3.87 \(\int \frac{F^{c (a+b x)} \log ^n(d x) (e+e n+b c e x \log (F) \log (d x))}{x} \, dx\)

Optimal. Leaf size=19 \[ e \log ^{n+1}(d x) F^{c (a+b x)} \]

[Out]

e*F^(c*(a + b*x))*Log[d*x]^(1 + n)

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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]

[Out]

e*F^(c*(a + b*x))*Log[d*x]^(1 + n)

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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)

[Out]

F**(c*(a + b*x))*e*log(d*x)**(n + 1)

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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]

[Out]

e*F^(a*c + b*c*x)*Log[d*x]^(1 + n)

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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)

[Out]

(-1/2*I*Pi*e*csgn(I*d)*csgn(I*x)*csgn(I*d*x)*F^(c*(b*x+a))+1/2*I*Pi*e*csgn(I*d)*
csgn(I*d*x)^2*F^(c*(b*x+a))+1/2*I*Pi*e*csgn(I*x)*csgn(I*d*x)^2*F^(c*(b*x+a))-1/2
*I*Pi*e*csgn(I*d*x)^3*F^(c*(b*x+a))+ln(d)*e*F^(c*(b*x+a))+e*F^(c*(b*x+a))*ln(x))
*(ln(d)+ln(x)-1/2*I*Pi*csgn(I*d*x)*(-csgn(I*d*x)+csgn(I*d))*(-csgn(I*d*x)+csgn(I
*x)))^n

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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")

[Out]

(F^(a*c)*e*log(d) + F^(a*c)*e*log(x))*e^(b*c*x*log(F) + n*log(log(d) + log(x)))

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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")

[Out]

Exception raised: TypeError

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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)

[Out]

Timed out

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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")

[Out]

Exception raised: RuntimeError