Optimal. Leaf size=22 \[ e x^2 \log ^{n+1}(d x) F^{c (a+b x)} \]
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Rubi [A] time = 0.132276, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.028 \[ e x^2 \log ^{n+1}(d x) F^{c (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[F^(c*(a + b*x))*x*Log[d*x]^n*(e + e*n + e*(2 + b*c*x*Log[F])*Log[d*x]),x]
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Rubi in Sympy [A] time = 8.15662, size = 20, normalized size = 0.91 \[ F^{c \left (a + b x\right )} e x^{2} \log{\left (d x \right )}^{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(F**(c*(b*x+a))*x*ln(d*x)**n*(e+e*n+e*(2+b*c*x*ln(F))*ln(d*x)),x)
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Mathematica [A] time = 0.0458321, size = 23, normalized size = 1.05 \[ e x^2 \log ^{n+1}(d x) F^{a c+b c x} \]
Antiderivative was successfully verified.
[In] Integrate[F^(c*(a + b*x))*x*Log[d*x]^n*(e + e*n + e*(2 + b*c*x*Log[F])*Log[d*x]),x]
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Maple [C] time = 0.116, size = 198, normalized size = 9. \[ \left ( -{\frac{i}{2}}\pi \,e{x}^{2}{\it csgn} \left ( id \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( idx \right ){F}^{c \left ( bx+a \right ) }+{\frac{i}{2}}\pi \,e{x}^{2}{\it csgn} \left ( id \right ) \left ({\it csgn} \left ( idx \right ) \right ) ^{2}{F}^{c \left ( bx+a \right ) }+{\frac{i}{2}}\pi \,e{x}^{2}{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( idx \right ) \right ) ^{2}{F}^{c \left ( bx+a \right ) }-{\frac{i}{2}}\pi \,e{x}^{2} \left ({\it csgn} \left ( idx \right ) \right ) ^{3}{F}^{c \left ( bx+a \right ) }+\ln \left ( d \right ) e{x}^{2}{F}^{c \left ( bx+a \right ) }+e{x}^{2}{F}^{c \left ( bx+a \right ) }\ln \left ( x \right ) \right ) \left ( \ln \left ( d \right ) +\ln \left ( x \right ) -{\frac{i}{2}}\pi \,{\it csgn} \left ( idx \right ) \left ( -{\it csgn} \left ( idx \right ) +{\it csgn} \left ( id \right ) \right ) \left ( -{\it csgn} \left ( idx \right ) +{\it csgn} \left ( ix \right ) \right ) \right ) ^{n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(F^(c*(b*x+a))*x*ln(d*x)^n*(e+e*n+e*(2+b*c*x*ln(F))*ln(d*x)),x)
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Maxima [A] time = 0.956811, size = 57, normalized size = 2.59 \[{\left (F^{a c} e x^{2} \log \left (d\right ) + F^{a c} e x^{2} \log \left (x\right )\right )} e^{\left (b c x \log \left (F\right ) + n \log \left (\log \left (d\right ) + \log \left (x\right )\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((b*c*x*log(F) + 2)*e*log(d*x) + e*n + e)*F^((b*x + a)*c)*x*log(d*x)^n,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*x*log(F) + 2)*e*log(d*x) + e*n + e)*F^((b*x + a)*c)*x*log(d*x)^n,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))*x*ln(d*x)**n*(e+e*n+e*(2+b*c*x*ln(F))*ln(d*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*x*log(F) + 2)*e*log(d*x) + e*n + e)*F^((b*x + a)*c)*x*log(d*x)^n,x, algorithm="giac")
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