Optimal. Leaf size=119 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}}{b-x}\right )}{d^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}}{b-x}\right )}{d^{3/4}} \]
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Rubi [F] time = 3.91, 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 {-a b+(-a+2 b) x}{\sqrt [4]{x (-a+x) (-b+x)^2} \left (-b^2+(2 b-a 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 {-a b+(-a+2 b) x}{\sqrt [4]{x (-a+x) (-b+x)^2} \left (-b^2+(2 b-a d) x+(-1+d) x^2\right )} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \frac {-a b+(-a+2 b) x}{\sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x} \left (-b^2+(2 b-a d) x+(-1+d) x^2\right )} \, dx}{\sqrt [4]{x (-a+x) (-b+x)^2}}\\ &=\frac {\left (\sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \left (\frac {-a+2 b-\frac {\sqrt {-4 a b+4 b^2+a^2 d}}{\sqrt {d}}}{\sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x} \left (2 b-a d-\sqrt {d} \sqrt {-4 a b+4 b^2+a^2 d}+2 (-1+d) x\right )}+\frac {-a+2 b+\frac {\sqrt {-4 a b+4 b^2+a^2 d}}{\sqrt {d}}}{\sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x} \left (2 b-a d+\sqrt {d} \sqrt {-4 a b+4 b^2+a^2 d}+2 (-1+d) x\right )}\right ) \, dx}{\sqrt [4]{x (-a+x) (-b+x)^2}}\\ &=\frac {\left (\left (-a+2 b-\frac {\sqrt {-4 a b+4 b^2+a^2 d}}{\sqrt {d}}\right ) \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \frac {1}{\sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x} \left (2 b-a d-\sqrt {d} \sqrt {-4 a b+4 b^2+a^2 d}+2 (-1+d) x\right )} \, dx}{\sqrt [4]{x (-a+x) (-b+x)^2}}+\frac {\left (\left (-a+2 b+\frac {\sqrt {-4 a b+4 b^2+a^2 d}}{\sqrt {d}}\right ) \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \frac {1}{\sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x} \left (2 b-a d+\sqrt {d} \sqrt {-4 a b+4 b^2+a^2 d}+2 (-1+d) x\right )} \, dx}{\sqrt [4]{x (-a+x) (-b+x)^2}}\\ \end {align*}
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Mathematica [F] time = 6.98, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-a b+(-a+2 b) x}{\sqrt [4]{x (-a+x) (-b+x)^2} \left (-b^2+(2 b-a d) x+(-1+d) x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.39, size = 119, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}}{b-x}\right )}{d^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}}{b-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 b + {\left (a - 2 \, b\right )} x}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {1}{4}} {\left ({\left (d - 1\right )} x^{2} - b^{2} - {\left (a d - 2 \, b\right )} x\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.65, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-a b +\left (-a +2 b \right ) x}{\left (x \left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{4}} \left (-b^{2}+\left (-a d +2 b \right ) x +\left (-1+d \right ) x^{2}\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 {a b + {\left (a - 2 \, b\right )} x}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {1}{4}} {\left ({\left (d - 1\right )} x^{2} - b^{2} - {\left (a d - 2 \, b\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\,\left (a-2\,b\right )}{{\left (-x\,\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/4}\,\left (x\,\left (2\,b-a\,d\right )-b^2+x^2\,\left (d-1\right )\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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