Optimal. Leaf size=125 \[ -\frac {4 \left (a b x^2+(-a-b) x^3+x^4\right )^{3/4}}{x (-a+x) (-b+x)}+2 \sqrt [4]{d} \text {ArcTan}\left (\frac {x}{\sqrt [4]{d} \sqrt [4]{a b x^2+(-a-b) x^3+x^4}}\right )+2 \sqrt [4]{d} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{d} \sqrt [4]{a b x^2+(-a-b) x^3+x^4}}\right ) \]
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Rubi [F]
time = 12.91, 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 x^2+(a+b) x^3}{(-a+x) (-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 x^2+(a+b) x^3}{(-a+x) (-b+x) \sqrt [4]{x^2 (-a+x) (-b+x)} \left (-a b d+(a+b) d x+(1-d) x^2\right )} \, dx &=\int \frac {x^2 (-2 a b+(a+b) x)}{(-a+x) (-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 {x^{3/2} (-2 a b+(a+b) x)}{(-a+x)^{5/4} (-b+x)^{5/4} \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 {\left (a+b-\frac {\sqrt {4 a b+a^2 d-2 a b d+b^2 d}}{\sqrt {d}}\right ) x^{3/2}}{(-a+x)^{5/4} (-b+x)^{5/4} \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 {\left (a+b+\frac {\sqrt {4 a b+a^2 d-2 a b d+b^2 d}}{\sqrt {d}}\right ) x^{3/2}}{(-a+x)^{5/4} (-b+x)^{5/4} \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 {x^{3/2}}{(-a+x)^{5/4} (-b+x)^{5/4} \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 {x^{3/2}}{(-a+x)^{5/4} (-b+x)^{5/4} \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 [A]
time = 39.52, size = 171, normalized size = 1.37 \begin {gather*} -\frac {2 x \left (2 \sqrt {\frac {x}{-a+x}}+\sqrt [4]{d} \sqrt [4]{\frac {b-x}{a-x}} \text {ArcTan}\left (\frac {\sqrt [4]{d} \sqrt [4]{\frac {b-x}{a-x}}}{\sqrt {\frac {x}{-a+x}}}\right )-\sqrt [4]{d} \sqrt [4]{\frac {b-x}{a-x}} \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{\frac {b-x}{a-x}}}{\sqrt {\frac {x}{-a+x}}}\right )\right )}{\sqrt {\frac {x}{-a+x}} \sqrt [4]{x^2 (-a+x) (-b+x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {-2 a b \,x^{2}+\left (a +b \right ) x^{3}}{\left (-a +x \right ) \left (-b +x \right ) \left (x^{2} \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{4}} \left (-a b d +\left (a +b \right ) d x +\left (1-d \right ) x^{2}\right )}\, dx\]
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*} \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 {x^3\,\left (a+b\right )-2\,a\,b\,x^2}{\left (a-x\right )\,\left (b-x\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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