Optimal. Leaf size=44 \[ -\frac {f \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} e} \]
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Rubi [A]
time = 0.05, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1156, 1121,
632, 212} \begin {gather*} -\frac {f \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{e \sqrt {b^2-4 a c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 1121
Rule 1156
Rubi steps
\begin {align*} \int \frac {d f+e f x}{a+b (d+e x)^2+c (d+e x)^4} \, dx &=\frac {f \text {Subst}\left (\int \frac {x}{a+b x^2+c x^4} \, dx,x,d+e x\right )}{e}\\ &=\frac {f \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{2 e}\\ &=-\frac {f \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c (d+e x)^2\right )}{e}\\ &=-\frac {f \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} e}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 47, normalized size = 1.07 \begin {gather*} \frac {f \tan ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c} e} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.04, size = 130, normalized size = 2.95
method | result | size |
default | \(\frac {f \left (\munderset {\textit {\_R} =\RootOf \left (e^{4} c \,\textit {\_Z}^{4}+4 d \,e^{3} c \,\textit {\_Z}^{3}+\left (6 d^{2} e^{2} c +e^{2} b \right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 d e b \right ) \textit {\_Z} +d^{4} c +d^{2} b +a \right )}{\sum }\frac {\left (\textit {\_R} e +d \right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 d \,e^{2} c \,\textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 c \,d^{3}+e b \textit {\_R} +b d}\right )}{2 e}\) | \(130\) |
risch | \(-\frac {f \ln \left (\left (\sqrt {-4 a c +b^{2}}\, e^{2}-e^{2} b \right ) x^{2}+\left (2 d e \sqrt {-4 a c +b^{2}}-2 d e b \right ) x +\sqrt {-4 a c +b^{2}}\, d^{2}-d^{2} b -2 a \right )}{2 \sqrt {-4 a c +b^{2}}\, e}+\frac {f \ln \left (\left (\sqrt {-4 a c +b^{2}}\, e^{2}+e^{2} b \right ) x^{2}+\left (2 d e \sqrt {-4 a c +b^{2}}+2 d e b \right ) x +\sqrt {-4 a c +b^{2}}\, d^{2}+d^{2} b +2 a \right )}{2 \sqrt {-4 a c +b^{2}}\, e}\) | \(176\) |
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 = 0.35, size = 268, normalized size = 6.09 \begin {gather*} \left [\frac {f e^{\left (-1\right )} \log \left (\frac {2 \, c^{2} x^{4} e^{4} + 8 \, c^{2} d x^{3} e^{3} + 2 \, c^{2} d^{4} + 2 \, b c d^{2} + 2 \, {\left (6 \, c^{2} d^{2} + b c\right )} x^{2} e^{2} + 4 \, {\left (2 \, c^{2} d^{3} + b c d\right )} x e + b^{2} - 2 \, a c - {\left (2 \, c x^{2} e^{2} + 4 \, c d x e + 2 \, c d^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} e^{4} + 4 \, c d x^{3} e^{3} + c d^{4} + {\left (6 \, c d^{2} + b\right )} x^{2} e^{2} + b d^{2} + 2 \, {\left (2 \, c d^{3} + b d\right )} x e + a}\right )}{2 \, \sqrt {b^{2} - 4 \, a c}}, -\frac {\sqrt {-b^{2} + 4 \, a c} f \arctan \left (-\frac {{\left (2 \, c x^{2} e^{2} + 4 \, c d x e + 2 \, c d^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) e^{\left (-1\right )}}{b^{2} - 4 \, a c}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 189 vs.
\(2 (41) = 82\).
time = 0.62, size = 189, normalized size = 4.30 \begin {gather*} - \frac {f \sqrt {- \frac {1}{4 a c - b^{2}}} \log {\left (\frac {2 d x}{e} + x^{2} + \frac {- 4 a c f \sqrt {- \frac {1}{4 a c - b^{2}}} + b^{2} f \sqrt {- \frac {1}{4 a c - b^{2}}} + b f + 2 c d^{2} f}{2 c e^{2} f} \right )}}{2 e} + \frac {f \sqrt {- \frac {1}{4 a c - b^{2}}} \log {\left (\frac {2 d x}{e} + x^{2} + \frac {4 a c f \sqrt {- \frac {1}{4 a c - b^{2}}} - b^{2} f \sqrt {- \frac {1}{4 a c - b^{2}}} + b f + 2 c d^{2} f}{2 c e^{2} f} \right )}}{2 e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.81, size = 62, normalized size = 1.41 \begin {gather*} \frac {f \arctan \left (\frac {2 \, c d^{2} f + 2 \, {\left (f x^{2} e + 2 \, d f x\right )} c e + b f}{\sqrt {-b^{2} + 4 \, a c} f}\right ) e^{\left (-1\right )}}{\sqrt {-b^{2} + 4 \, a c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.62, size = 477, normalized size = 10.84 \begin {gather*} \frac {f\,\mathrm {atan}\left (\frac {\frac {f\,\left (4\,c^2\,d^2\,e^7\,f+4\,c^2\,e^9\,f\,x^2-\frac {f\,\left (8\,b\,c^2\,d^2\,e^8+16\,b\,c^2\,d\,e^9\,x+8\,b\,c^2\,e^{10}\,x^2+16\,a\,c^2\,e^8\right )}{2\,e\,\sqrt {b^2-4\,a\,c}}+8\,c^2\,d\,e^8\,f\,x\right )\,1{}\mathrm {i}}{2\,e\,\sqrt {b^2-4\,a\,c}}+\frac {f\,\left (4\,c^2\,d^2\,e^7\,f+4\,c^2\,e^9\,f\,x^2+\frac {f\,\left (8\,b\,c^2\,d^2\,e^8+16\,b\,c^2\,d\,e^9\,x+8\,b\,c^2\,e^{10}\,x^2+16\,a\,c^2\,e^8\right )}{2\,e\,\sqrt {b^2-4\,a\,c}}+8\,c^2\,d\,e^8\,f\,x\right )\,1{}\mathrm {i}}{2\,e\,\sqrt {b^2-4\,a\,c}}}{\frac {f\,\left (4\,c^2\,d^2\,e^7\,f+4\,c^2\,e^9\,f\,x^2-\frac {f\,\left (8\,b\,c^2\,d^2\,e^8+16\,b\,c^2\,d\,e^9\,x+8\,b\,c^2\,e^{10}\,x^2+16\,a\,c^2\,e^8\right )}{2\,e\,\sqrt {b^2-4\,a\,c}}+8\,c^2\,d\,e^8\,f\,x\right )}{2\,e\,\sqrt {b^2-4\,a\,c}}-\frac {f\,\left (4\,c^2\,d^2\,e^7\,f+4\,c^2\,e^9\,f\,x^2+\frac {f\,\left (8\,b\,c^2\,d^2\,e^8+16\,b\,c^2\,d\,e^9\,x+8\,b\,c^2\,e^{10}\,x^2+16\,a\,c^2\,e^8\right )}{2\,e\,\sqrt {b^2-4\,a\,c}}+8\,c^2\,d\,e^8\,f\,x\right )}{2\,e\,\sqrt {b^2-4\,a\,c}}}\right )\,1{}\mathrm {i}}{e\,\sqrt {b^2-4\,a\,c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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