Optimal. Leaf size=83 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{x (-a-b)+a b+x^2}}{a-x}\right )}{d^{3/4}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{x (-a-b)+a b+x^2}}{a-x}\right )}{d^{3/4}} \]
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Rubi [F] time = 6.76, 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-3 b+2 x}{\sqrt [4]{(-a+x) (-b+x)} \left (-a^3+b d-\left (-3 a^2+d\right ) x-3 a x^2+x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {a-3 b+2 x}{\sqrt [4]{(-a+x) (-b+x)} \left (-a^3+b d-\left (-3 a^2+d\right ) x-3 a x^2+x^3\right )} \, dx &=\frac {\left (\sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \int \frac {a-3 b+2 x}{\sqrt [4]{-a+x} \sqrt [4]{-b+x} \left (-a^3+b d-\left (-3 a^2+d\right ) x-3 a x^2+x^3\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)}}\\ &=\frac {\left (\sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \int \frac {-a+3 b-2 x}{\sqrt [4]{-a+x} \sqrt [4]{-b+x} \left (a^3-b d-\left (3 a^2-d\right ) x+3 a x^2-x^3\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)}}\\ &=\frac {\left (\sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \int \left (\frac {3 \left (1-\frac {a}{3 b}\right ) b}{\sqrt [4]{-a+x} \sqrt [4]{-b+x} \left (a^3-b d-\left (3 a^2-d\right ) x+3 a x^2-x^3\right )}+\frac {2 x}{\sqrt [4]{-a+x} \sqrt [4]{-b+x} \left (-a^3+b d+\left (3 a^2-d\right ) x-3 a x^2+x^3\right )}\right ) \, dx}{\sqrt [4]{(-a+x) (-b+x)}}\\ &=\frac {\left (2 \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \int \frac {x}{\sqrt [4]{-a+x} \sqrt [4]{-b+x} \left (-a^3+b d+\left (3 a^2-d\right ) x-3 a x^2+x^3\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)}}+\frac {\left ((-a+3 b) \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \int \frac {1}{\sqrt [4]{-a+x} \sqrt [4]{-b+x} \left (a^3-b d-\left (3 a^2-d\right ) x+3 a x^2-x^3\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)}}\\ &=\frac {\left (8 \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (a+x^4\right )}{\sqrt [4]{a-b+x^4} \left (-a d+b d-d x^4+x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}+\frac {\left (4 (-a+3 b) \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{a-b+x^4} \left (a d-b d+x^4 \left (d-x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}\\ &=\frac {\left (8 \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-a-x^4\right )}{\sqrt [4]{a-b+x^4} \left (a \left (1-\frac {b}{a}\right ) d+d x^4-x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}+\frac {\left (4 (-a+3 b) \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{a-b+x^4} \left (a \left (1-\frac {b}{a}\right ) d+x^4 \left (d-x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}\\ &=\frac {\left (8 \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \left (\frac {a x^2}{\sqrt [4]{a-b+x^4} \left (-a \left (1-\frac {b}{a}\right ) d-d x^4+x^{12}\right )}+\frac {x^6}{\sqrt [4]{a-b+x^4} \left (-a \left (1-\frac {b}{a}\right ) d-d x^4+x^{12}\right )}\right ) \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}+\frac {\left (4 (-a+3 b) \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{a-b+x^4} \left (a \left (1-\frac {b}{a}\right ) d+x^4 \left (d-x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}\\ &=\frac {\left (8 \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\sqrt [4]{a-b+x^4} \left (-a \left (1-\frac {b}{a}\right ) d-d x^4+x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}+\frac {\left (8 a \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{a-b+x^4} \left (-a \left (1-\frac {b}{a}\right ) d-d x^4+x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}+\frac {\left (4 (-a+3 b) \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{a-b+x^4} \left (a \left (1-\frac {b}{a}\right ) d+x^4 \left (d-x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)}}\\ \end {align*}
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Mathematica [F] time = 1.84, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a-3 b+2 x}{\sqrt [4]{(-a+x) (-b+x)} \left (-a^3+b d-\left (-3 a^2+d\right ) x-3 a x^2+x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.36, size = 83, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a b+(-a-b) x+x^2}}{a-x}\right )}{d^{3/4}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a b+(-a-b) x+x^2}}{a-x}\right )}{d^{3/4}} \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 - 3 \, b + 2 \, x}{{\left (a^{3} + 3 \, a x^{2} - x^{3} - b d - {\left (3 \, a^{2} - d\right )} x\right )} \left ({\left (a - x\right )} {\left (b - x\right )}\right )^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {a -3 b +2 x}{\left (\left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{4}} \left (-a^{3}+b d -\left (-3 a^{2}+d \right ) x -3 a \,x^{2}+x^{3}\right )}\, dx\]
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 - 3 \, b + 2 \, x}{{\left (a^{3} + 3 \, a x^{2} - x^{3} - b d - {\left (3 \, a^{2} - d\right )} x\right )} \left ({\left (a - x\right )} {\left (b - x\right )}\right )^{\frac {1}{4}}}\,{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-3\,b+2\,x}{{\left (\left (a-x\right )\,\left (b-x\right )\right )}^{1/4}\,\left (3\,a\,x^2-b\,d+x\,\left (d-3\,a^2\right )+a^3-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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