Optimal. Leaf size=42 \[ \frac {\tanh ^{-1}\left (\frac {2 \sqrt {d} \sqrt {f} x^{1+m}}{e+2 f x^n}\right )}{2 \sqrt {d} \sqrt {f}} \]
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Rubi [A]
time = 0.16, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {2119, 214}
\begin {gather*} \frac {\tanh ^{-1}\left (\frac {2 \sqrt {d} \sqrt {f} x^{m+1}}{e+2 f x^n}\right )}{2 \sqrt {d} \sqrt {f}} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 2119
Rubi steps
\begin {align*} \int \frac {x^m \left (e (1+m)+2 f (1+m-n) x^n\right )}{e^2-4 d f x^{2+2 m}+4 e f x^n+4 f^2 x^{2 n}} \, dx &=\left (e^2 (1+m) (1+m-n)\right ) \text {Subst}\left (\int \frac {1}{e^2-4 d e^2 f (1+m)^2 (1+m-n)^2 x^2} \, dx,x,\frac {x^{1+m}}{e (1+m) (1+m-n)+2 f (1+m) (1+m-n) x^n}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {2 \sqrt {d} \sqrt {f} x^{1+m}}{e+2 f x^n}\right )}{2 \sqrt {d} \sqrt {f}}\\ \end {align*}
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Mathematica [F]
time = 0.44, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^m \left (e (1+m)+2 f (1+m-n) x^n\right )}{e^2-4 d f x^{2+2 m}+4 e f x^n+4 f^2 x^{2 n}} \, dx \end {gather*}
Verification is not applicable to the result.
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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 | \(\frac {\ln \left (x^{n}+\frac {2 d f x \,x^{m}+\sqrt {d f}\, e}{2 \sqrt {d f}\, f}\right )}{4 \sqrt {d f}}-\frac {\ln \left (x^{n}+\frac {-2 d f x \,x^{m}+\sqrt {d f}\, e}{2 \sqrt {d f}\, f}\right )}{4 \sqrt {d f}}\) | \(78\) |
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 = 167, normalized size = 3.98 \begin {gather*} \left [\frac {\sqrt {d f} \log \left (-\frac {4 \, d f x^{2} x^{2 \, m} + 4 \, \sqrt {d f} x x^{m} e + 4 \, f^{2} x^{2 \, n} + 4 \, {\left (2 \, \sqrt {d f} f x x^{m} + f e\right )} x^{n} + e^{2}}{4 \, d f x^{2} x^{2 \, m} - 4 \, f^{2} x^{2 \, n} - 4 \, f x^{n} e - e^{2}}\right )}{4 \, d f}, -\frac {\sqrt {-d f} \arctan \left (\frac {2 \, \sqrt {-d f} f x^{n} + \sqrt {-d f} e}{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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {e x^{m}}{4 d f x^{2} x^{2 m} - e^{2} - 4 e f x^{n} - 4 f^{2} x^{2 n}}\, dx - \int \frac {e m x^{m}}{4 d f x^{2} x^{2 m} - e^{2} - 4 e f x^{n} - 4 f^{2} x^{2 n}}\, dx - \int \frac {2 f x^{m} x^{n}}{4 d f x^{2} x^{2 m} - e^{2} - 4 e f x^{n} - 4 f^{2} x^{2 n}}\, dx - \int \frac {2 f m x^{m} x^{n}}{4 d f x^{2} x^{2 m} - e^{2} - 4 e f x^{n} - 4 f^{2} x^{2 n}}\, dx - \int \left (- \frac {2 f n x^{m} x^{n}}{4 d f x^{2} x^{2 m} - e^{2} - 4 e f x^{n} - 4 f^{2} x^{2 n}}\right )\, dx \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 (e\,\left (m+1\right )+2\,f\,x^n\,\left (m-n+1\right )\right )}{e^2+4\,f^2\,x^{2\,n}-4\,d\,f\,x^{2\,m+2}+4\,e\,f\,x^n} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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