Optimal. Leaf size=405 \[ \frac {\log \left (a^2 \sqrt [6]{d}-a \sqrt [3]{x (-a-b)+a b+x^2}-a \sqrt [6]{d} x\right )}{2 d^{2/3}}+\frac {\log \left (a^2 \sqrt [6]{d}+a \sqrt [3]{x (-a-b)+a b+x^2}-a \sqrt [6]{d} x\right )}{2 d^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {a^2}{\sqrt {3}}+\frac {2 \left (x (-a-b)+a b+x^2\right )^{2/3}}{\sqrt {3} \sqrt [3]{d}}-\frac {2 a x}{\sqrt {3}}+\frac {x^2}{\sqrt {3}}}{(a-x)^2}\right )}{2 d^{2/3}}-\frac {\log \left (a^4 \sqrt [3]{d}-2 a^3 \sqrt [3]{d} x+a^2 \left (x (-a-b)+a b+x^2\right )^{2/3}+a^2 \sqrt [3]{d} x^2+\sqrt [3]{x (-a-b)+a b+x^2} \left (a^3 \sqrt [6]{d}-a^2 \sqrt [6]{d} x\right )\right )}{4 d^{2/3}}-\frac {\log \left (a^4 \sqrt [3]{d}-2 a^3 \sqrt [3]{d} x+a^2 \left (x (-a-b)+a b+x^2\right )^{2/3}+a^2 \sqrt [3]{d} x^2+\sqrt [3]{x (-a-b)+a b+x^2} \left (a^2 \sqrt [6]{d} x-a^3 \sqrt [6]{d}\right )\right )}{4 d^{2/3}} \]
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Rubi [F] time = 9.11, 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) \left (a^2-2 a x+x^2\right )}{\sqrt [3]{(-a+x) (-b+x)} \left (-b^2+a^4 d+2 \left (b-2 a^3 d\right ) x+\left (-1+6 a^2 d\right ) x^2-4 a d x^3+d x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {(a-2 b+x) \left (a^2-2 a x+x^2\right )}{\sqrt [3]{(-a+x) (-b+x)} \left (-b^2+a^4 d+2 \left (b-2 a^3 d\right ) x+\left (-1+6 a^2 d\right ) x^2-4 a d x^3+d x^4\right )} \, dx &=\int \frac {(-a+x)^2 (a-2 b+x)}{\sqrt [3]{(-a+x) (-b+x)} \left (-b^2+a^4 d+2 \left (b-2 a^3 d\right ) x+\left (-1+6 a^2 d\right ) x^2-4 a d x^3+d x^4\right )} \, dx\\ &=\frac {\left (\sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {(-a+x)^{5/3} (a-2 b+x)}{\sqrt [3]{-b+x} \left (-b^2+a^4 d+2 \left (b-2 a^3 d\right ) x+\left (-1+6 a^2 d\right ) x^2-4 a d x^3+d x^4\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x)}}\\ &=\frac {\left (\sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {(-a+2 b-x) (-a+x)^{5/3}}{\sqrt [3]{-b+x} \left (b^2-a^4 d-2 \left (b-2 a^3 d\right ) x+\left (1-6 a^2 d\right ) x^2+4 a d x^3-d 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 (-a+x)^{5/3}}{\sqrt [3]{-b+x} \left (b^2-a^4 d-2 \left (b-2 a^3 d\right ) x+\left (1-6 a^2 d\right ) x^2+4 a d x^3-d x^4\right )}+\frac {x (-a+x)^{5/3}}{\sqrt [3]{-b+x} \left (-b^2+a^4 d+2 \left (b-2 a^3 d\right ) x-\left (1-6 a^2 d\right ) x^2-4 a d x^3+d 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 (-a+x)^{5/3}}{\sqrt [3]{-b+x} \left (-b^2+a^4 d+2 \left (b-2 a^3 d\right ) x-\left (1-6 a^2 d\right ) x^2-4 a d x^3+d 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 {(-a+x)^{5/3}}{\sqrt [3]{-b+x} \left (b^2-a^4 d-2 \left (b-2 a^3 d\right ) x+\left (1-6 a^2 d\right ) x^2+4 a d x^3-d 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^7 \left (a+x^3\right )}{\sqrt [3]{a-b+x^3} \left (a^2-2 a b+b^2+2 a x^3-2 b x^3+x^6-d 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^7}{\sqrt [3]{a-b+x^3} \left (a^2-2 a b+b^2+2 a x^3-2 b x^3+x^6-d x^{12}\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^7 \left (a+x^3\right )}{\sqrt [3]{a-b+x^3} \left (a^2-2 a b+b^2+(2 a-2 b) x^3+x^6-d 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^7}{\sqrt [3]{a-b+x^3} \left (a^2-2 a b+b^2+(2 a-2 b) x^3+x^6-d x^{12}\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^7 \left (a+x^3\right )}{\sqrt [3]{a-b+x^3} \left (a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right )+(2 a-2 b) x^3+x^6-d 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^7}{\sqrt [3]{a-b+x^3} \left (a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right )+(2 a-2 b) x^3+x^6-d x^{12}\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^7}{\sqrt [3]{a-b+x^3} \left (a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right )+2 a \left (1-\frac {b}{a}\right ) x^3+x^6-d x^{12}\right )}+\frac {x^{10}}{\sqrt [3]{a-b+x^3} \left (a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right )+2 a \left (1-\frac {b}{a}\right ) x^3+x^6-d x^{12}\right )}\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^7}{\sqrt [3]{a-b+x^3} \left (a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right )+(2 a-2 b) x^3+x^6-d x^{12}\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^{10}}{\sqrt [3]{a-b+x^3} \left (a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right )+2 a \left (1-\frac {b}{a}\right ) x^3+x^6-d x^{12}\right )} \, 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^7}{\sqrt [3]{a-b+x^3} \left (a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right )+2 a \left (1-\frac {b}{a}\right ) x^3+x^6-d 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^7}{\sqrt [3]{a-b+x^3} \left (a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right )+(2 a-2 b) x^3+x^6-d x^{12}\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)}}\\ \end {align*}
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Mathematica [F] time = 4.63, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a-2 b+x) \left (a^2-2 a x+x^2\right )}{\sqrt [3]{(-a+x) (-b+x)} \left (-b^2+a^4 d+2 \left (b-2 a^3 d\right ) x+\left (-1+6 a^2 d\right ) x^2-4 a d x^3+d x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 3.54, size = 405, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {a^2}{\sqrt {3}}-\frac {2 a x}{\sqrt {3}}+\frac {x^2}{\sqrt {3}}+\frac {2 \left (a b+(-a-b) x+x^2\right )^{2/3}}{\sqrt {3} \sqrt [3]{d}}}{(a-x)^2}\right )}{2 d^{2/3}}+\frac {\log \left (a^2 \sqrt [6]{d}-a \sqrt [6]{d} x-a \sqrt [3]{a b+(-a-b) x+x^2}\right )}{2 d^{2/3}}+\frac {\log \left (a^2 \sqrt [6]{d}-a \sqrt [6]{d} x+a \sqrt [3]{a b+(-a-b) x+x^2}\right )}{2 d^{2/3}}-\frac {\log \left (a^4 \sqrt [3]{d}-2 a^3 \sqrt [3]{d} x+a^2 \sqrt [3]{d} x^2+\left (a^3 \sqrt [6]{d}-a^2 \sqrt [6]{d} x\right ) \sqrt [3]{a b+(-a-b) x+x^2}+a^2 \left (a b+(-a-b) x+x^2\right )^{2/3}\right )}{4 d^{2/3}}-\frac {\log \left (a^4 \sqrt [3]{d}-2 a^3 \sqrt [3]{d} x+a^2 \sqrt [3]{d} x^2+\left (-a^3 \sqrt [6]{d}+a^2 \sqrt [6]{d} x\right ) \sqrt [3]{a b+(-a-b) x+x^2}+a^2 \left (a b+(-a-b) x+x^2\right )^{2/3}\right )}{4 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 (a^{2} - 2 \, a x + x^{2}\right )} {\left (a - 2 \, b + x\right )}}{{\left (a^{4} d - 4 \, a d x^{3} + d x^{4} + {\left (6 \, a^{2} d - 1\right )} x^{2} - b^{2} - 2 \, {\left (2 \, a^{3} d - b\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 = 0.14, size = 0, normalized size = 0.00 \[\int \frac {\left (a -2 b +x \right ) \left (a^{2}-2 a x +x^{2}\right )}{\left (\left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{3}} \left (-b^{2}+a^{4} d +2 \left (-2 a^{3} d +b \right ) x +\left (6 a^{2} d -1\right ) x^{2}-4 a d \,x^{3}+d \,x^{4}\right )}\, dx\]
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 (a^{2} - 2 \, a x + x^{2}\right )} {\left (a - 2 \, b + x\right )}}{{\left (a^{4} d - 4 \, a d x^{3} + d x^{4} + {\left (6 \, a^{2} d - 1\right )} x^{2} - b^{2} - 2 \, {\left (2 \, a^{3} d - b\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 (a^2-2\,a\,x+x^2\right )\,\left (a-2\,b+x\right )}{{\left (\left (a-x\right )\,\left (b-x\right )\right )}^{1/3}\,\left (x^2\,\left (6\,a^2\,d-1\right )+2\,x\,\left (b-2\,a^3\,d\right )+a^4\,d+d\,x^4-b^2-4\,a\,d\,x^3\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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