3.1.88 \(\int \frac {F^{c (a+b x)} \log ^n(d x) (e+e n+e (-1+b c x \log (F)) \log (d x))}{x^2} \, dx\) [88]

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

[Out]

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

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Rubi [A]
time = 0.10, 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 = {2233} \begin {gather*} \frac {e \log ^{n+1}(d x) F^{c (a+b x)}}{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^2,x]

[Out]

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

Rule 2233

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

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Mathematica [A]
time = 0.11, size = 23, normalized size = 1.05 \begin {gather*} \frac {e F^{a c+b c x} \log ^{1+n}(d x)}{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^2,x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.13, size = 136, normalized size = 6.18

method	result	size
risch	\(\frac {F^{c \left (b x +a \right )} e \left (2 \ln \left (d \right )+2 \ln \left (x \right )-i \pi \,\mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right )+i \pi \,\mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i d x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right )^{2}-i \pi \mathrm {csgn}\left (i d x \right )^{3}\right ) \left (\ln \left (d \right )+\ln \left (x \right )-\frac {i \pi \,\mathrm {csgn}\left (i d x \right ) \left (-\mathrm {csgn}\left (i d x \right )+\mathrm {csgn}\left (i d \right )\right ) \left (-\mathrm {csgn}\left (i d x \right )+\mathrm {csgn}\left (i x \right )\right )}{2}\right )^{n}}{2 x}\)	\(136\)

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^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.35, size = 41, normalized size = 1.86 \begin {gather*} \frac {{\left (F^{a c} e \log \left (d\right ) + F^{a c} 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 )}}{x} \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^2,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

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Fricas [A]
time = 0.46, size = 26, normalized size = 1.18 \begin {gather*} \frac {F^{b c x + a c} \log \left (d x\right )^{n} e \log \left (d x\right )}{x} \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^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e \left (\int \frac {F^{a c} F^{b c x} \log {\left (d x \right )}^{n}}{x^{2}}\, dx + \int \frac {F^{a c} F^{b c x} n \log {\left (d x \right )}^{n}}{x^{2}}\, dx + \int \left (- \frac {F^{a c} F^{b c x} \log {\left (d x \right )} \log {\left (d x \right )}^{n}}{x^{2}}\right )\, dx + \int \frac {F^{a c} F^{b c x} b c \log {\left (F \right )} \log {\left (d x \right )} \log {\left (d x \right )}^{n}}{x}\, 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**2,x)

[Out]

e*(Integral(F**(a*c)*F**(b*c*x)*log(d*x)**n/x**2, x) + Integral(F**(a*c)*F**(b*c*x)*n*log(d*x)**n/x**2, x) + I
ntegral(-F**(a*c)*F**(b*c*x)*log(d*x)*log(d*x)**n/x**2, x) + Integral(F**(a*c)*F**(b*c*x)*b*c*log(F)*log(d*x)*
log(d*x)**n/x, x))

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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^2,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

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Mupad [B]
time = 3.57, size = 23, normalized size = 1.05 \begin {gather*} \frac {F^{a\,c+b\,c\,x}\,e\,{\ln \left (d\,x\right )}^{n+1}}{x} \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^2,x)

[Out]

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

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