Optimal. Leaf size=81 \[ -\frac {2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{d} \sqrt [4]{x^3 (-a-b)+a b x^2+x^4}}\right )}{d^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{d} \sqrt [4]{x^3 (-a-b)+a b x^2+x^4}}\right )}{d^{3/4}} \]
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Rubi [F] time = 5.91, 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 {-2 a b+(a+b) x}{\sqrt [4]{x^2 (-a+x) (-b+x)} \left (a b d-(a+b) d x+(-1+d) x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {-2 a b+(a+b) x}{\sqrt [4]{x^2 (-a+x) (-b+x)} \left (a b d-(a+b) d x+(-1+d) x^2\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \int \frac {-2 a b+(a+b) x}{\sqrt {x} \sqrt [4]{-a+x} \sqrt [4]{-b+x} \left (a b d-(a+b) d x+(-1+d) x^2\right )} \, dx}{\sqrt [4]{x^2 (-a+x) (-b+x)}}\\ &=\frac {\left (\sqrt {x} \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \int \left (\frac {a+b+\frac {\sqrt {4 a b+a^2 d-2 a b d+b^2 d}}{\sqrt {d}}}{\sqrt {x} \sqrt [4]{-a+x} \sqrt [4]{-b+x} \left (-((a+b) d)-\sqrt {d} \sqrt {4 a b+a^2 d-2 a b d+b^2 d}+2 (-1+d) x\right )}+\frac {a+b-\frac {\sqrt {4 a b+a^2 d-2 a b d+b^2 d}}{\sqrt {d}}}{\sqrt {x} \sqrt [4]{-a+x} \sqrt [4]{-b+x} \left (-((a+b) d)+\sqrt {d} \sqrt {4 a b+a^2 d-2 a b d+b^2 d}+2 (-1+d) x\right )}\right ) \, dx}{\sqrt [4]{x^2 (-a+x) (-b+x)}}\\ &=\frac {\left (\left (a+b-\frac {\sqrt {2 a b (2-d)+a^2 d+b^2 d}}{\sqrt {d}}\right ) \sqrt {x} \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \int \frac {1}{\sqrt {x} \sqrt [4]{-a+x} \sqrt [4]{-b+x} \left (-((a+b) d)+\sqrt {d} \sqrt {4 a b+a^2 d-2 a b d+b^2 d}+2 (-1+d) x\right )} \, dx}{\sqrt [4]{x^2 (-a+x) (-b+x)}}+\frac {\left (\left (a+b+\frac {\sqrt {2 a b (2-d)+a^2 d+b^2 d}}{\sqrt {d}}\right ) \sqrt {x} \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \int \frac {1}{\sqrt {x} \sqrt [4]{-a+x} \sqrt [4]{-b+x} \left (-((a+b) d)-\sqrt {d} \sqrt {4 a b+a^2 d-2 a b d+b^2 d}+2 (-1+d) x\right )} \, dx}{\sqrt [4]{x^2 (-a+x) (-b+x)}}\\ \end {align*}
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Mathematica [F] time = 8.40, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-2 a b+(a+b) x}{\sqrt [4]{x^2 (-a+x) (-b+x)} \left (a b d-(a+b) d x+(-1+d) x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.37, size = 81, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{d} \sqrt [4]{a b x^2+(-a-b) x^3+x^4}}\right )}{d^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{d} \sqrt [4]{a b x^2+(-a-b) x^3+x^4}}\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 {2 \, a b - {\left (a + b\right )} x}{\left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {1}{4}} {\left (a b d - {\left (a + b\right )} d x + {\left (d - 1\right )} x^{2}\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {-2 a b +\left (a +b \right ) x}{\left (x^{2} \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{4}} \left (a b d -\left (a +b \right ) d x +\left (-1+d \right ) x^{2}\right )}\, dx\]
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 {2 \, a b - {\left (a + b\right )} x}{\left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {1}{4}} {\left (a b d - {\left (a + b\right )} d x + {\left (d - 1\right )} x^{2}\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 {2\,a\,b-x\,\left (a+b\right )}{\left (\left (d-1\right )\,x^2-d\,\left (a+b\right )\,x+a\,b\,d\right )\,{\left (x^2\,\left (a-x\right )\,\left (b-x\right )\right )}^{1/4}} \,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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