Optimal. Leaf size=119 \[ \frac {2 \text {ArcTan}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{b-x}\right )}{d^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{b-x}\right )}{d^{3/4}} \]
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Rubi [F]
time = 2.60, 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 [A]
time = 42.38, size = 163, normalized size = 1.37 \begin {gather*} -\frac {\sqrt [4]{-1+\frac {a}{x}} \sqrt {-2+\frac {2 b}{x}} x \left (-\text {ArcTan}\left (\frac {1+\sqrt {d} \sqrt {-1+\frac {a}{x}}-\frac {b}{x}}{\sqrt [4]{d} \sqrt [4]{-1+\frac {a}{x}} \sqrt {-2+\frac {2 b}{x}}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{-1+\frac {a}{x}} \sqrt {-2+\frac {2 b}{x}}}{-1+\sqrt {d} \sqrt {-1+\frac {a}{x}}+\frac {b}{x}}\right )\right )}{d^{3/4} \sqrt [4]{(b-x)^2 x (-a+x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \frac {-a b +\left (-a +2 b \right ) x}{\left (x \left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{4}} \left (-b^{2}+\left (-a d +2 b \right ) x +\left (-1+d \right ) x^{2}\right )}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
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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Sympy [F(-1)] Timed out
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*} \text {could not integrate} \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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