Optimal. Leaf size=88 \[ \frac {f^{\frac {1}{2} (2 a-3 e)+\frac {1}{2} (2 b x+e)}}{b d \log (f)}-\frac {\sqrt {c} f^{a-\frac {3 e}{2}} \tan ^{-1}\left (\frac {\sqrt {d} f^{\frac {1}{2} (2 b x+e)}}{\sqrt {c}}\right )}{b d^{3/2} \log (f)} \]
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Rubi [A] time = 0.07, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2248, 321, 205} \[ \frac {f^{\frac {1}{2} (2 a-3 e)+\frac {1}{2} (2 b x+e)}}{b d \log (f)}-\frac {\sqrt {c} f^{a-\frac {3 e}{2}} \tan ^{-1}\left (\frac {\sqrt {d} f^{\frac {1}{2} (2 b x+e)}}{\sqrt {c}}\right )}{b d^{3/2} \log (f)} \]
Antiderivative was successfully verified.
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Rule 205
Rule 321
Rule 2248
Rubi steps
\begin {align*} \int \frac {f^{a+3 b x}}{c+d f^{e+2 b x}} \, dx &=\frac {f^{a-\frac {3 e}{2}} \operatorname {Subst}\left (\int \frac {x^2}{c+d x^2} \, dx,x,f^{\frac {1}{2} (e+2 b x)}\right )}{b \log (f)}\\ &=\frac {f^{\frac {1}{2} (2 a-3 e)+\frac {1}{2} (e+2 b x)}}{b d \log (f)}-\frac {\left (c f^{a-\frac {3 e}{2}}\right ) \operatorname {Subst}\left (\int \frac {1}{c+d x^2} \, dx,x,f^{\frac {1}{2} (e+2 b x)}\right )}{b d \log (f)}\\ &=\frac {f^{\frac {1}{2} (2 a-3 e)+\frac {1}{2} (e+2 b x)}}{b d \log (f)}-\frac {\sqrt {c} f^{a-\frac {3 e}{2}} \tan ^{-1}\left (\frac {\sqrt {d} f^{\frac {1}{2} (e+2 b x)}}{\sqrt {c}}\right )}{b d^{3/2} \log (f)}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 67, normalized size = 0.76 \[ \frac {f^a \left (\frac {f^{b x-e}}{d}-\frac {\sqrt {c} f^{-3 e/2} \tan ^{-1}\left (\frac {\sqrt {d} f^{b x+\frac {e}{2}}}{\sqrt {c}}\right )}{d^{3/2}}\right )}{b \log (f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 166, normalized size = 1.89 \[ \left [\frac {f^{a - \frac {3}{2} \, e} \sqrt {-\frac {c}{d}} \log \left (-\frac {2 \, d f^{b x + \frac {1}{2} \, e} \sqrt {-\frac {c}{d}} - d f^{2 \, b x + e} + c}{d f^{2 \, b x + e} + c}\right ) + 2 \, f^{b x + \frac {1}{2} \, e} f^{a - \frac {3}{2} \, e}}{2 \, b d \log \relax (f)}, -\frac {f^{a - \frac {3}{2} \, e} \sqrt {\frac {c}{d}} \arctan \left (\frac {d f^{b x + \frac {1}{2} \, e} \sqrt {\frac {c}{d}}}{c}\right ) - f^{b x + \frac {1}{2} \, e} f^{a - \frac {3}{2} \, e}}{b d \log \relax (f)}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 77, normalized size = 0.88 \[ -f^{a} {\left (\frac {c \arctan \left (\frac {d f^{b x} f^{e}}{\sqrt {c d f^{e}}}\right )}{\sqrt {c d f^{e}} b d f^{e} \log \relax (f)} - \frac {f^{b x}}{b d f^{e} \log \relax (f)}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 171, normalized size = 1.94 \[ \frac {f^{\frac {2 a}{3}} f^{-e} f^{b x +\frac {a}{3}}}{b d \ln \relax (f )}+\frac {\sqrt {-c d}\, f^{a} f^{-\frac {3 e}{2}} \ln \left (-\frac {\sqrt {-c d}\, f^{\frac {a}{3}} f^{-\frac {e}{2}}}{d}+f^{b x +\frac {a}{3}}\right )}{2 b \,d^{2} \ln \relax (f )}-\frac {\sqrt {-c d}\, f^{a} f^{-\frac {3 e}{2}} \ln \left (\frac {\sqrt {-c d}\, f^{\frac {a}{3}} f^{-\frac {e}{2}}}{d}+f^{b x +\frac {a}{3}}\right )}{2 b \,d^{2} \ln \relax (f )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.00, size = 76, normalized size = 0.86 \[ -\frac {c f^{a - e} \arctan \left (\frac {d f^{b x + e}}{\sqrt {c d} f^{\frac {1}{2} \, e}}\right )}{\sqrt {c d} b d f^{\frac {1}{2} \, e} \log \relax (f)} + \frac {f^{b x + a - e}}{b d \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.55, size = 66, normalized size = 0.75 \[ -\frac {f^a\,{\mathrm {e}}^{-\frac {3\,e\,\ln \relax (f)}{2}}\,\left (c\,\mathrm {atan}\left (\frac {d\,f^{b\,x}\,{\mathrm {e}}^{\frac {e\,\ln \relax (f)}{2}}}{\sqrt {c\,d}}\right )-f^{b\,x}\,{\mathrm {e}}^{\frac {e\,\ln \relax (f)}{2}}\,\sqrt {c\,d}\right )}{b\,d\,\ln \relax (f)\,\sqrt {c\,d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.35, size = 110, normalized size = 1.25 \[ \operatorname {RootSum} {\left (4 z^{2} b^{2} d^{3} e^{3 e \log {\relax (f )}} \log {\relax (f )}^{2} + c e^{2 a \log {\relax (f )}}, \left (i \mapsto i \log {\left (- 2 i b d e^{- \frac {2 a \log {\relax (f )}}{3}} e^{e \log {\relax (f )}} \log {\relax (f )} + e^{\frac {\left (a + 3 b x\right ) \log {\relax (f )}}{3}} \right )} \right )\right )} + \frac {\left (\begin {cases} x & \text {for}\: b = 0 \vee f = 1 \\\frac {e^{b x \log {\relax (f )}}}{b \log {\relax (f )}} & \text {otherwise} \end {cases}\right ) e^{a \log {\relax (f )}} e^{- e \log {\relax (f )}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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