3.88 \(\int \frac{F^{c (a+b x)} \log ^n(d x) (e+e n+e (-1+b c x \log (F)) \log (d x))}{x^2} \, dx\)

Optimal. Leaf size=22 \[ \frac{e \log ^{n+1}(d x) F^{c (a+b x)}}{x} \]

[Out]

(e*F^(c*(a + b*x))*Log[d*x]^(1 + n))/x

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Rubi [A]  time = 0.131073, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.026, Rules used = {2202} \[ \frac{e \log ^{n+1}(d x) F^{c (a+b x)}}{x} \]

Antiderivative was successfully verified.

[In]

Int[(F^(c*(a + b*x))*Log[d*x]^n*(e + e*n + e*(-1 + b*c*x*Log[F])*Log[d*x]))/x^2,x]

[Out]

(e*F^(c*(a + b*x))*Log[d*x]^(1 + n))/x

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+e (-1+b c x \log (F)) \log (d x))}{x^2} \, dx &=\frac{e F^{c (a+b x)} \log ^{1+n}(d x)}{x}\\ \end{align*}

Mathematica [A]  time = 0.327632, size = 23, normalized size = 1.05 \[ \frac{e \log ^{n+1}(d x) F^{a c+b c x}}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(F^(c*(a + b*x))*Log[d*x]^n*(e + e*n + e*(-1 + b*c*x*Log[F])*Log[d*x]))/x^2,x]

[Out]

(e*F^(a*c + b*c*x)*Log[d*x]^(1 + n))/x

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Maple [C]  time = 0.105, size = 136, normalized size = 6.2 \begin{align*}{\frac{{F}^{c \left ( bx+a \right ) }e \left ( -i \left ({\it csgn} \left ( idx \right ) \right ) ^{3}\pi +i \left ({\it csgn} \left ( idx \right ) \right ) ^{2}{\it csgn} \left ( id \right ) \pi +i \left ({\it csgn} \left ( idx \right ) \right ) ^{2}{\it csgn} \left ( ix \right ) \pi -i{\it csgn} \left ( idx \right ){\it csgn} \left ( id \right ){\it csgn} \left ( ix \right ) \pi +2\,\ln \left ( x \right ) +2\,\ln \left ( d \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}}{2\,x}} \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+e*(-1+b*c*x*ln(F))*ln(d*x))/x^2,x)

[Out]

1/2*F^(c*(b*x+a))*e*(-I*csgn(I*d*x)^3*Pi+I*csgn(I*d*x)^2*csgn(I*d)*Pi+I*csgn(I*d*x)^2*csgn(I*x)*Pi-I*csgn(I*d*
x)*csgn(I*d)*csgn(I*x)*Pi+2*ln(x)+2*ln(d))/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 = 1.42827, size = 53, normalized size = 2.41 \begin{align*} \frac{{\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 )}}{x} \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+e*(-1+b*c*x*log(F))*log(d*x))/x^2,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)))/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+e*(-1+b*c*x*log(F))*log(d*x))/x^2,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+e*(-1+b*c*x*ln(F))*ln(d*x))/x**2,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+e*(-1+b*c*x*log(F))*log(d*x))/x^2,x, algorithm="giac")

[Out]

Exception raised: RuntimeError