Optimal. Leaf size=19 \[ e \log ^{n+1}(d x) F^{c (a+b x)} \]
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Rubi [A] time = 0.190229, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.029 \[ e \log ^{n+1}(d x) F^{c (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[(F^(c*(a + b*x))*Log[d*x]^n*(e + e*n + b*c*e*x*Log[F]*Log[d*x]))/x,x]
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Rubi in Sympy [A] time = 15.1638, size = 17, normalized size = 0.89 \[ F^{c \left (a + b x\right )} e \log{\left (d x \right )}^{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(F**(c*(b*x+a))*ln(d*x)**n*(e+e*n+b*c*e*x*ln(F)*ln(d*x))/x,x)
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Mathematica [A] time = 0.0392606, size = 20, normalized size = 1.05 \[ e \log ^{n+1}(d x) F^{a c+b c x} \]
Antiderivative was successfully verified.
[In] Integrate[(F^(c*(a + b*x))*Log[d*x]^n*(e + e*n + b*c*e*x*Log[F]*Log[d*x]))/x,x]
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Maple [C] time = 0.105, size = 180, normalized size = 9.5 \[ \left ( -{\frac{i}{2}}\pi \,e{\it csgn} \left ( id \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( idx \right ){F}^{c \left ( bx+a \right ) }+{\frac{i}{2}}\pi \,e{\it csgn} \left ( id \right ) \left ({\it csgn} \left ( idx \right ) \right ) ^{2}{F}^{c \left ( bx+a \right ) }+{\frac{i}{2}}\pi \,e{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( idx \right ) \right ) ^{2}{F}^{c \left ( bx+a \right ) }-{\frac{i}{2}}\pi \,e \left ({\it csgn} \left ( idx \right ) \right ) ^{3}{F}^{c \left ( bx+a \right ) }+\ln \left ( d \right ) e{F}^{c \left ( bx+a \right ) }+e{F}^{c \left ( bx+a \right ) }\ln \left ( x \right ) \right ) \left ( \ln \left ( d \right ) +\ln \left ( x \right ) -{\frac{i}{2}}\pi \,{\it csgn} \left ( idx \right ) \left ( -{\it csgn} \left ( idx \right ) +{\it csgn} \left ( id \right ) \right ) \left ( -{\it csgn} \left ( idx \right ) +{\it csgn} \left ( ix \right ) \right ) \right ) ^{n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(F^(c*(b*x+a))*ln(d*x)^n*(e+e*n+b*c*e*x*ln(F)*ln(d*x))/x,x)
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Maxima [A] time = 0.96519, size = 49, normalized size = 2.58 \[{\left (F^{a c} e \log \left (d\right ) + F^{a c} e \log \left (x\right )\right )} e^{\left (b c x \log \left (F\right ) + n \log \left (\log \left (d\right ) + \log \left (x\right )\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*c*e*x*log(d*x)*log(F) + e*n + e)*F^((b*x + a)*c)*log(d*x)^n/x,x, algorithm="maxima")
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*c*e*x*log(d*x)*log(F) + e*n + e)*F^((b*x + a)*c)*log(d*x)^n/x,x, algorithm="fricas")
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F**(c*(b*x+a))*ln(d*x)**n*(e+e*n+b*c*e*x*ln(F)*ln(d*x))/x,x)
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*c*e*x*log(d*x)*log(F) + e*n + e)*F^((b*x + a)*c)*log(d*x)^n/x,x, algorithm="giac")
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