Optimal. Leaf size=42 \[ \frac {\tan ^{-1}\left (\frac {2 \sqrt {d} \sqrt {f} x^{m+1}}{e+2 f x^3}\right )}{2 \sqrt {d} \sqrt {f}} \]
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Rubi [A] time = 0.22, antiderivative size = 42, normalized size of antiderivative = 1.00, 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 = {2094, 205} \[ \frac {\tan ^{-1}\left (\frac {2 \sqrt {d} \sqrt {f} x^{m+1}}{e+2 f x^3}\right )}{2 \sqrt {d} \sqrt {f}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 2094
Rubi steps
\begin {align*} \int \frac {x^m \left (e (1+m)+2 f (-2+m) x^3\right )}{e^2+4 e f x^3+4 f^2 x^6+4 d f x^{2+2 m}} \, dx &=-\left (\left (e^2 (2-m) (1+m)\right ) \operatorname {Subst}\left (\int \frac {1}{e^2+4 d e^2 f (-2+m)^2 (1+m)^2 x^2} \, dx,x,\frac {x^{1+m}}{e (-2+m) (1+m)+2 f (-2+m) (1+m) x^3}\right )\right )\\ &=\frac {\tan ^{-1}\left (\frac {2 \sqrt {d} \sqrt {f} x^{1+m}}{e+2 f x^3}\right )}{2 \sqrt {d} \sqrt {f}}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 42, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {2 \sqrt {d} \sqrt {f} x^{m+1}}{e+2 f x^3}\right )}{2 \sqrt {d} \sqrt {f}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 146, normalized size = 3.48 \[ \left [-\frac {\sqrt {-d f} \log \left (-\frac {4 \, f^{2} x^{6} - 4 \, d f x^{2} x^{2 \, m} + 4 \, e f x^{3} + 4 \, {\left (2 \, f x^{4} + e x\right )} \sqrt {-d f} x^{m} + e^{2}}{4 \, f^{2} x^{6} + 4 \, d f x^{2} x^{2 \, m} + 4 \, e f x^{3} + e^{2}}\right )}{4 \, d f}, -\frac {\sqrt {d f} \arctan \left (\frac {{\left (2 \, f x^{3} + e\right )} \sqrt {d f}}{2 \, d f x x^{m}}\right )}{2 \, d f}\right ] \]
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 \[ \int \frac {{\left (2 \, f {\left (m - 2\right )} x^{3} + e {\left (m + 1\right )}\right )} x^{m}}{4 \, f^{2} x^{6} + 4 \, e f x^{3} + 4 \, d f x^{2 \, m + 2} + e^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 78, normalized size = 1.86 \[ \frac {\ln \left (x^{m}-\frac {\left (2 f \,x^{3}+e \right ) \sqrt {-d f}}{2 d f x}\right )}{4 \sqrt {-d f}}-\frac {\ln \left (x^{m}+\frac {\left (2 f \,x^{3}+e \right ) \sqrt {-d f}}{2 d f x}\right )}{4 \sqrt {-d f}} \]
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 \[ \int \frac {{\left (2 \, f {\left (m - 2\right )} x^{3} + e {\left (m + 1\right )}\right )} x^{m}}{4 \, f^{2} x^{6} + 4 \, e f x^{3} + 4 \, d f x^{2 \, m + 2} + e^{2}}\,{d x} \]
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 \[ \int \frac {x^m\,\left (2\,f\,\left (m-2\right )\,x^3+e\,\left (m+1\right )\right )}{e^2+4\,f^2\,x^6+4\,e\,f\,x^3+4\,d\,f\,x^{2\,m+2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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