3.89 \(\int \frac {F^{c (a+b x)} \log ^n(d x) (e+e n+e (-2+b c x \log (F)) \log (d x))}{x^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.13, antiderivative size = 22, normalized size of antiderivative = 1.00, 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} \[ \frac {e \log ^{n+1}(d x) F^{c (a+b x)}}{x^2} \]

Antiderivative was successfully verified.

[In]

Int[(F^(c*(a + b*x))*Log[d*x]^n*(e + e*n + e*(-2 + b*c*x*Log[F])*Log[d*x]))/x^3,x]

[Out]

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

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 (-2+b c x \log (F)) \log (d x))}{x^3} \, dx &=\frac {e F^{c (a+b x)} \log ^{1+n}(d x)}{x^2}\\ \end {align*}

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Mathematica [A]  time = 0.34, size = 23, normalized size = 1.05 \[ \frac {e \log ^{n+1}(d x) F^{a c+b c x}}{x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(F^(c*(a + b*x))*Log[d*x]^n*(e + e*n + e*(-2 + b*c*x*Log[F])*Log[d*x]))/x^3,x]

[Out]

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

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fricas [A]  time = 0.49, size = 25, normalized size = 1.14 \[ \frac {F^{b c x + a c} e \log \left (d x\right )^{n} \log \left (d x\right )}{x^{2}} \]

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*(-2+b*c*x*log(F))*log(d*x))/x^3,x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left ({\left (b c x \log \relax (F) - 2\right )} e \log \left (d x\right ) + e n + e\right )} F^{{\left (b x + a\right )} c} \log \left (d x\right )^{n}}{x^{3}}\,{d x} \]

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*(-2+b*c*x*log(F))*log(d*x))/x^3,x, algorithm="giac")

[Out]

integrate(((b*c*x*log(F) - 2)*e*log(d*x) + e*n + e)*F^((b*x + a)*c)*log(d*x)^n/x^3, x)

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maple [C]  time = 0.23, size = 136, normalized size = 6.18 \[ \frac {\left (-i \pi \,\mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right )+i \pi \,\mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i d x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right )^{2}-i \pi \mathrm {csgn}\left (i d x \right )^{3}+2 \ln \relax (d )+2 \ln \relax (x )\right ) e \,F^{\left (b x +a \right ) c} \left (-\frac {i \pi \left (\mathrm {csgn}\left (i d \right )-\mathrm {csgn}\left (i d x \right )\right ) \left (\mathrm {csgn}\left (i x \right )-\mathrm {csgn}\left (i d x \right )\right ) \mathrm {csgn}\left (i d x \right )}{2}+\ln \relax (d )+\ln \relax (x )\right )^{n}}{2 x^{2}} \]

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*(-2+b*c*x*ln(F))*ln(d*x))/x^3,x)

[Out]

1/2*F^((b*x+a)*c)*e*(-I*Pi*csgn(I*d)*csgn(I*x)*csgn(I*d*x)+I*Pi*csgn(I*d)*csgn(I*d*x)^2+I*Pi*csgn(I*x)*csgn(I*
d*x)^2-I*Pi*csgn(I*d*x)^3+2*ln(d)+2*ln(x))/x^2*(-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 = 1.63, size = 39, normalized size = 1.77 \[ \frac {{\left (F^{a c} e \log \relax (d) + F^{a c} e \log \relax (x)\right )} e^{\left (b c x \log \relax (F) + n \log \left (\log \relax (d) + \log \relax (x)\right )\right )}}{x^{2}} \]

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*(-2+b*c*x*log(F))*log(d*x))/x^3,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^2

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mupad [B]  time = 3.58, size = 23, normalized size = 1.05 \[ \frac {F^{a\,c+b\,c\,x}\,e\,{\ln \left (d\,x\right )}^{n+1}}{x^2} \]

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) - 2)))/x^3,x)

[Out]

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

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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))*ln(d*x)**n*(e+e*n+e*(-2+b*c*x*ln(F))*ln(d*x))/x**3,x)

[Out]

Timed out

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