Optimal. Leaf size=52 \[ \sqrt [4]{2} \tan ^{-1}\left (\sqrt [4]{2} \sqrt [4]{a x+b+x^2}\right )-\sqrt [4]{2} \tanh ^{-1}\left (\sqrt [4]{2} \sqrt [4]{a x+b+x^2}\right ) \]
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Rubi [F] time = 0.02, 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+2 x}{\sqrt [4]{b+a x+x^2} \left (-1+2 b+2 a x+2 x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {a+2 x}{\sqrt [4]{b+a x+x^2} \left (-1+2 b+2 a x+2 x^2\right )} \, dx &=\int \frac {a+2 x}{\sqrt [4]{b+a x+x^2} \left (-1+2 b+2 a x+2 x^2\right )} \, dx\\ \end {align*}
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Mathematica [A] time = 0.28, size = 45, normalized size = 0.87 \begin {gather*} \sqrt [4]{2} \left (\tan ^{-1}\left (\sqrt [4]{2} \sqrt [4]{x (a+x)+b}\right )-\tanh ^{-1}\left (\sqrt [4]{2} \sqrt [4]{x (a+x)+b}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.13, size = 52, normalized size = 1.00 \begin {gather*} \sqrt [4]{2} \tan ^{-1}\left (\sqrt [4]{2} \sqrt [4]{b+a x+x^2}\right )-\sqrt [4]{2} \tanh ^{-1}\left (\sqrt [4]{2} \sqrt [4]{b+a x+x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 94, normalized size = 1.81 \begin {gather*} -\frac {1}{2} \cdot 8^{\frac {3}{4}} \arctan \left (\frac {1}{8} \cdot 8^{\frac {3}{4}} \sqrt {2 \, \sqrt {2} + 4 \, \sqrt {a x + x^{2} + b}} - \frac {1}{4} \cdot 8^{\frac {3}{4}} {\left (a x + x^{2} + b\right )}^{\frac {1}{4}}\right ) - \frac {1}{8} \cdot 8^{\frac {3}{4}} \log \left (8^{\frac {1}{4}} + 2 \, {\left (a x + x^{2} + b\right )}^{\frac {1}{4}}\right ) + \frac {1}{8} \cdot 8^{\frac {3}{4}} \log \left (-8^{\frac {1}{4}} + 2 \, {\left (a x + x^{2} + b\right )}^{\frac {1}{4}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 65, normalized size = 1.25 \begin {gather*} \frac {1}{4} \cdot 8^{\frac {3}{4}} \arctan \left (2 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (a x + x^{2} + b\right )}^{\frac {1}{4}}\right ) - \frac {1}{8} \cdot 8^{\frac {3}{4}} \log \left (\left (\frac {1}{2}\right )^{\frac {1}{4}} + {\left (a x + x^{2} + b\right )}^{\frac {1}{4}}\right ) + \frac {1}{8} \cdot 8^{\frac {3}{4}} \log \left ({\left | -\left (\frac {1}{2}\right )^{\frac {1}{4}} + {\left (a x + x^{2} + b\right )}^{\frac {1}{4}} \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {a +2 x}{\left (a x +x^{2}+b \right )^{\frac {1}{4}} \left (2 a x +2 x^{2}+2 b -1\right )}\, dx\]
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 + 2 \, x}{{\left (2 \, a x + 2 \, x^{2} + 2 \, b - 1\right )} {\left (a x + x^{2} + b\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.99, size = 39, normalized size = 0.75 \begin {gather*} 2^{1/4}\,\left (\mathrm {atan}\left ({\left (2\,x^2+2\,a\,x+2\,b\right )}^{1/4}\right )-\mathrm {atanh}\left ({\left (2\,x^2+2\,a\,x+2\,b\right )}^{1/4}\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + 2 x}{\sqrt [4]{a x + b + x^{2}} \left (2 a x + 2 b + 2 x^{2} - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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