3.83 \(\int F^{c (a+b x)} x^m \log ^n(d x) (e+e n+e (1+m+b c x \log (F)) \log (d x)) \, dx\)

Optimal. Leaf size=24 \[ e x^{m+1} \log ^{n+1}(d x) F^{c (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.145645, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.026, Rules used = {2202} \[ e x^{m+1} \log ^{n+1}(d x) F^{c (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.366007, size = 24, normalized size = 1. \[ e x^{m+1} \log ^{n+1}(d x) F^{c (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.2, size = 192, normalized size = 8. \begin{align*}{\frac{ \left ( 2\,ex{F}^{c \left ( bx+a \right ) }\ln \left ( x \right ) -ix{F}^{c \left ( bx+a \right ) }e\pi \,{\it csgn} \left ( id \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( idx \right ) +ix{F}^{c \left ( bx+a \right ) }e\pi \,{\it csgn} \left ( id \right ) \left ({\it csgn} \left ( idx \right ) \right ) ^{2}+ix{F}^{c \left ( bx+a \right ) }e\pi \,{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( idx \right ) \right ) ^{2}+2\,x{F}^{c \left ( bx+a \right ) }e\ln \left ( d \right ) -ix{F}^{c \left ( bx+a \right ) }e\pi \, \left ({\it csgn} \left ( idx \right ) \right ) ^{3} \right ){x}^{m} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(c*(b*x+a))*x^m*ln(d*x)^n*(e+e*n+e*(1+m+b*c*x*ln(F))*ln(d*x)),x)

[Out]

1/2*(2*e*x*F^(c*(b*x+a))*ln(x)-I*x*F^(c*(b*x+a))*e*Pi*csgn(I*d)*csgn(I*x)*csgn(I*d*x)+I*x*F^(c*(b*x+a))*e*Pi*c
sgn(I*d)*csgn(I*d*x)^2+I*x*F^(c*(b*x+a))*e*Pi*csgn(I*x)*csgn(I*d*x)^2+2*x*F^(c*(b*x+a))*e*ln(d)-I*x*F^(c*(b*x+
a))*e*Pi*csgn(I*d*x)^3)*x^m*(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 = 1.44309, size = 57, normalized size = 2.38 \begin{align*}{\left (F^{a c} e x \log \left (d\right ) + F^{a c} e x \log \left (x\right )\right )} e^{\left (b c x \log \left (F\right ) + m \log \left (x\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))*x^m*log(d*x)^n*(e+e*n+e*(1+m+b*c*x*log(F))*log(d*x)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.55793, size = 90, normalized size = 3.75 \begin{align*}{\left (e x \log \left (d\right ) + e x \log \left (x\right )\right )} F^{b c x + a c} x^{m}{\left (\log \left (d\right ) + \log \left (x\right )\right )}^{n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a))*x^m*log(d*x)^n*(e+e*n+e*(1+m+b*c*x*log(F))*log(d*x)),x, algorithm="fricas")

[Out]

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

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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))*x**m*ln(d*x)**n*(e+e*n+e*(1+m+b*c*x*ln(F))*ln(d*x)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left ({\left (b c x \log \left (F\right ) + m + 1\right )} e \log \left (d x\right ) + e n + e\right )} F^{{\left (b x + a\right )} c} x^{m} \log \left (d x\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a))*x^m*log(d*x)^n*(e+e*n+e*(1+m+b*c*x*log(F))*log(d*x)),x, algorithm="giac")

[Out]

integrate(((b*c*x*log(F) + m + 1)*e*log(d*x) + e*n + e)*F^((b*x + a)*c)*x^m*log(d*x)^n, x)