3.1.86 \(\int F^{c (a+b x)} \log ^n(d x) (e+e n+e (1+b c x \log (F)) \log (d x)) \, dx\) [86]

Optimal. Leaf size=20 \[ e F^{c (a+b x)} x \log ^{1+n}(d x) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 20, normalized size of antiderivative = 1.00, 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 = {2232} \begin {gather*} e x \log ^{n+1}(d x) F^{c (a+b x)} \end {gather*}

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]

[Out]

e*F^(c*(a + b*x))*x*Log[d*x]^(1 + n)

Rule 2232

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

________________________________________________________________________________________

Mathematica [A]
time = 0.09, size = 21, normalized size = 1.05 \begin {gather*} e F^{a c+b c x} x \log ^{1+n}(d x) \end {gather*}

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]

[Out]

e*F^(a*c + b*c*x)*x*Log[d*x]^(1 + n)

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.13, size = 186, normalized size = 9.30

method	result	size
risch	\(\left (-\frac {i x \,F^{c \left (b x +a \right )} e \pi \,\mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right )}{2}+\frac {i x \,F^{c \left (b x +a \right )} e \pi \,\mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i d x \right )^{2}}{2}+\frac {i x \,F^{c \left (b x +a \right )} e \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right )^{2}}{2}-\frac {i x \,F^{c \left (b x +a \right )} e \pi \mathrm {csgn}\left (i d x \right )^{3}}{2}+x \,F^{c \left (b x +a \right )} e \ln \left (d \right )+e x \,F^{c \left (b x +a \right )} \ln \left (x \right )\right ) \left (\ln \left (d \right )+\ln \left (x \right )-\frac {i \pi \,\mathrm {csgn}\left (i d x \right ) \left (-\mathrm {csgn}\left (i d x \right )+\mathrm {csgn}\left (i d \right )\right ) \left (-\mathrm {csgn}\left (i d x \right )+\mathrm {csgn}\left (i x \right )\right )}{2}\right )^{n}\)	\(186\)

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,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.36, size = 40, normalized size = 2.00 \begin {gather*} {\left (F^{a c} x e \log \left (d\right ) + F^{a c} x 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 {gather*}

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, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.50, size = 24, normalized size = 1.20 \begin {gather*} F^{b c x + a c} x \log \left (d x\right )^{n} e \log \left (d x\right ) \end {gather*}

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, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e \left (\int F^{a c} F^{b c x} \log {\left (d x \right )}^{n}\, dx + \int F^{a c} F^{b c x} n \log {\left (d x \right )}^{n}\, dx + \int F^{a c} F^{b c x} \log {\left (d x \right )} \log {\left (d x \right )}^{n}\, dx + \int F^{a c} F^{b c x} b c x \log {\left (F \right )} \log {\left (d x \right )} \log {\left (d x \right )}^{n}\, dx\right ) \end {gather*}

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)

[Out]

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

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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, algorithm="giac")

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Warning, infinity
is unsigned, perhaps you meant +infinityWarning, infinity is unsigned, perhaps you meant +infinityUnable to d
ivide, perhaps

________________________________________________________________________________________

Mupad [B]
time = 3.48, size = 21, normalized size = 1.05 \begin {gather*} F^{a\,c+b\,c\,x}\,e\,x\,{\ln \left (d\,x\right )}^{n+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

F^(a*c + b*c*x)*e*x*log(d*x)^(n + 1)

________________________________________________________________________________________