Optimal. Leaf size=44 \[ -\frac {f \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{e \sqrt {b^2-4 a c}} \]
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Rubi [A] time = 0.06, 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 = {1142, 1107, 618, 206} \[ -\frac {f \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{e \sqrt {b^2-4 a c}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 1107
Rule 1142
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 \operatorname {Subst}\left (\int \frac {x}{a+b x^2+c x^4} \, dx,x,d+e x\right )}{e}\\ &=\frac {f \operatorname {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{2 e}\\ &=-\frac {f \operatorname {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 = 0.02, size = 47, normalized size = 1.07 \[ \frac {f \tan ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {4 a c-b^2}}\right )}{e \sqrt {4 a c-b^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 274, normalized size = 6.23 \[ \left [\frac {f \log \left (\frac {2 \, c^{2} e^{4} x^{4} + 8 \, c^{2} d e^{3} x^{3} + 2 \, c^{2} d^{4} + 2 \, {\left (6 \, c^{2} d^{2} + b c\right )} e^{2} x^{2} + 2 \, b c d^{2} + 4 \, {\left (2 \, c^{2} d^{3} + b c d\right )} e x + b^{2} - 2 \, a c - {\left (2 \, c e^{2} x^{2} + 4 \, c d e x + 2 \, c d^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + c d^{4} + {\left (6 \, c d^{2} + b\right )} e^{2} x^{2} + b d^{2} + 2 \, {\left (2 \, c d^{3} + b d\right )} e x + a}\right )}{2 \, \sqrt {b^{2} - 4 \, a c} e}, -\frac {\sqrt {-b^{2} + 4 \, a c} f \arctan \left (-\frac {{\left (2 \, c e^{2} x^{2} + 4 \, c d e x + 2 \, c d^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right )}{{\left (b^{2} - 4 \, a c\right )} e}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 62, normalized size = 1.41 \[ \frac {f \arctan \left (\frac {2 \, c d^{2} f + 2 \, {\left (f x^{2} e + 2 \, d f x\right )} c e + b f}{\sqrt {-b^{2} + 4 \, a c} f}\right ) e^{\left (-1\right )}}{\sqrt {-b^{2} + 4 \, a c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.00, size = 130, normalized size = 2.95 \[ \frac {f \left (\RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right ) e +d \right ) \ln \left (-\RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right )+x \right )}{2 e \left (2 c \,e^{3} \RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right )^{3}+6 c d \,e^{2} \RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right )^{2}+6 e c \,d^{2} \RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right )+2 c \,d^{3}+b e \RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right )+b d \right )} \]
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 \[ \int \frac {e f x + d f}{{\left (e x + d\right )}^{4} c + {\left (e x + d\right )}^{2} b + a}\,{d x} \]
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 \[ \frac {f\,\mathrm {atan}\left (\frac {\frac {f\,\left (4\,c^2\,d^2\,e^7\,f+4\,c^2\,e^9\,f\,x^2-\frac {f\,\left (8\,b\,c^2\,d^2\,e^8+16\,b\,c^2\,d\,e^9\,x+8\,b\,c^2\,e^{10}\,x^2+16\,a\,c^2\,e^8\right )}{2\,e\,\sqrt {b^2-4\,a\,c}}+8\,c^2\,d\,e^8\,f\,x\right )\,1{}\mathrm {i}}{2\,e\,\sqrt {b^2-4\,a\,c}}+\frac {f\,\left (4\,c^2\,d^2\,e^7\,f+4\,c^2\,e^9\,f\,x^2+\frac {f\,\left (8\,b\,c^2\,d^2\,e^8+16\,b\,c^2\,d\,e^9\,x+8\,b\,c^2\,e^{10}\,x^2+16\,a\,c^2\,e^8\right )}{2\,e\,\sqrt {b^2-4\,a\,c}}+8\,c^2\,d\,e^8\,f\,x\right )\,1{}\mathrm {i}}{2\,e\,\sqrt {b^2-4\,a\,c}}}{\frac {f\,\left (4\,c^2\,d^2\,e^7\,f+4\,c^2\,e^9\,f\,x^2-\frac {f\,\left (8\,b\,c^2\,d^2\,e^8+16\,b\,c^2\,d\,e^9\,x+8\,b\,c^2\,e^{10}\,x^2+16\,a\,c^2\,e^8\right )}{2\,e\,\sqrt {b^2-4\,a\,c}}+8\,c^2\,d\,e^8\,f\,x\right )}{2\,e\,\sqrt {b^2-4\,a\,c}}-\frac {f\,\left (4\,c^2\,d^2\,e^7\,f+4\,c^2\,e^9\,f\,x^2+\frac {f\,\left (8\,b\,c^2\,d^2\,e^8+16\,b\,c^2\,d\,e^9\,x+8\,b\,c^2\,e^{10}\,x^2+16\,a\,c^2\,e^8\right )}{2\,e\,\sqrt {b^2-4\,a\,c}}+8\,c^2\,d\,e^8\,f\,x\right )}{2\,e\,\sqrt {b^2-4\,a\,c}}}\right )\,1{}\mathrm {i}}{e\,\sqrt {b^2-4\,a\,c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.19, size = 189, normalized size = 4.30 \[ - \frac {f \sqrt {- \frac {1}{4 a c - b^{2}}} \log {\left (\frac {2 d x}{e} + x^{2} + \frac {- 4 a c f \sqrt {- \frac {1}{4 a c - b^{2}}} + b^{2} f \sqrt {- \frac {1}{4 a c - b^{2}}} + b f + 2 c d^{2} f}{2 c e^{2} f} \right )}}{2 e} + \frac {f \sqrt {- \frac {1}{4 a c - b^{2}}} \log {\left (\frac {2 d x}{e} + x^{2} + \frac {4 a c f \sqrt {- \frac {1}{4 a c - b^{2}}} - b^{2} f \sqrt {- \frac {1}{4 a c - b^{2}}} + b f + 2 c d^{2} f}{2 c e^{2} f} \right )}}{2 e} \]
Verification of antiderivative is not currently implemented for this CAS.
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