Optimal. Leaf size=42 \[ \frac {\tan ^{-1}\left (\frac {2 \sqrt {d} \sqrt {f} x^{1+m}}{e+2 f x^2}\right )}{2 \sqrt {d} \sqrt {f}} \]
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Rubi [A]
time = 0.15, antiderivative size = 61, normalized size of antiderivative = 1.45, number of steps
used = 2, number of rules used = 2, integrand size = 51, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {2119, 211}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {2 \sqrt {d} \sqrt {f} \left (1-m^2\right ) x^{m+1}}{(1-m) (m+1) \left (e+2 f x^2\right )}\right )}{2 \sqrt {d} \sqrt {f}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 2119
Rubi steps
\begin {align*} \int \frac {x^m \left (e (1+m)+2 f (-1+m) x^2\right )}{e^2+4 e f x^2+4 f^2 x^4+4 d f x^{2+2 m}} \, dx &=-\left (\left (e^2 (1-m) (1+m)\right ) \text {Subst}\left (\int \frac {1}{e^2+4 d e^2 f (-1+m)^2 (1+m)^2 x^2} \, dx,x,\frac {x^{1+m}}{e (-1+m) (1+m)+2 f (-1+m) (1+m) x^2}\right )\right )\\ &=\frac {\tan ^{-1}\left (\frac {2 \sqrt {d} \sqrt {f} \left (1-m^2\right ) x^{1+m}}{(1-m) (1+m) \left (e+2 f x^2\right )}\right )}{2 \sqrt {d} \sqrt {f}}\\ \end {align*}
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Mathematica [A]
time = 0.94, size = 42, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {2 \sqrt {d} \sqrt {f} x^{1+m}}{e+2 f x^2}\right )}{2 \sqrt {d} \sqrt {f}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(77\) vs.
\(2(32)=64\).
time = 0.06, size = 78, normalized size = 1.86
method | result | size |
risch | \(-\frac {\ln \left (x^{m}+\frac {\left (2 f \,x^{2}+e \right ) \sqrt {-d f}}{2 d f x}\right )}{4 \sqrt {-d f}}+\frac {\ln \left (x^{m}-\frac {\left (2 f \,x^{2}+e \right ) \sqrt {-d f}}{2 d f x}\right )}{4 \sqrt {-d f}}\) | \(78\) |
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 [A]
time = 0.36, size = 148, normalized size = 3.52 \begin {gather*} \left [-\frac {\sqrt {-d f} \log \left (-\frac {4 \, f^{2} x^{4} - 4 \, d f x^{2} x^{2 \, m} + 4 \, f x^{2} e + 4 \, {\left (2 \, f x^{3} + x e\right )} \sqrt {-d f} x^{m} + e^{2}}{4 \, f^{2} x^{4} + 4 \, d f x^{2} x^{2 \, m} + 4 \, f x^{2} e + e^{2}}\right )}{4 \, d f}, -\frac {\sqrt {d f} \arctan \left (\frac {{\left (2 \, f x^{2} + e\right )} \sqrt {d f}}{2 \, d f x x^{m}}\right )}{2 \, d f}\right ] \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 = -0.02 \begin {gather*} \int \frac {x^m\,\left (2\,f\,\left (m-1\right )\,x^2+e\,\left (m+1\right )\right )}{e^2+4\,f^2\,x^4+4\,e\,f\,x^2+4\,d\,f\,x^{2\,m+2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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