Optimal. Leaf size=50 \[ \frac {f^{a-\frac {e}{2}} \tan ^{-1}\left (\frac {\sqrt {d} f^{b x+\frac {e}{2}}}{\sqrt {c}}\right )}{b \sqrt {c} \sqrt {d} \log (f)} \]
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Rubi [A] time = 0.08, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2249, 205} \[ \frac {f^{a-\frac {e}{2}} \tan ^{-1}\left (\frac {\sqrt {d} f^{b x+\frac {e}{2}}}{\sqrt {c}}\right )}{b \sqrt {c} \sqrt {d} \log (f)} \]
Antiderivative was successfully verified.
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Rule 205
Rule 2249
Rubi steps
\begin {align*} \int \frac {f^{a+b x}}{c+d f^{e+2 b x}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{c+d f^{-2 a+e} x^2} \, dx,x,f^{a+b x}\right )}{b \log (f)}\\ &=\frac {f^{a-\frac {e}{2}} \tan ^{-1}\left (\frac {\sqrt {d} f^{\frac {e}{2}+b x}}{\sqrt {c}}\right )}{b \sqrt {c} \sqrt {d} \log (f)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 50, normalized size = 1.00 \[ \frac {f^{a-\frac {e}{2}} \tan ^{-1}\left (\frac {\sqrt {d} f^{b x+\frac {e}{2}}}{\sqrt {c}}\right )}{b \sqrt {c} \sqrt {d} \log (f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 179, normalized size = 3.58 \[ \left [-\frac {\sqrt {-c d f^{-2 \, a + e}} \log \left (\frac {d f^{2 \, b x + 2 \, a} f^{-2 \, a + e} - 2 \, \sqrt {-c d f^{-2 \, a + e}} f^{b x + a} - c}{d f^{2 \, b x + 2 \, a} f^{-2 \, a + e} + c}\right )}{2 \, b c d f^{-2 \, a + e} \log \relax (f)}, -\frac {\sqrt {c d f^{-2 \, a + e}} \arctan \left (\frac {\sqrt {c d f^{-2 \, a + e}}}{d f^{b x + a} f^{-2 \, a + e}}\right )}{b c d f^{-2 \, a + e} \log \relax (f)}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 48, normalized size = 0.96 \[ \frac {f^{2 \, a} \arctan \left (\frac {d f^{b x} f^{e}}{\sqrt {c d f^{e}}}\right )}{\sqrt {c d f^{e}} b f^{a} \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 91, normalized size = 1.82 \[ -\frac {f^{a} \ln \left (-\frac {c \,f^{a}}{\sqrt {-c d \,f^{e}}}+f^{b x +a}\right )}{2 \sqrt {-c d \,f^{e}}\, b \ln \relax (f )}+\frac {f^{a} \ln \left (\frac {c \,f^{a}}{\sqrt {-c d \,f^{e}}}+f^{b x +a}\right )}{2 \sqrt {-c d \,f^{e}}\, b \ln \relax (f )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 45, normalized size = 0.90 \[ \frac {f^{a} \arctan \left (\frac {d f^{b x + e}}{\sqrt {c d} f^{\frac {1}{2} \, e}}\right )}{\sqrt {c d} b f^{\frac {1}{2} \, e} \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.85, size = 64, normalized size = 1.28 \[ \frac {\mathrm {atan}\left (\frac {f^{a+b\,x}\,\sqrt {b^2\,c\,d\,f^e\,{\ln \relax (f)}^2}}{b\,c\,\ln \relax (f)\,\sqrt {f^{2\,a}}}\right )\,\sqrt {f^{2\,a}}}{\sqrt {b^2\,c\,d\,f^e\,{\ln \relax (f)}^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.87, size = 51, normalized size = 1.02 \[ \operatorname {RootSum} {\left (4 z^{2} b^{2} c d e^{e \log {\relax (f )}} \log {\relax (f )}^{2} + e^{2 a \log {\relax (f )}}, \left (i \mapsto i \log {\left (2 i b c \log {\relax (f )} + f^{a + b x} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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