Optimal. Leaf size=122 \[ -\frac {\tan ^{-1}\left (1-\frac {2 \sqrt [4]{a x^2+b x+c}}{\sqrt [4]{c}}\right )}{2 \sqrt [4]{c}}+\frac {\tan ^{-1}\left (\frac {2 \sqrt [4]{a x^2+b x+c}}{\sqrt [4]{c}}+1\right )}{2 \sqrt [4]{c}}-\frac {\tanh ^{-1}\left (\frac {\frac {\sqrt {a x^2+b x+c}}{\sqrt [4]{c}}+\frac {\sqrt [4]{c}}{2}}{\sqrt [4]{a x^2+b x+c}}\right )}{2 \sqrt [4]{c}} \]
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Rubi [F] time = 0.03, 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 {b+2 a x}{\sqrt [4]{c+b x+a x^2} \left (5 c+4 b x+4 a x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {b+2 a x}{\sqrt [4]{c+b x+a x^2} \left (5 c+4 b x+4 a x^2\right )} \, dx &=\int \frac {b+2 a x}{\sqrt [4]{c+b x+a x^2} \left (5 c+4 b x+4 a x^2\right )} \, dx\\ \end {align*}
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Mathematica [F] time = 0.65, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {b+2 a x}{\sqrt [4]{c+b x+a x^2} \left (5 c+4 b x+4 a x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.30, size = 122, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (1-\frac {2 \sqrt [4]{c+b x+a x^2}}{\sqrt [4]{c}}\right )}{2 \sqrt [4]{c}}+\frac {\tan ^{-1}\left (1+\frac {2 \sqrt [4]{c+b x+a x^2}}{\sqrt [4]{c}}\right )}{2 \sqrt [4]{c}}-\frac {\tanh ^{-1}\left (\frac {\frac {\sqrt [4]{c}}{2}+\frac {\sqrt {c+b x+a x^2}}{\sqrt [4]{c}}}{\sqrt [4]{c+b x+a x^2}}\right )}{2 \sqrt [4]{c}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 154, normalized size = 1.26 \begin {gather*} -2 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{c}\right )^{\frac {1}{4}} \arctan \left (2 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \sqrt {-\frac {1}{2} \, c \sqrt {-\frac {1}{c}} + \sqrt {a x^{2} + b x + c}} \left (-\frac {1}{c}\right )^{\frac {1}{4}} - 2 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (a x^{2} + b x + c\right )}^{\frac {1}{4}} \left (-\frac {1}{c}\right )^{\frac {1}{4}}\right ) + \frac {1}{2} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{c}\right )^{\frac {1}{4}} \log \left (2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} c \left (-\frac {1}{c}\right )^{\frac {3}{4}} + {\left (a x^{2} + b x + c\right )}^{\frac {1}{4}}\right ) - \frac {1}{2} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{c}\right )^{\frac {1}{4}} \log \left (-2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} c \left (-\frac {1}{c}\right )^{\frac {3}{4}} + {\left (a x^{2} + b x + c\right )}^{\frac {1}{4}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.41, size = 202, normalized size = 1.66 \begin {gather*} \frac {4^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {2 \, \sqrt {2} \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (\sqrt {2} \left (\frac {1}{4}\right )^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, {\left (a x^{2} + b x + c\right )}^{\frac {1}{4}}\right )}}{c^{\frac {1}{4}}}\right )}{8 \, c^{\frac {1}{4}}} + \frac {4^{\frac {3}{4}} \sqrt {2} \arctan \left (-\frac {2 \, \sqrt {2} \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (\sqrt {2} \left (\frac {1}{4}\right )^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, {\left (a x^{2} + b x + c\right )}^{\frac {1}{4}}\right )}}{c^{\frac {1}{4}}}\right )}{8 \, c^{\frac {1}{4}}} - \frac {4^{\frac {3}{4}} \sqrt {2} \log \left (\sqrt {2} \left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (a x^{2} + b x + c\right )}^{\frac {1}{4}} c^{\frac {1}{4}} + \sqrt {a x^{2} + b x + c} + \frac {1}{2} \, \sqrt {c}\right )}{16 \, c^{\frac {1}{4}}} + \frac {4^{\frac {3}{4}} \sqrt {2} \log \left (-\sqrt {2} \left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (a x^{2} + b x + c\right )}^{\frac {1}{4}} c^{\frac {1}{4}} + \sqrt {a x^{2} + b x + c} + \frac {1}{2} \, \sqrt {c}\right )}{16 \, c^{\frac {1}{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.19, size = 0, normalized size = 0.00 \[\int \frac {2 a x +b}{\left (a \,x^{2}+b x +c \right )^{\frac {1}{4}} \left (4 a \,x^{2}+4 b x +5 c \right )}\, dx\]
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 x + b}{{\left (4 \, a x^{2} + 4 \, b x + 5 \, c\right )} {\left (a x^{2} + b x + c\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 = 1.33, size = 57, normalized size = 0.47 \begin {gather*} \frac {\sqrt {2}\,\left (\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (a\,x^2+b\,x+c\right )}^{1/4}}{{\left (-c\right )}^{1/4}}\right )-\mathrm {atanh}\left (\frac {\sqrt {2}\,{\left (a\,x^2+b\,x+c\right )}^{1/4}}{{\left (-c\right )}^{1/4}}\right )\right )}{2\,{\left (-c\right )}^{1/4}} \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 {2 a x + b}{\sqrt [4]{a x^{2} + b x + c} \left (4 a x^{2} + 4 b x + 5 c\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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