Optimal. Leaf size=107 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a x^2-b+c x^4}}\right )}{4 c^{5/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a x^2-b+c x^4}}\right )}{4 c^{5/4}}-\frac {x \left (a x^2-b+c x^4\right )^{3/4}}{2 c \left (b-a x^2\right )} \]
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Rubi [F] time = 0.15, 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 {x^4 \left (-2 b+a x^2\right )}{\left (-b+a x^2\right )^2 \sqrt [4]{-b+a x^2+c x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^4 \left (-2 b+a x^2\right )}{\left (-b+a x^2\right )^2 \sqrt [4]{-b+a x^2+c x^4}} \, dx &=\int \frac {x^4 \left (-2 b+a x^2\right )}{\left (-b+a x^2\right )^2 \sqrt [4]{-b+a x^2+c x^4}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.94, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4 \left (-2 b+a x^2\right )}{\left (-b+a x^2\right )^2 \sqrt [4]{-b+a x^2+c x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.04, size = 107, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a x^2-b+c x^4}}\right )}{4 c^{5/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a x^2-b+c x^4}}\right )}{4 c^{5/4}}-\frac {x \left (a x^2-b+c x^4\right )^{3/4}}{2 c \left (b-a x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 228, normalized size = 2.13 \begin {gather*} -\frac {4 \, {\left (a c x^{2} - b c\right )} \frac {1}{c^{5}}^{\frac {1}{4}} \arctan \left (\frac {c x \sqrt {\frac {c^{3} \sqrt {\frac {1}{c^{5}}} x^{2} + \sqrt {c x^{4} + a x^{2} - b}}{x^{2}}} \frac {1}{c^{5}}^{\frac {1}{4}} - {\left (c x^{4} + a x^{2} - b\right )}^{\frac {1}{4}} c \frac {1}{c^{5}}^{\frac {1}{4}}}{x}\right ) + {\left (a c x^{2} - b c\right )} \frac {1}{c^{5}}^{\frac {1}{4}} \log \left (\frac {c^{4} \frac {1}{c^{5}}^{\frac {3}{4}} x + {\left (c x^{4} + a x^{2} - b\right )}^{\frac {1}{4}}}{x}\right ) - {\left (a c x^{2} - b c\right )} \frac {1}{c^{5}}^{\frac {1}{4}} \log \left (-\frac {c^{4} \frac {1}{c^{5}}^{\frac {3}{4}} x - {\left (c x^{4} + a x^{2} - b\right )}^{\frac {1}{4}}}{x}\right ) - 4 \, {\left (c x^{4} + a x^{2} - b\right )}^{\frac {3}{4}} x}{8 \, {\left (a c x^{2} - b c\right )}} \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 {{\left (a x^{2} - 2 \, b\right )} x^{4}}{{\left (c x^{4} + a x^{2} - b\right )}^{\frac {1}{4}} {\left (a x^{2} - b\right )}^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.33, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} \left (a \,x^{2}-2 b \right )}{\left (a \,x^{2}-b \right )^{2} \left (c \,x^{4}+a \,x^{2}-b \right )^{\frac {1}{4}}}\, 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 {{\left (a x^{2} - 2 \, b\right )} x^{4}}{{\left (c x^{4} + a x^{2} - b\right )}^{\frac {1}{4}} {\left (a x^{2} - b\right )}^{2}}\,{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 {x^4\,\left (2\,b-a\,x^2\right )}{{\left (b-a\,x^2\right )}^2\,{\left (c\,x^4+a\,x^2-b\right )}^{1/4}} \,d x \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 {x^{4} \left (a x^{2} - 2 b\right )}{\left (a x^{2} - b\right )^{2} \sqrt [4]{a x^{2} - b + c x^{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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