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

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

[Out]

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

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Rubi [A]  time = 0.05, 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 = {2201} \[ e x \log ^{n+1}(d x) F^{c (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[F^(c*(a + b*x))*Log[d*x]^n*(e + e*n + 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 2201

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*}

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Mathematica [A]  time = 0.18, size = 21, normalized size = 1.05 \[ e x \log ^{n+1}(d x) F^{a c+b c x} \]

Antiderivative was successfully verified.

[In]

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

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fricas [A]  time = 0.46, size = 23, normalized size = 1.15 \[ F^{b c x + a c} e x \log \left (d x\right )^{n} \log \left (d x\right ) \]

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)*e*x*log(d*x)^n*log(d*x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

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 due to rounding error%%%{1,[0,2,0,0,0,2,1]%%%}+%%%{2,[0,2,0,0,0,1,1]%%%}+%%%{1,[0,2,0,0,0,0,1]%
%%} / %%%{1,[0,3,0,0,0,2,0]%%%}+%%%{2,[0,3,0,0,0,1,0]%%%}+%%%{1,[0,3,0,0,0,0,0]%%%} Error: Bad Argument Value

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maple [C]  time = 0.24, size = 186, normalized size = 9.30 \[ \left (-\frac {i \pi e x \,F^{\left (b x +a \right ) c} \mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right )}{2}+\frac {i \pi e x \,F^{\left (b x +a \right ) c} \mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i d x \right )^{2}}{2}+\frac {i \pi e x \,F^{\left (b x +a \right ) c} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right )^{2}}{2}-\frac {i \pi e x \,F^{\left (b x +a \right ) c} \mathrm {csgn}\left (i d x \right )^{3}}{2}+e x \,F^{\left (b x +a \right ) c} \ln \relax (d )+e x \,F^{\left (b x +a \right ) c} \ln \relax (x )\right ) \left (-\frac {i \pi \left (\mathrm {csgn}\left (i d \right )-\mathrm {csgn}\left (i d x \right )\right ) \left (\mathrm {csgn}\left (i x \right )-\mathrm {csgn}\left (i d x \right )\right ) \mathrm {csgn}\left (i d x \right )}{2}+\ln \relax (d )+\ln \relax (x )\right )^{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.60, size = 38, normalized size = 1.90 \[ {\left (F^{a c} e x \log \relax (d) + F^{a c} e x \log \relax (x)\right )} e^{\left (b c x \log \relax (F) + n \log \left (\log \relax (d) + \log \relax (x)\right )\right )} \]

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

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

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)

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

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
))

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