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

Optimal. Leaf size=22 \[ e x^2 \log ^{n+1}(d x) F^{c (a+b x)} \]

[Out]

e*F^(c*(b*x+a))*x^2*ln(d*x)^(1+n)

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Rubi [A]  time = 0.09, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {2202} \[ e x^2 \log ^{n+1}(d x) F^{c (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.27, size = 23, normalized size = 1.05 \[ e x^2 \log ^{n+1}(d x) F^{a c+b c x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.48, size = 25, normalized size = 1.14 \[ F^{b c x + a c} e x^{2} \log \left (d x\right )^{n} \log \left (d x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a))*x*log(d*x)^n*(e+e*n+e*(2+b*c*x*log(F))*log(d*x)),x, algorithm="fricas")

[Out]

F^(b*c*x + a*c)*e*x^2*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))*x*log(d*x)^n*(e+e*n+e*(2+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 = 0.21, size = 198, normalized size = 9.00 \[ \left (-\frac {i \pi e \,x^{2} F^{\left (b x +a \right ) c} \mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right )}{2}+\frac {i \pi e \,x^{2} F^{\left (b x +a \right ) c} \mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i d x \right )^{2}}{2}+\frac {i \pi e \,x^{2} F^{\left (b x +a \right ) c} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right )^{2}}{2}-\frac {i \pi e \,x^{2} F^{\left (b x +a \right ) c} \mathrm {csgn}\left (i d x \right )^{3}}{2}+e \,x^{2} F^{\left (b x +a \right ) c} \ln \relax (d )+e \,x^{2} F^{\left (b x +a \right ) c} \ln \relax (x )\right ) \left (-\frac {i \pi \left (\mathrm {csgn}\left (i d \right )-\mathrm {csgn}\left (i d x \right )\right ) \left (\mathrm {csgn}\left (i x \right )-\mathrm {csgn}\left (i d x \right )\right ) \mathrm {csgn}\left (i d x \right )}{2}+\ln \relax (d )+\ln \relax (x )\right )^{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^((b*x+a)*c)*x*ln(d*x)^n*(e+e*n+e*(2+b*c*x*ln(F))*ln(d*x)),x)

[Out]

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

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maxima [A]  time = 1.59, size = 42, normalized size = 1.91 \[ {\left (F^{a c} e x^{2} \log \relax (d) + F^{a c} e x^{2} \log \relax (x)\right )} e^{\left (b c x \log \relax (F) + n \log \left (\log \relax (d) + \log \relax (x)\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a))*x*log(d*x)^n*(e+e*n+e*(2+b*c*x*log(F))*log(d*x)),x, algorithm="maxima")

[Out]

(F^(a*c)*e*x^2*log(d) + F^(a*c)*e*x^2*log(x))*e^(b*c*x*log(F) + n*log(log(d) + log(x)))

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mupad [B]  time = 3.39, size = 23, normalized size = 1.05 \[ F^{a\,c+b\,c\,x}\,e\,x^2\,{\ln \left (d\,x\right )}^{n+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(c*(a + b*x))*x*log(d*x)^n*(e + e*n + e*log(d*x)*(b*c*x*log(F) + 2)),x)

[Out]

F^(a*c + b*c*x)*e*x^2*log(d*x)^(n + 1)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(c*(b*x+a))*x*ln(d*x)**n*(e+e*n+e*(2+b*c*x*ln(F))*ln(d*x)),x)

[Out]

Timed out

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