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 = 0.129157, 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, Rules used = {2202} \[ e \log ^{n+1}(d x) F^{c (a+b x)} \]

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)

Rule 2202

Int[Log[(d_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_)))*(x_)^(m_.)*((e_) + Log[(d_.)*(x_)]*(h_.)*((f_.) +
(g_.)*(x_))), x_Symbol] :> Simp[(e*x^(m + 1)*F^(c*(a + b*x))*Log[d*x]^(n + 1))/(n + 1), x] /; FreeQ[{F, a, b,
c, d, e, f, g, h, m, n}, x] && EqQ[e*(m + 1) - f*h*(n + 1), 0] && EqQ[g*h*(n + 1) - b*c*e*Log[F], 0] && NeQ[n,
-1]

Rubi steps

\begin{align*} \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 &=e F^{c (a+b x)} \log ^{1+n}(d x)\\ \end{align*}

Mathematica [A]  time = 0.032266, size = 19, normalized size = 1. \[ e \log ^{n+1}(d x) F^{c (a+b x)} \]

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^(c*(a + b*x))*Log[d*x]^(1 + n)

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Maple [C]  time = 0.088, size = 180, normalized size = 9.5 \begin{align*} \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} \end{align*}

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 = 1.38625, size = 49, normalized size = 2.58 \begin{align*}{\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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a))*log(d*x)^n*(e+e*n+b*c*e*x*log(F)*log(d*x))/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. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a))*log(d*x)^n*(e+e*n+b*c*e*x*log(F)*log(d*x))/x,x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a))*log(d*x)^n*(e+e*n+b*c*e*x*log(F)*log(d*x))/x,x, algorithm="giac")

[Out]

Exception raised: RuntimeError