx &=\frac {\left (\sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {(a-2 b+x) (-b+x)^{2/3}}{\sqrt [3]{-a+x} \left (a^4-b^2 d-2 \left (2 a^3-b d\right ) x+\left (6 a^2-d\right ) x^2-4 a x^3+x^4\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x)}}\\ &=\frac {\left (\sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \left (\frac {2 \left (1-\frac {a}{2 b}\right ) b (-b+x)^{2/3}}{\sqrt [3]{-a+x} \left (-a^4+b^2 d+2 \left (2 a^3-b d\right ) x-\left (6 a^2-d\right ) x^2+4 a x^3-x^4\right )}+\frac {x (-b+x)^{2/3}}{\sqrt [3]{-a+x} \left (a^4-b^2 d-2 \left (2 a^3-b d\right ) x+\left (6 a^2-d\right ) x^2-4 a x^3+x^4\right )}\right ) \, dx}{\sqrt [3]{(-a+x) (-b+x)}}\\ &=\frac {\left (\sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {x (-b+x)^{2/3}}{\sqrt [3]{-a+x} \left (a^4-b^2 d-2 \left (2 a^3-b d\right ) x+\left (6 a^2-d\right ) x^2-4 a x^3+x^4\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x)}}+\frac {\left ((-a+2 b) \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {(-b+x)^{2/3}}{\sqrt [3]{-a+x} \left (-a^4+b^2 d+2 \left (2 a^3-b d\right ) x-\left (6 a^2-d\right ) x^2+4 a x^3-x^4\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x)}}\\ &=\frac {\left (3 \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x \left (a+x^3\right ) \left (a-b+x^3\right )^{2/3}}{-a^2 d-b^2 d+2 b d x^3-d x^6+x^{12}+2 a d \left (b-x^3\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)}}+\frac {\left (3 (-a+2 b) \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x \left (a-b+x^3\right )^{2/3}}{a^2 d+b^2 d-2 b d x^3+2 a d \left (-b+x^3\right )+x^6 \left (d-x^6\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)}}\\ &=\frac {\left (3 \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x \left (-a-x^3\right ) \left (a-b+x^3\right )^{2/3}}{a^2 \left (1+\frac {b^2}{a^2}\right ) d-2 b d x^3+d x^6-x^{12}-2 a d \left (b-x^3\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)}}+\frac {\left (3 (-a+2 b) \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x \left (a-b+x^3\right )^{2/3}}{a^2 \left (1+\frac {b^2}{a^2}\right ) d-2 b d x^3+2 a d \left (-b+x^3\right )+x^6 \left (d-x^6\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)}}\\ &=\frac {\left (3 \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \left (\frac {a x \left (a-b+x^3\right )^{2/3}}{-a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) d-2 a \left (1-\frac {b}{a}\right ) d x^3-d x^6+x^{12}}+\frac {x^4 \left (a-b+x^3\right )^{2/3}}{-a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) d-2 a \left (1-\frac {b}{a}\right ) d x^3-d x^6+x^{12}}\right ) \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)}}+\frac {\left (3 (-a+2 b) \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x \left (a-b+x^3\right )^{2/3}}{a^2 \left (1+\frac {b^2}{a^2}\right ) d-2 b d x^3+2 a d \left (-b+x^3\right )+x^6 \left (d-x^6\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)}}\\ &=\frac {\left (3 \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^4 \left (a-b+x^3\right )^{2/3}}{-a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) d-2 a \left (1-\frac {b}{a}\right ) d x^3-d x^6+x^{12}} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)}}+\frac {\left (3 a \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x \left (a-b+x^3\right )^{2/3}}{-a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) d-2 a \left (1-\frac {b}{a}\right ) d x^3-d x^6+x^{12}} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)}}+\frac {\left (3 (-a+2 b) \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x \left (a-b+x^3\right )^{2/3}}{a^2 \left (1+\frac {b^2}{a^2}\right ) d-2 b d x^3+2 a d \left (-b+x^3\right )+x^6 \left (d-x^6\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)}}\\ \end {align*}
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Mathematica [F] time = 2.58, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a-2 b+x) (-b+x)}{\sqrt [3]{(-a+x) (-b+x)} \left (a^4-b^2 d-2 \left (2 a^3-b d\right ) x+\left (6 a^2-d\right ) x^2-4 a x^3+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.24, size = 289, normalized size = 1.00 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt [3]{x (-a-b)+a b+x^2} \left (a \sqrt [6]{d}-\sqrt [6]{d} x\right )}{a^2+\sqrt [3]{d} \left (x (-a-b)+a b+x^2\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 (-a-b)+a b+x^2}}{\sqrt [6]{d} \sqrt [3]{x (-a-b)+a b+x^2}+2 a-2 x}\right )}{2 d^{5/6}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{x (-a-b)+a b+x^2}}{\sqrt [6]{d} \sqrt [3]{x (-a-b)+a b+x^2}-2 a+2 x}\right )}{2 d^{5/6}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [3]{x (-a-b)+a b+x^2}}{a-x}\right )}{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 {{\left (a - 2 \, b + x\right )} {\left (b - x\right )}}{{\left (a^{4} - 4 \, a x^{3} + x^{4} - b^{2} d + {\left (6 \, a^{2} - d\right )} x^{2} - 2 \, {\left (2 \, a^{3} - b d\right )} x\right )} \left ({\left (a - x\right )} {\left (b - x\right )}\right )^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.55, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a -2 b +x \right ) \left (-b +x \right )}{\left (\left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{3}} \left (a^{4}-b^{2} d -2 \left (2 a^{3}-b d \right ) x +\left (6 a^{2}-d \right ) x^{2}-4 a \,x^{3}+x^{4}\right )}\, dx \end {gather*}
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 (a - 2 \, b + x\right )} {\left (b - x\right )}}{{\left (a^{4} - 4 \, a x^{3} + x^{4} - b^{2} d + {\left (6 \, a^{2} - d\right )} x^{2} - 2 \, {\left (2 \, a^{3} - b d\right )} x\right )} \left ({\left (a - x\right )} {\left (b - x\right )}\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 {\left (b-x\right )\,\left (a-2\,b+x\right )}{{\left (\left (a-x\right )\,\left (b-x\right )\right )}^{1/3}\,\left (x^2\,\left (d-6\,a^2\right )-2\,x\,\left (b\,d-2\,a^3\right )+b^2\,d+4\,a\,x^3-a^4-x^4\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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