Optimal. Leaf size=42 \[ \frac {\tan ^{-1}\left (\frac {2 \sqrt {d} \sqrt {f} x^{1+m}}{e+2 f x^3}\right )}{2 \sqrt {d} \sqrt {f}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.15, 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 = {2119, 211}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {2 \sqrt {d} \sqrt {f} x^{m+1}}{e+2 f x^3}\right )}{2 \sqrt {d} \sqrt {f}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 211
Rule 2119
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 ) \text {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*}
________________________________________________________________________________________
Mathematica [A]
time = 2.28, 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^3}\right )}{2 \sqrt {d} \sqrt {f}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(77\) vs.
\(2(32)=64\).
time = 0.03, size = 78, normalized size = 1.86
method | result | size |
risch | \(-\frac {\ln \left (x^{m}+\frac {\left (2 f \,x^{3}+e \right ) \sqrt {-d f}}{2 d f x}\right )}{4 \sqrt {-d f}}+\frac {\ln \left (x^{m}-\frac {\left (2 f \,x^{3}+e \right ) \sqrt {-d f}}{2 d f x}\right )}{4 \sqrt {-d f}}\) | \(78\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.37, size = 148, normalized size = 3.52 \begin {gather*} \left [-\frac {\sqrt {-d f} \log \left (-\frac {4 \, f^{2} x^{6} - 4 \, d f x^{2} x^{2 \, m} + 4 \, f x^{3} e + 4 \, {\left (2 \, f x^{4} + x e\right )} \sqrt {-d f} x^{m} + e^{2}}{4 \, f^{2} x^{6} + 4 \, d f x^{2} x^{2 \, m} + 4 \, f x^{3} e + 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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________